Research Article An Algorithm for Computing Geometric …downloads.hindawi.com/journals/aaa/2014/978629.pdf · An Algorithm for Computing Geometric Mean of Two Hermitian Positive

Research ArticleAn Algorithm for Computing Geometric Mean of TwoHermitian Positive Definite Matrices via Matrix Sign

F Soleymani1 M Sharifi1 S Shateyi2 and F Khaksar Haghani1

1 Department of Mathematics Islamic Azad University Shahrekord Branch Shahrekord Iran2Department of Mathematics and Applied Mathematics School of Mathematical and Natural Sciences University of VendaPrivate Bag X5050 Thohoyandou 0950 South Africa

Correspondence should be addressed to S Shateyi stanfordshateyiunivenacza

Received 7 May 2014 Revised 12 July 2014 Accepted 23 July 2014 Published 6 August 2014

Academic Editor Alicia Cordero

Copyright copy 2014 F Soleymani 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

Using the relation between a principal matrix square root and its inverse with the geometric mean we present a fast algorithmfor computing the geometric mean of two Hermitian positive definite matrices The algorithm is stable and possesses a highconvergence order Some experiments are included to support the proposed computational algorithm

1 Introduction

This paper tries to provide a fast algorithm for finding thegeometric mean of two Hermitian positive definite matricesvia the application of matrix sign function It is knownthat the scalar arithmetic-geometric mean agm(119886 119887) of two(nonnegative) numbers 119886 and 119887 is defined by starting with1198860= 119886 and 119887

0= 119887 and then iterating

119886119896+1

=1

2[119886119896+ 119887119896]

119887119896+1

= radic119886119896119887119896 119896 = 0 1 2

(1)

until 119886119896

= 119887119896to the desired precision The scalar sequences

of 119886119896 119887119896 converge to each other Note that in (1) radic119886

119896119887119896

is the geometric mean of two positive numbers 119886119896and 119887119896per

computing cycle Although this iterative formula is quite easyand reliable for two scalars its extension for general squareand nonsingular matrices is not an easy task In this work weare concerned with the matrix case of two square Hermitianpositive definite matrices

A right definition of the matrix geometric meanGM(119860 119861) of two positive definite matrices 119860 and 119861 can beexpressed by

GM (119860 119861) = 119860(119860minus1

119861)12

(2)

where given a square matrix 119872 having no nonpositivereal eigenvalues 119872

12 denotes the unique solution of thequadratic matrix equation

1198832= 119872 (3)

whose eigenvalues lie in the right half plane The definition(2) was given in the seventies by Pusz and Woronowicz [1]There are some other important definitions for computing thematrix geometric mean of two matrices see for example thework of Lawson-Lim [2]Note thatwhen just twomatrices areinvolved the theory is well developed but in case of findingthe matrix geometric mean of more than two matrices theformulation is kind of hard see for more [3]

A variant formulation for the scalar case of geometricmean could be expressed by

gm (119886 119887) = radic119886119887

= (119886119887119886

119886)

12

= 11988612

(119887

119886)

12

11988612

= 11988612

(119886minus12

119887119886minus12

)12

11988612

(4)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 978629 6 pageshttpdxdoiorg1011552014978629

2 Abstract and Applied Analysis

As could be seen in (4) the computation of scalargeometric mean is fully related to the square root of thepositive scalar 119886 and its inverse A similar and significantdefinition for a matrix geometric mean has been developedand suggested by Bhatia in [4] as follows

119860119861 = 11986012

(119860minus12

119861119860minus12

)12

11986012

(5)

for two Hermitian positive definite (HPD) matrices 119860 and 119861We use the notation to show the geometric mean of twoHPD matrices

Note that the formulas (2) and (5) coincide and are equalFor more refer to [5]

The rest of this paper is organized as follows InSection 2 we combine the root-finding Newtonrsquos method anda Chebyshev-Halley-type method to devise a fast iterativeformula for solving nonlinear equations In Section 3 weprovide the link between nonlinear equation solvers andthe calculation of some special matrix functions This willillustrate how the new algorithms could be constructed andimplemented An implementation of the proposed iterativeformula in symbolic softwareMathematica [6] along a discus-sion about the stability of the schemewill be provided thereinNote that the idea of computing the geometric mean usingthe sign function can also be found in [7] In Section 4 weshow the numerical results and highlight the benefit of theproposed technique Conclusions are drawn in Section 5

2 Construction of a New Method

It is known that a common way for improving the con-vergence speed of iterative methods for solving nonlinearscalar smooth equations is to combine the already developedschemes Hence let us first combine the well-known methodof Newton into a special method of Chebyshev-Halley-typescheme [8] to derive a new iterative scheme as follows

119910119896= 119909119896minus

119891 (119909119896)

1198911015840 (119909119896)

119909119896+1

= 119910119896minus (1 +

1

2(

119871119896

1 minus ((45) 119871119896)))

119891 (119910119896)

1198911015840 (119910119896)

(6)

wherein 119871119896

= 11989110158401015840(119910119896)119891(119910119896)1198911015840(119910

119896)2 For obtaining a back-

ground about nonlinear equations one may consult [9 10]

Theorem 1 Let 120572 isin 119863 be a simple zero of a sufficientlydifferentiable function 119891 119863 sube R rarr R for an open interval119863 which contains 119909

0as an initial approximation of 120572 Then

the iterative expression (6) without memory satisfies the errorequation below

119890119896+1

= (211988852

5minus 1198883

21198883) 1198906

119896+ 119874 (119890

7

119896) (7)

wherein 119888119895= 119891(119895)(120572)(1198951198911015840(120572)) 119890

119896= 119909119896minus 120572

Proof The proof of this theorem is based on Taylorrsquos seriesexpansion of the iterative method (6) around the solution in

the 119896th iterate To save the space and not to be distracted fromthe main topic we here exclude the proof The steps of theproof are similar to those taken in [11]

The iterative method (6) reaches sixth-order convergenceusing five functional evaluations and thus achieves theefficiency index 615 asymp 1430 which is higher than that ofNewton that is 212 asymp 1414 Furthermore applying (6) forsolving the polynomial equation 119892(119909) equiv 1199092 minus1 = 0 has globalconvergence (except the points lying on the imaginary axis)This global behavior which could easily be shown by drawingthe basins of attraction (6) for 119892(119909) is useful in practicalmatrix problems so as to allow us deal with all kinds of HPDmatrices with any spectra

Remark 2 Note that since the global behavior can beobserved from the basins of attraction then we do not pursuethis fact by theoretical analysis for (6)

A recent discussion about the relation between matrixsign and nonlinear equation solvers is given in [12 13] Westate that all of such extensions in the scalar case and studyingtheir orders are of symbolic computational nature Such anextension to the matrix environment will also be done in thenext section using symbolic computations

3 Construction of an Algorithmfor Geometric Mean

The relation between Section 2 and our main aim for com-puting matrix geometric mean is not clear at the first sightTo illustrate this we recall that many of the important matrixfunctions such as matrix square root that is the solution tothematrix equation (3) andmatrix sign functionwhich is thesolution to the following matrix equation

1198832= 119868 (8)

can be calculated approximately by matrix iteration func-tions Such fixed-point type methods are convergent underspecial conditions to the aimed matrix function and arebasically originated from root-finding methods [7]

On the other hand the important definition of thematrixgeometric mean (5) requires the computation of the matrixsquare root of 119860 and its inverse Hence in order to design anefficient algorithm for (5) we wish to construct an iterativeexpression in which we compute both 119860

12 and 119860minus12 at thesame time

Toward this aim we apply the new nonlinear equationsolver (6) for solving thematrix equation (8)This applicationwould yield the following matrix iteration in its reciprocalform

119883119896+1

= (48119883119896+ 272119883

3

119896+ 272119883

5

119896+ 48119883

7

119896)

times [7119868 + 1481198832

119896+ 330119883

4

119896+ 148119883

6

119896+ 71198838

119896]minus1

(9)

with a proper1198830for finding the sign matrix

Abstract and Applied Analysis 3

Remark 3 The iteration (9) is not a member of the Padefamily of iterations introduced in [14] for matrix sign

In order to connect (9) with our aim we remind animportant identity as follows [7]

sign([0 119860

119868 0]) = [

0 11986012

Aminus12 0] (10)

which indicates an important relationship between principalmatrix square root 11986012 and the matrix sign function

The identity (10) has an advantage which is the computa-tion of the principal inversematrix square root alongwith theprincipal matrix square root at the same time Let us in whatfollows first study the stability behavior of (9)

Lemma 4 Let 119860 = [119886119894119895]119899times119899

have no eigenvalues onRminus Thenthe sequence 119883

119896119896=infin

119896=0generated by (9) using 119883

0= [ 0 119860119868 0

] isasymptotically stable

Proof Let Δ119896be a numerical perturbation introduced at the

119896th iterate of (9) We obtain

119883119896= 119883119896+ Δ119896 (11)

All terms that are quadratic in their errors are removed fromour analysis This formal manipulation is meaningful if Δ

119896is

sufficiently small We have

119883119896+1

= (48119883119896+ 272119883

3

119896+ 272119883

5

119896+ 48119883

7

119896)

times [7119868 + 1481198832

119896+ 330119883

4

119896+ 148119883

6

119896+ 71198838

119896]minus1

= (48 (119883119896+ Δ119896) + 272(119883

119896+ Δ119896)3

+ 272(119883119896+ Δ119896)5+ 48(119883

119896+ Δ119896)7)

times [7119868 + 148(119883119896+ Δ119896)2+ 330(119883

119896+ Δ119896)4

+ 148(119883119896+ Δ119896)6+ 7(119883

119896+ Δ119896)8]minus1

(12)

Note that the commutativity between 119883119896and Δ

119896is not

used throughout this lemma To simplify this the followingidentity will be used (for any nonsingular matrix 119861 and thematrix 119862)

(119861 + 119862)minus1

asymp 119861minus1

minus 119861minus1

119862119861minus1

(13)

Simplifying yields

119883119896+1

asymp (119878 +5

2Δ119896+

3

2119878Δ119896119878) (119868 + 2119878Δ

119896+ 2Δ119896119878)minus1

asymp (119878 +5

2Δ119896+

3

2119878Δ119896119878) (119868 minus 2119878Δ

119896minus 2Δ119896119878)

(14)

where 1198782 = 119868 119878minus1 = 119878 and for enough large 119896 we assumed

119883119896asymp 119878 After some algebraicmanipulations and usingΔ

119896+1=

119883119896+1

minus 119883119896+1

asymp 119883119896+1

minus 119878 we conclude that

Δ119896+1

asymp1

2Δ119896minus

1

2119878Δ119896119878 (15)

Applying (15) recursively we have

1003817100381710038171003817Δ 119896+11003817100381710038171003817 le

1

2119896+11003817100381710038171003817Δ 0 minus 119878Δ

01198781003817100381710038171003817 (16)

From (16) we can conclude that the perturbation at the iterate119896+1 is boundedThis allows to conclude that a perturbation ata given step is bounded at the subsequent stepsTherefore thesequence 119883

119896 generated by (9) is asymptotically stable

Note that Lemma 4 just shows the asymptotical stabilityIn general the fixed-point iteration 119883

119896+1= 119892(119883

119896) is stable

in a neighborhood of a fixed point 119883 if Frechet derivative119889119892119883has bounded powers [7] Furthermore if 119883

119896+1= 119892(119883

119896)

is any super-linearly convergent iteration for 119878 = sign(1198830)

then 119889119892119878(119864) = 119871

119878(119864) = (12)(119864 minus 119878119864119878) where 119871

119878is the

Frechet derivative of the matrix sign function at 119878 Hence 119889119892119878

is idempotent (119889119892119878∘ 119889119892119878

= 119889119892119878) and the iteration is stable

thus all sign iterations such as (9) are automatically stableIt is now easy to deduce the following convergence

theorem for (9)

Theorem 5 Let 119860 = [119886119894119895]119899times119899

have no eigenvalues on Rminus If1198830

= [ 0 119860119868 0

] then the iterative method (9) converges to 119878 =

sign([ 0 119860119868 0

])

Proof The convergence of rational iterations can be analyzedin terms of the convergence of the eigenvalues of thematrices119883119896The reason for this is that if119883has a Jordan decomposition

119883 = 119885119869119885minus1 then 119877(119883) = 119885119877(119869)119885minus1 Let 119860 have a Jordancanonical form arranged as 119885minus1119860119885 = Λ where 119885 is anonsingular matrix It is also known that [7]

sign (Λ) = sign (119885minus1

119860119885) = 119885minus1 sign (119860)119885 (17)

If we define 119863119896

= 119885minus1119883119896119885 then from the method (9) we

obtain

119863119896+1

= (48119863119896+ 272119863

3

119896+ 272119863

5

119896+ 48119863

7

119896)

times [7119868 + 1481198632

119896+ 330119863

4

119896+ 148119863

6

119896+ 71198638

119896]minus1

(18)

Notice that if 1198630is a diagonal matrix then based on an

inductive proof all successive119863119896are diagonal too From (18)

it is enough to prove that 119863119896 converges to sign(Λ) in order

to ensure the convergence of the sequence generated by (9)We can write (18) as 119899 uncoupled scalar iterations to solve

119892(119909) = 1199092 minus 1 = 0 given by

119889119894

119896+1=

48119889119894119896+ 272119889119894

119896

3

+ 272119889119894119896

5

+ 48119889119894119896

7

7 + 148119889119894119896

2

+ 330119889119894119896

4

+ 148119889119894119896

6

+ 7119889119894119896

8 (19)

where 119889119894119896

= (119863119896)119894119894and 1 le 119894 le 119899 From (18) and (19) it is

enough to study the convergence of 119889119894119896 to sign(120582

119894) for all

1 le 119894 le 119899From (19) and since the eigenvalues of 119860 are not pure

imaginary we have that sign(120582119894) = 119904119894= plusmn1 Thus we attain

119889119894119896+1

minus 1

119889119894119896+1

+ 1= minus

(minus1 + 119889119894119896)6

(7 + 119889119894119896(minus6 + 7119889119894

119896))

(1 + 119889119894119896)6

(7 + 119889119894119896(6 + 7119889119894

119896))

(20)


Since |1198891198940| = |120582

119894| gt 0 we have

lim119896rarrinfin

1003816100381610038161003816100381610038161003816100381610038161003816

119889119894119896+1

minus 1

119889119894119896+1

+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

= 0 (21)

and lim119896rarrinfin

|119889119894119896| = 1 = | sign(120582

119894)| This shows that

119889119894119896 is convergent Now it could be easy to conclude that

lim119896rarrinfin

119863119896= sign(Λ) Finally we have

lim119896rarrinfin

119883119896= 119885( lim

119896rarrinfin

119863119896)119885minus1

= 119885 sign (Λ) 119885minus1

= sign (119860)

(22)

The proof is complete

The iteration (9) requires one matrix inversion beforecomputing step and obtains both 11986012 and 119860minus12 which areof interest in (5) The implementation of (9) for computingprincipal square roots requires a sharp attention so as to savemuch effort Since the intermediate matrices are all sparse (atleast half of the entries are zero) then one could simply usesparse approximation techniques to save up the memory andtime

An implementation of (9) to compute 119860119861 for twoHPD matrices in the programming package Mathematica isbrought forward as in (Algorithm 1)

In Algorithm 1 the four-argument function MGM com-putes119860119861 by entering the four arguments ldquomatrix119860rdquo ldquomatrix119861rdquo ldquothe maximum number of iterations that (9) is allowed totakerdquo and the ldquotolerancerdquo of the stopping termination in theinfinity norm 119883

119896+1minus 119883119896infin119883119896+1

infin

le toleranceNote that for computing the principal matrix square

root of (119860minus12119861119860minus12)12 we have used the Jordan Canonicalapproach which has been provided in the general form forcomputing matrix functions in the code FunM

4 Experiments

We test the contributed method (9) denoted by PM usingMathematica 8 inmachine precision Apart from this schemeanother iterative method which is known as DB method [15]and given by

1198840= 119860 119885

0= 119868 119896 = 0 1

119884119896+1

=1

2[119884119896+ 119885minus1

119896]

119885119896+1

=1

2[119885119896+ 119884minus1

119896]

(23)

is considered This method generates the sequences 119884119896 and

119885119896 which converge to 11986012 and 119860minus12 respectivelyWe remark that the first and the second substeps of (6)

result in the quadratic Newtonrsquos method (NM) and cubicChebyshev-Halleyrsquos method (CHM) for matrix sign [7] inwhat follows

119883119896+1

=1

2(119883119896+ 119883minus1

119896)

119883119896+1

= (16119883119896+ 24119883

3

119896) [3119868 + 30119883

2

119896+ 71198834

119896]minus1

(24)

Example 6 As the first example we consider the matrix 119860 =

( 2 11 2

) 119861 = ( 38 11 2

) whereas its exact matrix geometric mean isgiven by [16] 119860119861 = ( 8 1

1 2)

The proposed approach converges to the solution matrixin 2 iterations which shows a completely fast convergence

Example 7 We now consider the two HPD matrices asfollows

119860 = (

2 1

1 d dd d 1

1 2

)

119899times119899

119861 =

(((((

(

3

2

2

5

2

5d d

d d2

5

2

5

3

2

)))))

)119899times119899

(25)

when 119899 = 250 and compute 119860119861

We compare the behavior of differentmethods and reportthe numerical results using 119897

infinfor all norms involved with

the stopping criterion 119883119896+1

minus 119883119896infin119883119896+1

infin

le 120598 = 10minus5

in Figure 1 As could be seen the numerical results are inharmony with the theoretical aspects of Section 3 and show afast convergence for the proposed method (9) instead of DBNM and CHM

5 Conclusions

Based on the quadratical Newtonrsquos scheme and the cubicalmethod of Chebyshev-Halley we have developed an iterativemethod with sixth order of convergence for solving nonlin-ear scalar smooth equations The computational efficiencyshowed its superiority in contrast to NM Then the methodwith global behavior for finding matrix sign function hasbeen extended for computing the principal matrix squareroot and its inverse This procedure was followed so as tobuild a fast algorithm for finding the matrix geometric meanof two HPD matrices We also have studied the asymptoticalstability for the proposed technique

To illustrate the new technique some numerical exam-ples were presented Computational results have justifiedrobust and efficient convergence behavior of the proposedmethod Similar numerical experimentations carried out fora number of problems of different types confirmed the aboveconclusions to a great extent

We conclude the paper with the remark that in manynumerical applications high precision in computations isrequiredThe results of numerical experiments show that thehigh order efficient methods such as (9) associated with amultiple-precision arithmetic are very useful because theyyield a clear reduction in the number of iterates


ClearAll[Globalrsquolowast]FunM[fun X ]= Module[faux dim mataux JordanD sim JordanF eps

fdiag diagQ fauxD (dim = LengthX

faux[xx i j ]=

Which[i lt= j 1Abs[i - j] (D[fun x Abs[i - j]]) x -gt xx True 0]

mataux[Y ]= Table[faux[Y[[i j]] i j] i 1 dim j 1 dim]

JordanD = JordanDecomposition[X] N sim = JordanD[[1]]

JordanF = JordanD[[2]] eps = 1lowast10 ⋀ -10

diagQ = Norm[JordanF - DiagonalMatrix[Diagonal[JordanF]]]

fauxD[xx ]= (fun) x -gt xx

fdiag= DiagonalMatrix[Map[fauxD Diagonal[JordanF]]]

Which[diagQ lt eps simfdiagInverse[sim] True

simmataux[JordanF]Inverse[sim]])]

MGM[A B maxIterations tolerance ]= Module[k = 0

n n = Dimensions[A] Id = SparseArray[i i -gt 1 n n]

A1 = SparseArrayArrayFlatten[0 NA Id 0] Y[0] = A1

R[0] = 1 Id2 = SparseArray[i i -gt 1 2 n 2 n]

QuietWhile[k lt maxIterations ampamp R[k] gt= tolerance

Y2 = Y[k]Y[k] Y3 = Y2Y[k] Y4 = Y3Y[k]

Y5 = Y4Y[k] Y6 = Y5Y[k] Y7 = Y6Y[k] Y8 = Y7Y[k]

l1 = SparseArray[7 Id2 + 148 Y2 + 330 Y4 + 148 Y6 + 7 Y8]

l2 = SparseArrayArrayFlatten[Inversel1[[1 n 1 n]] 0

0 Inversel1[[n + 1 2 n n + 1 2 n]]]

Y[k + 1] = SparseArray[(48 Y[k] + 272 Y3 + 272 Y5 + 48 Y7)l2]

R[k + 1] = Norm[Y[k + 1] - Y[k] Infinity] Norm[Y[k + 1] Infinity] k++]

AS = Y[k][[1 n n + 1 2 n]] IAS = Y[k][[n + 1 2 n 1 n]]

Mat = (IASBIAS)

ASFunM[Sqrt[x] Mat]AS]

Algorithm 1

2 4 6 8 10

001

1

CHMNM

DBPM

10minus4

10minus6

Figure 1The log plot for the comparison of number of iterations inExample 7

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to anonymous reviewers for carefullyreading the paper and helping to improve the presentation

References

[1] W Pusz and S L Woronowicz ldquoFunctional calculus forsesquilinear forms and the purification maprdquo Reports on Math-ematical Physics vol 8 no 2 pp 159ndash170 1975

[2] J D Lawson and Y Lim ldquoThe geometric mean matricesmetrics and morerdquo The American Mathematical Monthly vol108 no 9 pp 797ndash812 2001

[3] D A Bini and B Iannazzo ldquoA note on computing matrixgeometric meansrdquo Advances in Computational Mathematicsvol 35 no 2ndash4 pp 175ndash192 2011

[4] R Bhatia Positive Definite Matrices Princeton Series in AppliedMathematics Princeton University Press Princeton NJ USA2007

[5] B Iannazzo and B Meini ldquoThe palindromic cyclic reductionand related algorithmsrdquo Calcolo 2014

[6] J Hoste Mathematica Demystified McGraw-Hill New YorkNY USA 2009

[7] N J Higham Functions of Matrices Theory and ComputationSociety for Industrial and Applied Mathematics PhiladelphiaPa USA 2008

[8] J M Gutierrez and M A Hernandez ldquoA family of Chebyshev-Halley type methods in Banach spacesrdquo Bulletin of the Aus-tralian Mathematical Society vol 55 no 1 pp 113ndash130 1997

[9] R F Lin H M Ren Z Smarda Q B Wu Y Khan andJ L Hu ldquoNew families of third-order iterative methods forfinding multiple rootsrdquo Journal of Applied Mathematics vol2014 Article ID 812072 9 pages 2014


[10] F Soleymani ldquoSome high-order iterative methods for findingall the real zerosrdquoThai Journal of Mathematics vol 12 no 2 pp313ndash327 2014

[11] F Soleymani S Shateyi and G Ozkum ldquoAn iterative solverin the presence and absence of multiplicity for nonlinearequationsrdquo The Scientific World Journal vol 2013 Article ID837243 9 pages 2013

[12] F Soleymani E Tohidi S Shateyi and F K Haghani ldquoSomematrix iterations for computing matrix sign functionrdquo Journalof Applied Mathematics vol 2014 Article ID 425654 9 pages2014

[13] F Soleymani P S Stanimirovic S Shateyi and F KhaksarHaghani ldquoApproximating the matrix sign function using anovel iterativemethodrdquoAbstract andApplied Analysis vol 2014Article ID 105301 9 pages 2014

[14] C Kenney and A J Laub ldquoRational iterative methods for thematrix sign functionrdquo SIAM Journal on Matrix Analysis andApplications vol 12 no 2 pp 273ndash291 1991

[15] E D Denman and A N Beavers Jr ldquoThe matrix sign functionand computations in systemsrdquo Applied Mathematics and Com-putation vol 2 no 1 pp 63ndash94 1976

[16] B Iannazzo ldquoThe geometric mean of two matrices from acomputational viewpointrdquo httparxivorgabs12010101

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



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Stochastic AnalysisInternational Journal of


As could be seen in (4) the computation of scalargeometric mean is fully related to the square root of thepositive scalar 119886 and its inverse A similar and significantdefinition for a matrix geometric mean has been developedand suggested by Bhatia in [4] as follows

119860119861 = 11986012

(119860minus12

119861119860minus12

)12

11986012

(5)

for two Hermitian positive definite (HPD) matrices 119860 and 119861We use the notation to show the geometric mean of twoHPD matrices

Note that the formulas (2) and (5) coincide and are equalFor more refer to [5]

The rest of this paper is organized as follows InSection 2 we combine the root-finding Newtonrsquos method anda Chebyshev-Halley-type method to devise a fast iterativeformula for solving nonlinear equations In Section 3 weprovide the link between nonlinear equation solvers andthe calculation of some special matrix functions This willillustrate how the new algorithms could be constructed andimplemented An implementation of the proposed iterativeformula in symbolic softwareMathematica [6] along a discus-sion about the stability of the schemewill be provided thereinNote that the idea of computing the geometric mean usingthe sign function can also be found in [7] In Section 4 weshow the numerical results and highlight the benefit of theproposed technique Conclusions are drawn in Section 5

2 Construction of a New Method

It is known that a common way for improving the con-vergence speed of iterative methods for solving nonlinearscalar smooth equations is to combine the already developedschemes Hence let us first combine the well-known methodof Newton into a special method of Chebyshev-Halley-typescheme [8] to derive a new iterative scheme as follows

119910119896= 119909119896minus

119891 (119909119896)

1198911015840 (119909119896)

119909119896+1

= 119910119896minus (1 +

1

2(

119871119896

1 minus ((45) 119871119896)))

119891 (119910119896)

1198911015840 (119910119896)

(6)

wherein 119871119896

= 11989110158401015840(119910119896)119891(119910119896)1198911015840(119910

119896)2 For obtaining a back-

ground about nonlinear equations one may consult [9 10]

Theorem 1 Let 120572 isin 119863 be a simple zero of a sufficientlydifferentiable function 119891 119863 sube R rarr R for an open interval119863 which contains 119909

0as an initial approximation of 120572 Then

the iterative expression (6) without memory satisfies the errorequation below

119890119896+1

= (211988852

5minus 1198883

21198883) 1198906

119896+ 119874 (119890

7

119896) (7)

wherein 119888119895= 119891(119895)(120572)(1198951198911015840(120572)) 119890

119896= 119909119896minus 120572

Proof The proof of this theorem is based on Taylorrsquos seriesexpansion of the iterative method (6) around the solution in

the 119896th iterate To save the space and not to be distracted fromthe main topic we here exclude the proof The steps of theproof are similar to those taken in [11]

The iterative method (6) reaches sixth-order convergenceusing five functional evaluations and thus achieves theefficiency index 615 asymp 1430 which is higher than that ofNewton that is 212 asymp 1414 Furthermore applying (6) forsolving the polynomial equation 119892(119909) equiv 1199092 minus1 = 0 has globalconvergence (except the points lying on the imaginary axis)This global behavior which could easily be shown by drawingthe basins of attraction (6) for 119892(119909) is useful in practicalmatrix problems so as to allow us deal with all kinds of HPDmatrices with any spectra

Remark 2 Note that since the global behavior can beobserved from the basins of attraction then we do not pursuethis fact by theoretical analysis for (6)

A recent discussion about the relation between matrixsign and nonlinear equation solvers is given in [12 13] Westate that all of such extensions in the scalar case and studyingtheir orders are of symbolic computational nature Such anextension to the matrix environment will also be done in thenext section using symbolic computations

3 Construction of an Algorithmfor Geometric Mean

The relation between Section 2 and our main aim for com-puting matrix geometric mean is not clear at the first sightTo illustrate this we recall that many of the important matrixfunctions such as matrix square root that is the solution tothematrix equation (3) andmatrix sign functionwhich is thesolution to the following matrix equation

1198832= 119868 (8)

can be calculated approximately by matrix iteration func-tions Such fixed-point type methods are convergent underspecial conditions to the aimed matrix function and arebasically originated from root-finding methods [7]

On the other hand the important definition of thematrixgeometric mean (5) requires the computation of the matrixsquare root of 119860 and its inverse Hence in order to design anefficient algorithm for (5) we wish to construct an iterativeexpression in which we compute both 119860

12 and 119860minus12 at thesame time

Toward this aim we apply the new nonlinear equationsolver (6) for solving thematrix equation (8)This applicationwould yield the following matrix iteration in its reciprocalform

119883119896+1

= (48119883119896+ 272119883

3

119896+ 272119883

5

119896+ 48119883

7

119896)

times [7119868 + 1481198832

119896+ 330119883

4

119896+ 148119883

6

119896+ 71198838

119896]minus1

(9)

with a proper1198830for finding the sign matrix




sign([0 119860

119868 0]) = [

0 11986012

Aminus12 0] (10)



Lemma 4 Let 119860 = [119886119894119895]119899times119899


119896119896=infin


0= [ 0 119860119868 0




119883119896= 119883119896+ Δ119896 (11)


119896is


119883119896+1

= (48119883119896+ 272119883

3

119896+ 272119883

5

119896+ 48119883

7

119896)

times [7119868 + 1481198832

119896+ 330119883

4

119896+ 148119883

6

119896+ 71198838

119896]minus1

= (48 (119883119896+ Δ119896) + 272(119883

119896+ Δ119896)3

+ 272(119883119896+ Δ119896)5+ 48(119883

119896+ Δ119896)7)

times [7119868 + 148(119883119896+ Δ119896)2+ 330(119883

119896+ Δ119896)4

+ 148(119883119896+ Δ119896)6+ 7(119883

119896+ Δ119896)8]minus1

(12)


119896is not


(119861 + 119862)minus1

asymp 119861minus1

minus 119861minus1

119862119861minus1

(13)

Simplifying yields

119883119896+1

asymp (119878 +5

2Δ119896+

3

2119878Δ119896119878) (119868 + 2119878Δ

119896+ 2Δ119896119878)minus1

asymp (119878 +5

2Δ119896+

3

2119878Δ119896119878) (119868 minus 2119878Δ

119896minus 2Δ119896119878)

(14)



119896+1=

119883119896+1

minus 119883119896+1

asymp 119883119896+1


Δ119896+1

asymp1

2Δ119896minus

1

2119878Δ119896119878 (15)


1003817100381710038171003817Δ 119896+11003817100381710038171003817 le

1

2119896+11003817100381710038171003817Δ 0 minus 119878Δ

01198781003817100381710038171003817 (16)




119896+1= 119892(119883

119896) is stable


119896+1= 119892(119883

119896)


then 119889119892119878(119864) = 119871

119878(119864) = (12)(119864 minus 119878119864119878) where 119871

119878is the


is idempotent (119889119892119878∘ 119889119892119878



theorem for (9)



= [ 0 119860119868 0


sign([ 0 119860119868 0

])




119860119885) = 119885minus1 sign (119860)119885 (17)



obtain

119863119896+1

= (48119863119896+ 272119863

3

119896+ 272119863

5

119896+ 48119863

7

119896)

times [7119868 + 1481198632

119896+ 330119863

4

119896+ 148119863

6

119896+ 71198638

119896]minus1

(18)





119892(119909) = 1199092 minus 1 = 0 given by

119889119894

119896+1=

48119889119894119896+ 272119889119894

119896

3

+ 272119889119894119896

5

+ 48119889119894119896

7

7 + 148119889119894119896

2

+ 330119889119894119896

4

+ 148119889119894119896

6

+ 7119889119894119896

8 (19)

where 119889119894119896



119894) for all



119889119894119896+1

minus 1

119889119894119896+1

+ 1= minus

(minus1 + 119889119894119896)6

(7 + 119889119894119896(minus6 + 7119889119894

119896))

(1 + 119889119894119896)6

(7 + 119889119894119896(6 + 7119889119894

119896))

(20)


Since |1198891198940| = |120582


lim119896rarrinfin

1003816100381610038161003816100381610038161003816100381610038161003816

119889119894119896+1

minus 1

119889119894119896+1

+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

= 0 (21)


|119889119894119896| = 1 = | sign(120582



lim119896rarrinfin


lim119896rarrinfin

119883119896= 119885( lim

119896rarrinfin

119863119896)119885minus1

= 119885 sign (Λ) 119885minus1

= sign (119860)

(22)





119896+1minus 119883119896infin119883119896+1

infin



4 Experiments


1198840= 119860 119885

0= 119868 119896 = 0 1

119884119896+1

=1

2[119884119896+ 119885minus1

119896]

119885119896+1

=1

2[119885119896+ 119884minus1

119896]

(23)




119883119896+1

=1

2(119883119896+ 119883minus1

119896)

119883119896+1

= (16119883119896+ 24119883

3

119896) [3119868 + 30119883

2

119896+ 71198834

119896]minus1

(24)


( 2 11 2

) 119861 = ( 38 11 2


1 2)



119860 = (

2 1

1 d dd d 1

1 2

)

119899times119899

119861 =

(((((

(

3

2

2

5

2

5d d

d d2

5

2

5

3

2

)))))

)119899times119899

(25)

when 119899 = 250 and compute 119860119861




minus 119883119896infin119883119896+1

infin

le 120598 = 10minus5


5 Conclusions







faux[xx i j ]=















Y2 = Y[k]Y[k] Y3 = Y2Y[k] Y4 = Y3Y[k]




0 Inversel1[[n + 1 2 n n + 1 2 n]]]




Mat = (IASBIAS)


Algorithm 1

2 4 6 8 10

001

1

CHMNM

DBPM

10minus4

10minus6




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












sign([0 119860

119868 0]) = [

0 11986012

Aminus12 0] (10)



Lemma 4 Let 119860 = [119886119894119895]119899times119899


119896119896=infin


0= [ 0 119860119868 0




119883119896= 119883119896+ Δ119896 (11)


119896is


119883119896+1

= (48119883119896+ 272119883

3

119896+ 272119883

5

119896+ 48119883

7

119896)

times [7119868 + 1481198832

119896+ 330119883

4

119896+ 148119883

6

119896+ 71198838

119896]minus1

= (48 (119883119896+ Δ119896) + 272(119883

119896+ Δ119896)3

+ 272(119883119896+ Δ119896)5+ 48(119883

119896+ Δ119896)7)

times [7119868 + 148(119883119896+ Δ119896)2+ 330(119883

119896+ Δ119896)4

+ 148(119883119896+ Δ119896)6+ 7(119883

119896+ Δ119896)8]minus1

(12)


119896is not


(119861 + 119862)minus1

asymp 119861minus1

minus 119861minus1

119862119861minus1

(13)

Simplifying yields

119883119896+1

asymp (119878 +5

2Δ119896+

3

2119878Δ119896119878) (119868 + 2119878Δ

119896+ 2Δ119896119878)minus1

asymp (119878 +5

2Δ119896+

3

2119878Δ119896119878) (119868 minus 2119878Δ

119896minus 2Δ119896119878)

(14)



119896+1=

119883119896+1

minus 119883119896+1

asymp 119883119896+1


Δ119896+1

asymp1

2Δ119896minus

1

2119878Δ119896119878 (15)


1003817100381710038171003817Δ 119896+11003817100381710038171003817 le

1

2119896+11003817100381710038171003817Δ 0 minus 119878Δ

01198781003817100381710038171003817 (16)




119896+1= 119892(119883

119896) is stable


119896+1= 119892(119883

119896)


then 119889119892119878(119864) = 119871

119878(119864) = (12)(119864 minus 119878119864119878) where 119871

119878is the


is idempotent (119889119892119878∘ 119889119892119878



theorem for (9)



= [ 0 119860119868 0


sign([ 0 119860119868 0

])




119860119885) = 119885minus1 sign (119860)119885 (17)



obtain

119863119896+1

= (48119863119896+ 272119863

3

119896+ 272119863

5

119896+ 48119863

7

119896)

times [7119868 + 1481198632

119896+ 330119863

4

119896+ 148119863

6

119896+ 71198638

119896]minus1

(18)





119892(119909) = 1199092 minus 1 = 0 given by

119889119894

119896+1=

48119889119894119896+ 272119889119894

119896

3

+ 272119889119894119896

5

+ 48119889119894119896

7

7 + 148119889119894119896

2

+ 330119889119894119896

4

+ 148119889119894119896

6

+ 7119889119894119896

8 (19)

where 119889119894119896



119894) for all



119889119894119896+1

minus 1

119889119894119896+1

+ 1= minus

(minus1 + 119889119894119896)6

(7 + 119889119894119896(minus6 + 7119889119894

119896))

(1 + 119889119894119896)6

(7 + 119889119894119896(6 + 7119889119894

119896))

(20)


Since |1198891198940| = |120582


lim119896rarrinfin

1003816100381610038161003816100381610038161003816100381610038161003816

119889119894119896+1

minus 1

119889119894119896+1

+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

= 0 (21)


|119889119894119896| = 1 = | sign(120582



lim119896rarrinfin


lim119896rarrinfin

119883119896= 119885( lim

119896rarrinfin

119863119896)119885minus1

= 119885 sign (Λ) 119885minus1

= sign (119860)

(22)





119896+1minus 119883119896infin119883119896+1

infin



4 Experiments


1198840= 119860 119885

0= 119868 119896 = 0 1

119884119896+1

=1

2[119884119896+ 119885minus1

119896]

119885119896+1

=1

2[119885119896+ 119884minus1

119896]

(23)




119883119896+1

=1

2(119883119896+ 119883minus1

119896)

119883119896+1

= (16119883119896+ 24119883

3

119896) [3119868 + 30119883

2

119896+ 71198834

119896]minus1

(24)


( 2 11 2

) 119861 = ( 38 11 2


1 2)



119860 = (

2 1

1 d dd d 1

1 2

)

119899times119899

119861 =

(((((

(

3

2

2

5

2

5d d

d d2

5

2

5

3

2

)))))

)119899times119899

(25)

when 119899 = 250 and compute 119860119861




minus 119883119896infin119883119896+1

infin

le 120598 = 10minus5


5 Conclusions







faux[xx i j ]=















Y2 = Y[k]Y[k] Y3 = Y2Y[k] Y4 = Y3Y[k]




0 Inversel1[[n + 1 2 n n + 1 2 n]]]




Mat = (IASBIAS)


Algorithm 1

2 4 6 8 10

001

1

CHMNM

DBPM

10minus4

10minus6




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Since |1198891198940| = |120582


lim119896rarrinfin

1003816100381610038161003816100381610038161003816100381610038161003816

119889119894119896+1

minus 1

119889119894119896+1

+ 1

1003816100381610038161003816100381610038161003816100381610038161003816

= 0 (21)


|119889119894119896| = 1 = | sign(120582



lim119896rarrinfin


lim119896rarrinfin

119883119896= 119885( lim

119896rarrinfin

119863119896)119885minus1

= 119885 sign (Λ) 119885minus1

= sign (119860)

(22)





119896+1minus 119883119896infin119883119896+1

infin



4 Experiments


1198840= 119860 119885

0= 119868 119896 = 0 1

119884119896+1

=1

2[119884119896+ 119885minus1

119896]

119885119896+1

=1

2[119885119896+ 119884minus1

119896]

(23)




119883119896+1

=1

2(119883119896+ 119883minus1

119896)

119883119896+1

= (16119883119896+ 24119883

3

119896) [3119868 + 30119883

2

119896+ 71198834

119896]minus1

(24)


( 2 11 2

) 119861 = ( 38 11 2


1 2)



119860 = (

2 1

1 d dd d 1

1 2

)

119899times119899

119861 =

(((((

(

3

2

2

5

2

5d d

d d2

5

2

5

3

2

)))))

)119899times119899

(25)

when 119899 = 250 and compute 119860119861




minus 119883119896infin119883119896+1

infin

le 120598 = 10minus5


5 Conclusions







faux[xx i j ]=















Y2 = Y[k]Y[k] Y3 = Y2Y[k] Y4 = Y3Y[k]




0 Inversel1[[n + 1 2 n n + 1 2 n]]]




Mat = (IASBIAS)


Algorithm 1

2 4 6 8 10

001

1

CHMNM

DBPM

10minus4

10minus6




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












faux[xx i j ]=















Y2 = Y[k]Y[k] Y3 = Y2Y[k] Y4 = Y3Y[k]




0 Inversel1[[n + 1 2 n n + 1 2 n]]]




Mat = (IASBIAS)


Algorithm 1

2 4 6 8 10

001

1

CHMNM

DBPM

10minus4

10minus6




Acknowledgment


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

Research Article An Algorithm for Computing Geometric …downloads.hindawi.com/journals/aaa/2014/978629.pdf · An Algorithm for Computing Geometric Mean of Two Hermitian Positive