University of Groningen Exploiting the Composite Step

University of Groningen

Exploiting the Composite Step Strategy to the Biconjugate A-Orthogonal Residual Method forNon-Hermitian Linear SystemsJing Yan-Fei Huang Ting-Zhu Carpentieri Bruno Duan Yong

Published inJournal of applied mathematics

DOI1011552013408167

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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 408167 16 pageshttpdxdoiorg1011552013408167

Research ArticleExploiting the Composite Step Strategy tothe Biconjugate 119860-Orthogonal Residual Method forNon-Hermitian Linear Systems

Yan-Fei Jing1 Ting-Zhu Huang1 Bruno Carpentieri2 and Yong Duan1

1 School of Mathematical Sciences Institute of Computational Science University of Electronic Science and Technology of ChinaChengdu Sichuan 611731 China

2 Institute of Mathematics and Computing Science University of Groningen Nijenborgh 9 PO Box 4079700 AK Groningen The Netherlands

Correspondence should be addressed to Yan-Fei Jing 00jyfvictory163com

Received 15 October 2012 Accepted 19 December 2012

Academic Editor Zhongxiao Jia

Copyright copy 2013 Yan-Fei Jing et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The Biconjugate 119860-Orthogonal Residual (BiCOR) method carried out in finite precision arithmetic by means of the biconjugate119860-orthonormalization procedure may possibly tend to suffer from two sources of numerical instability known as two kinds ofbreakdowns similarly to those of the Biconjugate Gradient (BCG)methodThis paper naturally exploits the composite step strategyemployed in the development of the composite step BCG (CSBCG)method into the BiCORmethod to cure one of the breakdownscalled as pivot breakdown Analogously to the CSBCGmethod the resulting interesting variant with only a minor modification tothe usual implementation of the BiCOR method is able to avoid near pivot breakdowns and compute all the well-defined BiCORiterates stably on the assumption that the underlying biconjugate 119860-orthonormalization procedure does not break down Anotherbenefit acquired is that it seems to be a viable algorithm providing some further practically desired smoothing of the convergencehistory of the norm of the residuals which is justified by numerical experiments In addition the exhibited method inherits thepromising advantages of the empirically observed stability and fast convergence rate of the BiCOR method over the BCG methodso that it outperforms the CSBCG method to some extent

1 Introduction

The computational cost of many simulations with integralequations or partial differential equations that model theprocess is dominated by the solution of systems of linearequations of the form 119860119909 = 119887 The field of iterativemethods for solving linear systems has been observed as anexplosion of activity spurred by demand due to extraordinarytechnological advances in engineering and sciences in thepast half century [1] Krylov subspace methods belong to oneof the most widespread and extensively accepted techniquesfor iterative solution of todayrsquos large-scale linear systems [2]With respect to ldquothe greatest influence on the developmentand practice of science and engineering in the 20th centuryrdquoas written by Dongarra and Sullivan [3] Krylov subspace

methods are considered as one of the ldquoTop Ten Algorithmsof the Centuryrdquo

Many advances in the development of Krylov subspacemethods have been inspired and made by the study of evenmore effective approaches to linear systems Variousmethodsdiffer in the way they extract information from Krylovspaces A basis for the underlying Krylov subspace to formiterative recurrences involved is usually constructed based onthe state-of-the-art Arnoldi orthogonalization procedure orLanczos biorthogonalization procedure see recent excellentand thorough review articles by Simoncini and Szyld [2]and by Philippe and Reichel [4] or monographs such asthose of Greenbaum [5] and Saad [6] A novel Lanczos-typebiconjugate 119860-orthonormalization procedure has recentlybeen established to give birth to a new family of efficient

2 Journal of Applied Mathematics

short-recurrence methods for large real nonsymmetric andcomplex non-Hermitian systems of linear equations namedas the Lanczos biconjugate 119860-orthonormalization methods[7]

As observed from numerous numerical experiments car-ried out with the Lanczos biconjugate 119860-orthonormalizationmethods it has been numerically demonstrated that thisfamily of solvers shows competitive convergence propertiesis cheap in memory as it is derived from short-term vectorrecurrences is parameter-free and does not require a sym-metric preconditioner like other methods if 119860 is symmetricindefinite [8ndash10] The efficiency of the BiCOR method ishighlighted by the performance profiles on a set of 14 sparsematrix problems of size up to 15M unknowns shown in [10]

On the other hand this family of solvers is often facedwith apparently irregular convergence behaviors appearingas ldquospikesrdquo in the convergence history of the norm ofthe residuals possibly leading to substantial build-up ofrounding errors andworse approximate solutions or possiblyeven overflow [7 9] Therefore it is quite necessary totackle their irregular convergence properties to obtain morestabilized variants so as to improve the accuracy of the desirednumerical physical solutions It is emphasized that our mainattention in this paper is focused on the straightforward nat-ural enhancement of the Biconjugate119860-Orthogonal Residual(BiCOR) method which is the basic underlying variant ofthe Lanczos biconjugate119860-Orthonormalizationmethods [7]The improvement of the BiCOR method will simultaneouslyresult in analogous improvements for the other two evolvingvariants known as the Conjugate 119860-orthogonal ResidualSquared (CORS) method and the Biconjugate 119860-OrthogonalResidual Stabilized (BiCORSTAB) method

By comparison of the biconjugate 119860-orthonormalizationprocedure [7] and the Lanczos biorthogonalization proce-dure [6] respectively for the BiCOR method [7] and theBiconjugate Gradient (BCG) method [11] as well as theircorresponding implementation algorithms it is obviouslyobserved that when carried out in finite precision arithmeticthe BiCOR method does tend to suffer from two possiblesources of numerical instability known as two kinds of break-downs exactly similarly to those of the BCGmethodWe stillcall such two kinds of breakdowns as Lanczos breakdown andpivot breakdown which will be analyzed and discussed inthe following sections As far as reported a large number ofstrategies designed to handle such two kinds of breakdownshave been proposed in the literature [12ndash16] includingcomposite step techniques [11 17ndash21] for pivot breakdownand look-ahead techniques [22ndash30] for Lanczos breakdownor both types For comprehensive reviews and discussionsabout the two kinds of breakdownsmentioned above refer to[2 6 16] and the references therein As shown and analyzeddetailedly by Bank and Chan [18 19] the 2 times 2 compositestep strategy employed in the development of the compositestep biconjugate gradient method (CSBCG) can eliminate(near) pivot breakdowns of the BCG method caused by(near) singularity of principal submatrices of the tridiagonalmatrix generated by the underlying Lanczos biorthogonal-ization procedure assuming that Lanczos breakdowns donot occur Furthermore besides simplifying implementations

a great deal in comparison with those existing look-aheadtechniques the composite step strategy used there is ableto provide some smoothing of the convergence history ofthe norm of the residuals without involving any arbitrarymachine-dependent or user-supplied tolerances

This paper revisits the composite step strategy taken forthe CSBCG method [18 19] to naturally stabilize the con-vergence performance of the BiCOR method from the pivot-breakdown treatment point of view Therefore suppose thatthe underlying biconjugate119860-orthonormalization proceduredoes not break down that is to assume that there is nooccurrence of the so-called Lanczos breakdown encounteredduring algorithm implementations throughoutThe resultinginteresting algorithm given a name as the composite stepBiCOR (CSBiCOR) method could be also considered as arelatively simple modification of the regular BiCOR method

The main objectives are twofold First the CSBiCORmethod is devised to be able to avoid near pivot breakdownsand compute all the well-defined BiCOR iterates stably withonly minor modifications inspired by the advantages ofthe CSBCG method over the BCG method Second theCSBiCOR method can reduce the number of spikes in theconvergence history of the norm of the residuals to thegreatest extent providing some further practically desiredsmoothing behavior towards stabilizing the behavior of theBiCOR method when it has erratic convergence behaviorsAdditional purpose is that the CSBiCORmethod inherits thepromising advantages of the empirically observed stabilityand fast convergence rate of the BiCORmethod over the BCGmethod so that it outperforms the CSBCG method to someextent

The remainder of this work is organized as follows A briefreview of the biconjugate 119860-orthonormalization procedureis made and the factorization of the resulted nonsingulartridiagonal matrices in the aforementioned procedure isconsidered in Section 2 for the preparation of the theoreticalbasis and relevant background of the CSBiCORmethod Andthen the breakdowns of the BiCOR method are presentedin Section 3 while the CSBiCOR method is derived inSection 4 Section 5 analyzes the properties of the CSBiCORmethod as well as providing a best approximation resultfor the convergence of the CSBiCOR method In order toconveniently and simply present the CSBiCORmethod someimplementation issues on both algorithm simplification andstepping strategy are discussed detailedly in Section 6 Theimproved performance of the CSBiCOR method will beillustrated in Section 7 in the perspective that the CSBiCORmethod could hopefully smooth the convergence historythrough the reduction of the number of spikes when theBiCOR method has irregular convergence behavior Finallyconcluding remarks are made in the last section

Throughout the paper we denote the overbar ldquondashrdquo theconjugate complex of a scalar vector or matrix and thesuperscript ldquo119879rdquo the transpose of a vector or matrix For anon-Hermitian matrix 119860 = (119886

119894119895)119873times119873

isin C119873times119873 the Hermitianconjugate of 119860 is denoted as

119860119867equiv 119860119879

= (119886119895119894)119873times119873 (1)

Journal of Applied Mathematics 3

The standard Hermitian inner product of two complexvectors 119906 119907 isin C119873 is defined as

⟨119906 119907⟩ = 119906119867119907 =

119873

sum

119894=1

119906119894119907119894 (2)

The nested Krylov subspace of dimension 119905 generated by 119860from 119907 is of the form

K119905 (119860 119907) = span 119907 119860119907 119860

2119907 119860

119905minus1119907 (3)

In addition 119890119894denotes the 119894th column of the appropriate

identity matrixWhen it will be helpful we will use the word ldquoideallyrdquo

(or ldquomathematicallyrdquo) to refer to a result that could holdin exact arithmetic ignoring effects of rounding errors andldquonumericallyrdquo (or ldquocomputationallyrdquo) to a result of a finiteprecision computation

2 Theoretical Basis for the CSBiCOR Method

For the sake of discussing the theoretical basis and relevantbackground of the CSBiCOR method the biconjugate 119860-orthonormalization procedure [7] is first briefly recalled asin Algorithm 1 which can ideally build up a pair of bicon-jugate 119860-orthonormal bases for the dual Krylov subspacesK119898(119860 1199071) and K

119898(119860119867 1199081) where 119907

1and 119908

1are chosen

initially to satisfy certain conditionsObserve that the above algorithm is possible to have

Lanczos-type breakdown whenever 120575119895+1

vanishes while 119908119895+1

and 119860119907119895+1

are not equal to 0 isin C119873 appearing in line 8 Inthe interest of counteraction against such breakdowns referoneself to remedies such as the so-called look-ahead strate-gies [22ndash30] which can enhance stability while increasingcost modestly But that is outside the scope of this paperand we will not pursue that here For more details pleaserefer to [2 6] and the references therein Throughout the restof the present paper suppose there is no such Lanczos-typebreakdown encountered during algorithm implementationsbecause most of our considerations concern the explorationof the composite step strategy [18 19] to handle the pivotbreakdown occurring in the BiCORmethod for solving non-Hermitian linear systems

Next some properties of the vectors produced byAlgorithm 1 are reviewed [7] in the following proposition forthe preparation of the theoretical basis of the composite stepmethod

Proposition 1 If Algorithm 1 proceeds 119898 steps then the rightand left Lanczos-type vectors 119907

119895 119895 = 1 2 119898 and 119908

119894 119894 =

1 2 119898 form a biconjugate 119860-orthonormal system in exactarithmetic that is

⟨120596119894 119860119907119895⟩ = 120575119894119895 1 le 119894 119895 le 119898 (4)

Furthermore denote by 119881119898= [1199071 1199072 119907

119898] and 119882

119898=

[1199081 1199082 119908

119898] the 119873 times 119898 matrices and by 119879

119898the extended

tridiagonal matrix of the form

119879119898= [

119879119898

120575119898+1119890119879

119898

] (5)

(1) Choose 1199071 1205961 such that ⟨120596

1 1198601199071⟩ = 1

(2) Set 1205731= 1205751equiv 0 120596

0= 1199070equiv 0 isin C119873

(3) for 119895 = 1 2 do(4) 120572

119895= ⟨120596119895 119860(119860119907

119895)⟩

(5) 119907119895+1= 119860119907119895minus 120572119895119907119895minus 120573119895119907119895minus1


(7) 120575119895+1=10038161003816100381610038161003816⟨119895+1 119860119907119895+1⟩10038161003816100381610038161003816

12

(8) 120573119895+1= ⟨119895+1 119860119907119895+1⟩ 120575119895+1

(9) 119907119895+1= 119907119895+1120575119895+1

(10) 120596119895+1= 119895+1120573119895+1

(11) end for

Algorithm 1 Biconjugate 119860-orthonormalization procedure

where

119879119898=

[[[[[[

[

12057211205732

120575212057221205733

120575119898minus1

120572119898minus1

120573119898

120575119898120572119898

]]]]]]

]

(6)

whose entries are the coefficients generated during the algo-rithm implementation and in which 120572

1 120572

119898 1205732 120573

119898

are complex while 1205752 120575

119898are positive Then with the

biconjugate 119860-orthonormalization procedure the followingfour relations hold

119860119881119898= 119881119898119879119898+ 120575119898+1119907119898+1119890119879

119898 (7)

119860119867119882119898= 119882119898119879119867

119898+ 120573119898+1120596119898+1119890119879

119898 (8)

119882119867

119898119860119881119898= 119868119898 (9)

119882119867

1198981198602119881119898= 119879119898 (10)

It is well known that the Lanczos biorthogonalizationprocedure canmathematically build up a pair of biorthogonalbases for the dual Krylov subspaces K

119898(119860 1199071) and

K119898(119860119867 1199081) where 119907

1and 119908

1are initially selected

such that ⟨1205961 1199071⟩ = 1 [6] While the biconjugate 119860-

orthonormalization procedure presented as in Algorithm 1can ideally build up a pair of biconjugate 119860-orthonormalbases for the dual Krylov subspaces K

119898(119860 1199071) and

K119898(119860119867 1199081) where 119907

1and 119908

1are chosen initially to satisfy

⟨1205961 1198601199071⟩ = 1 Keeping in mind the above differences in

terms of different Krylov subspace bases constructed bythe two different underlying procedures we will in parallelborrow some of the results of the CSBCG method [18] forestablishing the following counterparts for the CSBiCORmethod see [18] for relevant proofs

It should be emphasized that the present CSBiCORmethod will be derived in exactly the same way the CSBCGmethod is obtained [19] Hence the analysis and theoreticalbackground for the CSBiCOR method in this section are the


counterparts of the CSBCG method in a different contextthat is the biconjugate 119860-orthonormalization procedure aspresented in Algorithm 1 They are included as follows forthe purpose of theoretical completeness and can be provedin the same way for the corresponding results of the CSBCGmethod analyzed in [18]

The following proposition states the tridiagonal matrix119879119898formed in Proposition 1 cannot have two successive sin-

gular leading principal submatrices if the coefficient matrix119860 is nonsingular

Proposition 2 Let 119879119896(1 le 119896 le 119898) be the upper left principal

submatrices of the nonsingular tridiagonalmatrix119879119898appeared

as in Proposition 1 Then 119879119896minus1

and 119879119896cannot be both singular

It is noted that there may exist possible breakdown in thefactorization without pivoting of the tridiagonal matrix 119879

119898

formed in Proposition 1 The following results illustrate howto correct such problemwith the occasional use of 2 times 2 blockpivots

Proposition 3 Let 119879119898be the nonsingular tridiagonal matrix

formed in Proposition 1 Then 119879119898can be factored as

119879119898= 119871119898119863119898119880119898 (11)

where 119871119898is unit lower block bidiagonal119880

119898is unit upper block

bidiagonal and 119863119898is block diagonal with 1 times 1 and 2 times 2

diagonal blocks

With the above two propositions one can prove thefollowing result insuring that if the upper left principalsubmatrices 119879

119896(1 le 119896 le 119898) of 119879

119898is singular it can still be

factored

Corollary 4 Suppose one upper left principal submatrix119879119896(1 le 119896 le 119898) of the nonsingular tridiagonal matrix

119879119898

formed in Proposition 1 is singular but 119879119896minus1

must benonsingular as stated in Proposition 2 Then

119879119896= 119871119896119863119896119880119896 (12)

where 119871119896is unit lower block bidiagonal 119880



diagonal blocks and in particular the last block of 119863119896is the

1 times 1 zero matrix

From Corollary 4 the singularity of 119879119896(1 le 119896 le 119898) can

be recognized by only examining the last diagonal element(and next potential pivot) 119889

119896 If 119889119896= 0 then the 2 times 2 block

[119889119896120573119896+1

120575119896+1120572119896+1

] (13)

will be nonsingular and can be used as a 2 times 2 pivot where120573119896+1 120575119896+1

and 120572119896+1

are the corresponding elements of 119879119898as

shown in Proposition 1It can be concluded that appropriate combinations of

1 times 1 and 2 times 2 steps for the BiCOR method can skip overthe breakdowns caused by the singular principal submatricesof the tridiagonal matrix 119879

119898generated by the underlying

biconjugate 119860-orthonormalization procedure presented asin Algorithm 1 since the BiCOR method pivots implicitlywith those submatrices The next section will have a detailedinvestigation of breakdowns possibly occurring in the BiCORmethod

3 Breakdowns of the BiCOR Method

Given an initial guess 1199090to the non-Hermitian linear system

119860119909 = 119887 associated with the initial residual 1199030= 119887 minus

1198601199090 define a Krylov subspace L

119898equiv 119860119867 span(119882

119898) =

119860119867K119898(119860119867 1199081) where119882

119898is defined in Proposition 1 119907

1=

1199030||1199030||2and 119908

1is chosen arbitrarily such that ⟨119908

1 1198601199071⟩ = 0

But 1199081is often chosen to be equal to 119860119907

1||1198601199071||2

2subjecting

to ⟨1199081 1198601199071⟩ = 1 It is worth noting that this choice for

1199081plays a significant role in establishing the empirically

observed superiority of the BiCOR method to the BiCR [31]method as well as to the BCG method [7] Thus runningAlgorithm 1 119898 steps we can seek an 119898th approximate solu-tion119909

119898from the affine subspace 119909

0+K119898(119860 1199071) of dimension

119898 by imposing the Petrov-Galerkin condition

119887 minus 119860119909119898perpL119898 (14)

which can be mathematically written in matrix formulationas

(119860119867119882119898)119867

(119887 minus 119860119909119898) = 0 (15)

Analogously an 119898th dual approximation 119909lowast119898

of thecorresponding dual system 119860119867119909lowast = 119887

lowast is sought fromthe affine subspace 119909lowast

0+ K119898(119860119867 1199081) of dimension 119898 by

satisfying

119887lowastminus 119860119867119909lowast

119898perp 119860K

119898(119860 1199071) (16)


(119860119881119898)119867(119887lowastminus 119860119867119909119898) = 0 (17)

where 119909lowast0is an initial dual approximate solution and 119881

119898is

defined in Proposition 1 with 1199071= 1199030||1199030||2

Consequently the BiCOR iterates 119909119895rsquos can be computed

by the coming Algorithm 2 which is just the unprecondi-tionedBiCORmethodwith the preconditioner119872 there takenas the identity matrix [7] and has been rewritten with thealgorithmic scheme of the unpreconditioned BCGmethod aspresented in [6 19]

Suppose Algorithm 2 runs successfully to step 119899 that is120590119894= 0 120588119894= 0 119894 = 0 1 119899 minus 1 The BiCOR iterates satisfy the

following properties [7]

Proposition 5 Let 119877119899+1

= [1199030 1199031 119903

119899] 119877lowast119899+1

=

[119903lowast

0 119903lowast

1 119903

lowast

119899] and 119875

119899+1= [119901

0 1199011 119901

119899] 119875lowast119899+1

=

[119901lowast

0 119901lowast

1 119901

lowast

119899] One has the following

(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)

Range(119877lowast119899+1) = Range(119875lowast

119899+1) =K

119899+1(119860119867 119903lowast

0)


(1) Compute 1199030= 119887 minus 119860119909

0for some initial guess 119909

0

(2) Choose 119903lowast0= 119875 (119860) 119903

0such that ⟨119903lowast

0 1198601199030⟩ = 0 where 119875(119905) is a polynomial in 119905

(eg 119903lowast0= 1198601199030)

(3) Set 1199010= 1199030 119901lowast0= 119903lowast

0 1199020= 1198601199010 119902lowast0= 119860119867119901lowast

0 1199030= 1198601199030 1205880= ⟨119903lowast

0 1199030⟩

(4) for 119899 = 0 1 do(5) 120590

119899= ⟨119902lowast

119899 119902119899⟩

(6) 120572119899= 120588119899120590119899

(7) 119909119899+1= 119909119899+ 120572119899119901119899

(8) 119903119899+1= 119903119899minus 120572119899119902119899

(9) 119909lowast

119899+1= 119909lowast

119899+ 120572119899119901lowast

119899

(10) 119903lowast119899+1= 119903lowast

119899minus 120572119899119902lowast

119899

(11) 119903119899+1= 119860119903119899+1

(12) 120588119899+1= ⟨119903lowast

119899+1 119903119899+1⟩

(13) if 120588119899+1= 0method fails

(14) 120573119899+1= 120588119899+1120588119899

(15) 119901119899+1= 119903119899+1+ 120573119899+1119901119899

(16) 119901lowast119899+1= 119903lowast

119899+1+ 120573119899+1119901lowast

119899

(17) 119902119899+1= 119903119899+1+ 120573119899+1119902119899

(18) 119902lowast119899+1= 119860119867119901lowast

119899

(19) check convergence continue if necessary(20) end for

Algorithm 2 Algorithm BiCOR

(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1

is diagonal

Similarly to the breakdowns of the BCGmethod [19] it isobserved from Algorithm 2 that there also exist two possiblekinds of breakdowns for the BiCOR method

(1) 120588119899equiv ⟨119903lowast

119899 119903119899⟩ equiv ⟨119903

lowast

119899 119860119903119899⟩ = 0 but 119903lowast

119899and 119860119903

119899are not

equal to 0 isin C119873 appearing in line 14

(2) 120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast

119899 119860119901119899⟩ = 0 appearing in line 6

Although the computational formulae for the quantitieswhere the breakdowns reside are different between theBiCORmethod and the BCGmethod we do not have a bettername for them And therefore we still call the two casesof breakdowns described above as Lanczos breakdown andpivot breakdown respectively

The Lanczos breakdown can be cured using look-aheadtechniques [22ndash30] asmentioned in the first section but suchtechniques require a careful and sophisticated way so as tomake them become necessarily quite complicated to applyThis aspect of applying look-ahead techniques to the BiCORmethod demands further research

In this paper we attempt to resort to the composite stepidea employed for the CSBCG method [18 19] to handle thepivot breakdown of the BiCOR method with the assumptionthat the underlying biconjugate 119860-orthonormalization pro-cedure depicted as in Algorithm 1 does not break down thatis the situation where 120590

119899= 0 while 120588

119899= 0

4 The Composite Step BiCOR Method

Suppose Algorithm 2 comes across a situation where 120590119899=

0 after successful algorithm implementation up to step 119899with the assumption that 120588

119899= 0 which indicates that the

updates of 119909119899+1 119903119899+1 119909lowast

119899+1 119903lowast

119899+1are not well defined Taking

the composite step idea we will avoid division by 120590119899= 0

via skipping this (119899 + 1)th update and exploiting a compositestep update to directly obtain the quantities in step (119899 + 2)with scaled versions of 119903

119899+1and 119903lowast

119899+1as well as with the

previous primary search direction vector 119901119899and shadow

search direction vector 119901lowast119899The following process for deriving

the CSBiCOR method is the same as that of the derivation ofthe CSBCG method [19] except for the different underlyingprocedures involved to correspondingly generate differentKrylov subspace bases

Analogously define auxiliary vectors 119911119899+1isin K119899+2(119860 1199030)

and 119911lowast119899+1isinK119899+2(119860119867 119903lowast

0) as follows

119911119899+1= 120590119899119903119899+1

= 120590119899119903119899minus 120588119899119860119901119899

(18)

119911lowast

119899+1= 120590119899119903lowast

119899+1

= 120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899

(19)

which are then used to look for the iterates 119909119899+2isin 1199090+

K119899+2(119860 1199030) and 119909lowast

119899+2isin 119909lowast

0+ K119899+2(119860119867 119903lowast

0) in step (119899 + 2)

as follows

119909119899+2= 119909119899+ [119901119899 119911119899+1] 119891119899

119909lowast

119899+2= 119909lowast

119899+ [119901lowast

119899 119911lowast

119899+1] 119891lowast

119899

(20)


where 119891119899 119891lowast

119899isin C2 Correspondingly the (119899 + 2)th primary

residual 119903119899+2isin K

119899+3(119860 1199030) and shadow residual 119903lowast

119899+2isin

K119899+3(119860119867 119903lowast

0) are respectively computed as

119903119899+2= 119903119899minus 119860 [119901

119899 119911119899+1] 119891119899 (21)

119903lowast

119899+2= 119903lowast

119899minus 119860119867[119901lowast

119899 119911lowast

119899+1] 119891lowast

119899 (22)

The biconjugate 119860-orthogonality condition between theBiCOR primary residuals and shadow residuals shown asProperty (2) in Proposition 5 requires

⟨[119901lowast

119899 119911lowast

119899+1] 119860119903119899+2⟩ = 0

⟨[119901119899 119911119899+1] 119860119867119903lowast

119899+2⟩ = 0

(23)

combining with (21) and (22) gives rise to the following two2 times 2 systems of linear equations for respectively solving 119891

119899

and 119891lowast119899

[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119891(1)

119899

119891(2)

119899

]

= [⟨119901lowast

119899 119860119903119899⟩

⟨119911lowast

119899+1 119860119903119899⟩]

(24)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119891lowast(1)

119899

119891lowast(2)

119899

]

= [⟨119860119901119899 119903lowast

119899⟩

⟨119860119911119899+1 119903lowast

119899⟩]

(25)

Similarly the (119899 + 2)th primary search direction vector119901119899+2

isin K119899+3(119860 1199030) and shadow search direction vector

119901lowast

119899+2isin K119899+3(119860119867 119903lowast

0) in a composite step are computed with

the following form

119901119899+2= 119903119899+2+ [119901119899 119911119899+1] 119892119899 (26)

119901lowast

119899+2= 119903lowast

119899+2+ [119901lowast

119899

119911lowast

119899+1] 119892lowast

119899 (27)

where 119892119899 119892lowast

119899isin C2

The biconjugate 1198602-orthogonality condition between theBiCOR primary search direction vectors and shadow searchdirection vectors shown as Property (3) in Proposition 5requires

⟨[119901lowast

119899 119911lowast

119899+1] 1198602119901119899+2⟩ = 0

⟨[119901119899 119911119899+1] (119860119867)2

119901lowast

119899+2⟩ = 0

(28)

combining with (26) and (27) results in the following two 2 times2 systems of linear equations for respectively solving 119892

119899and

119892lowast

119899

[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119892(1)

119899

119892(2)

119899

]

= minus[⟨119860119867119901lowast

119899 119860119903119899+2⟩

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩]

(29)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119892lowast(1)

119899

119892lowast(2)

119899

]

= minus[⟨119860119901119899 119860119867119903lowast

119899+2⟩

⟨119860119911119899+1 119860119867119903lowast

119899+2⟩]

(30)

Therefore it could be able to advance from step 119899 tostep (119899 + 2) to provide 119909

119899+2 119903119899+2

119909lowast119899+2

119903lowast119899+2

119901119899+2

119901lowast119899+2

bysolving the above four 2 times 2 linear systems represented as in(24) (25) (29) and (30) With an appropriate combinationof 1 times 1 and 2 times 2 steps the CSBiCOR method can besimply obtained with only a minor modification to theusual implementation of theBiCORmethod Before explicitlypresenting the algorithm some properties of the CSBiCORmethod will be given in the following section

5 Some Properties of the CSBiCOR Method

In order to show the use of 2 times 2 steps is sufficientfor CSBiCOR method to compute exactly all those well-defined BiCOR iterates stably provided that the underlyingbiconjugate 119860-orthonormalization procedure depicted as inAlgorithm 1 does not break down it is necessary to establishsome lemmas similar to those for the CSBCG method [19]

Lemma 6 Using either a 1times 1 or a 2times 2 step to obtain 119901119899 119901lowast119899

the following hold

⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119903119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ = ⟨119902

lowast

119899 119902119899⟩ equiv 120590119899

(31)

Proof If 119901119899 119901lowast

119899are obtained via a 1 times 1 step that is

119901119899= 119903119899+ 120573119899119901119899minus1

119901lowast119899

= 119903lowast

119899+ 120573119899119901lowast

119899minus1as shown in lines

15 and 16 of Algorithm 2 then with the biconjugate 1198602-orthogonality condition between the BiCOR primary searchdirection vectors and shadow search direction vectors shownas Property (3) in Proposition 5 it follows that

⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast

119899minus 120573119899119901lowast

119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899


⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901119899minus 120573119899119901119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899 119860119901119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(32)

If 119901119899 119901lowast

119899are obtained via a 2 times 2 step that is 119901

119899=

119903119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)

119899minus2119911119899minus1

119901lowast119899= 119903lowast

119899+ 119892lowast(1)

119899minus2119901lowast

119899minus2+ 119892lowast(2)


119899minus1as

presented in (26) and (27) then according to the biconjugate1198602-orthogonality conditions mentioned above for deriving119901119899 119901lowast

119899with a 2 times 2 step it follows that

⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast

119899minus 119892lowast(1)


119899minus2minus 119892lowast(2)


119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892lowast(1)

119899minus2⟨119860119867119901lowast

119899minus2 119860119901119899⟩

minus 119892lowast(2)

119899minus2⟨119860119867119911lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901

119899minus 119892(1)

119899minus2119901119899minus2minus 119892(2)

119899minus2119911119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892(1)

119899minus2⟨119860119867119901lowast

119899 119860119901119899minus2⟩

minus 119892(2)

119899minus2⟨119860119867119901lowast

119899 119860119911119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(33)

With the above lemma we can show the relationshipsamong the four 2 times 2 coefficientmatrices of the linear systemsrepresented as in (24) (25) (29) and (30)

Lemma 7 The four 2 times 2 coefficient matrices of the linearsystems represented as in (24) (25) (29) and (30) have thefollowing properties and relationships

(1) The coefficient matrices in (24) and (29) are identicalso are those matrices in (25) and (30)

(2) All the coefficient matrices involved are symmetric

(3) The coefficient matrices in (24) and (25) (correspond-ingly in (29) and (30)) are Hermitian to each other

Proof The first relation is obvious by observation of thecorresponding coefficient matrices in (24) and (29) (corre-spondingly in (25) and (30))

For proving the second property it suffices to show⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119911119899+1⟩ From the presentations of

119911119899+1

and 119911lowast119899+1

as respectively defined in (18) and (19) andfollowing from Lemma 6 we have

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867(120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899) 119860119901119899⟩

= 120590119899⟨119860119867119903lowast

119899 119860119901119899⟩ minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

⟨119860119867119901lowast

119899 119860119911119899+1⟩ = ⟨119860

119867119901lowast

119899 119860 (120590119899119903119899minus 120588119899119860119901119899)⟩

= 120590119899⟨119860119867119901lowast

119899 119860119903119899⟩ minus 120588119899⟨119860119867119901lowast

119899 1198602119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

(34)

The third fact directly comes from the Hermitian prop-erty of the standard Hermitian inner product of two complexvectors defined at the end of the first section see for instance[6]

It is noticed that similarHermitian relationships also holdfor the correspondingmatriced involved in (1) (2) (3) and (4)in [19] when the CSBCGmethod is applied in complex cases

Now an alternative result stating that a 2 times 2 step is alwayssufficient to skip over the pivot breakdown of the BiCORmethod when 120590

119899= 0 just as stated at the end of Section 2

can be shown in the next lemma

Lemma 8 Suppose that the biconjugate 119860-orthonormalization procedure depicted as in Algorithm 1underlying the BiCOR method does not breakdown that is thesituation where 120588

119894= 0 119894 = 0 1 119899 and ⟨119911lowast

119899+1 119860119911119899+1⟩ = 0 If

120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast

119899 119860119901119899⟩ = 0 then the 2 times 2 coefficient

matrices in (24) (25) (29) and (30) are nonsingular

Proof It suffices to show that if 120590119899equiv ⟨119902

lowast

119899 119902119899⟩ equiv

⟨119860119867119901lowast

119899 119860119901119899⟩ = 0 then ⟨119860119867119911lowast

119899+1 119860119901119899⟩ = 0 From Lemma 7

and (18) we have

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119911lowast

119899+1 minus119911119899+1

120588119899

⟩

= minus1

120588119899

⟨119911lowast

119899+1 119860119911119899+1⟩ = 0

(35)

The next proposition shows some properties of theCSBiCOR iterates similar to those of the BiCOR methodpresented in Proposition 5 with some minor changes

Proposition 9 For the CSBiCOR algorithm let 119877119899+1

=

[1199030 1199031 119903

119899]119877lowast119899+1= [119903lowast

0 119903lowast

1 119903

lowast

119899] where 119903

119894 119903lowast119894are replaced

by 119911119894 119911lowast119894appropriately if the 119894th step is a composite 2 times 2

step and let 119875119899+1= [1199010 1199011 119901

119899] 119875lowast119899+1= [119901lowast

0 119901lowast

1 119901

lowast

119899]

Assuming the underlying biconjugate 119860-orthonormalizationprocedure does not break down one has the following proper-ties


(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)

(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1= diag(119863

119896) where 119863

119896is either of order

1 times 1 or 2 times 2 and the sum of the dimensions of 119863119896rsquos is

(119899 + 1)

Proof The properties can be proved with the same frame forthose of Theorem 44 in [19]

Therefore we can show with the above properties thatthe use of 2 times 2 steps is sufficient for the CSBiCORmethod to compute exactly all those well-defined BiCORiterates stably provided that the underlying biconjugate 119860-orthonormalization procedure depicted as in Algorithm 1does not break down

Proposition 10 The CSBiCOR method is able to computeexactly all those well-defined BiCOR iterates stably withoutpivot breakdown assuming that the underlying biconjugate 119860-orthonormalization procedure does not break down

Proof By construction 119909119899+2= 119909119899+ 119891(1)

119899119901119899+ 119891(2)

119899119911119899+1

byinduction 119909

119899isin 1199090+ K119899(119860 1199030) and 119901

119899isin K

119899+1(119860 1199030)

by definition 119911119899+1

isin K119899+2(119860 1199030) Then it follows that

119909119899+2isin 1199090+K119899+2(119860 1199030) From Proposition 9 we have 119903

119899+2perp

119860119867K119899+2(119860119867 119903lowast

0)Therefore 119909

119899+2exactly satisfies the Petrov-

Galerkin condition defining the BiCOR iterates

Before ending this section we present a best approx-imation result for the CSBiCOR method which uses thesame framework as that for the CSBCG method [19] Fordiscussion about convergence results for Krylov subspacesmethods refer to [2 6 32 33] and the references therein Tothe best of our knowledge this is the first attempt to study theconvergence result for the BiCOR method [7]

First let V119898= span(119907

1 1199072 119907

119898) and W

119898=

span(1199081 1199082 119908

119898) denote the Krylov subspaces generated

by the biconjugate 119860-orthonormalization procedure afterrunning Algorithm 1 119898 steps The norms associated with thespacesV

119873equiv C119873 andW

119873equiv C119873 are defined as

|119907|2

119903= 119907119867119872119903119907 |119908|

2

119897= 119908119867119872119897119908 (36)

where 119907 119908 isin C119873 while119872119903and119872

119897are symmetric positive

definite matrices Then we have the following propositionwith the initial guess 119909

0of the linear system being 0 isin C119873

The casewith a nonzero initial guess can be treated adaptively

Proposition 11 Suppose that for all 119907 isin V119873and for all 119908 isin

W119873 one has

10038161003816100381610038161003816119908119867119860211990710038161003816100381610038161003816le Γ|119907|119903|119908|119897 (37)

where Γ is a constant independent of 119907 and 119908 Furthermoreassuming that for those steps in the composite step Biconjugate

119860-Orthogonal Residual (CSBiCOR) method in which we com-pute an approximation 119909

119898 we have

inf119907isinV119898

|119907|119903=1

sup119908isinW

119898

|119908|119897le1

1199081198671198602119907 ge 120574119896ge 120574 gt 0

(38)

Then

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903 le (1 +

Γ

120574) inf119907isinV119898

|119909 minus 119907|119903 (39)

Proof From the Petrov-Galerkin condition (15) we have for119908 isinW

119898

(119860119867119908)119867

(119860119909 minus 119860119909119898) = 119908

1198671198602(119909 minus 119909

119898) = 0 (40)

yielding with arbitrary 119907 isinV119898

1199081198671198602(119909119898minus 119907) = 119908

1198671198602(119909 minus 119907) (41)

It is noted that 119909119898minus 119907 isin V

119898because of the assumption

of the zero initial guess Then taking the sup of both sidesof the above equation for all |119908|

119897le 1 with the inf-sup

condition (38) to bound the left-hand side and the continuityassumption (37) to bound the right-hand side results in

120574119896

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907

10038171003817100381710038171003816100381610038161003816119903le Γ|119909 minus 119907|119903|119908|119897 le Γ|119909 minus 119907|119903 (42)

which yields

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903le |119909 minus 119907|119903 +

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907

10038171003817100381710038171003816100381610038161003816119903le (1 +

Γ

120574) |119909 minus 119907|119903

(43)

with the triangle inequality The final assertion (39) followsimmediately since 119907 isinV

119898is arbitrary

6 Implementation Issues

For the purpose of conveniently and simply presenting theCSBiCOR method we first prove some relations simplifyingthe solutions of the four 2 times 2 linear systems represented asin (24) (25) (29) and (30)

Lemma 12 For the four 2 times 2 linear systems represented as in(24) (25) (29) and (30) the following relations hold

(1) ⟨119901lowast119899 119860119903119899⟩ = ⟨119860119901

119899 119903lowast119899⟩ = ⟨119903

lowast

119899 119860119903119899⟩ = ⟨119903

lowast

119899 119903119899⟩ equiv 120588

119899

and ⟨119911lowast119899+1 119860119903119899⟩ = ⟨119860119911

119899+1 119903lowast

119899⟩ = 0 119891

119899= 119891lowast

119899

(2) ⟨119860119867119901lowast119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast

119899+2⟩ = 0 and

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩ = ⟨119860119911

119899+1 119860119867119903lowast

119899+2⟩ = minus120588

119899+2119891(2)

119899

where 120588119899+2equiv ⟨119903lowast

119899+2 119860119903119899+2⟩ 119892119899= 119892lowast

119899

(3) ⟨119860119867119901lowast119899 119860119911119899+1⟩ = ⟨119860119901

119899 119860119867119911lowast

119899+1⟩ = minus120579

119899+1120588119899 where

120579119899+1equiv ⟨119911lowast

119899+1 119860119911119899+1⟩

Proof (1) If a 1 times 1 step is taken then 119901lowast119899

= 119903lowast

119899+

120573119899119901lowast

119899minus1 giving that ⟨119901lowast

119899 119860119903119899⟩ = ⟨119903

lowast

119899+ 120573119899119901lowast

119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩+120573119899⟨119901lowast

119899minus1 119860119903119899⟩ = ⟨119903

lowast

119899 119860119903119899⟩ because of the last-term


vanishment due to Proposition 9 Analogously the iteration119901119899= 119903119899+ 120573119899119901119899minus1

similarly gives ⟨119860119901119899 119903lowast

119899⟩ = ⟨119860119903

119899+

120573119899119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ + 120573

119899⟨119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ =

⟨119903lowast119899 119860119903119899⟩ from Proposition 9 and the Hermitian property of

the standard Hermitian inner product If a 2times2 step is takenthen by construction as shown in (27) and Proposition 9we have ⟨119901lowast

119899 119860119903119899⟩ = ⟨119903lowast

119899+ 119892lowast(1)


119899minus2

+ 119892lowast(2)


119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩ + 119892

lowast(1)

119899minus2⟨119901lowast

119899minus2

119860119903119899⟩ + 119892

lowast(2)


119899minus1 119860119903119899⟩ = ⟨119903lowast

119899 119860119903119899⟩

Similarly with Proposition 9 and the Hermitian property ofthe standard Hermitian inner product it can be shown that⟨119860119901119899 119903lowast

119899⟩= ⟨119860(119903

119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)

119899minus2119911119899minus1) 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ +

119892(1)

119899minus2⟨119860119901119899minus2 119903lowast

119899⟩+119892(2)


119899⟩= ⟨119860119903

119899 119903lowast

119899⟩= ⟨119903lowast119899 119860119903119899⟩The

equalities ⟨119911lowast119899+1 119860119903119899⟩ = ⟨119860119911

119899+1 119903lowast

119899⟩ = 0 directly follow from

Proposition 9 Finally because the coefficient matrices of thesymmetric systems in (24) and (25) respectively governing119891

119899

and 119891lowast119899are Hermitian to each other as shown in Lemma 7

while the corresponding right-hand sides are conjugate toeach other as shown here we can deduce that 119891

119899= 119891lowast

119899

(2) From Proposition 9 it can directly verify that⟨119860119867119901lowast

119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast

119899+2⟩ = 0 with the fact

that 119860119901119899isin K

119899+2(119860 1199030) and 119860119867119901lowast

119899isin K

119899+2(119860119867 119903lowast

0)

Next it can be noted that if 120590119899= 0 then 119891(2)

119899= 0

Otherwise the first equation in the linear system (24)cannot hold for ⟨119860119867119901lowast

119899 119860119911119899+1⟩ = 0 as proved in Lemma 8

and ⟨119901lowast119899 119860119903119899⟩ = 120588

119899= 0 as shown in the above (1)

Then by construction of (22) and Proposition 9 we have⟨119860119867119911lowast

119899+1 119860119903119899+2⟩ = ⟨(1119891lowast(2)

119899)(119903lowast

119899minus 119903lowast

119899+2minus 119891lowast(1)

119899119860119867119901lowast

119899) 119860119903119899+2⟩

= (1119891lowast(2)119899 )(⟨119903lowast119899 119860119903119899+2⟩minus ⟨119903lowast

119899+2 119860119903119899+2⟩minus119891lowast(1)

119899 ⟨119860119867119901lowast

119899 119860119903119899+2⟩)

= (minus1119891lowast(2)119899 )⟨119903lowast119899+2 119860119903119899+2⟩ = (minus1119891lowast(2)

119899 )120588119899+2 = (minus1119891(2)

119899)120588119899+2

with the fact that 119891119899= 119891lowast

119899proved in the above (1) By

construction of (21) and Proposition 9 we can also show that⟨119860119911119899+1 119860119867119903lowast

119899+2⟩ = minus120588

119899+2119891(2)

119899 Similarly to the relationshipsbetween 119891

119899and 119891lowast

119899 we can deduce that 119892

119899= 119892lowast

119899 because

the coefficient matrices of the symmetric systems in (29) and(30) respectively governing 119892


119899are Hermitian to each

other as shown in Lemma 7 while the corresponding right-hand sides are conjugate to each other as shown here

(3) By construction of (19) and Proposition 9 it fol-lows that ⟨119860119867119901lowast

119899 119860119911119899+1⟩ = ⟨(1120588

119899)(120590119899119903lowast


119899+1) 119860119911119899+1⟩ =

(1120588119899)(120590119899⟨119903lowast

119899 119860119911119899+1⟩ minus ⟨119911

lowast

119899+1 119860119911119899+1⟩) = (minus1120588

119899)⟨119911lowast

119899+1 119860119911119899+1⟩

= minus120579119899+1120588119899 where 120579

119899+1equiv ⟨119911

lowast

119899+1 119860119911119899+1⟩ Because it is

already known that ⟨119860119867119911lowast119899+1 119860119901119899⟩ = ⟨119860119867119901lowast

119899 119860119911119899+1⟩ as

proved in Lemma 7 we have ⟨119860119901119899 119860119867119911lowast

119899+1⟩ = ⟨119860119867119911lowast

119899+1 119860119901119899⟩

= ⟨119860119867119901lowast119899 119860119911119899+1⟩ = minus120579

119899+1120588119899 The proof is completed

As a result of Lemmas 6 and 12 when a 2times2 step is takenthe solutions for119891

119899and 119892119899involved in the four linear systems

represented as in (24) (25) (29) and (30) can be simplifiedonly by solving the following two linear systems

[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119891(1)

119899

119891(2)

119899

]

]

= [120588119899

0]

[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119892(1)

119899

119892(2)

119899

]

]

= [

[

0120588119899+2

119891(2)

119899

]

]

(44)

where 120577119899+1= ⟨119860119867119911lowast

119899+1 119860119911119899+1⟩

It should be emphasized that these above two systemshave the same representations for the CSBCG method [19]but differ in detailed computational formulae for the cor-responding quantities because of the different underlyingprocedures mentioned in Section 1 As a result 119891

119899and 119892

119899can

be explicitly represented as

119891119899=(120577119899+1120588119899 120579119899+1) 1205882

119899

120578119899

119892119899= (120588119899+2

120588119899

120590119899120588119899+2

120579119899+1

)

(45)

where 120578119899equiv 120590119899120577119899+11205882

119899minus 1205792

119899+1

Combining all the recurrences discussed above for eithera 1 times 1 step or a 2 times 2 step and taking the strategy ofreducing the number of matrix-vector multiplications byintroducing an auxiliary vector recurrence and changingvariables adopted for the BiCORmethod [7] as well as for theBiCR method [31] together lead to the CSBiCOR methodThe pseudocode for the preconditioned CSBiCOR with aleft preconditioner 119861 can be represented by Algorithm 3where 119904

119899+1and 119911

119899+1are used to respectively denote the

unpreconditioned and the preconditioned residuals It isobvious to note that the number of matrix-vector multipli-cations in this algorithm remains the same per step as in theBiCOR algorithm [7] Therefore the cost of the use of 2 times 2composite steps is negligible That is 2 times 2 composite stepscost approximately twice as much as 1 times 1 steps just as statedfor the CSBCG method [18] Thus from the point of view ofchoosing 1times 1 or 2times 2 steps there is no significant differencewith respect to algorithm cost However the presentedalgorithm is with the motivation of obtaining smootherand hopefully faster convergence behavior in compari-son with the BiCOR method besides eliminating pivotbreakdowns

For the issue of deciding between 1 times 1 and 2 times2 updates the heuristic based on the magnitudes of theresiduals developed for the CSBCG method [18] is bor-rowed for the CSBiCOR method in order to choose thestep size which maximizes numerical stability as well asto avoid overflow The principle is to avoid ldquospikesrdquo inthe convergence history of the norm of the residuals A2 times 2 update will be chosen if the following circumstancesatisfies

1003817100381710038171003817119903119899+11003817100381710038171003817 gt max 1003817100381710038171003817119903119899

1003817100381710038171003817 1003817100381710038171003817119903119899+2

1003817100381710038171003817 (46)

For completeness of algorithm implementation we recallthe test procedure as follows Define 120587

119899+2equiv 120578119899119903119899+2= 120578119899119903119899minus

120577119899+11205883

119899119860119901119899minus 120579119899+11205882

119899119860119911119899+1

Then with the scaled versions ofthe corresponding quantities the tests 119903

119899+1 le 119903

119899 and


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030) Set 119901

0= 1199030 1199010= 1199030 1199020= 1198601199010 1199020= 1198601198671199010

(3) Compute 1205880= ⟨1199030 1198601199030⟩

(4) Begin LOOP (119899 = 0 1 2 )(5) 120590119899= ⟨119902119899 119902119899⟩

(6) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(7) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(8) 119910119899+1= 119860119904119899+1

(9) 119910119899+1= 119860119867119904119899+1

(10) 120579119899+1= ⟨119904119899+1 119910119899+1⟩

(11) 120577119899+1= ⟨119910119899+1 119910119899+1⟩

(12) if 1 times 1 step then(13) 120572

119899= 120588119899120590119899

(14) 120588119899+1= 120579119899+11205902

119899

(15) 120573119899+1= 120588119899+1120588119899

(16) 119909119899+1= 119909119899+ 120572119899119901119899

(17) 119903119899+1= 119903119899minus 120572119899119902119899

(18) 119903119899+1= 119903119899minus 120572119899119902119899

(19) 119901119899+1= 119904119899+1120590119899+ 120573119899+1119901119899

(20) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(21) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(22) 119899 larr 119899 + 1

(23) else(24) 120575

119899= 120590119899120577119899+11205882

119899minus 1205792

119899+1

(25) 120572119899= 120577119899+11205883

119899120575119899

(26) 120572119899+1= 120579119899+11205882

119899120575119899

(27) 119909119899+2= 119909119899+ 120572119899119901119899+ 120572119899+1119904119899+1

(28) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(29) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(30) solve 119861119911119899+2= 119903119899+2

(31) solve 119861119867119899+2= 119903119899+2

(32) 119899+2= 119860119911119899+2

(33) 119911119899+2= 119860119867119899+2

(34) 120588119899+2= ⟨119911119899+2 119903119899+2⟩

(35) 120573119899+1= 120588119899+2120588119899

(36) 120573119899+2= 120588119899+2120590119899120579119899+1

(37) 119901119899+2= 119911119899+2+ 120573119899+1119901119899+ 120573119899+2119904119899+1

(38) 119902119899+2= 119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(39) 119902119899+2= 119911119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(40) 119899 larr 119899 + 2

(41) end if(42) Check convergence continue if necessary(43) End LOOP

Algorithm 3 Left preconditioned CSBiCOR method

119903119899+2 le 119903

119899+1 can be respectively replaced by 119911

119899+1 le

|120590119899|119903119899 and |120590

119899|120587119899+2 le |120578

119899|119911119899+1 Consequently the

test can be implemented with the code fragment shown inAlgorithm 4

Analogously to the effect obtained by the CSBCGmethod[19] in circumstances where (46) is satisfied the use of2 times 2 updates for the CSBiCOR method is also able to cutoff spikes in the convergence history of the norm of theresiduals possibly caused by taking two 1 times 1 steps in suchcircumstances The resulting CSBiCOR algorithm inheritsthe promising properties of the CSBCG method [18 19]

That is the composite strategy employed can eliminatenear pivot breakdowns while can provide some smoothingof the convergence history of the norm of the residualswithout involving any arbitrary machine-dependent or user-supplied tolerances However it should be emphasized thatthe CSBiCOR method could only eliminate those spikes dueto small pivots in the upper left principal submatrices119879

119896(1 le

119896 le 119898) of 119879119898formed in Proposition 1 but not all spikes

So that the residual norm of the CSBiCOR method doesnot decrease monotonically By the way learning from themathematically equivalent but numerically different variants


If ||119911119899+1|| le |120590

119899|||119903119899|| Then

1 times 1 StepElse||120587119899+2|| = ||120578

119899119903119899minus 120577119899+11205883

119899119902119899minus 120579119899+11205882

119899119910119899+1||

If |120590119899|||120587119899+2|| le |120578

119899|||119911119899+1|| Then

2 times 2 StepElse1 times 1 Step

End IfEnd If

Algorithm 4

Table 1 Structures of the three sets of test problems

Ex Group and name id rows cols Nonzeros Sym1 Dehghanilight in tissue 1873 29282 29282 406084 02 Kimkim1 862 38415 38415 933195 03 HByoung1c 278 841 841 4089 85

of the CSBCGmethod [18] we could also develop such kindsof variants of the CSBiCORmethod which will be taken intofurther account

7 Examples and Numerical Experiments

This section mainly focuses on the analysis of differentnumerical effects of the composite step strategy employedinto the BiCOR method with far from being exhaustive afew typical circumstances of test problems as arising fromelectromagnetics discretizations of 2D3D physical domainsand acoustics which are described in Table 1 All of themare borrowed in the MATLAB format from the Universityof Florida Sparse Matrix Collection provided by Davis [34]in which the meanings of the column headers of Table 1can be found The effect analysis part may provide somesuggestions that when to make use of the composite stepstrategy should make significant progress towards an exactsolution according to the convergence history of the residualnorms It is with the hope that the stabilizing effect ofthe CSBiCOR method could make the residual norm plotbecome smoother and hopefully faster decreasing since faroutlying iterates and residuals are avoided in the smoothedsequences

In a realistic setting one would use a precondi-tioner With appropriate preconditioning techniques suchas those existing well-established preconditioning method-ologies [35ndash37] based on approximate inverses all the fol-lowing involved methods for numerical comparison arevery attractive for solving relevant classes of non-Hermitianlinear systems But this is not the point we pursue hereAll the experiments are performed without preconditioningtechniques in default if without other clarification That isthe preconditioner 119861 in Algorithm 3 will be taken as theidentity matrix except otherwise clarified Refer to the surveyby Benzi [38] and the book by Saad [6] on preconditioning

techniques for improving the performance and reliability ofKrylov subspace methods

For all these test problems below the BiCORmethod willbe implemented using the same code as for the CSBiCORmethodwith the pivot test beingmodified to always choose 1-step update steps for the purpose of conveniently showing thestabilizing effect of the composite step strategy on the BiCORmethod So does for the BCG method as implementated in[18 19]

The experiments have been carried out with machineprecision 10minus16 in double precision floating point arithmeticin MATLAB 704 with a PC-Pentium (R) D CPU 300GHz1 GB of RAMWemake comparisons in four aspects numberof iterations (referred to as Iters) CPU consuming timein seconds (referred to as CPU) log

10of the updated and

final true relative residual 2-norms defined respectively aslog10119903119899211990302and log

10119887 minus 119860119909

119899211990302(referred to as

Relres and TRR) Iters takes the form of ldquolowastlowastrdquo recordingthe total number of iteration steps and the number of 2 times2 iteration steps involved in the corresponding compositestep methods Numerical results in terms of Iters CPU andTRR are reported by means of tables while convergencehistories involved are shown in figures with Iters (on thehorizontal axis) versus Relres (on the vertical axis) Thestopping criterion used here is that the 2-norm of the residualbe reduced by a factor (referred to as TOL) of the 2-normof the initial residual that is 119903

119899211990302lt TOL or when

Iters exceeded the maximal iteration number (referred toas MAXIT) Here we take 119872119860119883119868119879 = 500 All thesetests are started with an initial guess equal to 0 isin C119899Whenever the considered problem contains no right-handside to the original linear system 119860119909 = 119887 let 119887 = 119860119890where 119890 is the 119899 times 1 vector whose elements are all equalto unity such that 119909 = (1 1 1)119879 is the exact solutionA symbol ldquolowastrdquo is used to indicate that the method did notmeet the required TOL beforeMAXIT or did not converge atall


Table 2 Comparison results of Example 1 with TOL = 10minus6

Method BCG BiCOR CSBCG CSBiCORIters 330 318 3273 3180CPU 407807 390256 401300 392116TRR minus60318 minus60150 minus60318 minus60150

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)

Figure 1 Convergence histories of Example 1 with TOL = 10minus6

71 Example 1 DehghaniLight in Tissue The first example isone in which the BCG method and the BiCOR method havealmost quite smooth convergence behaviors and as few asnear breakdownsTheCSBiCORmethodhas exactly the sameconvergence behavior as the BiCOR method by investigatingTable 2 and Figure 1(b) Under such circumstances when theconvergence history of the norm of the residuals is smoothenough so that the composite step strategy does not gain alot for the CSBiCOR method the BiCOR method is muchpreferred and there is no need to employ the compositestep strategy because the MATLAB implementation maybe probably a little penalizing for the CSBiCOR code withrespect to CPU By the way from Figure 1(c) the CSBCGmethod works as predicted in [18 19] smoothing a little theconvergence history of the BCG method In particular theCSBCG method computes a subset of the BCG iterates byclipping three ldquospikesrdquo in the BCG history by means of thecomposite 2times2 steps from the particular investigations givenin the bottom first two plots of Figures 1(d) 1(e) and Table 2

Another interesting observation is that the good prop-erties of the BiCOR method in terms of fast convergencespeed and smooth convergence behavior in comparison with

the BCGmethod are instinctually inherited by the CSBiCORmethod so that the CSBiCOR method outperforms theCSBCGmethod in aspects of Iters andCPU as well as smootheffect as shown in Table 2 and Figure 1(f)

72 Example 2 KimKim1 This is a good test example tovividly demonstrate the efficacy of the composite step strategyas depicted in Figure 2 The BCG method as observed inFigure 2(a) takes on many local ldquospikesrdquo in the convergencecurve In such a case the CSBCG method does a wonderfulattempt to smooth the convergence history of the norm ofthe residuals as shown in Figure 2(c) although not leadingto a monotonic decreasing convergence behavior So doesthe CSBiCOR method to the BiCOR method by observa-tion of Figure 2(d) Moreover the CSBiCOR method seemsto take less CPU than the BiCOR method while keepingapproximately the same TRR when the BiCOR method isimplemented using the same code as for the CSBiCORmethod and the pivot test is modified to always choose 1-stepupdate steps Finally the CSBiCORmethod is again shown tobe competitive to the CSBCG method as seen in Table 3 andFigure 2(b)


0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)




73 Example 3 HBYoung1c In the third example the BCGand BiCOR methods have small irregularities and do notconverge superlinearly as reflected in Figure 3(a) The twomethods above seem to have almost the same ldquoasymptoticalrdquospeed of convergence while the BiCOR method seems a littlesmoother than the BCG method From Figures 3(c) and3(d) the composite step strategy seems to play a surprisinglygood stabilizing effect to the BCG and BiCOR methodFurthermore the CSBCG and CSBiCOR methods take lessCPU respectively than the BCG and BiCOR methods asreported in Table 4 It is stressed that although the CSBiCORmethod consumes a little more Iters and CPU than theCSBCGmethod it presents a little more smooth convergencebehavior than the latter method as displayed in Figure 3(b)This is still thanks to the promising advantages of the

empirically observed stability and fast convergence rate of theBiCOR method over the BCG method

8 Concluding Remarks

We have presented a new interesting variant of the BiCORmethod for solving non-Hermitian systems of linear equa-tions Our approach is naturally based on and inspired by thecomposite step strategy taken for the CSBCGmethod [18 19]It is noted that exact pivot breakdowns are rare in practicehowever near breakdowns could cause severe numericalinstability The resulting CSBiCOR method is both theo-retically and numerically demonstrated to avoid near pivotbreakdowns and compute all the well-defined BiCOR iterates


0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)




stably with only minor modifications with the assumptionthat the underlying biconjugate 119860-orthonormalization pro-cedure does not break down Besides reducing the number ofspikes in the convergence history of the norm of the residualsto the greatest extent the CSBiCOR method could providesome further practically desired smoothing behavior towardsstabilizing the behavior of the BiCOR method when it haserratic convergence behaviors Additionally the CSBiCORmethod seems to be superior to the CSBCGmethod to someextent because of the inherited promising advantages of theempirically observed stability and fast convergence rate of theBiCOR method over the BCG method

Since the BiCOR method is the most basic variant ofthe family of Lanczos biconjugate 119860-orthonormalizationmethods its improvement will analogously lead to similar

improvements for the CORS and BiCORSTAB methodswhich is under investigation

It is important to note the nonnegligible added complex-ity when deciding the composite step strategyThat is we haveto make some extra computation before deciding to take a2 times 2 step Therefore when the 2 times 2 step will not be takenafter the decision test it will lead to extra computation as wellas CPU consuming time It seems critical for us to optimizethe code to implement the CSBiCOR method

Acknowledgments

The authors would like to thank Professor Randolph EBank for showing us the ldquoPLTMGrdquo software package andfor his insightful and beneficial discussions and suggestions


Furthermore the authors wish to acknowledge ProfessorZhongxiao Jia and the anonymous referees for their sug-gestions in the presentation of this article This research issupported by NSFC (60973015 61170311 11126103 1120105511171054) the Specialized Research Fund for the DoctoralProgram of Higher Education (20120185120026) SichuanProvince Sci amp Tech Research Project (2011JY0002) and theFundamental Research Funds for the Central Universities

References

[1] Y Saad and H A van der Vorst ldquoIterative solution of linearsystems in the 20th centuryrdquo Journal of Computational andApplied Mathematics vol 43 pp 1155ndash1174 2005

[2] V Simoncini and D B Szyld ldquoRecent computational devel-opments in Krylov subspace methods for linear systemsrdquoNumerical Linear Algebra with Applications vol 14 no 1 pp1ndash59 2007

[3] J Dongarra and F Sullivan ldquoGuest editorsrsquointroduction to thetop 10 algorithmsrdquoComputer Science and Engineering vol 2 pp22ndash23 2000

[4] B Philippe and L Reichel ldquoOn the generation of Krylovsubspace basesrdquo Applied Numerical Mathematics vol 62 no 9pp 1171ndash1186 2012

[5] A Greenbaum IterativeMethods for Solving Linear Systems vol17 of Frontiers in Applied Mathematics SIAM Philadelphia PaUSA 1997

[6] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2nd edition 2003

[7] Y-F Jing T-ZHuang Y Zhang et al ldquoLanczos-type variants ofthe COCR method for complex nonsymmetric linear systemsrdquoJournal of Computational Physics vol 228 no 17 pp 6376ndash63942009

[8] Y-F Jing B Carpentieri and T-Z Huang ldquoExperiments withLanczos biconjugate 119860-orthonormalization methods for MoMdiscretizations of Maxwellrsquos equationsrdquo Progress in Electromag-netics Research vol 99 pp 427ndash451 2009

[9] Y-F Jing T-Z Huang Y Duan and B Carpentieri ldquoAcomparative studyof iterative solutions to linear systems arisingin quantum mechanicsrdquo Journal of Computational Physics vol229 no 22 pp 8511ndash8520 2010

[10] B Carpentieri Y-F Jing and T-Z Huang ldquoThe BICOR andCORS iterative algorithms for solving nonsymmetric linearsystemsrdquo SIAM Journal on Scientific Computing vol 33 no 5pp 3020ndash3036 2011

[11] R Fletcher ldquoConjugate gradient methods for indefinite sys-temsrdquo in Numerical Analysis (Proc 6th Biennial Dundee ConfUniv Dundee Dundee 1975) vol 506 of Lecture Notes inMathematics pp 73ndash89 Springer Berlin Germany 1976

[12] W Joubert Generalized conjugate gradient and lanczos methodsfor the solution of nonsymmetric systems of linear equations[PhD thesis] University of Texas Austin Tex USA 1990

[13] W Joubert ldquoLanczosmethods for the solution of nonsymmetricsystems of linear equationsrdquo SIAM Journal on Matrix Analysisand Applications vol 13 no 3 pp 926ndash943 1992

[14] M H Gutknecht ldquoA completed theory of the unsymmetricLanczos process and related algorithms Irdquo SIAM Journal onMatrix Analysis and Applications vol 13 no 2 pp 594ndash6391992

[15] M H Gutknecht ldquoA completed theory of the unsymmetricLanczos process and related algorithms IIrdquo SIAM Journal onMatrix Analysis and Applications vol 15 no 1 pp 15ndash58 1994

[16] M H Gutknecht ldquoBlock Krylov space methods for linearsystems withmultiple right-hand sides an introductionrdquo inModern Mathematical Models Methods and Algorithms for RealWorld Systems A H Siddiqi I S Duff and O ChristensenEds Anamaya Publishers New Delhi India 2006

[17] D G Luenberger ldquoHyperbolic pairs in the method of conjugategradientsrdquo SIAM Journal on Applied Mathematics vol 17 pp1263ndash1267 1969

[18] R E Bank and T F Chan ldquoAn analysis of the composite stepbiconjugate gradient methodrdquoNumerischeMathematik vol 66no 3 pp 295ndash319 1993

[19] R E Bank and T F Chan ldquoA composite step bi-conjugate gra-dient algorithm for nonsymmetric linear systemsrdquo NumericalAlgorithms vol 7 no 1 pp 1ndash16 1994

[20] R W Freund and N M Nachtigal ldquoQMR a quasi-minimalresidualmethod for non-Hermitian linear systemsrdquoNumerischeMathematik vol 60 no 3 pp 315ndash339 1991

[21] M H Gutknecht ldquoThe unsymmetric Lanczos algorithms andtheir relations to Pade approximation continued fraction andthe QD algorithmrdquo in Proc Copper Mountain Conference onIterative Methods Book 2 Breckenridge Co BreckenridgeColo USA 1990

[22] B N Parlett D R Taylor and Z A Liu ldquoA look-aheadLanczos algorithm for unsymmetric matricesrdquo Mathematics ofComputation vol 44 no 169 pp 105ndash124 1985

[23] B N Parlett ldquoReduction to tridiagonal form and minimalrealizationsrdquo SIAM Journal onMatrix Analysis andApplicationsvol 13 no 2 pp 567ndash593 1992

[24] C Brezinski M Redivo Zaglia and H Sadok ldquoAvoidingbreakdown and near-breakdown in Lanczos type algorithmsrdquoNumerical Algorithms vol 1 no 3 pp 261ndash284 1991

[25] C Brezinski M Redivo Zaglia and H Sadok ldquoA breakdown-free Lanczos type algorithm for solving linear systemsrdquoNumerische Mathematik vol 63 no 1 pp 29ndash38 1992

[26] C Brezinski M Redivo-Zaglia and H Sadok ldquoBreakdowns inthe implementation of the Lanczos method for solving linearsystemsrdquo Computers amp Mathematics with Applications vol 33no 1-2 pp 31ndash44 1997

[27] C Brezinski M R Zaglia and H Sadok ldquoNew look-aheadLanczos-type algorithms for linear systemsrdquoNumerische Math-ematik vol 83 no 1 pp 53ndash85 1999

[28] N M Nachtigal A look-ahead variant of the lanczos algorithmand its application to the quasi-minimal residual method for non-Hermitian linear systems [PhD thesis] Massachusettes Instituteof Technology Cambridge Mass USA 1991

[29] R W Freund M H Gutknecht and N M Nachtigal ldquoAnimplementation of the look-ahead Lanczos algorithm for non-Hermitianmatricesrdquo SIAM Journal on Scientific Computing vol14 no 1 pp 137ndash158 1993

[30] M H Gutknecht ldquoLanczos-type solvers for nonsymmetriclinear systems of equationsrdquo in Acta Numerica 1997 vol 6of Acta Numerica pp 271ndash397 Cambridge University PressCambridge UK 1997

[31] T Sogabe M Sugihara and S-L Zhang ldquoAn extension of theconjugate residual method to nonsymmetric linear systemsrdquoJournal of Computational andAppliedMathematics vol 226 no1 pp 103ndash113 2009


[32] P Concus G H Golub and D P OrsquoLeary ldquoA generalizedconjugate gradient method for the numerical solution of ellipticpartial differential equationsrdquo in Sparse Matrix Computations(Proc Sympos Argonne Nat Lab Lemont Ill 1975) pp 309ndash332 Academic Press New York NY USA 1976

[33] A B J Kuijlaars ldquoConvergence analysis of Krylov subspaceiterations with methods from potential theoryrdquo SIAM Reviewvol 48 no 1 pp 3ndash40 2006

[34] T Davis ldquoThe university of Florida sparse matrix collectionrdquoNA Digest vol 97 no 23 1997

[35] T Huckle ldquoApproximate sparsity patterns for the inverse of amatrix and preconditioningrdquo Applied Numerical Mathematicsvol 30 no 2-3 pp 291ndash303 1999

[36] G A Gravvanis ldquoExplicit approximate inverse preconditioningtechniquesrdquo Archives of Computational Methods in Engineeringvol 9 no 4 pp 371ndash402 2002

[37] B Carpentieri I S Duff L Giraud and G Sylvand ldquoCom-bining fast multipole techniques and an approximate inversepreconditioner for large electromagnetism calculationsrdquo SIAMJournal on Scientific Computing vol 27 no 3 pp 774ndash792 2005

[38] M Benzi ldquoPreconditioning techniques for large linear systemsa surveyrdquo Journal of Computational Physics vol 182 no 2 pp418ndash477 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 408167 16 pageshttpdxdoiorg1011552013408167

Research ArticleExploiting the Composite Step Strategy tothe Biconjugate 119860-Orthogonal Residual Method forNon-Hermitian Linear Systems

Yan-Fei Jing1 Ting-Zhu Huang1 Bruno Carpentieri2 and Yong Duan1

1 School of Mathematical Sciences Institute of Computational Science University of Electronic Science and Technology of ChinaChengdu Sichuan 611731 China

2 Institute of Mathematics and Computing Science University of Groningen Nijenborgh 9 PO Box 4079700 AK Groningen The Netherlands

Correspondence should be addressed to Yan-Fei Jing 00jyfvictory163com

Received 15 October 2012 Accepted 19 December 2012

Academic Editor Zhongxiao Jia

Copyright copy 2013 Yan-Fei Jing et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The Biconjugate 119860-Orthogonal Residual (BiCOR) method carried out in finite precision arithmetic by means of the biconjugate119860-orthonormalization procedure may possibly tend to suffer from two sources of numerical instability known as two kinds ofbreakdowns similarly to those of the Biconjugate Gradient (BCG)methodThis paper naturally exploits the composite step strategyemployed in the development of the composite step BCG (CSBCG)method into the BiCORmethod to cure one of the breakdownscalled as pivot breakdown Analogously to the CSBCGmethod the resulting interesting variant with only a minor modification tothe usual implementation of the BiCOR method is able to avoid near pivot breakdowns and compute all the well-defined BiCORiterates stably on the assumption that the underlying biconjugate 119860-orthonormalization procedure does not break down Anotherbenefit acquired is that it seems to be a viable algorithm providing some further practically desired smoothing of the convergencehistory of the norm of the residuals which is justified by numerical experiments In addition the exhibited method inherits thepromising advantages of the empirically observed stability and fast convergence rate of the BiCOR method over the BCG methodso that it outperforms the CSBCG method to some extent

1 Introduction

The computational cost of many simulations with integralequations or partial differential equations that model theprocess is dominated by the solution of systems of linearequations of the form 119860119909 = 119887 The field of iterativemethods for solving linear systems has been observed as anexplosion of activity spurred by demand due to extraordinarytechnological advances in engineering and sciences in thepast half century [1] Krylov subspace methods belong to oneof the most widespread and extensively accepted techniquesfor iterative solution of todayrsquos large-scale linear systems [2]With respect to ldquothe greatest influence on the developmentand practice of science and engineering in the 20th centuryrdquoas written by Dongarra and Sullivan [3] Krylov subspace

methods are considered as one of the ldquoTop Ten Algorithmsof the Centuryrdquo

Many advances in the development of Krylov subspacemethods have been inspired and made by the study of evenmore effective approaches to linear systems Variousmethodsdiffer in the way they extract information from Krylovspaces A basis for the underlying Krylov subspace to formiterative recurrences involved is usually constructed based onthe state-of-the-art Arnoldi orthogonalization procedure orLanczos biorthogonalization procedure see recent excellentand thorough review articles by Simoncini and Szyld [2]and by Philippe and Reichel [4] or monographs such asthose of Greenbaum [5] and Saad [6] A novel Lanczos-typebiconjugate 119860-orthonormalization procedure has recentlybeen established to give birth to a new family of efficient











119894119895)119873times119873


119860119867equiv 119860119879

= (119886119895119894)119873times119873 (1)



⟨119906 119907⟩ = 119906119867119907 =

119873

sum

119894=1

119906119894119907119894 (2)


K119905 (119860 119907) = span 119907 119860119907 119860

2119907 119860

119905minus1119907 (3)






119898(119860119867 1199081) where 119907

1and 119908

1are chosen




and 119860119907119895+1




119895 119895 = 1 2 119898 and 119908

119894 119894 =


⟨120596119894 119860119907119895⟩ = 120575119894119895 1 le 119894 119895 le 119898 (4)


119898] and 119882

119898=

[1199081 1199082 119908


119898the extended


119879119898= [

119879119898

120575119898+1119890119879

119898

] (5)

(1) Choose 1199071 1205961 such that ⟨120596

1 1198601199071⟩ = 1

(2) Set 1205731= 1205751equiv 0 120596

0= 1199070equiv 0 isin C119873

(3) for 119895 = 1 2 do(4) 120572

119895= ⟨120596119895 119860(119860119907

119895)⟩



(7) 120575119895+1=10038161003816100381610038161003816⟨119895+1 119860119907119895+1⟩10038161003816100381610038161003816

12

(8) 120573119895+1= ⟨119895+1 119860119907119895+1⟩ 120575119895+1

(9) 119907119895+1= 119907119895+1120575119895+1

(10) 120596119895+1= 119895+1120573119895+1

(11) end for


where

119879119898=

[[[[[[

[

12057211205732

120575212057221205733

120575119898minus1

120572119898minus1

120573119898

120575119898120572119898

]]]]]]

]

(6)


1 120572

119898 1205732 120573

119898




119860119881119898= 119881119898119879119898+ 120575119898+1119907119898+1119890119879

119898 (7)

119860119867119882119898= 119882119898119879119867

119898+ 120573119898+1120596119898+1119890119879

119898 (8)

119882119867

119898119860119881119898= 119868119898 (9)

119882119867

1198981198602119881119898= 119879119898 (10)


119898(119860 1199071) and

K119898(119860119867 1199081) where 119907

1and 119908




119898(119860 1199071) and

K119898(119860119867 1199081) where 119907

1and 119908














119898




119879119898= 119871119898119863119898119880119898 (11)




diagonal blocks


119896(1 le 119896 le 119898) of 119879


factored


119879119898



119879119896= 119871119896119863119896119880119896 (12)









[119889119896120573119896+1

120575119896+1120572119896+1

] (13)


and 120572119896+1










119898equiv 119860119867 span(119882

119898) =

119860119867K119898(119860119867 1199081) where119882


1=

1199030||1199030||2and 119908


1 1198601199071⟩ = 0


1||1198601199071||2

2subjecting





0+K119898(119860 1199071) of dimension


119887 minus 119860119909119898perpL119898 (14)


(119860119867119882119898)119867

(119887 minus 119860119909119898) = 0 (15)




0+ K119898(119860119867 1199081) of dimension 119898 by

satisfying


119898perp 119860K

119898(119860 1199071) (16)


(119860119881119898)119867(119887lowastminus 119860119867119909119898) = 0 (17)


119898is







= [1199030 1199031 119903

119899] 119877lowast119899+1

=

[119903lowast

0 119903lowast

1 119903

lowast

119899] and 119875

119899+1= [119901

0 1199011 119901

119899] 119875lowast119899+1

=

[119901lowast

0 119901lowast

1 119901

lowast


(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030)


0 1199020= 1198601199010 119902lowast0= 119860119867119901lowast

0 1199030= 1198601199030 1205880= ⟨119903lowast

0 1199030⟩

(4) for 119899 = 0 1 do(5) 120590

119899= ⟨119902lowast

119899 119902119899⟩

(6) 120572119899= 120588119899120590119899

(7) 119909119899+1= 119909119899+ 120572119899119901119899

(8) 119903119899+1= 119903119899minus 120572119899119902119899

(9) 119909lowast

119899+1= 119909lowast

119899+ 120572119899119901lowast

119899

(10) 119903lowast119899+1= 119903lowast

119899minus 120572119899119902lowast

119899

(11) 119903119899+1= 119860119903119899+1

(12) 120588119899+1= ⟨119903lowast

119899+1 119903119899+1⟩

(13) if 120588119899+1= 0method fails

(14) 120573119899+1= 120588119899+1120588119899

(15) 119901119899+1= 119903119899+1+ 120573119899+1119901119899

(16) 119901lowast119899+1= 119903lowast

119899+1+ 120573119899+1119901lowast

119899

(17) 119902119899+1= 119903119899+1+ 120573119899+1119902119899

(18) 119902lowast119899+1= 119860119867119901lowast

119899



(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1

is diagonal


(1) 120588119899equiv ⟨119903lowast

119899 119903119899⟩ equiv ⟨119903

lowast

119899 119860119903119899⟩ = 0 but 119903lowast

119899and 119860119903

119899are not


(2) 120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast





119899= 0 while 120588

119899= 0






119899+1 119903lowast




119899+1and 119903lowast







0) as follows

119911119899+1= 120590119899119903119899+1

= 120590119899119903119899minus 120588119899119860119901119899

(18)

119911lowast

119899+1= 120590119899119903lowast

119899+1

= 120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899

(19)


K119899+2(119860 1199030) and 119909lowast


0+ K119899+2(119860119867 119903lowast

0) in step (119899 + 2)

as follows

119909119899+2= 119909119899+ [119901119899 119911119899+1] 119891119899

119909lowast

119899+2= 119909lowast

119899+ [119901lowast

119899 119911lowast

119899+1] 119891lowast

119899

(20)


where 119891119899 119891lowast




119899+2isin

K119899+3(119860119867 119903lowast


119903119899+2= 119903119899minus 119860 [119901

119899 119911119899+1] 119891119899 (21)

119903lowast

119899+2= 119903lowast

119899minus 119860119867[119901lowast

119899 119911lowast

119899+1] 119891lowast

119899 (22)


⟨[119901lowast

119899 119911lowast

119899+1] 119860119903119899+2⟩ = 0

⟨[119901119899 119911119899+1] 119860119867119903lowast

119899+2⟩ = 0

(23)


119899


[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119891(1)

119899

119891(2)

119899

]

= [⟨119901lowast

119899 119860119903119899⟩

⟨119911lowast

119899+1 119860119903119899⟩]

(24)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119891lowast(1)

119899

119891lowast(2)

119899

]

= [⟨119860119901119899 119903lowast

119899⟩

⟨119860119911119899+1 119903lowast

119899⟩]

(25)



119901lowast

119899+2isin K119899+3(119860119867 119903lowast


the following form

119901119899+2= 119903119899+2+ [119901119899 119911119899+1] 119892119899 (26)

119901lowast

119899+2= 119903lowast

119899+2+ [119901lowast

119899

119911lowast

119899+1] 119892lowast

119899 (27)

where 119892119899 119892lowast

119899isin C2


⟨[119901lowast

119899 119911lowast

119899+1] 1198602119901119899+2⟩ = 0

⟨[119901119899 119911119899+1] (119860119867)2

119901lowast

119899+2⟩ = 0

(28)


119899and

119892lowast

119899

[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119892(1)

119899

119892(2)

119899

]

= minus[⟨119860119867119901lowast

119899 119860119903119899+2⟩

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩]

(29)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119892lowast(1)

119899

119892lowast(2)

119899

]

= minus[⟨119860119901119899 119860119867119903lowast

119899+2⟩

⟨119860119911119899+1 119860119867119903lowast

119899+2⟩]

(30)


119899+2 119903119899+2

119909lowast119899+2

119903lowast119899+2

119901119899+2

119901lowast119899+2





the following hold

⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119903119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ = ⟨119902

lowast

119899 119902119899⟩ equiv 120590119899

(31)

Proof If 119901119899 119901lowast


119901119899= 119903119899+ 120573119899119901119899minus1

119901lowast119899

= 119903lowast

119899+ 120573119899119901lowast



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast

119899minus 120573119899119901lowast

119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899


⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901119899minus 120573119899119901119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899 119860119901119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(32)

If 119901119899 119901lowast


119899=

119903119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)

119899minus2119911119899minus1

119901lowast119899= 119903lowast

119899+ 119892lowast(1)


119899minus2+ 119892lowast(2)


119899minus1as



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast





119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892lowast(1)

119899minus2⟨119860119867119901lowast

119899minus2 119860119901119899⟩


119899minus2⟨119860119867119911lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901

119899minus 119892(1)


119899minus2119911119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892(1)

119899minus2⟨119860119867119901lowast

119899 119860119901119899minus2⟩

minus 119892(2)

119899minus2⟨119860119867119901lowast

119899 119860119911119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(33)








119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast


119911119899+1

and 119911lowast119899+1


⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867(120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899) 119860119901119899⟩

= 120590119899⟨119860119867119903lowast

119899 119860119901119899⟩ minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

⟨119860119867119901lowast

119899 119860119911119899+1⟩ = ⟨119860

119867119901lowast

119899 119860 (120590119899119903119899minus 120588119899119860119901119899)⟩

= 120590119899⟨119860119867119901lowast

119899 119860119903119899⟩ minus 120588119899⟨119860119867119901lowast

119899 1198602119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

(34)







119894= 0 119894 = 0 1 119899 and ⟨119911lowast

119899+1 119860119911119899+1⟩ = 0 If

120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast




lowast

119899 119902119899⟩ equiv

⟨119860119867119901lowast

119899 119860119901119899⟩ = 0 then ⟨119860119867119911lowast

119899+1 119860119901119899⟩ = 0 From Lemma 7

and (18) we have

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119911lowast

119899+1 minus119911119899+1

120588119899

⟩

= minus1

120588119899

⟨119911lowast

119899+1 119860119911119899+1⟩ = 0

(35)



=

[1199030 1199031 119903

119899]119877lowast119899+1= [119903lowast

0 119903lowast

1 119903

lowast

119899] where 119903



step and let 119875119899+1= [1199010 1199011 119901

119899] 119875lowast119899+1= [119901lowast

0 119901lowast

1 119901

lowast

119899]



(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)

(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1= diag(119863

119896) where 119863



(119899 + 1)





119899119901119899+ 119891(2)

119899119911119899+1

byinduction 119909

119899isin 1199090+ K119899(119860 1199030) and 119901

119899isin K

119899+1(119860 1199030)




119899+2perp

119860119867K119899+2(119860119867 119903lowast

0)Therefore 119909





1 1199072 119907

119898) and W

119898=

span(1199081 1199082 119908





|119907|2

119903= 119907119867119872119903119907 |119908|

2

119897= 119908119867119872119897119908 (36)







W119873 one has

10038161003816100381610038161003816119908119867119860211990710038161003816100381610038161003816le Γ|119907|119903|119908|119897 (37)



119898 we have


|119907|119903=1

sup119908isinW

119898

|119908|119897le1

1199081198671198602119907 ge 120574119896ge 120574 gt 0

(38)

Then

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903 le (1 +

Γ

120574) inf119907isinV119898

|119909 minus 119907|119903 (39)


119898

(119860119867119908)119867

(119860119909 minus 119860119909119898) = 119908

1198671198602(119909 minus 119909

119898) = 0 (40)


1199081198671198602(119909119898minus 119907) = 119908

1198671198602(119909 minus 119907) (41)






120574119896

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907


which yields

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903le |119909 minus 119907|119903 +

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907

10038171003817100381710038171003816100381610038161003816119903le (1 +

Γ

120574) |119909 minus 119907|119903

(43)


119898is arbitrary




(1) ⟨119901lowast119899 119860119903119899⟩ = ⟨119860119901

119899 119903lowast119899⟩ = ⟨119903

lowast

119899 119860119903119899⟩ = ⟨119903

lowast

119899 119903119899⟩ equiv 120588

119899

and ⟨119911lowast119899+1 119860119903119899⟩ = ⟨119860119911

119899+1 119903lowast

119899⟩ = 0 119891

119899= 119891lowast

119899

(2) ⟨119860119867119901lowast119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast

119899+2⟩ = 0 and

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩ = ⟨119860119911

119899+1 119860119867119903lowast

119899+2⟩ = minus120588

119899+2119891(2)

119899


119899+2 119860119903119899+2⟩ 119892119899= 119892lowast

119899

(3) ⟨119860119867119901lowast119899 119860119911119899+1⟩ = ⟨119860119901

119899 119860119867119911lowast

119899+1⟩ = minus120579

119899+1120588119899 where

120579119899+1equiv ⟨119911lowast

119899+1 119860119911119899+1⟩


= 119903lowast

119899+

120573119899119901lowast


119899 119860119903119899⟩ = ⟨119903

lowast

119899+ 120573119899119901lowast

119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩+120573119899⟨119901lowast

119899minus1 119860119903119899⟩ = ⟨119903

lowast





119899⟩ = ⟨119860119903

119899+

120573119899119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ + 120573

119899⟨119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ =



119899 119860119903119899⟩ = ⟨119903lowast

119899+ 119892lowast(1)


119899minus2

+ 119892lowast(2)


119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩ + 119892

lowast(1)


119899minus2

119860119903119899⟩ + 119892

lowast(2)


119899minus1 119860119903119899⟩ = ⟨119903lowast

119899 119860119903119899⟩


119899⟩= ⟨119860(119903

119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)


119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ +

119892(1)


119899⟩+119892(2)


119899⟩= ⟨119860119903

119899 119903lowast

119899⟩= ⟨119903lowast119899 119860119903119899⟩The


119899+1 119903lowast



119899



119899= 119891lowast

119899


119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast


that 119860119901119899isin K

119899+2(119860 1199030) and 119860119867119901lowast

119899isin K

119899+2(119860119867 119903lowast

0)


119899= 0



and ⟨119901lowast119899 119860119903119899⟩ = 120588



119899+1 119860119903119899+2⟩ = ⟨(1119891lowast(2)

119899)(119903lowast


119899+2minus 119891lowast(1)

119899119860119867119901lowast

119899) 119860119903119899+2⟩


119899+2 119860119903119899+2⟩minus119891lowast(1)

119899 ⟨119860119867119901lowast

119899 119860119903119899+2⟩)


119899 )120588119899+2 = (minus1119891(2)

119899)120588119899+2




119899+2⟩ = minus120588

119899+2119891(2)




119899= 119892lowast

119899 because






119899 119860119911119899+1⟩ = ⟨(1120588

119899)(120590119899119903lowast


119899+1) 119860119911119899+1⟩ =

(1120588119899)(120590119899⟨119903lowast

119899 119860119911119899+1⟩ minus ⟨119911

lowast

119899+1 119860119911119899+1⟩) = (minus1120588

119899)⟨119911lowast

119899+1 119860119911119899+1⟩

= minus120579119899+1120588119899 where 120579

119899+1equiv ⟨119911

lowast

119899+1 119860119911119899+1⟩ Because it is


119899 119860119911119899+1⟩ as


119899+1⟩ = ⟨119860119867119911lowast

119899+1 119860119901119899⟩

= ⟨119860119867119901lowast119899 119860119911119899+1⟩ = minus120579





[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119891(1)

119899

119891(2)

119899

]

]

= [120588119899

0]

[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119892(1)

119899

119892(2)

119899

]

]

= [

[

0120588119899+2

119891(2)

119899

]

]

(44)

where 120577119899+1= ⟨119860119867119911lowast

119899+1 119860119911119899+1⟩


119899and 119892

119899can


119891119899=(120577119899+1120588119899 120579119899+1) 1205882

119899

120578119899

119892119899= (120588119899+2

120588119899

120590119899120588119899+2

120579119899+1

)

(45)

where 120578119899equiv 120590119899120577119899+11205882

119899minus 1205792

119899+1


119899+1and 119911




1003817100381710038171003817119903119899+11003817100381710038171003817 gt max 1003817100381710038171003817119903119899

1003817100381710038171003817 1003817100381710038171003817119903119899+2

1003817100381710038171003817 (46)


119899+2equiv 120578119899119903119899+2= 120578119899119903119899minus

120577119899+11205883

119899119860119901119899minus 120579119899+11205882

119899119860119911119899+1


119899+1 le 119903

119899 and


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030) Set 119901

0= 1199030 1199010= 1199030 1199020= 1198601199010 1199020= 1198601198671199010

(3) Compute 1205880= ⟨1199030 1198601199030⟩

(4) Begin LOOP (119899 = 0 1 2 )(5) 120590119899= ⟨119902119899 119902119899⟩

(6) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(7) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(8) 119910119899+1= 119860119904119899+1

(9) 119910119899+1= 119860119867119904119899+1

(10) 120579119899+1= ⟨119904119899+1 119910119899+1⟩

(11) 120577119899+1= ⟨119910119899+1 119910119899+1⟩


119899= 120588119899120590119899

(14) 120588119899+1= 120579119899+11205902

119899

(15) 120573119899+1= 120588119899+1120588119899

(16) 119909119899+1= 119909119899+ 120572119899119901119899

(17) 119903119899+1= 119903119899minus 120572119899119902119899

(18) 119903119899+1= 119903119899minus 120572119899119902119899

(19) 119901119899+1= 119904119899+1120590119899+ 120573119899+1119901119899

(20) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(21) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(22) 119899 larr 119899 + 1

(23) else(24) 120575

119899= 120590119899120577119899+11205882

119899minus 1205792

119899+1

(25) 120572119899= 120577119899+11205883

119899120575119899

(26) 120572119899+1= 120579119899+11205882

119899120575119899

(27) 119909119899+2= 119909119899+ 120572119899119901119899+ 120572119899+1119904119899+1

(28) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(29) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(30) solve 119861119911119899+2= 119903119899+2

(31) solve 119861119867119899+2= 119903119899+2

(32) 119899+2= 119860119911119899+2

(33) 119911119899+2= 119860119867119899+2

(34) 120588119899+2= ⟨119911119899+2 119903119899+2⟩

(35) 120573119899+1= 120588119899+2120588119899

(36) 120573119899+2= 120588119899+2120590119899120579119899+1

(37) 119901119899+2= 119911119899+2+ 120573119899+1119901119899+ 120573119899+2119904119899+1

(38) 119902119899+2= 119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(39) 119902119899+2= 119911119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(40) 119899 larr 119899 + 2



119903119899+2 le 119903


119899+1 le

|120590119899|119903119899 and |120590

119899|120587119899+2 le |120578





119896(1 le




If ||119911119899+1|| le |120590

119899|||119903119899|| Then

1 times 1 StepElse||120587119899+2|| = ||120578

119899119903119899minus 120577119899+11205883

119899119902119899minus 120579119899+11205882

119899119910119899+1||

If |120590119899|||120587119899+2|| le |120578

119899|||119911119899+1|| Then


End IfEnd If

Algorithm 4












10119887 minus 119860119909



119899211990302lt TOL or when





0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)







0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















119894119895)119873times119873


119860119867equiv 119860119879

= (119886119895119894)119873times119873 (1)



⟨119906 119907⟩ = 119906119867119907 =

119873

sum

119894=1

119906119894119907119894 (2)


K119905 (119860 119907) = span 119907 119860119907 119860

2119907 119860

119905minus1119907 (3)






119898(119860119867 1199081) where 119907

1and 119908

1are chosen




and 119860119907119895+1




119895 119895 = 1 2 119898 and 119908

119894 119894 =


⟨120596119894 119860119907119895⟩ = 120575119894119895 1 le 119894 119895 le 119898 (4)


119898] and 119882

119898=

[1199081 1199082 119908


119898the extended


119879119898= [

119879119898

120575119898+1119890119879

119898

] (5)

(1) Choose 1199071 1205961 such that ⟨120596

1 1198601199071⟩ = 1

(2) Set 1205731= 1205751equiv 0 120596

0= 1199070equiv 0 isin C119873

(3) for 119895 = 1 2 do(4) 120572

119895= ⟨120596119895 119860(119860119907

119895)⟩



(7) 120575119895+1=10038161003816100381610038161003816⟨119895+1 119860119907119895+1⟩10038161003816100381610038161003816

12

(8) 120573119895+1= ⟨119895+1 119860119907119895+1⟩ 120575119895+1

(9) 119907119895+1= 119907119895+1120575119895+1

(10) 120596119895+1= 119895+1120573119895+1

(11) end for


where

119879119898=

[[[[[[

[

12057211205732

120575212057221205733

120575119898minus1

120572119898minus1

120573119898

120575119898120572119898

]]]]]]

]

(6)


1 120572

119898 1205732 120573

119898




119860119881119898= 119881119898119879119898+ 120575119898+1119907119898+1119890119879

119898 (7)

119860119867119882119898= 119882119898119879119867

119898+ 120573119898+1120596119898+1119890119879

119898 (8)

119882119867

119898119860119881119898= 119868119898 (9)

119882119867

1198981198602119881119898= 119879119898 (10)


119898(119860 1199071) and

K119898(119860119867 1199081) where 119907

1and 119908




119898(119860 1199071) and

K119898(119860119867 1199081) where 119907

1and 119908














119898




119879119898= 119871119898119863119898119880119898 (11)




diagonal blocks


119896(1 le 119896 le 119898) of 119879


factored


119879119898



119879119896= 119871119896119863119896119880119896 (12)









[119889119896120573119896+1

120575119896+1120572119896+1

] (13)


and 120572119896+1










119898equiv 119860119867 span(119882

119898) =

119860119867K119898(119860119867 1199081) where119882


1=

1199030||1199030||2and 119908


1 1198601199071⟩ = 0


1||1198601199071||2

2subjecting





0+K119898(119860 1199071) of dimension


119887 minus 119860119909119898perpL119898 (14)


(119860119867119882119898)119867

(119887 minus 119860119909119898) = 0 (15)




0+ K119898(119860119867 1199081) of dimension 119898 by

satisfying


119898perp 119860K

119898(119860 1199071) (16)


(119860119881119898)119867(119887lowastminus 119860119867119909119898) = 0 (17)


119898is







= [1199030 1199031 119903

119899] 119877lowast119899+1

=

[119903lowast

0 119903lowast

1 119903

lowast

119899] and 119875

119899+1= [119901

0 1199011 119901

119899] 119875lowast119899+1

=

[119901lowast

0 119901lowast

1 119901

lowast


(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030)


0 1199020= 1198601199010 119902lowast0= 119860119867119901lowast

0 1199030= 1198601199030 1205880= ⟨119903lowast

0 1199030⟩

(4) for 119899 = 0 1 do(5) 120590

119899= ⟨119902lowast

119899 119902119899⟩

(6) 120572119899= 120588119899120590119899

(7) 119909119899+1= 119909119899+ 120572119899119901119899

(8) 119903119899+1= 119903119899minus 120572119899119902119899

(9) 119909lowast

119899+1= 119909lowast

119899+ 120572119899119901lowast

119899

(10) 119903lowast119899+1= 119903lowast

119899minus 120572119899119902lowast

119899

(11) 119903119899+1= 119860119903119899+1

(12) 120588119899+1= ⟨119903lowast

119899+1 119903119899+1⟩

(13) if 120588119899+1= 0method fails

(14) 120573119899+1= 120588119899+1120588119899

(15) 119901119899+1= 119903119899+1+ 120573119899+1119901119899

(16) 119901lowast119899+1= 119903lowast

119899+1+ 120573119899+1119901lowast

119899

(17) 119902119899+1= 119903119899+1+ 120573119899+1119902119899

(18) 119902lowast119899+1= 119860119867119901lowast

119899



(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1

is diagonal


(1) 120588119899equiv ⟨119903lowast

119899 119903119899⟩ equiv ⟨119903

lowast

119899 119860119903119899⟩ = 0 but 119903lowast

119899and 119860119903

119899are not


(2) 120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast





119899= 0 while 120588

119899= 0






119899+1 119903lowast




119899+1and 119903lowast







0) as follows

119911119899+1= 120590119899119903119899+1

= 120590119899119903119899minus 120588119899119860119901119899

(18)

119911lowast

119899+1= 120590119899119903lowast

119899+1

= 120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899

(19)


K119899+2(119860 1199030) and 119909lowast


0+ K119899+2(119860119867 119903lowast

0) in step (119899 + 2)

as follows

119909119899+2= 119909119899+ [119901119899 119911119899+1] 119891119899

119909lowast

119899+2= 119909lowast

119899+ [119901lowast

119899 119911lowast

119899+1] 119891lowast

119899

(20)


where 119891119899 119891lowast




119899+2isin

K119899+3(119860119867 119903lowast


119903119899+2= 119903119899minus 119860 [119901

119899 119911119899+1] 119891119899 (21)

119903lowast

119899+2= 119903lowast

119899minus 119860119867[119901lowast

119899 119911lowast

119899+1] 119891lowast

119899 (22)


⟨[119901lowast

119899 119911lowast

119899+1] 119860119903119899+2⟩ = 0

⟨[119901119899 119911119899+1] 119860119867119903lowast

119899+2⟩ = 0

(23)


119899


[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119891(1)

119899

119891(2)

119899

]

= [⟨119901lowast

119899 119860119903119899⟩

⟨119911lowast

119899+1 119860119903119899⟩]

(24)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119891lowast(1)

119899

119891lowast(2)

119899

]

= [⟨119860119901119899 119903lowast

119899⟩

⟨119860119911119899+1 119903lowast

119899⟩]

(25)



119901lowast

119899+2isin K119899+3(119860119867 119903lowast


the following form

119901119899+2= 119903119899+2+ [119901119899 119911119899+1] 119892119899 (26)

119901lowast

119899+2= 119903lowast

119899+2+ [119901lowast

119899

119911lowast

119899+1] 119892lowast

119899 (27)

where 119892119899 119892lowast

119899isin C2


⟨[119901lowast

119899 119911lowast

119899+1] 1198602119901119899+2⟩ = 0

⟨[119901119899 119911119899+1] (119860119867)2

119901lowast

119899+2⟩ = 0

(28)


119899and

119892lowast

119899

[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119892(1)

119899

119892(2)

119899

]

= minus[⟨119860119867119901lowast

119899 119860119903119899+2⟩

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩]

(29)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119892lowast(1)

119899

119892lowast(2)

119899

]

= minus[⟨119860119901119899 119860119867119903lowast

119899+2⟩

⟨119860119911119899+1 119860119867119903lowast

119899+2⟩]

(30)


119899+2 119903119899+2

119909lowast119899+2

119903lowast119899+2

119901119899+2

119901lowast119899+2





the following hold

⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119903119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ = ⟨119902

lowast

119899 119902119899⟩ equiv 120590119899

(31)

Proof If 119901119899 119901lowast


119901119899= 119903119899+ 120573119899119901119899minus1

119901lowast119899

= 119903lowast

119899+ 120573119899119901lowast



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast

119899minus 120573119899119901lowast

119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899


⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901119899minus 120573119899119901119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899 119860119901119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(32)

If 119901119899 119901lowast


119899=

119903119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)

119899minus2119911119899minus1

119901lowast119899= 119903lowast

119899+ 119892lowast(1)


119899minus2+ 119892lowast(2)


119899minus1as



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast





119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892lowast(1)

119899minus2⟨119860119867119901lowast

119899minus2 119860119901119899⟩


119899minus2⟨119860119867119911lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901

119899minus 119892(1)


119899minus2119911119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892(1)

119899minus2⟨119860119867119901lowast

119899 119860119901119899minus2⟩

minus 119892(2)

119899minus2⟨119860119867119901lowast

119899 119860119911119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(33)








119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast


119911119899+1

and 119911lowast119899+1


⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867(120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899) 119860119901119899⟩

= 120590119899⟨119860119867119903lowast

119899 119860119901119899⟩ minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

⟨119860119867119901lowast

119899 119860119911119899+1⟩ = ⟨119860

119867119901lowast

119899 119860 (120590119899119903119899minus 120588119899119860119901119899)⟩

= 120590119899⟨119860119867119901lowast

119899 119860119903119899⟩ minus 120588119899⟨119860119867119901lowast

119899 1198602119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

(34)







119894= 0 119894 = 0 1 119899 and ⟨119911lowast

119899+1 119860119911119899+1⟩ = 0 If

120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast




lowast

119899 119902119899⟩ equiv

⟨119860119867119901lowast

119899 119860119901119899⟩ = 0 then ⟨119860119867119911lowast

119899+1 119860119901119899⟩ = 0 From Lemma 7

and (18) we have

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119911lowast

119899+1 minus119911119899+1

120588119899

⟩

= minus1

120588119899

⟨119911lowast

119899+1 119860119911119899+1⟩ = 0

(35)



=

[1199030 1199031 119903

119899]119877lowast119899+1= [119903lowast

0 119903lowast

1 119903

lowast

119899] where 119903



step and let 119875119899+1= [1199010 1199011 119901

119899] 119875lowast119899+1= [119901lowast

0 119901lowast

1 119901

lowast

119899]



(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)

(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1= diag(119863

119896) where 119863



(119899 + 1)





119899119901119899+ 119891(2)

119899119911119899+1

byinduction 119909

119899isin 1199090+ K119899(119860 1199030) and 119901

119899isin K

119899+1(119860 1199030)




119899+2perp

119860119867K119899+2(119860119867 119903lowast

0)Therefore 119909





1 1199072 119907

119898) and W

119898=

span(1199081 1199082 119908





|119907|2

119903= 119907119867119872119903119907 |119908|

2

119897= 119908119867119872119897119908 (36)







W119873 one has

10038161003816100381610038161003816119908119867119860211990710038161003816100381610038161003816le Γ|119907|119903|119908|119897 (37)



119898 we have


|119907|119903=1

sup119908isinW

119898

|119908|119897le1

1199081198671198602119907 ge 120574119896ge 120574 gt 0

(38)

Then

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903 le (1 +

Γ

120574) inf119907isinV119898

|119909 minus 119907|119903 (39)


119898

(119860119867119908)119867

(119860119909 minus 119860119909119898) = 119908

1198671198602(119909 minus 119909

119898) = 0 (40)


1199081198671198602(119909119898minus 119907) = 119908

1198671198602(119909 minus 119907) (41)






120574119896

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907


which yields

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903le |119909 minus 119907|119903 +

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907

10038171003817100381710038171003816100381610038161003816119903le (1 +

Γ

120574) |119909 minus 119907|119903

(43)


119898is arbitrary




(1) ⟨119901lowast119899 119860119903119899⟩ = ⟨119860119901

119899 119903lowast119899⟩ = ⟨119903

lowast

119899 119860119903119899⟩ = ⟨119903

lowast

119899 119903119899⟩ equiv 120588

119899

and ⟨119911lowast119899+1 119860119903119899⟩ = ⟨119860119911

119899+1 119903lowast

119899⟩ = 0 119891

119899= 119891lowast

119899

(2) ⟨119860119867119901lowast119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast

119899+2⟩ = 0 and

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩ = ⟨119860119911

119899+1 119860119867119903lowast

119899+2⟩ = minus120588

119899+2119891(2)

119899


119899+2 119860119903119899+2⟩ 119892119899= 119892lowast

119899

(3) ⟨119860119867119901lowast119899 119860119911119899+1⟩ = ⟨119860119901

119899 119860119867119911lowast

119899+1⟩ = minus120579

119899+1120588119899 where

120579119899+1equiv ⟨119911lowast

119899+1 119860119911119899+1⟩


= 119903lowast

119899+

120573119899119901lowast


119899 119860119903119899⟩ = ⟨119903

lowast

119899+ 120573119899119901lowast

119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩+120573119899⟨119901lowast

119899minus1 119860119903119899⟩ = ⟨119903

lowast





119899⟩ = ⟨119860119903

119899+

120573119899119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ + 120573

119899⟨119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ =



119899 119860119903119899⟩ = ⟨119903lowast

119899+ 119892lowast(1)


119899minus2

+ 119892lowast(2)


119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩ + 119892

lowast(1)


119899minus2

119860119903119899⟩ + 119892

lowast(2)


119899minus1 119860119903119899⟩ = ⟨119903lowast

119899 119860119903119899⟩


119899⟩= ⟨119860(119903

119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)


119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ +

119892(1)


119899⟩+119892(2)


119899⟩= ⟨119860119903

119899 119903lowast

119899⟩= ⟨119903lowast119899 119860119903119899⟩The


119899+1 119903lowast



119899



119899= 119891lowast

119899


119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast


that 119860119901119899isin K

119899+2(119860 1199030) and 119860119867119901lowast

119899isin K

119899+2(119860119867 119903lowast

0)


119899= 0



and ⟨119901lowast119899 119860119903119899⟩ = 120588



119899+1 119860119903119899+2⟩ = ⟨(1119891lowast(2)

119899)(119903lowast


119899+2minus 119891lowast(1)

119899119860119867119901lowast

119899) 119860119903119899+2⟩


119899+2 119860119903119899+2⟩minus119891lowast(1)

119899 ⟨119860119867119901lowast

119899 119860119903119899+2⟩)


119899 )120588119899+2 = (minus1119891(2)

119899)120588119899+2




119899+2⟩ = minus120588

119899+2119891(2)




119899= 119892lowast

119899 because






119899 119860119911119899+1⟩ = ⟨(1120588

119899)(120590119899119903lowast


119899+1) 119860119911119899+1⟩ =

(1120588119899)(120590119899⟨119903lowast

119899 119860119911119899+1⟩ minus ⟨119911

lowast

119899+1 119860119911119899+1⟩) = (minus1120588

119899)⟨119911lowast

119899+1 119860119911119899+1⟩

= minus120579119899+1120588119899 where 120579

119899+1equiv ⟨119911

lowast

119899+1 119860119911119899+1⟩ Because it is


119899 119860119911119899+1⟩ as


119899+1⟩ = ⟨119860119867119911lowast

119899+1 119860119901119899⟩

= ⟨119860119867119901lowast119899 119860119911119899+1⟩ = minus120579





[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119891(1)

119899

119891(2)

119899

]

]

= [120588119899

0]

[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119892(1)

119899

119892(2)

119899

]

]

= [

[

0120588119899+2

119891(2)

119899

]

]

(44)

where 120577119899+1= ⟨119860119867119911lowast

119899+1 119860119911119899+1⟩


119899and 119892

119899can


119891119899=(120577119899+1120588119899 120579119899+1) 1205882

119899

120578119899

119892119899= (120588119899+2

120588119899

120590119899120588119899+2

120579119899+1

)

(45)

where 120578119899equiv 120590119899120577119899+11205882

119899minus 1205792

119899+1


119899+1and 119911




1003817100381710038171003817119903119899+11003817100381710038171003817 gt max 1003817100381710038171003817119903119899

1003817100381710038171003817 1003817100381710038171003817119903119899+2

1003817100381710038171003817 (46)


119899+2equiv 120578119899119903119899+2= 120578119899119903119899minus

120577119899+11205883

119899119860119901119899minus 120579119899+11205882

119899119860119911119899+1


119899+1 le 119903

119899 and


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030) Set 119901

0= 1199030 1199010= 1199030 1199020= 1198601199010 1199020= 1198601198671199010

(3) Compute 1205880= ⟨1199030 1198601199030⟩

(4) Begin LOOP (119899 = 0 1 2 )(5) 120590119899= ⟨119902119899 119902119899⟩

(6) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(7) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(8) 119910119899+1= 119860119904119899+1

(9) 119910119899+1= 119860119867119904119899+1

(10) 120579119899+1= ⟨119904119899+1 119910119899+1⟩

(11) 120577119899+1= ⟨119910119899+1 119910119899+1⟩


119899= 120588119899120590119899

(14) 120588119899+1= 120579119899+11205902

119899

(15) 120573119899+1= 120588119899+1120588119899

(16) 119909119899+1= 119909119899+ 120572119899119901119899

(17) 119903119899+1= 119903119899minus 120572119899119902119899

(18) 119903119899+1= 119903119899minus 120572119899119902119899

(19) 119901119899+1= 119904119899+1120590119899+ 120573119899+1119901119899

(20) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(21) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(22) 119899 larr 119899 + 1

(23) else(24) 120575

119899= 120590119899120577119899+11205882

119899minus 1205792

119899+1

(25) 120572119899= 120577119899+11205883

119899120575119899

(26) 120572119899+1= 120579119899+11205882

119899120575119899

(27) 119909119899+2= 119909119899+ 120572119899119901119899+ 120572119899+1119904119899+1

(28) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(29) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(30) solve 119861119911119899+2= 119903119899+2

(31) solve 119861119867119899+2= 119903119899+2

(32) 119899+2= 119860119911119899+2

(33) 119911119899+2= 119860119867119899+2

(34) 120588119899+2= ⟨119911119899+2 119903119899+2⟩

(35) 120573119899+1= 120588119899+2120588119899

(36) 120573119899+2= 120588119899+2120590119899120579119899+1

(37) 119901119899+2= 119911119899+2+ 120573119899+1119901119899+ 120573119899+2119904119899+1

(38) 119902119899+2= 119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(39) 119902119899+2= 119911119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(40) 119899 larr 119899 + 2



119903119899+2 le 119903


119899+1 le

|120590119899|119903119899 and |120590

119899|120587119899+2 le |120578





119896(1 le




If ||119911119899+1|| le |120590

119899|||119903119899|| Then

1 times 1 StepElse||120587119899+2|| = ||120578

119899119903119899minus 120577119899+11205883

119899119902119899minus 120579119899+11205882

119899119910119899+1||

If |120590119899|||120587119899+2|| le |120578

119899|||119911119899+1|| Then


End IfEnd If

Algorithm 4












10119887 minus 119860119909



119899211990302lt TOL or when





0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)







0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











⟨119906 119907⟩ = 119906119867119907 =

119873

sum

119894=1

119906119894119907119894 (2)


K119905 (119860 119907) = span 119907 119860119907 119860

2119907 119860

119905minus1119907 (3)






119898(119860119867 1199081) where 119907

1and 119908

1are chosen




and 119860119907119895+1




119895 119895 = 1 2 119898 and 119908

119894 119894 =


⟨120596119894 119860119907119895⟩ = 120575119894119895 1 le 119894 119895 le 119898 (4)


119898] and 119882

119898=

[1199081 1199082 119908


119898the extended


119879119898= [

119879119898

120575119898+1119890119879

119898

] (5)

(1) Choose 1199071 1205961 such that ⟨120596

1 1198601199071⟩ = 1

(2) Set 1205731= 1205751equiv 0 120596

0= 1199070equiv 0 isin C119873

(3) for 119895 = 1 2 do(4) 120572

119895= ⟨120596119895 119860(119860119907

119895)⟩



(7) 120575119895+1=10038161003816100381610038161003816⟨119895+1 119860119907119895+1⟩10038161003816100381610038161003816

12

(8) 120573119895+1= ⟨119895+1 119860119907119895+1⟩ 120575119895+1

(9) 119907119895+1= 119907119895+1120575119895+1

(10) 120596119895+1= 119895+1120573119895+1

(11) end for


where

119879119898=

[[[[[[

[

12057211205732

120575212057221205733

120575119898minus1

120572119898minus1

120573119898

120575119898120572119898

]]]]]]

]

(6)


1 120572

119898 1205732 120573

119898




119860119881119898= 119881119898119879119898+ 120575119898+1119907119898+1119890119879

119898 (7)

119860119867119882119898= 119882119898119879119867

119898+ 120573119898+1120596119898+1119890119879

119898 (8)

119882119867

119898119860119881119898= 119868119898 (9)

119882119867

1198981198602119881119898= 119879119898 (10)


119898(119860 1199071) and

K119898(119860119867 1199081) where 119907

1and 119908




119898(119860 1199071) and

K119898(119860119867 1199081) where 119907

1and 119908














119898




119879119898= 119871119898119863119898119880119898 (11)




diagonal blocks


119896(1 le 119896 le 119898) of 119879


factored


119879119898



119879119896= 119871119896119863119896119880119896 (12)









[119889119896120573119896+1

120575119896+1120572119896+1

] (13)


and 120572119896+1










119898equiv 119860119867 span(119882

119898) =

119860119867K119898(119860119867 1199081) where119882


1=

1199030||1199030||2and 119908


1 1198601199071⟩ = 0


1||1198601199071||2

2subjecting





0+K119898(119860 1199071) of dimension


119887 minus 119860119909119898perpL119898 (14)


(119860119867119882119898)119867

(119887 minus 119860119909119898) = 0 (15)




0+ K119898(119860119867 1199081) of dimension 119898 by

satisfying


119898perp 119860K

119898(119860 1199071) (16)


(119860119881119898)119867(119887lowastminus 119860119867119909119898) = 0 (17)


119898is







= [1199030 1199031 119903

119899] 119877lowast119899+1

=

[119903lowast

0 119903lowast

1 119903

lowast

119899] and 119875

119899+1= [119901

0 1199011 119901

119899] 119875lowast119899+1

=

[119901lowast

0 119901lowast

1 119901

lowast


(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030)


0 1199020= 1198601199010 119902lowast0= 119860119867119901lowast

0 1199030= 1198601199030 1205880= ⟨119903lowast

0 1199030⟩

(4) for 119899 = 0 1 do(5) 120590

119899= ⟨119902lowast

119899 119902119899⟩

(6) 120572119899= 120588119899120590119899

(7) 119909119899+1= 119909119899+ 120572119899119901119899

(8) 119903119899+1= 119903119899minus 120572119899119902119899

(9) 119909lowast

119899+1= 119909lowast

119899+ 120572119899119901lowast

119899

(10) 119903lowast119899+1= 119903lowast

119899minus 120572119899119902lowast

119899

(11) 119903119899+1= 119860119903119899+1

(12) 120588119899+1= ⟨119903lowast

119899+1 119903119899+1⟩

(13) if 120588119899+1= 0method fails

(14) 120573119899+1= 120588119899+1120588119899

(15) 119901119899+1= 119903119899+1+ 120573119899+1119901119899

(16) 119901lowast119899+1= 119903lowast

119899+1+ 120573119899+1119901lowast

119899

(17) 119902119899+1= 119903119899+1+ 120573119899+1119902119899

(18) 119902lowast119899+1= 119860119867119901lowast

119899



(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1

is diagonal


(1) 120588119899equiv ⟨119903lowast

119899 119903119899⟩ equiv ⟨119903

lowast

119899 119860119903119899⟩ = 0 but 119903lowast

119899and 119860119903

119899are not


(2) 120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast





119899= 0 while 120588

119899= 0






119899+1 119903lowast




119899+1and 119903lowast







0) as follows

119911119899+1= 120590119899119903119899+1

= 120590119899119903119899minus 120588119899119860119901119899

(18)

119911lowast

119899+1= 120590119899119903lowast

119899+1

= 120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899

(19)


K119899+2(119860 1199030) and 119909lowast


0+ K119899+2(119860119867 119903lowast

0) in step (119899 + 2)

as follows

119909119899+2= 119909119899+ [119901119899 119911119899+1] 119891119899

119909lowast

119899+2= 119909lowast

119899+ [119901lowast

119899 119911lowast

119899+1] 119891lowast

119899

(20)


where 119891119899 119891lowast




119899+2isin

K119899+3(119860119867 119903lowast


119903119899+2= 119903119899minus 119860 [119901

119899 119911119899+1] 119891119899 (21)

119903lowast

119899+2= 119903lowast

119899minus 119860119867[119901lowast

119899 119911lowast

119899+1] 119891lowast

119899 (22)


⟨[119901lowast

119899 119911lowast

119899+1] 119860119903119899+2⟩ = 0

⟨[119901119899 119911119899+1] 119860119867119903lowast

119899+2⟩ = 0

(23)


119899


[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119891(1)

119899

119891(2)

119899

]

= [⟨119901lowast

119899 119860119903119899⟩

⟨119911lowast

119899+1 119860119903119899⟩]

(24)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119891lowast(1)

119899

119891lowast(2)

119899

]

= [⟨119860119901119899 119903lowast

119899⟩

⟨119860119911119899+1 119903lowast

119899⟩]

(25)



119901lowast

119899+2isin K119899+3(119860119867 119903lowast


the following form

119901119899+2= 119903119899+2+ [119901119899 119911119899+1] 119892119899 (26)

119901lowast

119899+2= 119903lowast

119899+2+ [119901lowast

119899

119911lowast

119899+1] 119892lowast

119899 (27)

where 119892119899 119892lowast

119899isin C2


⟨[119901lowast

119899 119911lowast

119899+1] 1198602119901119899+2⟩ = 0

⟨[119901119899 119911119899+1] (119860119867)2

119901lowast

119899+2⟩ = 0

(28)


119899and

119892lowast

119899

[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119892(1)

119899

119892(2)

119899

]

= minus[⟨119860119867119901lowast

119899 119860119903119899+2⟩

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩]

(29)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119892lowast(1)

119899

119892lowast(2)

119899

]

= minus[⟨119860119901119899 119860119867119903lowast

119899+2⟩

⟨119860119911119899+1 119860119867119903lowast

119899+2⟩]

(30)


119899+2 119903119899+2

119909lowast119899+2

119903lowast119899+2

119901119899+2

119901lowast119899+2





the following hold

⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119903119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ = ⟨119902

lowast

119899 119902119899⟩ equiv 120590119899

(31)

Proof If 119901119899 119901lowast


119901119899= 119903119899+ 120573119899119901119899minus1

119901lowast119899

= 119903lowast

119899+ 120573119899119901lowast



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast

119899minus 120573119899119901lowast

119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899


⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901119899minus 120573119899119901119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899 119860119901119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(32)

If 119901119899 119901lowast


119899=

119903119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)

119899minus2119911119899minus1

119901lowast119899= 119903lowast

119899+ 119892lowast(1)


119899minus2+ 119892lowast(2)


119899minus1as



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast





119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892lowast(1)

119899minus2⟨119860119867119901lowast

119899minus2 119860119901119899⟩


119899minus2⟨119860119867119911lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901

119899minus 119892(1)


119899minus2119911119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892(1)

119899minus2⟨119860119867119901lowast

119899 119860119901119899minus2⟩

minus 119892(2)

119899minus2⟨119860119867119901lowast

119899 119860119911119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(33)








119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast


119911119899+1

and 119911lowast119899+1


⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867(120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899) 119860119901119899⟩

= 120590119899⟨119860119867119903lowast

119899 119860119901119899⟩ minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

⟨119860119867119901lowast

119899 119860119911119899+1⟩ = ⟨119860

119867119901lowast

119899 119860 (120590119899119903119899minus 120588119899119860119901119899)⟩

= 120590119899⟨119860119867119901lowast

119899 119860119903119899⟩ minus 120588119899⟨119860119867119901lowast

119899 1198602119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

(34)







119894= 0 119894 = 0 1 119899 and ⟨119911lowast

119899+1 119860119911119899+1⟩ = 0 If

120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast




lowast

119899 119902119899⟩ equiv

⟨119860119867119901lowast

119899 119860119901119899⟩ = 0 then ⟨119860119867119911lowast

119899+1 119860119901119899⟩ = 0 From Lemma 7

and (18) we have

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119911lowast

119899+1 minus119911119899+1

120588119899

⟩

= minus1

120588119899

⟨119911lowast

119899+1 119860119911119899+1⟩ = 0

(35)



=

[1199030 1199031 119903

119899]119877lowast119899+1= [119903lowast

0 119903lowast

1 119903

lowast

119899] where 119903



step and let 119875119899+1= [1199010 1199011 119901

119899] 119875lowast119899+1= [119901lowast

0 119901lowast

1 119901

lowast

119899]



(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)

(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1= diag(119863

119896) where 119863



(119899 + 1)





119899119901119899+ 119891(2)

119899119911119899+1

byinduction 119909

119899isin 1199090+ K119899(119860 1199030) and 119901

119899isin K

119899+1(119860 1199030)




119899+2perp

119860119867K119899+2(119860119867 119903lowast

0)Therefore 119909





1 1199072 119907

119898) and W

119898=

span(1199081 1199082 119908





|119907|2

119903= 119907119867119872119903119907 |119908|

2

119897= 119908119867119872119897119908 (36)







W119873 one has

10038161003816100381610038161003816119908119867119860211990710038161003816100381610038161003816le Γ|119907|119903|119908|119897 (37)



119898 we have


|119907|119903=1

sup119908isinW

119898

|119908|119897le1

1199081198671198602119907 ge 120574119896ge 120574 gt 0

(38)

Then

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903 le (1 +

Γ

120574) inf119907isinV119898

|119909 minus 119907|119903 (39)


119898

(119860119867119908)119867

(119860119909 minus 119860119909119898) = 119908

1198671198602(119909 minus 119909

119898) = 0 (40)


1199081198671198602(119909119898minus 119907) = 119908

1198671198602(119909 minus 119907) (41)






120574119896

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907


which yields

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903le |119909 minus 119907|119903 +

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907

10038171003817100381710038171003816100381610038161003816119903le (1 +

Γ

120574) |119909 minus 119907|119903

(43)


119898is arbitrary




(1) ⟨119901lowast119899 119860119903119899⟩ = ⟨119860119901

119899 119903lowast119899⟩ = ⟨119903

lowast

119899 119860119903119899⟩ = ⟨119903

lowast

119899 119903119899⟩ equiv 120588

119899

and ⟨119911lowast119899+1 119860119903119899⟩ = ⟨119860119911

119899+1 119903lowast

119899⟩ = 0 119891

119899= 119891lowast

119899

(2) ⟨119860119867119901lowast119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast

119899+2⟩ = 0 and

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩ = ⟨119860119911

119899+1 119860119867119903lowast

119899+2⟩ = minus120588

119899+2119891(2)

119899


119899+2 119860119903119899+2⟩ 119892119899= 119892lowast

119899

(3) ⟨119860119867119901lowast119899 119860119911119899+1⟩ = ⟨119860119901

119899 119860119867119911lowast

119899+1⟩ = minus120579

119899+1120588119899 where

120579119899+1equiv ⟨119911lowast

119899+1 119860119911119899+1⟩


= 119903lowast

119899+

120573119899119901lowast


119899 119860119903119899⟩ = ⟨119903

lowast

119899+ 120573119899119901lowast

119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩+120573119899⟨119901lowast

119899minus1 119860119903119899⟩ = ⟨119903

lowast





119899⟩ = ⟨119860119903

119899+

120573119899119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ + 120573

119899⟨119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ =



119899 119860119903119899⟩ = ⟨119903lowast

119899+ 119892lowast(1)


119899minus2

+ 119892lowast(2)


119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩ + 119892

lowast(1)


119899minus2

119860119903119899⟩ + 119892

lowast(2)


119899minus1 119860119903119899⟩ = ⟨119903lowast

119899 119860119903119899⟩


119899⟩= ⟨119860(119903

119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)


119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ +

119892(1)


119899⟩+119892(2)


119899⟩= ⟨119860119903

119899 119903lowast

119899⟩= ⟨119903lowast119899 119860119903119899⟩The


119899+1 119903lowast



119899



119899= 119891lowast

119899


119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast


that 119860119901119899isin K

119899+2(119860 1199030) and 119860119867119901lowast

119899isin K

119899+2(119860119867 119903lowast

0)


119899= 0



and ⟨119901lowast119899 119860119903119899⟩ = 120588



119899+1 119860119903119899+2⟩ = ⟨(1119891lowast(2)

119899)(119903lowast


119899+2minus 119891lowast(1)

119899119860119867119901lowast

119899) 119860119903119899+2⟩


119899+2 119860119903119899+2⟩minus119891lowast(1)

119899 ⟨119860119867119901lowast

119899 119860119903119899+2⟩)


119899 )120588119899+2 = (minus1119891(2)

119899)120588119899+2




119899+2⟩ = minus120588

119899+2119891(2)




119899= 119892lowast

119899 because






119899 119860119911119899+1⟩ = ⟨(1120588

119899)(120590119899119903lowast


119899+1) 119860119911119899+1⟩ =

(1120588119899)(120590119899⟨119903lowast

119899 119860119911119899+1⟩ minus ⟨119911

lowast

119899+1 119860119911119899+1⟩) = (minus1120588

119899)⟨119911lowast

119899+1 119860119911119899+1⟩

= minus120579119899+1120588119899 where 120579

119899+1equiv ⟨119911

lowast

119899+1 119860119911119899+1⟩ Because it is


119899 119860119911119899+1⟩ as


119899+1⟩ = ⟨119860119867119911lowast

119899+1 119860119901119899⟩

= ⟨119860119867119901lowast119899 119860119911119899+1⟩ = minus120579





[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119891(1)

119899

119891(2)

119899

]

]

= [120588119899

0]

[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119892(1)

119899

119892(2)

119899

]

]

= [

[

0120588119899+2

119891(2)

119899

]

]

(44)

where 120577119899+1= ⟨119860119867119911lowast

119899+1 119860119911119899+1⟩


119899and 119892

119899can


119891119899=(120577119899+1120588119899 120579119899+1) 1205882

119899

120578119899

119892119899= (120588119899+2

120588119899

120590119899120588119899+2

120579119899+1

)

(45)

where 120578119899equiv 120590119899120577119899+11205882

119899minus 1205792

119899+1


119899+1and 119911




1003817100381710038171003817119903119899+11003817100381710038171003817 gt max 1003817100381710038171003817119903119899

1003817100381710038171003817 1003817100381710038171003817119903119899+2

1003817100381710038171003817 (46)


119899+2equiv 120578119899119903119899+2= 120578119899119903119899minus

120577119899+11205883

119899119860119901119899minus 120579119899+11205882

119899119860119911119899+1


119899+1 le 119903

119899 and


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030) Set 119901

0= 1199030 1199010= 1199030 1199020= 1198601199010 1199020= 1198601198671199010

(3) Compute 1205880= ⟨1199030 1198601199030⟩

(4) Begin LOOP (119899 = 0 1 2 )(5) 120590119899= ⟨119902119899 119902119899⟩

(6) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(7) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(8) 119910119899+1= 119860119904119899+1

(9) 119910119899+1= 119860119867119904119899+1

(10) 120579119899+1= ⟨119904119899+1 119910119899+1⟩

(11) 120577119899+1= ⟨119910119899+1 119910119899+1⟩


119899= 120588119899120590119899

(14) 120588119899+1= 120579119899+11205902

119899

(15) 120573119899+1= 120588119899+1120588119899

(16) 119909119899+1= 119909119899+ 120572119899119901119899

(17) 119903119899+1= 119903119899minus 120572119899119902119899

(18) 119903119899+1= 119903119899minus 120572119899119902119899

(19) 119901119899+1= 119904119899+1120590119899+ 120573119899+1119901119899

(20) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(21) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(22) 119899 larr 119899 + 1

(23) else(24) 120575

119899= 120590119899120577119899+11205882

119899minus 1205792

119899+1

(25) 120572119899= 120577119899+11205883

119899120575119899

(26) 120572119899+1= 120579119899+11205882

119899120575119899

(27) 119909119899+2= 119909119899+ 120572119899119901119899+ 120572119899+1119904119899+1

(28) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(29) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(30) solve 119861119911119899+2= 119903119899+2

(31) solve 119861119867119899+2= 119903119899+2

(32) 119899+2= 119860119911119899+2

(33) 119911119899+2= 119860119867119899+2

(34) 120588119899+2= ⟨119911119899+2 119903119899+2⟩

(35) 120573119899+1= 120588119899+2120588119899

(36) 120573119899+2= 120588119899+2120590119899120579119899+1

(37) 119901119899+2= 119911119899+2+ 120573119899+1119901119899+ 120573119899+2119904119899+1

(38) 119902119899+2= 119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(39) 119902119899+2= 119911119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(40) 119899 larr 119899 + 2



119903119899+2 le 119903


119899+1 le

|120590119899|119903119899 and |120590

119899|120587119899+2 le |120578





119896(1 le




If ||119911119899+1|| le |120590

119899|||119903119899|| Then

1 times 1 StepElse||120587119899+2|| = ||120578

119899119903119899minus 120577119899+11205883

119899119902119899minus 120579119899+11205882

119899119910119899+1||

If |120590119899|||120587119899+2|| le |120578

119899|||119911119899+1|| Then


End IfEnd If

Algorithm 4












10119887 minus 119860119909



119899211990302lt TOL or when





0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)







0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119898




119879119898= 119871119898119863119898119880119898 (11)




diagonal blocks


119896(1 le 119896 le 119898) of 119879


factored


119879119898



119879119896= 119871119896119863119896119880119896 (12)









[119889119896120573119896+1

120575119896+1120572119896+1

] (13)


and 120572119896+1










119898equiv 119860119867 span(119882

119898) =

119860119867K119898(119860119867 1199081) where119882


1=

1199030||1199030||2and 119908


1 1198601199071⟩ = 0


1||1198601199071||2

2subjecting





0+K119898(119860 1199071) of dimension


119887 minus 119860119909119898perpL119898 (14)


(119860119867119882119898)119867

(119887 minus 119860119909119898) = 0 (15)




0+ K119898(119860119867 1199081) of dimension 119898 by

satisfying


119898perp 119860K

119898(119860 1199071) (16)


(119860119881119898)119867(119887lowastminus 119860119867119909119898) = 0 (17)


119898is







= [1199030 1199031 119903

119899] 119877lowast119899+1

=

[119903lowast

0 119903lowast

1 119903

lowast

119899] and 119875

119899+1= [119901

0 1199011 119901

119899] 119875lowast119899+1

=

[119901lowast

0 119901lowast

1 119901

lowast


(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030)


0 1199020= 1198601199010 119902lowast0= 119860119867119901lowast

0 1199030= 1198601199030 1205880= ⟨119903lowast

0 1199030⟩

(4) for 119899 = 0 1 do(5) 120590

119899= ⟨119902lowast

119899 119902119899⟩

(6) 120572119899= 120588119899120590119899

(7) 119909119899+1= 119909119899+ 120572119899119901119899

(8) 119903119899+1= 119903119899minus 120572119899119902119899

(9) 119909lowast

119899+1= 119909lowast

119899+ 120572119899119901lowast

119899

(10) 119903lowast119899+1= 119903lowast

119899minus 120572119899119902lowast

119899

(11) 119903119899+1= 119860119903119899+1

(12) 120588119899+1= ⟨119903lowast

119899+1 119903119899+1⟩

(13) if 120588119899+1= 0method fails

(14) 120573119899+1= 120588119899+1120588119899

(15) 119901119899+1= 119903119899+1+ 120573119899+1119901119899

(16) 119901lowast119899+1= 119903lowast

119899+1+ 120573119899+1119901lowast

119899

(17) 119902119899+1= 119903119899+1+ 120573119899+1119902119899

(18) 119902lowast119899+1= 119860119867119901lowast

119899



(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1

is diagonal


(1) 120588119899equiv ⟨119903lowast

119899 119903119899⟩ equiv ⟨119903

lowast

119899 119860119903119899⟩ = 0 but 119903lowast

119899and 119860119903

119899are not


(2) 120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast





119899= 0 while 120588

119899= 0






119899+1 119903lowast




119899+1and 119903lowast







0) as follows

119911119899+1= 120590119899119903119899+1

= 120590119899119903119899minus 120588119899119860119901119899

(18)

119911lowast

119899+1= 120590119899119903lowast

119899+1

= 120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899

(19)


K119899+2(119860 1199030) and 119909lowast


0+ K119899+2(119860119867 119903lowast

0) in step (119899 + 2)

as follows

119909119899+2= 119909119899+ [119901119899 119911119899+1] 119891119899

119909lowast

119899+2= 119909lowast

119899+ [119901lowast

119899 119911lowast

119899+1] 119891lowast

119899

(20)


where 119891119899 119891lowast




119899+2isin

K119899+3(119860119867 119903lowast


119903119899+2= 119903119899minus 119860 [119901

119899 119911119899+1] 119891119899 (21)

119903lowast

119899+2= 119903lowast

119899minus 119860119867[119901lowast

119899 119911lowast

119899+1] 119891lowast

119899 (22)


⟨[119901lowast

119899 119911lowast

119899+1] 119860119903119899+2⟩ = 0

⟨[119901119899 119911119899+1] 119860119867119903lowast

119899+2⟩ = 0

(23)


119899


[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119891(1)

119899

119891(2)

119899

]

= [⟨119901lowast

119899 119860119903119899⟩

⟨119911lowast

119899+1 119860119903119899⟩]

(24)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119891lowast(1)

119899

119891lowast(2)

119899

]

= [⟨119860119901119899 119903lowast

119899⟩

⟨119860119911119899+1 119903lowast

119899⟩]

(25)



119901lowast

119899+2isin K119899+3(119860119867 119903lowast


the following form

119901119899+2= 119903119899+2+ [119901119899 119911119899+1] 119892119899 (26)

119901lowast

119899+2= 119903lowast

119899+2+ [119901lowast

119899

119911lowast

119899+1] 119892lowast

119899 (27)

where 119892119899 119892lowast

119899isin C2


⟨[119901lowast

119899 119911lowast

119899+1] 1198602119901119899+2⟩ = 0

⟨[119901119899 119911119899+1] (119860119867)2

119901lowast

119899+2⟩ = 0

(28)


119899and

119892lowast

119899

[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119892(1)

119899

119892(2)

119899

]

= minus[⟨119860119867119901lowast

119899 119860119903119899+2⟩

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩]

(29)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119892lowast(1)

119899

119892lowast(2)

119899

]

= minus[⟨119860119901119899 119860119867119903lowast

119899+2⟩

⟨119860119911119899+1 119860119867119903lowast

119899+2⟩]

(30)


119899+2 119903119899+2

119909lowast119899+2

119903lowast119899+2

119901119899+2

119901lowast119899+2





the following hold

⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119903119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ = ⟨119902

lowast

119899 119902119899⟩ equiv 120590119899

(31)

Proof If 119901119899 119901lowast


119901119899= 119903119899+ 120573119899119901119899minus1

119901lowast119899

= 119903lowast

119899+ 120573119899119901lowast



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast

119899minus 120573119899119901lowast

119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899


⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901119899minus 120573119899119901119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899 119860119901119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(32)

If 119901119899 119901lowast


119899=

119903119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)

119899minus2119911119899minus1

119901lowast119899= 119903lowast

119899+ 119892lowast(1)


119899minus2+ 119892lowast(2)


119899minus1as



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast





119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892lowast(1)

119899minus2⟨119860119867119901lowast

119899minus2 119860119901119899⟩


119899minus2⟨119860119867119911lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901

119899minus 119892(1)


119899minus2119911119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892(1)

119899minus2⟨119860119867119901lowast

119899 119860119901119899minus2⟩

minus 119892(2)

119899minus2⟨119860119867119901lowast

119899 119860119911119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(33)








119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast


119911119899+1

and 119911lowast119899+1


⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867(120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899) 119860119901119899⟩

= 120590119899⟨119860119867119903lowast

119899 119860119901119899⟩ minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

⟨119860119867119901lowast

119899 119860119911119899+1⟩ = ⟨119860

119867119901lowast

119899 119860 (120590119899119903119899minus 120588119899119860119901119899)⟩

= 120590119899⟨119860119867119901lowast

119899 119860119903119899⟩ minus 120588119899⟨119860119867119901lowast

119899 1198602119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

(34)







119894= 0 119894 = 0 1 119899 and ⟨119911lowast

119899+1 119860119911119899+1⟩ = 0 If

120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast




lowast

119899 119902119899⟩ equiv

⟨119860119867119901lowast

119899 119860119901119899⟩ = 0 then ⟨119860119867119911lowast

119899+1 119860119901119899⟩ = 0 From Lemma 7

and (18) we have

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119911lowast

119899+1 minus119911119899+1

120588119899

⟩

= minus1

120588119899

⟨119911lowast

119899+1 119860119911119899+1⟩ = 0

(35)



=

[1199030 1199031 119903

119899]119877lowast119899+1= [119903lowast

0 119903lowast

1 119903

lowast

119899] where 119903



step and let 119875119899+1= [1199010 1199011 119901

119899] 119875lowast119899+1= [119901lowast

0 119901lowast

1 119901

lowast

119899]



(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)

(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1= diag(119863

119896) where 119863



(119899 + 1)





119899119901119899+ 119891(2)

119899119911119899+1

byinduction 119909

119899isin 1199090+ K119899(119860 1199030) and 119901

119899isin K

119899+1(119860 1199030)




119899+2perp

119860119867K119899+2(119860119867 119903lowast

0)Therefore 119909





1 1199072 119907

119898) and W

119898=

span(1199081 1199082 119908





|119907|2

119903= 119907119867119872119903119907 |119908|

2

119897= 119908119867119872119897119908 (36)







W119873 one has

10038161003816100381610038161003816119908119867119860211990710038161003816100381610038161003816le Γ|119907|119903|119908|119897 (37)



119898 we have


|119907|119903=1

sup119908isinW

119898

|119908|119897le1

1199081198671198602119907 ge 120574119896ge 120574 gt 0

(38)

Then

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903 le (1 +

Γ

120574) inf119907isinV119898

|119909 minus 119907|119903 (39)


119898

(119860119867119908)119867

(119860119909 minus 119860119909119898) = 119908

1198671198602(119909 minus 119909

119898) = 0 (40)


1199081198671198602(119909119898minus 119907) = 119908

1198671198602(119909 minus 119907) (41)






120574119896

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907


which yields

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903le |119909 minus 119907|119903 +

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907

10038171003817100381710038171003816100381610038161003816119903le (1 +

Γ

120574) |119909 minus 119907|119903

(43)


119898is arbitrary




(1) ⟨119901lowast119899 119860119903119899⟩ = ⟨119860119901

119899 119903lowast119899⟩ = ⟨119903

lowast

119899 119860119903119899⟩ = ⟨119903

lowast

119899 119903119899⟩ equiv 120588

119899

and ⟨119911lowast119899+1 119860119903119899⟩ = ⟨119860119911

119899+1 119903lowast

119899⟩ = 0 119891

119899= 119891lowast

119899

(2) ⟨119860119867119901lowast119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast

119899+2⟩ = 0 and

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩ = ⟨119860119911

119899+1 119860119867119903lowast

119899+2⟩ = minus120588

119899+2119891(2)

119899


119899+2 119860119903119899+2⟩ 119892119899= 119892lowast

119899

(3) ⟨119860119867119901lowast119899 119860119911119899+1⟩ = ⟨119860119901

119899 119860119867119911lowast

119899+1⟩ = minus120579

119899+1120588119899 where

120579119899+1equiv ⟨119911lowast

119899+1 119860119911119899+1⟩


= 119903lowast

119899+

120573119899119901lowast


119899 119860119903119899⟩ = ⟨119903

lowast

119899+ 120573119899119901lowast

119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩+120573119899⟨119901lowast

119899minus1 119860119903119899⟩ = ⟨119903

lowast





119899⟩ = ⟨119860119903

119899+

120573119899119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ + 120573

119899⟨119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ =



119899 119860119903119899⟩ = ⟨119903lowast

119899+ 119892lowast(1)


119899minus2

+ 119892lowast(2)


119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩ + 119892

lowast(1)


119899minus2

119860119903119899⟩ + 119892

lowast(2)


119899minus1 119860119903119899⟩ = ⟨119903lowast

119899 119860119903119899⟩


119899⟩= ⟨119860(119903

119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)


119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ +

119892(1)


119899⟩+119892(2)


119899⟩= ⟨119860119903

119899 119903lowast

119899⟩= ⟨119903lowast119899 119860119903119899⟩The


119899+1 119903lowast



119899



119899= 119891lowast

119899


119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast


that 119860119901119899isin K

119899+2(119860 1199030) and 119860119867119901lowast

119899isin K

119899+2(119860119867 119903lowast

0)


119899= 0



and ⟨119901lowast119899 119860119903119899⟩ = 120588



119899+1 119860119903119899+2⟩ = ⟨(1119891lowast(2)

119899)(119903lowast


119899+2minus 119891lowast(1)

119899119860119867119901lowast

119899) 119860119903119899+2⟩


119899+2 119860119903119899+2⟩minus119891lowast(1)

119899 ⟨119860119867119901lowast

119899 119860119903119899+2⟩)


119899 )120588119899+2 = (minus1119891(2)

119899)120588119899+2




119899+2⟩ = minus120588

119899+2119891(2)




119899= 119892lowast

119899 because






119899 119860119911119899+1⟩ = ⟨(1120588

119899)(120590119899119903lowast


119899+1) 119860119911119899+1⟩ =

(1120588119899)(120590119899⟨119903lowast

119899 119860119911119899+1⟩ minus ⟨119911

lowast

119899+1 119860119911119899+1⟩) = (minus1120588

119899)⟨119911lowast

119899+1 119860119911119899+1⟩

= minus120579119899+1120588119899 where 120579

119899+1equiv ⟨119911

lowast

119899+1 119860119911119899+1⟩ Because it is


119899 119860119911119899+1⟩ as


119899+1⟩ = ⟨119860119867119911lowast

119899+1 119860119901119899⟩

= ⟨119860119867119901lowast119899 119860119911119899+1⟩ = minus120579





[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119891(1)

119899

119891(2)

119899

]

]

= [120588119899

0]

[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119892(1)

119899

119892(2)

119899

]

]

= [

[

0120588119899+2

119891(2)

119899

]

]

(44)

where 120577119899+1= ⟨119860119867119911lowast

119899+1 119860119911119899+1⟩


119899and 119892

119899can


119891119899=(120577119899+1120588119899 120579119899+1) 1205882

119899

120578119899

119892119899= (120588119899+2

120588119899

120590119899120588119899+2

120579119899+1

)

(45)

where 120578119899equiv 120590119899120577119899+11205882

119899minus 1205792

119899+1


119899+1and 119911




1003817100381710038171003817119903119899+11003817100381710038171003817 gt max 1003817100381710038171003817119903119899

1003817100381710038171003817 1003817100381710038171003817119903119899+2

1003817100381710038171003817 (46)


119899+2equiv 120578119899119903119899+2= 120578119899119903119899minus

120577119899+11205883

119899119860119901119899minus 120579119899+11205882

119899119860119911119899+1


119899+1 le 119903

119899 and


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030) Set 119901

0= 1199030 1199010= 1199030 1199020= 1198601199010 1199020= 1198601198671199010

(3) Compute 1205880= ⟨1199030 1198601199030⟩

(4) Begin LOOP (119899 = 0 1 2 )(5) 120590119899= ⟨119902119899 119902119899⟩

(6) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(7) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(8) 119910119899+1= 119860119904119899+1

(9) 119910119899+1= 119860119867119904119899+1

(10) 120579119899+1= ⟨119904119899+1 119910119899+1⟩

(11) 120577119899+1= ⟨119910119899+1 119910119899+1⟩


119899= 120588119899120590119899

(14) 120588119899+1= 120579119899+11205902

119899

(15) 120573119899+1= 120588119899+1120588119899

(16) 119909119899+1= 119909119899+ 120572119899119901119899

(17) 119903119899+1= 119903119899minus 120572119899119902119899

(18) 119903119899+1= 119903119899minus 120572119899119902119899

(19) 119901119899+1= 119904119899+1120590119899+ 120573119899+1119901119899

(20) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(21) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(22) 119899 larr 119899 + 1

(23) else(24) 120575

119899= 120590119899120577119899+11205882

119899minus 1205792

119899+1

(25) 120572119899= 120577119899+11205883

119899120575119899

(26) 120572119899+1= 120579119899+11205882

119899120575119899

(27) 119909119899+2= 119909119899+ 120572119899119901119899+ 120572119899+1119904119899+1

(28) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(29) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(30) solve 119861119911119899+2= 119903119899+2

(31) solve 119861119867119899+2= 119903119899+2

(32) 119899+2= 119860119911119899+2

(33) 119911119899+2= 119860119867119899+2

(34) 120588119899+2= ⟨119911119899+2 119903119899+2⟩

(35) 120573119899+1= 120588119899+2120588119899

(36) 120573119899+2= 120588119899+2120590119899120579119899+1

(37) 119901119899+2= 119911119899+2+ 120573119899+1119901119899+ 120573119899+2119904119899+1

(38) 119902119899+2= 119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(39) 119902119899+2= 119911119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(40) 119899 larr 119899 + 2



119903119899+2 le 119903


119899+1 le

|120590119899|119903119899 and |120590

119899|120587119899+2 le |120578





119896(1 le




If ||119911119899+1|| le |120590

119899|||119903119899|| Then

1 times 1 StepElse||120587119899+2|| = ||120578

119899119903119899minus 120577119899+11205883

119899119902119899minus 120579119899+11205882

119899119910119899+1||

If |120590119899|||120587119899+2|| le |120578

119899|||119911119899+1|| Then


End IfEnd If

Algorithm 4












10119887 minus 119860119909



119899211990302lt TOL or when





0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)







0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030)


0 1199020= 1198601199010 119902lowast0= 119860119867119901lowast

0 1199030= 1198601199030 1205880= ⟨119903lowast

0 1199030⟩

(4) for 119899 = 0 1 do(5) 120590

119899= ⟨119902lowast

119899 119902119899⟩

(6) 120572119899= 120588119899120590119899

(7) 119909119899+1= 119909119899+ 120572119899119901119899

(8) 119903119899+1= 119903119899minus 120572119899119902119899

(9) 119909lowast

119899+1= 119909lowast

119899+ 120572119899119901lowast

119899

(10) 119903lowast119899+1= 119903lowast

119899minus 120572119899119902lowast

119899

(11) 119903119899+1= 119860119903119899+1

(12) 120588119899+1= ⟨119903lowast

119899+1 119903119899+1⟩

(13) if 120588119899+1= 0method fails

(14) 120573119899+1= 120588119899+1120588119899

(15) 119901119899+1= 119903119899+1+ 120573119899+1119901119899

(16) 119901lowast119899+1= 119903lowast

119899+1+ 120573119899+1119901lowast

119899

(17) 119902119899+1= 119903119899+1+ 120573119899+1119902119899

(18) 119902lowast119899+1= 119860119867119901lowast

119899



(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1

is diagonal


(1) 120588119899equiv ⟨119903lowast

119899 119903119899⟩ equiv ⟨119903

lowast

119899 119860119903119899⟩ = 0 but 119903lowast

119899and 119860119903

119899are not


(2) 120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast





119899= 0 while 120588

119899= 0






119899+1 119903lowast




119899+1and 119903lowast







0) as follows

119911119899+1= 120590119899119903119899+1

= 120590119899119903119899minus 120588119899119860119901119899

(18)

119911lowast

119899+1= 120590119899119903lowast

119899+1

= 120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899

(19)


K119899+2(119860 1199030) and 119909lowast


0+ K119899+2(119860119867 119903lowast

0) in step (119899 + 2)

as follows

119909119899+2= 119909119899+ [119901119899 119911119899+1] 119891119899

119909lowast

119899+2= 119909lowast

119899+ [119901lowast

119899 119911lowast

119899+1] 119891lowast

119899

(20)


where 119891119899 119891lowast




119899+2isin

K119899+3(119860119867 119903lowast


119903119899+2= 119903119899minus 119860 [119901

119899 119911119899+1] 119891119899 (21)

119903lowast

119899+2= 119903lowast

119899minus 119860119867[119901lowast

119899 119911lowast

119899+1] 119891lowast

119899 (22)


⟨[119901lowast

119899 119911lowast

119899+1] 119860119903119899+2⟩ = 0

⟨[119901119899 119911119899+1] 119860119867119903lowast

119899+2⟩ = 0

(23)


119899


[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119891(1)

119899

119891(2)

119899

]

= [⟨119901lowast

119899 119860119903119899⟩

⟨119911lowast

119899+1 119860119903119899⟩]

(24)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119891lowast(1)

119899

119891lowast(2)

119899

]

= [⟨119860119901119899 119903lowast

119899⟩

⟨119860119911119899+1 119903lowast

119899⟩]

(25)



119901lowast

119899+2isin K119899+3(119860119867 119903lowast


the following form

119901119899+2= 119903119899+2+ [119901119899 119911119899+1] 119892119899 (26)

119901lowast

119899+2= 119903lowast

119899+2+ [119901lowast

119899

119911lowast

119899+1] 119892lowast

119899 (27)

where 119892119899 119892lowast

119899isin C2


⟨[119901lowast

119899 119911lowast

119899+1] 1198602119901119899+2⟩ = 0

⟨[119901119899 119911119899+1] (119860119867)2

119901lowast

119899+2⟩ = 0

(28)


119899and

119892lowast

119899

[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119892(1)

119899

119892(2)

119899

]

= minus[⟨119860119867119901lowast

119899 119860119903119899+2⟩

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩]

(29)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119892lowast(1)

119899

119892lowast(2)

119899

]

= minus[⟨119860119901119899 119860119867119903lowast

119899+2⟩

⟨119860119911119899+1 119860119867119903lowast

119899+2⟩]

(30)


119899+2 119903119899+2

119909lowast119899+2

119903lowast119899+2

119901119899+2

119901lowast119899+2





the following hold

⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119903119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ = ⟨119902

lowast

119899 119902119899⟩ equiv 120590119899

(31)

Proof If 119901119899 119901lowast


119901119899= 119903119899+ 120573119899119901119899minus1

119901lowast119899

= 119903lowast

119899+ 120573119899119901lowast



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast

119899minus 120573119899119901lowast

119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899


⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901119899minus 120573119899119901119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899 119860119901119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(32)

If 119901119899 119901lowast


119899=

119903119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)

119899minus2119911119899minus1

119901lowast119899= 119903lowast

119899+ 119892lowast(1)


119899minus2+ 119892lowast(2)


119899minus1as



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast





119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892lowast(1)

119899minus2⟨119860119867119901lowast

119899minus2 119860119901119899⟩


119899minus2⟨119860119867119911lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901

119899minus 119892(1)


119899minus2119911119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892(1)

119899minus2⟨119860119867119901lowast

119899 119860119901119899minus2⟩

minus 119892(2)

119899minus2⟨119860119867119901lowast

119899 119860119911119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(33)








119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast


119911119899+1

and 119911lowast119899+1


⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867(120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899) 119860119901119899⟩

= 120590119899⟨119860119867119903lowast

119899 119860119901119899⟩ minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

⟨119860119867119901lowast

119899 119860119911119899+1⟩ = ⟨119860

119867119901lowast

119899 119860 (120590119899119903119899minus 120588119899119860119901119899)⟩

= 120590119899⟨119860119867119901lowast

119899 119860119903119899⟩ minus 120588119899⟨119860119867119901lowast

119899 1198602119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

(34)







119894= 0 119894 = 0 1 119899 and ⟨119911lowast

119899+1 119860119911119899+1⟩ = 0 If

120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast




lowast

119899 119902119899⟩ equiv

⟨119860119867119901lowast

119899 119860119901119899⟩ = 0 then ⟨119860119867119911lowast

119899+1 119860119901119899⟩ = 0 From Lemma 7

and (18) we have

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119911lowast

119899+1 minus119911119899+1

120588119899

⟩

= minus1

120588119899

⟨119911lowast

119899+1 119860119911119899+1⟩ = 0

(35)



=

[1199030 1199031 119903

119899]119877lowast119899+1= [119903lowast

0 119903lowast

1 119903

lowast

119899] where 119903



step and let 119875119899+1= [1199010 1199011 119901

119899] 119875lowast119899+1= [119901lowast

0 119901lowast

1 119901

lowast

119899]



(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)

(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1= diag(119863

119896) where 119863



(119899 + 1)





119899119901119899+ 119891(2)

119899119911119899+1

byinduction 119909

119899isin 1199090+ K119899(119860 1199030) and 119901

119899isin K

119899+1(119860 1199030)




119899+2perp

119860119867K119899+2(119860119867 119903lowast

0)Therefore 119909





1 1199072 119907

119898) and W

119898=

span(1199081 1199082 119908





|119907|2

119903= 119907119867119872119903119907 |119908|

2

119897= 119908119867119872119897119908 (36)







W119873 one has

10038161003816100381610038161003816119908119867119860211990710038161003816100381610038161003816le Γ|119907|119903|119908|119897 (37)



119898 we have


|119907|119903=1

sup119908isinW

119898

|119908|119897le1

1199081198671198602119907 ge 120574119896ge 120574 gt 0

(38)

Then

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903 le (1 +

Γ

120574) inf119907isinV119898

|119909 minus 119907|119903 (39)


119898

(119860119867119908)119867

(119860119909 minus 119860119909119898) = 119908

1198671198602(119909 minus 119909

119898) = 0 (40)


1199081198671198602(119909119898minus 119907) = 119908

1198671198602(119909 minus 119907) (41)






120574119896

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907


which yields

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903le |119909 minus 119907|119903 +

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907

10038171003817100381710038171003816100381610038161003816119903le (1 +

Γ

120574) |119909 minus 119907|119903

(43)


119898is arbitrary




(1) ⟨119901lowast119899 119860119903119899⟩ = ⟨119860119901

119899 119903lowast119899⟩ = ⟨119903

lowast

119899 119860119903119899⟩ = ⟨119903

lowast

119899 119903119899⟩ equiv 120588

119899

and ⟨119911lowast119899+1 119860119903119899⟩ = ⟨119860119911

119899+1 119903lowast

119899⟩ = 0 119891

119899= 119891lowast

119899

(2) ⟨119860119867119901lowast119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast

119899+2⟩ = 0 and

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩ = ⟨119860119911

119899+1 119860119867119903lowast

119899+2⟩ = minus120588

119899+2119891(2)

119899


119899+2 119860119903119899+2⟩ 119892119899= 119892lowast

119899

(3) ⟨119860119867119901lowast119899 119860119911119899+1⟩ = ⟨119860119901

119899 119860119867119911lowast

119899+1⟩ = minus120579

119899+1120588119899 where

120579119899+1equiv ⟨119911lowast

119899+1 119860119911119899+1⟩


= 119903lowast

119899+

120573119899119901lowast


119899 119860119903119899⟩ = ⟨119903

lowast

119899+ 120573119899119901lowast

119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩+120573119899⟨119901lowast

119899minus1 119860119903119899⟩ = ⟨119903

lowast





119899⟩ = ⟨119860119903

119899+

120573119899119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ + 120573

119899⟨119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ =



119899 119860119903119899⟩ = ⟨119903lowast

119899+ 119892lowast(1)


119899minus2

+ 119892lowast(2)


119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩ + 119892

lowast(1)


119899minus2

119860119903119899⟩ + 119892

lowast(2)


119899minus1 119860119903119899⟩ = ⟨119903lowast

119899 119860119903119899⟩


119899⟩= ⟨119860(119903

119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)


119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ +

119892(1)


119899⟩+119892(2)


119899⟩= ⟨119860119903

119899 119903lowast

119899⟩= ⟨119903lowast119899 119860119903119899⟩The


119899+1 119903lowast



119899



119899= 119891lowast

119899


119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast


that 119860119901119899isin K

119899+2(119860 1199030) and 119860119867119901lowast

119899isin K

119899+2(119860119867 119903lowast

0)


119899= 0



and ⟨119901lowast119899 119860119903119899⟩ = 120588



119899+1 119860119903119899+2⟩ = ⟨(1119891lowast(2)

119899)(119903lowast


119899+2minus 119891lowast(1)

119899119860119867119901lowast

119899) 119860119903119899+2⟩


119899+2 119860119903119899+2⟩minus119891lowast(1)

119899 ⟨119860119867119901lowast

119899 119860119903119899+2⟩)


119899 )120588119899+2 = (minus1119891(2)

119899)120588119899+2




119899+2⟩ = minus120588

119899+2119891(2)




119899= 119892lowast

119899 because






119899 119860119911119899+1⟩ = ⟨(1120588

119899)(120590119899119903lowast


119899+1) 119860119911119899+1⟩ =

(1120588119899)(120590119899⟨119903lowast

119899 119860119911119899+1⟩ minus ⟨119911

lowast

119899+1 119860119911119899+1⟩) = (minus1120588

119899)⟨119911lowast

119899+1 119860119911119899+1⟩

= minus120579119899+1120588119899 where 120579

119899+1equiv ⟨119911

lowast

119899+1 119860119911119899+1⟩ Because it is


119899 119860119911119899+1⟩ as


119899+1⟩ = ⟨119860119867119911lowast

119899+1 119860119901119899⟩

= ⟨119860119867119901lowast119899 119860119911119899+1⟩ = minus120579





[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119891(1)

119899

119891(2)

119899

]

]

= [120588119899

0]

[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119892(1)

119899

119892(2)

119899

]

]

= [

[

0120588119899+2

119891(2)

119899

]

]

(44)

where 120577119899+1= ⟨119860119867119911lowast

119899+1 119860119911119899+1⟩


119899and 119892

119899can


119891119899=(120577119899+1120588119899 120579119899+1) 1205882

119899

120578119899

119892119899= (120588119899+2

120588119899

120590119899120588119899+2

120579119899+1

)

(45)

where 120578119899equiv 120590119899120577119899+11205882

119899minus 1205792

119899+1


119899+1and 119911




1003817100381710038171003817119903119899+11003817100381710038171003817 gt max 1003817100381710038171003817119903119899

1003817100381710038171003817 1003817100381710038171003817119903119899+2

1003817100381710038171003817 (46)


119899+2equiv 120578119899119903119899+2= 120578119899119903119899minus

120577119899+11205883

119899119860119901119899minus 120579119899+11205882

119899119860119911119899+1


119899+1 le 119903

119899 and


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030) Set 119901

0= 1199030 1199010= 1199030 1199020= 1198601199010 1199020= 1198601198671199010

(3) Compute 1205880= ⟨1199030 1198601199030⟩

(4) Begin LOOP (119899 = 0 1 2 )(5) 120590119899= ⟨119902119899 119902119899⟩

(6) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(7) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(8) 119910119899+1= 119860119904119899+1

(9) 119910119899+1= 119860119867119904119899+1

(10) 120579119899+1= ⟨119904119899+1 119910119899+1⟩

(11) 120577119899+1= ⟨119910119899+1 119910119899+1⟩


119899= 120588119899120590119899

(14) 120588119899+1= 120579119899+11205902

119899

(15) 120573119899+1= 120588119899+1120588119899

(16) 119909119899+1= 119909119899+ 120572119899119901119899

(17) 119903119899+1= 119903119899minus 120572119899119902119899

(18) 119903119899+1= 119903119899minus 120572119899119902119899

(19) 119901119899+1= 119904119899+1120590119899+ 120573119899+1119901119899

(20) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(21) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(22) 119899 larr 119899 + 1

(23) else(24) 120575

119899= 120590119899120577119899+11205882

119899minus 1205792

119899+1

(25) 120572119899= 120577119899+11205883

119899120575119899

(26) 120572119899+1= 120579119899+11205882

119899120575119899

(27) 119909119899+2= 119909119899+ 120572119899119901119899+ 120572119899+1119904119899+1

(28) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(29) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(30) solve 119861119911119899+2= 119903119899+2

(31) solve 119861119867119899+2= 119903119899+2

(32) 119899+2= 119860119911119899+2

(33) 119911119899+2= 119860119867119899+2

(34) 120588119899+2= ⟨119911119899+2 119903119899+2⟩

(35) 120573119899+1= 120588119899+2120588119899

(36) 120573119899+2= 120588119899+2120590119899120579119899+1

(37) 119901119899+2= 119911119899+2+ 120573119899+1119901119899+ 120573119899+2119904119899+1

(38) 119902119899+2= 119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(39) 119902119899+2= 119911119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(40) 119899 larr 119899 + 2



119903119899+2 le 119903


119899+1 le

|120590119899|119903119899 and |120590

119899|120587119899+2 le |120578





119896(1 le




If ||119911119899+1|| le |120590

119899|||119903119899|| Then

1 times 1 StepElse||120587119899+2|| = ||120578

119899119903119899minus 120577119899+11205883

119899119902119899minus 120579119899+11205882

119899119910119899+1||

If |120590119899|||120587119899+2|| le |120578

119899|||119911119899+1|| Then


End IfEnd If

Algorithm 4












10119887 minus 119860119909



119899211990302lt TOL or when





0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)







0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 119891119899 119891lowast




119899+2isin

K119899+3(119860119867 119903lowast


119903119899+2= 119903119899minus 119860 [119901

119899 119911119899+1] 119891119899 (21)

119903lowast

119899+2= 119903lowast

119899minus 119860119867[119901lowast

119899 119911lowast

119899+1] 119891lowast

119899 (22)


⟨[119901lowast

119899 119911lowast

119899+1] 119860119903119899+2⟩ = 0

⟨[119901119899 119911119899+1] 119860119867119903lowast

119899+2⟩ = 0

(23)


119899


[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119891(1)

119899

119891(2)

119899

]

= [⟨119901lowast

119899 119860119903119899⟩

⟨119911lowast

119899+1 119860119903119899⟩]

(24)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119891lowast(1)

119899

119891lowast(2)

119899

]

= [⟨119860119901119899 119903lowast

119899⟩

⟨119860119911119899+1 119903lowast

119899⟩]

(25)



119901lowast

119899+2isin K119899+3(119860119867 119903lowast


the following form

119901119899+2= 119903119899+2+ [119901119899 119911119899+1] 119892119899 (26)

119901lowast

119899+2= 119903lowast

119899+2+ [119901lowast

119899

119911lowast

119899+1] 119892lowast

119899 (27)

where 119892119899 119892lowast

119899isin C2


⟨[119901lowast

119899 119911lowast

119899+1] 1198602119901119899+2⟩ = 0

⟨[119901119899 119911119899+1] (119860119867)2

119901lowast

119899+2⟩ = 0

(28)


119899and

119892lowast

119899

[⟨119860119867119901lowast

119899 119860119901119899⟩ ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ ⟨119860

119867119911lowast

119899+1 119860119911119899+1⟩][119892(1)

119899

119892(2)

119899

]

= minus[⟨119860119867119901lowast

119899 119860119903119899+2⟩

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩]

(29)

[⟨119860119901119899 119860119867119901lowast

119899⟩ ⟨119860119901

119899 119860119867119911lowast

119899+1⟩

⟨119860119911119899+1 119860119867119901lowast

119899⟩ ⟨119860119911

119899+1 119860119867119911lowast

119899+1⟩][119892lowast(1)

119899

119892lowast(2)

119899

]

= minus[⟨119860119901119899 119860119867119903lowast

119899+2⟩

⟨119860119911119899+1 119860119867119903lowast

119899+2⟩]

(30)


119899+2 119903119899+2

119909lowast119899+2

119903lowast119899+2

119901119899+2

119901lowast119899+2





the following hold

⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119903119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ = ⟨119902

lowast

119899 119902119899⟩ equiv 120590119899

(31)

Proof If 119901119899 119901lowast


119901119899= 119903119899+ 120573119899119901119899minus1

119901lowast119899

= 119903lowast

119899+ 120573119899119901lowast



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast

119899minus 120573119899119901lowast

119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899


⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901119899minus 120573119899119901119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899 119860119901119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(32)

If 119901119899 119901lowast


119899=

119903119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)

119899minus2119911119899minus1

119901lowast119899= 119903lowast

119899+ 119892lowast(1)


119899minus2+ 119892lowast(2)


119899minus1as



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast





119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892lowast(1)

119899minus2⟨119860119867119901lowast

119899minus2 119860119901119899⟩


119899minus2⟨119860119867119911lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901

119899minus 119892(1)


119899minus2119911119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892(1)

119899minus2⟨119860119867119901lowast

119899 119860119901119899minus2⟩

minus 119892(2)

119899minus2⟨119860119867119901lowast

119899 119860119911119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(33)








119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast


119911119899+1

and 119911lowast119899+1


⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867(120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899) 119860119901119899⟩

= 120590119899⟨119860119867119903lowast

119899 119860119901119899⟩ minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

⟨119860119867119901lowast

119899 119860119911119899+1⟩ = ⟨119860

119867119901lowast

119899 119860 (120590119899119903119899minus 120588119899119860119901119899)⟩

= 120590119899⟨119860119867119901lowast

119899 119860119903119899⟩ minus 120588119899⟨119860119867119901lowast

119899 1198602119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

(34)







119894= 0 119894 = 0 1 119899 and ⟨119911lowast

119899+1 119860119911119899+1⟩ = 0 If

120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast




lowast

119899 119902119899⟩ equiv

⟨119860119867119901lowast

119899 119860119901119899⟩ = 0 then ⟨119860119867119911lowast

119899+1 119860119901119899⟩ = 0 From Lemma 7

and (18) we have

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119911lowast

119899+1 minus119911119899+1

120588119899

⟩

= minus1

120588119899

⟨119911lowast

119899+1 119860119911119899+1⟩ = 0

(35)



=

[1199030 1199031 119903

119899]119877lowast119899+1= [119903lowast

0 119903lowast

1 119903

lowast

119899] where 119903



step and let 119875119899+1= [1199010 1199011 119901

119899] 119875lowast119899+1= [119901lowast

0 119901lowast

1 119901

lowast

119899]



(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)

(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1= diag(119863

119896) where 119863



(119899 + 1)





119899119901119899+ 119891(2)

119899119911119899+1

byinduction 119909

119899isin 1199090+ K119899(119860 1199030) and 119901

119899isin K

119899+1(119860 1199030)




119899+2perp

119860119867K119899+2(119860119867 119903lowast

0)Therefore 119909





1 1199072 119907

119898) and W

119898=

span(1199081 1199082 119908





|119907|2

119903= 119907119867119872119903119907 |119908|

2

119897= 119908119867119872119897119908 (36)







W119873 one has

10038161003816100381610038161003816119908119867119860211990710038161003816100381610038161003816le Γ|119907|119903|119908|119897 (37)



119898 we have


|119907|119903=1

sup119908isinW

119898

|119908|119897le1

1199081198671198602119907 ge 120574119896ge 120574 gt 0

(38)

Then

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903 le (1 +

Γ

120574) inf119907isinV119898

|119909 minus 119907|119903 (39)


119898

(119860119867119908)119867

(119860119909 minus 119860119909119898) = 119908

1198671198602(119909 minus 119909

119898) = 0 (40)


1199081198671198602(119909119898minus 119907) = 119908

1198671198602(119909 minus 119907) (41)






120574119896

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907


which yields

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903le |119909 minus 119907|119903 +

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907

10038171003817100381710038171003816100381610038161003816119903le (1 +

Γ

120574) |119909 minus 119907|119903

(43)


119898is arbitrary




(1) ⟨119901lowast119899 119860119903119899⟩ = ⟨119860119901

119899 119903lowast119899⟩ = ⟨119903

lowast

119899 119860119903119899⟩ = ⟨119903

lowast

119899 119903119899⟩ equiv 120588

119899

and ⟨119911lowast119899+1 119860119903119899⟩ = ⟨119860119911

119899+1 119903lowast

119899⟩ = 0 119891

119899= 119891lowast

119899

(2) ⟨119860119867119901lowast119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast

119899+2⟩ = 0 and

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩ = ⟨119860119911

119899+1 119860119867119903lowast

119899+2⟩ = minus120588

119899+2119891(2)

119899


119899+2 119860119903119899+2⟩ 119892119899= 119892lowast

119899

(3) ⟨119860119867119901lowast119899 119860119911119899+1⟩ = ⟨119860119901

119899 119860119867119911lowast

119899+1⟩ = minus120579

119899+1120588119899 where

120579119899+1equiv ⟨119911lowast

119899+1 119860119911119899+1⟩


= 119903lowast

119899+

120573119899119901lowast


119899 119860119903119899⟩ = ⟨119903

lowast

119899+ 120573119899119901lowast

119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩+120573119899⟨119901lowast

119899minus1 119860119903119899⟩ = ⟨119903

lowast





119899⟩ = ⟨119860119903

119899+

120573119899119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ + 120573

119899⟨119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ =



119899 119860119903119899⟩ = ⟨119903lowast

119899+ 119892lowast(1)


119899minus2

+ 119892lowast(2)


119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩ + 119892

lowast(1)


119899minus2

119860119903119899⟩ + 119892

lowast(2)


119899minus1 119860119903119899⟩ = ⟨119903lowast

119899 119860119903119899⟩


119899⟩= ⟨119860(119903

119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)


119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ +

119892(1)


119899⟩+119892(2)


119899⟩= ⟨119860119903

119899 119903lowast

119899⟩= ⟨119903lowast119899 119860119903119899⟩The


119899+1 119903lowast



119899



119899= 119891lowast

119899


119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast


that 119860119901119899isin K

119899+2(119860 1199030) and 119860119867119901lowast

119899isin K

119899+2(119860119867 119903lowast

0)


119899= 0



and ⟨119901lowast119899 119860119903119899⟩ = 120588



119899+1 119860119903119899+2⟩ = ⟨(1119891lowast(2)

119899)(119903lowast


119899+2minus 119891lowast(1)

119899119860119867119901lowast

119899) 119860119903119899+2⟩


119899+2 119860119903119899+2⟩minus119891lowast(1)

119899 ⟨119860119867119901lowast

119899 119860119903119899+2⟩)


119899 )120588119899+2 = (minus1119891(2)

119899)120588119899+2




119899+2⟩ = minus120588

119899+2119891(2)




119899= 119892lowast

119899 because






119899 119860119911119899+1⟩ = ⟨(1120588

119899)(120590119899119903lowast


119899+1) 119860119911119899+1⟩ =

(1120588119899)(120590119899⟨119903lowast

119899 119860119911119899+1⟩ minus ⟨119911

lowast

119899+1 119860119911119899+1⟩) = (minus1120588

119899)⟨119911lowast

119899+1 119860119911119899+1⟩

= minus120579119899+1120588119899 where 120579

119899+1equiv ⟨119911

lowast

119899+1 119860119911119899+1⟩ Because it is


119899 119860119911119899+1⟩ as


119899+1⟩ = ⟨119860119867119911lowast

119899+1 119860119901119899⟩

= ⟨119860119867119901lowast119899 119860119911119899+1⟩ = minus120579





[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119891(1)

119899

119891(2)

119899

]

]

= [120588119899

0]

[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119892(1)

119899

119892(2)

119899

]

]

= [

[

0120588119899+2

119891(2)

119899

]

]

(44)

where 120577119899+1= ⟨119860119867119911lowast

119899+1 119860119911119899+1⟩


119899and 119892

119899can


119891119899=(120577119899+1120588119899 120579119899+1) 1205882

119899

120578119899

119892119899= (120588119899+2

120588119899

120590119899120588119899+2

120579119899+1

)

(45)

where 120578119899equiv 120590119899120577119899+11205882

119899minus 1205792

119899+1


119899+1and 119911




1003817100381710038171003817119903119899+11003817100381710038171003817 gt max 1003817100381710038171003817119903119899

1003817100381710038171003817 1003817100381710038171003817119903119899+2

1003817100381710038171003817 (46)


119899+2equiv 120578119899119903119899+2= 120578119899119903119899minus

120577119899+11205883

119899119860119901119899minus 120579119899+11205882

119899119860119911119899+1


119899+1 le 119903

119899 and


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030) Set 119901

0= 1199030 1199010= 1199030 1199020= 1198601199010 1199020= 1198601198671199010

(3) Compute 1205880= ⟨1199030 1198601199030⟩

(4) Begin LOOP (119899 = 0 1 2 )(5) 120590119899= ⟨119902119899 119902119899⟩

(6) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(7) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(8) 119910119899+1= 119860119904119899+1

(9) 119910119899+1= 119860119867119904119899+1

(10) 120579119899+1= ⟨119904119899+1 119910119899+1⟩

(11) 120577119899+1= ⟨119910119899+1 119910119899+1⟩


119899= 120588119899120590119899

(14) 120588119899+1= 120579119899+11205902

119899

(15) 120573119899+1= 120588119899+1120588119899

(16) 119909119899+1= 119909119899+ 120572119899119901119899

(17) 119903119899+1= 119903119899minus 120572119899119902119899

(18) 119903119899+1= 119903119899minus 120572119899119902119899

(19) 119901119899+1= 119904119899+1120590119899+ 120573119899+1119901119899

(20) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(21) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(22) 119899 larr 119899 + 1

(23) else(24) 120575

119899= 120590119899120577119899+11205882

119899minus 1205792

119899+1

(25) 120572119899= 120577119899+11205883

119899120575119899

(26) 120572119899+1= 120579119899+11205882

119899120575119899

(27) 119909119899+2= 119909119899+ 120572119899119901119899+ 120572119899+1119904119899+1

(28) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(29) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(30) solve 119861119911119899+2= 119903119899+2

(31) solve 119861119867119899+2= 119903119899+2

(32) 119899+2= 119860119911119899+2

(33) 119911119899+2= 119860119867119899+2

(34) 120588119899+2= ⟨119911119899+2 119903119899+2⟩

(35) 120573119899+1= 120588119899+2120588119899

(36) 120573119899+2= 120588119899+2120590119899120579119899+1

(37) 119901119899+2= 119911119899+2+ 120573119899+1119901119899+ 120573119899+2119904119899+1

(38) 119902119899+2= 119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(39) 119902119899+2= 119911119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(40) 119899 larr 119899 + 2



119903119899+2 le 119903


119899+1 le

|120590119899|119903119899 and |120590

119899|120587119899+2 le |120578





119896(1 le




If ||119911119899+1|| le |120590

119899|||119903119899|| Then

1 times 1 StepElse||120587119899+2|| = ||120578

119899119903119899minus 120577119899+11205883

119899119902119899minus 120579119899+11205882

119899119910119899+1||

If |120590119899|||120587119899+2|| le |120578

119899|||119911119899+1|| Then


End IfEnd If

Algorithm 4












10119887 minus 119860119909



119899211990302lt TOL or when





0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)







0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901119899minus 120573119899119901119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 120573119899⟨119860119867119901lowast

119899 119860119901119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(32)

If 119901119899 119901lowast


119899=

119903119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)

119899minus2119911119899minus1

119901lowast119899= 119903lowast

119899+ 119892lowast(1)


119899minus2+ 119892lowast(2)


119899minus1as



⟨119860119867119903lowast

119899 119860119901119899⟩ = ⟨119860

119867(119901lowast





119899minus1) 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892lowast(1)

119899minus2⟨119860119867119901lowast

119899minus2 119860119901119899⟩


119899minus2⟨119860119867119911lowast

119899minus1 119860119901119899⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

⟨119860119867119901lowast

119899 119860119903119899⟩ = ⟨119860

119867119901lowast

119899 119860 (119901

119899minus 119892(1)


119899minus2119911119899minus1)⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩ minus 119892(1)

119899minus2⟨119860119867119901lowast

119899 119860119901119899minus2⟩

minus 119892(2)

119899minus2⟨119860119867119901lowast

119899 119860119911119899minus1⟩

= ⟨119860119867119901lowast

119899 119860119901119899⟩

= ⟨119902lowast

119899 119902119899⟩ equiv 120590119899

(33)








119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast


119911119899+1

and 119911lowast119899+1


⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867(120590119899119903lowast

119899minus 120588119899119860119867119901lowast

119899) 119860119901119899⟩

= 120590119899⟨119860119867119903lowast

119899 119860119901119899⟩ minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

⟨119860119867119901lowast

119899 119860119911119899+1⟩ = ⟨119860

119867119901lowast

119899 119860 (120590119899119903119899minus 120588119899119860119901119899)⟩

= 120590119899⟨119860119867119901lowast

119899 119860119903119899⟩ minus 120588119899⟨119860119867119901lowast

119899 1198602119901119899⟩

= 1205902

119899minus 120588119899⟨(119860119867)2

119901lowast

119899 119860119901119899⟩

(34)







119894= 0 119894 = 0 1 119899 and ⟨119911lowast

119899+1 119860119911119899+1⟩ = 0 If

120590119899equiv ⟨119902lowast

119899 119902119899⟩ equiv ⟨119860

119867119901lowast




lowast

119899 119902119899⟩ equiv

⟨119860119867119901lowast

119899 119860119901119899⟩ = 0 then ⟨119860119867119911lowast

119899+1 119860119901119899⟩ = 0 From Lemma 7

and (18) we have

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119901lowast

119899 119860119911119899+1⟩

⟨119860119867119911lowast

119899+1 119860119901119899⟩ = ⟨119860

119867119911lowast

119899+1 minus119911119899+1

120588119899

⟩

= minus1

120588119899

⟨119911lowast

119899+1 119860119911119899+1⟩ = 0

(35)



=

[1199030 1199031 119903

119899]119877lowast119899+1= [119903lowast

0 119903lowast

1 119903

lowast

119899] where 119903



step and let 119875119899+1= [1199010 1199011 119901

119899] 119875lowast119899+1= [119901lowast

0 119901lowast

1 119901

lowast

119899]



(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)

(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1= diag(119863

119896) where 119863



(119899 + 1)





119899119901119899+ 119891(2)

119899119911119899+1

byinduction 119909

119899isin 1199090+ K119899(119860 1199030) and 119901

119899isin K

119899+1(119860 1199030)




119899+2perp

119860119867K119899+2(119860119867 119903lowast

0)Therefore 119909





1 1199072 119907

119898) and W

119898=

span(1199081 1199082 119908





|119907|2

119903= 119907119867119872119903119907 |119908|

2

119897= 119908119867119872119897119908 (36)







W119873 one has

10038161003816100381610038161003816119908119867119860211990710038161003816100381610038161003816le Γ|119907|119903|119908|119897 (37)



119898 we have


|119907|119903=1

sup119908isinW

119898

|119908|119897le1

1199081198671198602119907 ge 120574119896ge 120574 gt 0

(38)

Then

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903 le (1 +

Γ

120574) inf119907isinV119898

|119909 minus 119907|119903 (39)


119898

(119860119867119908)119867

(119860119909 minus 119860119909119898) = 119908

1198671198602(119909 minus 119909

119898) = 0 (40)


1199081198671198602(119909119898minus 119907) = 119908

1198671198602(119909 minus 119907) (41)






120574119896

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907


which yields

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903le |119909 minus 119907|119903 +

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907

10038171003817100381710038171003816100381610038161003816119903le (1 +

Γ

120574) |119909 minus 119907|119903

(43)


119898is arbitrary




(1) ⟨119901lowast119899 119860119903119899⟩ = ⟨119860119901

119899 119903lowast119899⟩ = ⟨119903

lowast

119899 119860119903119899⟩ = ⟨119903

lowast

119899 119903119899⟩ equiv 120588

119899

and ⟨119911lowast119899+1 119860119903119899⟩ = ⟨119860119911

119899+1 119903lowast

119899⟩ = 0 119891

119899= 119891lowast

119899

(2) ⟨119860119867119901lowast119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast

119899+2⟩ = 0 and

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩ = ⟨119860119911

119899+1 119860119867119903lowast

119899+2⟩ = minus120588

119899+2119891(2)

119899


119899+2 119860119903119899+2⟩ 119892119899= 119892lowast

119899

(3) ⟨119860119867119901lowast119899 119860119911119899+1⟩ = ⟨119860119901

119899 119860119867119911lowast

119899+1⟩ = minus120579

119899+1120588119899 where

120579119899+1equiv ⟨119911lowast

119899+1 119860119911119899+1⟩


= 119903lowast

119899+

120573119899119901lowast


119899 119860119903119899⟩ = ⟨119903

lowast

119899+ 120573119899119901lowast

119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩+120573119899⟨119901lowast

119899minus1 119860119903119899⟩ = ⟨119903

lowast





119899⟩ = ⟨119860119903

119899+

120573119899119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ + 120573

119899⟨119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ =



119899 119860119903119899⟩ = ⟨119903lowast

119899+ 119892lowast(1)


119899minus2

+ 119892lowast(2)


119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩ + 119892

lowast(1)


119899minus2

119860119903119899⟩ + 119892

lowast(2)


119899minus1 119860119903119899⟩ = ⟨119903lowast

119899 119860119903119899⟩


119899⟩= ⟨119860(119903

119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)


119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ +

119892(1)


119899⟩+119892(2)


119899⟩= ⟨119860119903

119899 119903lowast

119899⟩= ⟨119903lowast119899 119860119903119899⟩The


119899+1 119903lowast



119899



119899= 119891lowast

119899


119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast


that 119860119901119899isin K

119899+2(119860 1199030) and 119860119867119901lowast

119899isin K

119899+2(119860119867 119903lowast

0)


119899= 0



and ⟨119901lowast119899 119860119903119899⟩ = 120588



119899+1 119860119903119899+2⟩ = ⟨(1119891lowast(2)

119899)(119903lowast


119899+2minus 119891lowast(1)

119899119860119867119901lowast

119899) 119860119903119899+2⟩


119899+2 119860119903119899+2⟩minus119891lowast(1)

119899 ⟨119860119867119901lowast

119899 119860119903119899+2⟩)


119899 )120588119899+2 = (minus1119891(2)

119899)120588119899+2




119899+2⟩ = minus120588

119899+2119891(2)




119899= 119892lowast

119899 because






119899 119860119911119899+1⟩ = ⟨(1120588

119899)(120590119899119903lowast


119899+1) 119860119911119899+1⟩ =

(1120588119899)(120590119899⟨119903lowast

119899 119860119911119899+1⟩ minus ⟨119911

lowast

119899+1 119860119911119899+1⟩) = (minus1120588

119899)⟨119911lowast

119899+1 119860119911119899+1⟩

= minus120579119899+1120588119899 where 120579

119899+1equiv ⟨119911

lowast

119899+1 119860119911119899+1⟩ Because it is


119899 119860119911119899+1⟩ as


119899+1⟩ = ⟨119860119867119911lowast

119899+1 119860119901119899⟩

= ⟨119860119867119901lowast119899 119860119911119899+1⟩ = minus120579





[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119891(1)

119899

119891(2)

119899

]

]

= [120588119899

0]

[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119892(1)

119899

119892(2)

119899

]

]

= [

[

0120588119899+2

119891(2)

119899

]

]

(44)

where 120577119899+1= ⟨119860119867119911lowast

119899+1 119860119911119899+1⟩


119899and 119892

119899can


119891119899=(120577119899+1120588119899 120579119899+1) 1205882

119899

120578119899

119892119899= (120588119899+2

120588119899

120590119899120588119899+2

120579119899+1

)

(45)

where 120578119899equiv 120590119899120577119899+11205882

119899minus 1205792

119899+1


119899+1and 119911




1003817100381710038171003817119903119899+11003817100381710038171003817 gt max 1003817100381710038171003817119903119899

1003817100381710038171003817 1003817100381710038171003817119903119899+2

1003817100381710038171003817 (46)


119899+2equiv 120578119899119903119899+2= 120578119899119903119899minus

120577119899+11205883

119899119860119901119899minus 120579119899+11205882

119899119860119911119899+1


119899+1 le 119903

119899 and


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030) Set 119901

0= 1199030 1199010= 1199030 1199020= 1198601199010 1199020= 1198601198671199010

(3) Compute 1205880= ⟨1199030 1198601199030⟩

(4) Begin LOOP (119899 = 0 1 2 )(5) 120590119899= ⟨119902119899 119902119899⟩

(6) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(7) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(8) 119910119899+1= 119860119904119899+1

(9) 119910119899+1= 119860119867119904119899+1

(10) 120579119899+1= ⟨119904119899+1 119910119899+1⟩

(11) 120577119899+1= ⟨119910119899+1 119910119899+1⟩


119899= 120588119899120590119899

(14) 120588119899+1= 120579119899+11205902

119899

(15) 120573119899+1= 120588119899+1120588119899

(16) 119909119899+1= 119909119899+ 120572119899119901119899

(17) 119903119899+1= 119903119899minus 120572119899119902119899

(18) 119903119899+1= 119903119899minus 120572119899119902119899

(19) 119901119899+1= 119904119899+1120590119899+ 120573119899+1119901119899

(20) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(21) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(22) 119899 larr 119899 + 1

(23) else(24) 120575

119899= 120590119899120577119899+11205882

119899minus 1205792

119899+1

(25) 120572119899= 120577119899+11205883

119899120575119899

(26) 120572119899+1= 120579119899+11205882

119899120575119899

(27) 119909119899+2= 119909119899+ 120572119899119901119899+ 120572119899+1119904119899+1

(28) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(29) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(30) solve 119861119911119899+2= 119903119899+2

(31) solve 119861119867119899+2= 119903119899+2

(32) 119899+2= 119860119911119899+2

(33) 119911119899+2= 119860119867119899+2

(34) 120588119899+2= ⟨119911119899+2 119903119899+2⟩

(35) 120573119899+1= 120588119899+2120588119899

(36) 120573119899+2= 120588119899+2120590119899120579119899+1

(37) 119901119899+2= 119911119899+2+ 120573119899+1119901119899+ 120573119899+2119904119899+1

(38) 119902119899+2= 119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(39) 119902119899+2= 119911119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(40) 119899 larr 119899 + 2



119903119899+2 le 119903


119899+1 le

|120590119899|119903119899 and |120590

119899|120587119899+2 le |120578





119896(1 le




If ||119911119899+1|| le |120590

119899|||119903119899|| Then

1 times 1 StepElse||120587119899+2|| = ||120578

119899119903119899minus 120577119899+11205883

119899119902119899minus 120579119899+11205882

119899119910119899+1||

If |120590119899|||120587119899+2|| le |120578

119899|||119911119899+1|| Then


End IfEnd If

Algorithm 4












10119887 minus 119860119909



119899211990302lt TOL or when





0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)







0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(1) Range(119877119899+1) = Range(119875

119899+1) = K

119899+1(119860 1199030)


119899+1) =K

119899+1(119860119867 119903lowast

0)

(2) 119877lowast119867119899+1119860119877119899+1

is diagonal

(3) 119875lowast119867119899+11198602119875119899+1= diag(119863

119896) where 119863



(119899 + 1)





119899119901119899+ 119891(2)

119899119911119899+1

byinduction 119909

119899isin 1199090+ K119899(119860 1199030) and 119901

119899isin K

119899+1(119860 1199030)




119899+2perp

119860119867K119899+2(119860119867 119903lowast

0)Therefore 119909





1 1199072 119907

119898) and W

119898=

span(1199081 1199082 119908





|119907|2

119903= 119907119867119872119903119907 |119908|

2

119897= 119908119867119872119897119908 (36)







W119873 one has

10038161003816100381610038161003816119908119867119860211990710038161003816100381610038161003816le Γ|119907|119903|119908|119897 (37)



119898 we have


|119907|119903=1

sup119908isinW

119898

|119908|119897le1

1199081198671198602119907 ge 120574119896ge 120574 gt 0

(38)

Then

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903 le (1 +

Γ

120574) inf119907isinV119898

|119909 minus 119907|119903 (39)


119898

(119860119867119908)119867

(119860119909 minus 119860119909119898) = 119908

1198671198602(119909 minus 119909

119898) = 0 (40)


1199081198671198602(119909119898minus 119907) = 119908

1198671198602(119909 minus 119907) (41)






120574119896

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907


which yields

10038161003816100381610038161003817100381710038171003817119909 minus 119909119898

10038171003817100381710038171003816100381610038161003816119903le |119909 minus 119907|119903 +

10038161003816100381610038161003817100381710038171003817119909119898 minus 119907

10038171003817100381710038171003816100381610038161003816119903le (1 +

Γ

120574) |119909 minus 119907|119903

(43)


119898is arbitrary




(1) ⟨119901lowast119899 119860119903119899⟩ = ⟨119860119901

119899 119903lowast119899⟩ = ⟨119903

lowast

119899 119860119903119899⟩ = ⟨119903

lowast

119899 119903119899⟩ equiv 120588

119899

and ⟨119911lowast119899+1 119860119903119899⟩ = ⟨119860119911

119899+1 119903lowast

119899⟩ = 0 119891

119899= 119891lowast

119899

(2) ⟨119860119867119901lowast119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast

119899+2⟩ = 0 and

⟨119860119867119911lowast

119899+1 119860119903119899+2⟩ = ⟨119860119911

119899+1 119860119867119903lowast

119899+2⟩ = minus120588

119899+2119891(2)

119899


119899+2 119860119903119899+2⟩ 119892119899= 119892lowast

119899

(3) ⟨119860119867119901lowast119899 119860119911119899+1⟩ = ⟨119860119901

119899 119860119867119911lowast

119899+1⟩ = minus120579

119899+1120588119899 where

120579119899+1equiv ⟨119911lowast

119899+1 119860119911119899+1⟩


= 119903lowast

119899+

120573119899119901lowast


119899 119860119903119899⟩ = ⟨119903

lowast

119899+ 120573119899119901lowast

119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩+120573119899⟨119901lowast

119899minus1 119860119903119899⟩ = ⟨119903

lowast





119899⟩ = ⟨119860119903

119899+

120573119899119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ + 120573

119899⟨119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ =



119899 119860119903119899⟩ = ⟨119903lowast

119899+ 119892lowast(1)


119899minus2

+ 119892lowast(2)


119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩ + 119892

lowast(1)


119899minus2

119860119903119899⟩ + 119892

lowast(2)


119899minus1 119860119903119899⟩ = ⟨119903lowast

119899 119860119903119899⟩


119899⟩= ⟨119860(119903

119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)


119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ +

119892(1)


119899⟩+119892(2)


119899⟩= ⟨119860119903

119899 119903lowast

119899⟩= ⟨119903lowast119899 119860119903119899⟩The


119899+1 119903lowast



119899



119899= 119891lowast

119899


119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast


that 119860119901119899isin K

119899+2(119860 1199030) and 119860119867119901lowast

119899isin K

119899+2(119860119867 119903lowast

0)


119899= 0



and ⟨119901lowast119899 119860119903119899⟩ = 120588



119899+1 119860119903119899+2⟩ = ⟨(1119891lowast(2)

119899)(119903lowast


119899+2minus 119891lowast(1)

119899119860119867119901lowast

119899) 119860119903119899+2⟩


119899+2 119860119903119899+2⟩minus119891lowast(1)

119899 ⟨119860119867119901lowast

119899 119860119903119899+2⟩)


119899 )120588119899+2 = (minus1119891(2)

119899)120588119899+2




119899+2⟩ = minus120588

119899+2119891(2)




119899= 119892lowast

119899 because






119899 119860119911119899+1⟩ = ⟨(1120588

119899)(120590119899119903lowast


119899+1) 119860119911119899+1⟩ =

(1120588119899)(120590119899⟨119903lowast

119899 119860119911119899+1⟩ minus ⟨119911

lowast

119899+1 119860119911119899+1⟩) = (minus1120588

119899)⟨119911lowast

119899+1 119860119911119899+1⟩

= minus120579119899+1120588119899 where 120579

119899+1equiv ⟨119911

lowast

119899+1 119860119911119899+1⟩ Because it is


119899 119860119911119899+1⟩ as


119899+1⟩ = ⟨119860119867119911lowast

119899+1 119860119901119899⟩

= ⟨119860119867119901lowast119899 119860119911119899+1⟩ = minus120579





[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119891(1)

119899

119891(2)

119899

]

]

= [120588119899

0]

[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119892(1)

119899

119892(2)

119899

]

]

= [

[

0120588119899+2

119891(2)

119899

]

]

(44)

where 120577119899+1= ⟨119860119867119911lowast

119899+1 119860119911119899+1⟩


119899and 119892

119899can


119891119899=(120577119899+1120588119899 120579119899+1) 1205882

119899

120578119899

119892119899= (120588119899+2

120588119899

120590119899120588119899+2

120579119899+1

)

(45)

where 120578119899equiv 120590119899120577119899+11205882

119899minus 1205792

119899+1


119899+1and 119911




1003817100381710038171003817119903119899+11003817100381710038171003817 gt max 1003817100381710038171003817119903119899

1003817100381710038171003817 1003817100381710038171003817119903119899+2

1003817100381710038171003817 (46)


119899+2equiv 120578119899119903119899+2= 120578119899119903119899minus

120577119899+11205883

119899119860119901119899minus 120579119899+11205882

119899119860119911119899+1


119899+1 le 119903

119899 and


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030) Set 119901

0= 1199030 1199010= 1199030 1199020= 1198601199010 1199020= 1198601198671199010

(3) Compute 1205880= ⟨1199030 1198601199030⟩

(4) Begin LOOP (119899 = 0 1 2 )(5) 120590119899= ⟨119902119899 119902119899⟩

(6) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(7) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(8) 119910119899+1= 119860119904119899+1

(9) 119910119899+1= 119860119867119904119899+1

(10) 120579119899+1= ⟨119904119899+1 119910119899+1⟩

(11) 120577119899+1= ⟨119910119899+1 119910119899+1⟩


119899= 120588119899120590119899

(14) 120588119899+1= 120579119899+11205902

119899

(15) 120573119899+1= 120588119899+1120588119899

(16) 119909119899+1= 119909119899+ 120572119899119901119899

(17) 119903119899+1= 119903119899minus 120572119899119902119899

(18) 119903119899+1= 119903119899minus 120572119899119902119899

(19) 119901119899+1= 119904119899+1120590119899+ 120573119899+1119901119899

(20) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(21) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(22) 119899 larr 119899 + 1

(23) else(24) 120575

119899= 120590119899120577119899+11205882

119899minus 1205792

119899+1

(25) 120572119899= 120577119899+11205883

119899120575119899

(26) 120572119899+1= 120579119899+11205882

119899120575119899

(27) 119909119899+2= 119909119899+ 120572119899119901119899+ 120572119899+1119904119899+1

(28) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(29) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(30) solve 119861119911119899+2= 119903119899+2

(31) solve 119861119867119899+2= 119903119899+2

(32) 119899+2= 119860119911119899+2

(33) 119911119899+2= 119860119867119899+2

(34) 120588119899+2= ⟨119911119899+2 119903119899+2⟩

(35) 120573119899+1= 120588119899+2120588119899

(36) 120573119899+2= 120588119899+2120590119899120579119899+1

(37) 119901119899+2= 119911119899+2+ 120573119899+1119901119899+ 120573119899+2119904119899+1

(38) 119902119899+2= 119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(39) 119902119899+2= 119911119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(40) 119899 larr 119899 + 2



119903119899+2 le 119903


119899+1 le

|120590119899|119903119899 and |120590

119899|120587119899+2 le |120578





119896(1 le




If ||119911119899+1|| le |120590

119899|||119903119899|| Then

1 times 1 StepElse||120587119899+2|| = ||120578

119899119903119899minus 120577119899+11205883

119899119902119899minus 120579119899+11205882

119899119910119899+1||

If |120590119899|||120587119899+2|| le |120578

119899|||119911119899+1|| Then


End IfEnd If

Algorithm 4












10119887 minus 119860119909



119899211990302lt TOL or when





0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)







0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119899⟩ = ⟨119860119903

119899+

120573119899119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ + 120573

119899⟨119860119901119899minus1 119903lowast

119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ =



119899 119860119903119899⟩ = ⟨119903lowast

119899+ 119892lowast(1)


119899minus2

+ 119892lowast(2)


119899minus1 119860119903119899⟩ =

⟨119903lowast

119899 119860119903119899⟩ + 119892

lowast(1)


119899minus2

119860119903119899⟩ + 119892

lowast(2)


119899minus1 119860119903119899⟩ = ⟨119903lowast

119899 119860119903119899⟩


119899⟩= ⟨119860(119903

119899+ 119892(1)

119899minus2119901119899minus2+ 119892(2)


119899⟩ = ⟨119860119903

119899 119903lowast

119899⟩ +

119892(1)


119899⟩+119892(2)


119899⟩= ⟨119860119903

119899 119903lowast

119899⟩= ⟨119903lowast119899 119860119903119899⟩The


119899+1 119903lowast



119899



119899= 119891lowast

119899


119899 119860119903119899+2⟩ = ⟨119860119901

119899 119860119867119903lowast


that 119860119901119899isin K

119899+2(119860 1199030) and 119860119867119901lowast

119899isin K

119899+2(119860119867 119903lowast

0)


119899= 0



and ⟨119901lowast119899 119860119903119899⟩ = 120588



119899+1 119860119903119899+2⟩ = ⟨(1119891lowast(2)

119899)(119903lowast


119899+2minus 119891lowast(1)

119899119860119867119901lowast

119899) 119860119903119899+2⟩


119899+2 119860119903119899+2⟩minus119891lowast(1)

119899 ⟨119860119867119901lowast

119899 119860119903119899+2⟩)


119899 )120588119899+2 = (minus1119891(2)

119899)120588119899+2




119899+2⟩ = minus120588

119899+2119891(2)




119899= 119892lowast

119899 because






119899 119860119911119899+1⟩ = ⟨(1120588

119899)(120590119899119903lowast


119899+1) 119860119911119899+1⟩ =

(1120588119899)(120590119899⟨119903lowast

119899 119860119911119899+1⟩ minus ⟨119911

lowast

119899+1 119860119911119899+1⟩) = (minus1120588

119899)⟨119911lowast

119899+1 119860119911119899+1⟩

= minus120579119899+1120588119899 where 120579

119899+1equiv ⟨119911

lowast

119899+1 119860119911119899+1⟩ Because it is


119899 119860119911119899+1⟩ as


119899+1⟩ = ⟨119860119867119911lowast

119899+1 119860119901119899⟩

= ⟨119860119867119901lowast119899 119860119911119899+1⟩ = minus120579





[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119891(1)

119899

119891(2)

119899

]

]

= [120588119899

0]

[[[

[

120590119899minus120579119899+1

120588119899

minus120579119899+1

120588119899

120577119899+1

]]]

]

[

[

119892(1)

119899

119892(2)

119899

]

]

= [

[

0120588119899+2

119891(2)

119899

]

]

(44)

where 120577119899+1= ⟨119860119867119911lowast

119899+1 119860119911119899+1⟩


119899and 119892

119899can


119891119899=(120577119899+1120588119899 120579119899+1) 1205882

119899

120578119899

119892119899= (120588119899+2

120588119899

120590119899120588119899+2

120579119899+1

)

(45)

where 120578119899equiv 120590119899120577119899+11205882

119899minus 1205792

119899+1


119899+1and 119911




1003817100381710038171003817119903119899+11003817100381710038171003817 gt max 1003817100381710038171003817119903119899

1003817100381710038171003817 1003817100381710038171003817119903119899+2

1003817100381710038171003817 (46)


119899+2equiv 120578119899119903119899+2= 120578119899119903119899minus

120577119899+11205883

119899119860119901119899minus 120579119899+11205882

119899119860119911119899+1


119899+1 le 119903

119899 and


(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030) Set 119901

0= 1199030 1199010= 1199030 1199020= 1198601199010 1199020= 1198601198671199010

(3) Compute 1205880= ⟨1199030 1198601199030⟩

(4) Begin LOOP (119899 = 0 1 2 )(5) 120590119899= ⟨119902119899 119902119899⟩

(6) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(7) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(8) 119910119899+1= 119860119904119899+1

(9) 119910119899+1= 119860119867119904119899+1

(10) 120579119899+1= ⟨119904119899+1 119910119899+1⟩

(11) 120577119899+1= ⟨119910119899+1 119910119899+1⟩


119899= 120588119899120590119899

(14) 120588119899+1= 120579119899+11205902

119899

(15) 120573119899+1= 120588119899+1120588119899

(16) 119909119899+1= 119909119899+ 120572119899119901119899

(17) 119903119899+1= 119903119899minus 120572119899119902119899

(18) 119903119899+1= 119903119899minus 120572119899119902119899

(19) 119901119899+1= 119904119899+1120590119899+ 120573119899+1119901119899

(20) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(21) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(22) 119899 larr 119899 + 1

(23) else(24) 120575

119899= 120590119899120577119899+11205882

119899minus 1205792

119899+1

(25) 120572119899= 120577119899+11205883

119899120575119899

(26) 120572119899+1= 120579119899+11205882

119899120575119899

(27) 119909119899+2= 119909119899+ 120572119899119901119899+ 120572119899+1119904119899+1

(28) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(29) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(30) solve 119861119911119899+2= 119903119899+2

(31) solve 119861119867119899+2= 119903119899+2

(32) 119899+2= 119860119911119899+2

(33) 119911119899+2= 119860119867119899+2

(34) 120588119899+2= ⟨119911119899+2 119903119899+2⟩

(35) 120573119899+1= 120588119899+2120588119899

(36) 120573119899+2= 120588119899+2120590119899120579119899+1

(37) 119901119899+2= 119911119899+2+ 120573119899+1119901119899+ 120573119899+2119904119899+1

(38) 119902119899+2= 119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(39) 119902119899+2= 119911119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(40) 119899 larr 119899 + 2



119903119899+2 le 119903


119899+1 le

|120590119899|119903119899 and |120590

119899|120587119899+2 le |120578





119896(1 le




If ||119911119899+1|| le |120590

119899|||119903119899|| Then

1 times 1 StepElse||120587119899+2|| = ||120578

119899119903119899minus 120577119899+11205883

119899119902119899minus 120579119899+11205882

119899119910119899+1||

If |120590119899|||120587119899+2|| le |120578

119899|||119911119899+1|| Then


End IfEnd If

Algorithm 4












10119887 minus 119860119909



119899211990302lt TOL or when





0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)







0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(1) Compute 1199030= 119887 minus 119860119909


0

(2) Choose 119903lowast0= 119875 (119860) 119903



(eg 119903lowast0= 1198601199030) Set 119901

0= 1199030 1199010= 1199030 1199020= 1198601199010 1199020= 1198601198671199010

(3) Compute 1205880= ⟨1199030 1198601199030⟩

(4) Begin LOOP (119899 = 0 1 2 )(5) 120590119899= ⟨119902119899 119902119899⟩

(6) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(7) 119904119899+1= 120590119899119903119899minus 120588119899119902119899

(8) 119910119899+1= 119860119904119899+1

(9) 119910119899+1= 119860119867119904119899+1

(10) 120579119899+1= ⟨119904119899+1 119910119899+1⟩

(11) 120577119899+1= ⟨119910119899+1 119910119899+1⟩


119899= 120588119899120590119899

(14) 120588119899+1= 120579119899+11205902

119899

(15) 120573119899+1= 120588119899+1120588119899

(16) 119909119899+1= 119909119899+ 120572119899119901119899

(17) 119903119899+1= 119903119899minus 120572119899119902119899

(18) 119903119899+1= 119903119899minus 120572119899119902119899

(19) 119901119899+1= 119904119899+1120590119899+ 120573119899+1119901119899

(20) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(21) 119902119899+1= 119910119899+1120590119899+ 120573119899+1119902119899

(22) 119899 larr 119899 + 1

(23) else(24) 120575

119899= 120590119899120577119899+11205882

119899minus 1205792

119899+1

(25) 120572119899= 120577119899+11205883

119899120575119899

(26) 120572119899+1= 120579119899+11205882

119899120575119899

(27) 119909119899+2= 119909119899+ 120572119899119901119899+ 120572119899+1119904119899+1

(28) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(29) 119903119899+2= 119903119899minus 120572119899119902119899minus 120572119899+1119910119899+1

(30) solve 119861119911119899+2= 119903119899+2

(31) solve 119861119867119899+2= 119903119899+2

(32) 119899+2= 119860119911119899+2

(33) 119911119899+2= 119860119867119899+2

(34) 120588119899+2= ⟨119911119899+2 119903119899+2⟩

(35) 120573119899+1= 120588119899+2120588119899

(36) 120573119899+2= 120588119899+2120590119899120579119899+1

(37) 119901119899+2= 119911119899+2+ 120573119899+1119901119899+ 120573119899+2119904119899+1

(38) 119902119899+2= 119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(39) 119902119899+2= 119911119899+2+ 120573119899+1119902119899+ 120573119899+2119910119899+1

(40) 119899 larr 119899 + 2



119903119899+2 le 119903


119899+1 le

|120590119899|119903119899 and |120590

119899|120587119899+2 le |120578





119896(1 le




If ||119911119899+1|| le |120590

119899|||119903119899|| Then

1 times 1 StepElse||120587119899+2|| = ||120578

119899119903119899minus 120577119899+11205883

119899119902119899minus 120579119899+11205882

119899119910119899+1||

If |120590119899|||120587119899+2|| le |120578

119899|||119911119899+1|| Then


End IfEnd If

Algorithm 4












10119887 minus 119860119909



119899211990302lt TOL or when





0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)







0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










If ||119911119899+1|| le |120590

119899|||119903119899|| Then

1 times 1 StepElse||120587119899+2|| = ||120578

119899119903119899minus 120577119899+11205883

119899119902119899minus 120579119899+11205882

119899119910119899+1||

If |120590119899|||120587119899+2|| le |120578

119899|||119911119899+1|| Then


End IfEnd If

Algorithm 4












10119887 minus 119860119909



119899211990302lt TOL or when





0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)







0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

BiCORCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(b)

BCGCSBCG

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(c)

0 10 20 30 40 50minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Iters

Relre

s

BCGCSBCG

(d)

250 260 270 280 290 300minus52

minus5

minus48

minus46

minus44

minus42

minus4

minus38

minus36

Iters

Relre

s

BCGCSBCG

(e)

CSBCGCSBiCOR

0 100 200 300Iters

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Relre

s

(f)







0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Iters

Relre

s

BiCORCSBiCOR

(d)









0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGBiCOR

CSBCGCSBiCOR

(a)

0 20 40 60 80 100 120 140 160 180minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

CSBCGCSBiCOR

(b)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BCGCSBCG

(c)

0 50 100 150 200 250minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

Iters

Relre

s

BiCORCSBiCOR

(d)








Acknowledgments




References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

University of Groningen Exploiting the Composite Step