13 Methods for the solution of AXD − BXC = E

8/11/2019 13 Methods for the solution of AXD BXC = E

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BIT 20 11980 , 34l 345

METHODS FOR THE SOLUTION OF A X D B X C = E

ND ITS PPLIC TIO N

IN THE NUMERIC L SOLUTION OF

IMPLIC IT ORDIN RY DIFFER ENTI L EQU TIONS

MICHAEL A. EPTON

Abstract

The solution of the equation

A X D - B X C= E

is discussed, partly in terms of the

generalized eigenproblem. Useful applications arise in connection with the numerical

solution of implicit differential equations.

Introduction

We consider here the l inear equat ion for X

(1)

AXD -BX C = E

where

A,B~R mxm,

C , D e R ~ x n an d X , E ~ R m xn

By us ing tensor prod ucts together wi th the appro pria te def ini t ion of ~ and 5,

equat ion (1) can be wri t ten

(1)

[ ADT) -

(BCT)]3c = ~.

This equat ion is a general izat ion of the sys tem

(2)

AX -X C = E

s tudied by Gantm acher [1] , Bar te l s and S tewar t [2] a nd o the rs . Gan tmac her has

shown that (2) has a unique solut ion i f

spectrum (A) N spectrum (C) = ~ ,

and has fur ther given an expl ici t solut ion provided the Jordan decom posi t ions are

given for A and C. Barte ls and Stewart have sho wn how (2) ma y be solved i f us t a

Schur type decomposi t ion is avai lable for A and C. Recent ly, Enright [3] has

observed tha t (2) m ay be quite efficiently solved if one m atrix, say C, is redu ced to

Schu r form while A is redu ced to H essenberg. Th e essential business of this paper

is to extend these ideas to the system (1).

Received May 16, 1979. Revised August 1, 1980.


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342

MICHAE L A EPTON

Discussion

The existence and u niquen ess of a solutio n to (1) is deter min ed by the spectral

decomposi t ions of the matr ix penci ls

A - 2 B , C - 2 D .

In fact, (1) may be

transformed into (2) using a shift technique (cf. [1], V.2, p. 28).

E = A X D - B X C = (A - )~B + )~B)XD - B X C

= (A -2 B )X D -B X (C- )~D ) .

Assuming tha t

A - 2 B

a nd

C - 2 D

are

regular

penci ls and that 2 is not an

eigenvalue of e i ther , we may pre-mul t iply by (A -2 B ) -1, pos t -mu l t iply by

(C - 2D ) - 1 a nd ob t a in

(3)

(A -2 B ) - IE (C- 2 D ) -1 = X [ D (C- )~D ) -a ] - [ (A -2 B ) - IB ] X

which has essentially the same form as equation (2).

The t ransfo rma t ion indicated by eq uat ion (3) represents a perfect ly acceptable

appro ach to the solut ion of (1). However , i ts success depends upon a choice for

the shift 2 that makes the matrices

(A-)~B)

a nd

( C - 2 D )

well conditioned. In

what follows we shal l show how equ at ion (1) ma y be app roac hed di rect ly, us ing

techniques associa ted wi th the numerical t reatment of the general ized

eigenproblem.

Without loss of general i ty we assume that the row dimension m of X and E

exceeds or equals the column dimension n. If this is not originally true, i t can be

achieved s imply by t ransposing eq uat ion (1). Wh en this assump tion is made, the

algorithm to be described presently is essentially optimal with respect to

ar i thmet ic operat ions .

Using the theory for the general ized e igenproblem we know that there exis t

t ransfo rmat io ns Q and Z ~ R such that

Q CZ

a nd

QDZ

are upper t r iangular .

(Such trans format ions a re provided by the Q -z code of S tewar t and M ole r [4]

and i t s ref inement by Ward [5] ; a lso per t inent i s the

L - Z

a lgor i thm of

Kau fma nn [6]) . Indeed , Gantm acher shows how to f ind t rans format ions Q and Z

such that

Q CZ

a nd

QDZ

are in Jordan normal form, but we usual ly avoid this

t ransformat ion because i t s computat ion tends to be a numerical ly poorly

condi t ioned process*. Set t ing

(4.a), (4.b)

QC Z = M, QDZ = A

and defining X' , E ' by

(5.a), (5.b) X ' =

xo- 1 , E = EZ

we see that (1) takes the form

A X Q -1 Q D Z -B X Q -1 Q C Z = E Z

or

(6)

AX A-BX M = E .

* Counterexamples to this advice do exist. See

Applications

below.


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METHODS FOR THE SOLUTION OF

A X D B X C = E . . . 343

De no tin g the elem ents of A and M by -;~ij,/~ij, the co lum ns of X , E by x~, e~ wh ere

i= l(1)n, and using the fact that A and M a re uppe r t r iangular, the j th c olumn

of equa tion (6) reads

J J

(7) A ~ x'i2 ~j-B ~ x'~pij = e~.

i = i = l

These equa tions may be solved sequential ly for x ~ i=1 (1) n by solving the n

systems

(8) (,~jjA - jiB)x) = e'j - ~, (2oA - I~,jB)xl.

i < j

Moreo ver, systems of equa tions of the form (8) may b e quite efficiently solved if

t ransformations R and S are found such that RAS and RBS are both upper

triangular or even upper Hessenberg. The sim ultaneous reduction of A and B to

upper Hessenberg form may be performed most efficiently by the prel iminary

rout ines in Kaufmann s L - Z general ized eigenvalue package [6], and less

efficiently but more stably by the prel iminary routines for Stewa rt and Moler s

(2-Z general ized eigenvalue package.

With R and S computed such that RAS and RBS are Hessenberg, the result ing

systems

(9) (2~jRAS-t~j~RBS)(S-lx'~)= R Ie ~- ~ ()~jA-p~fl)x'~l

i < j

may be solved for S-~x~ quite readily. Because RAS and RBS are He ssenberg, so is

2~jRAS-I~jjRBS, and such matrices can be factored in about (n2/2) multiplies.

Since S is general ly of simple form (in the L- Z algori thm, S is a produc t of

elementary permutat ions and elementary lower tr iangular t ransformations), xj is

readi ly computed by

(10) x} = S ( S - I x ) .

Applications

The applicat ions of most immed iate interest to the author and ul t imately the

st imulus of these remarks arise in the implementat ion of implici t Rung e-K utta

integrat ion formulae

[7

and block mult istep formulae [8] for the numerical

solut ion of implicit differential equa tions (i.e., equa tions of the form g(2 ,x )= 0,

[93.

Because implici t Run ge- Ku tta formulae are a subcase of the block mult istep

formulae, we treat this lat ter case in detai l and conclude with some remarks about

the simplifications possible for IRK formulae.

As used to obtain a solut ion x(t) of g(2 ,x) =0 , the b lock mul t is tep methods

work as fol lows. Suppose one has available quanti t ies (2j , ,_p,

xj, n_~), j=

1(1)v,

p = 1 (1)k that app rox ima te 2(t), x(t) at past t imes tj, n_ p = tn_p_ ~ + hcj. (Usually the

num ber s c~ satisfy O~=cj~_1.) The essential idea here is that for fixed p, time points


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MICHAEL A E PTON

t~,._p, lie in the interval [t ._ p_ 1, t ._p ] and

x j , ._p 5%._p

represent the solut ion

x ( t )

and its derivative ~(t) at these time points. The integ er v, called the b lock size of

the method, is the num ber of est imates of x and ~ generated at each integrat ion

step. To advance the method from t ime t ._a to t ime t . , one then requires that

quan tities x~.., ~j.. assoc iated with the interval [t ._ 1, t .] satisfy the differential

equation,

(11)

g(5:j,., xj,.) = 0

and simultaneously, the block muit istep discret izat ion formulae

0 = l v .

p = 0

To solve the pair of equ atio ns (11) and (12) for x~.,, 2~.,, a varian t of Ne wto n s

meth od is used that at stage l of the i terat ion, enforces upo n p erturb ations

.s,n -- Xj , n 'L n j,n

the condit ions

(13) g(2}t,).,

x j,.,) + [Og/~Yc] 65:(],).,+ [~g /Ox ] ~x}l). = 0 j = 1 . . . . v

14) rl t) + ~, er A )~ ) -- ht~ )~a) ~ = 0

x- ' i j ~oj , n r ' i j ~j ,n , t

3=1

where

r ~

is defined as the amount by which

x},,, x(~),

fail to satisfy (12):

r~t)

~ t.(o)y0) _ hfl(ob?(o

= ~' i j ~j , n -r i j - - j, nJ

j = l

15)

+ p~l [ j=~

( ~ x J ' - P - h f l l z ] ) ~ J ' - P ) l

In (13),

[Og/OS:]

and [0g/0x] den ote the Ja cobia ns of g with respect to i ts f i rst and

second argum ent evaluated at some c haracterist ic value of (2, x). In general , the

solut ion of the pair (13), (14) ma y be ac complish ed by solving two systems of the

form (1), as follows.

Define matrices

A, B, C, D, G (z), R (, 6 X (

and 6X a) by

A = (1/h)[Og/c35c],

C = - rM )l T

Lt l J J

GO) = gr ~(/)~,., x~,))],

Then (13) and (14) can be written

(13)

(14)

B = [ag/c3x],

D (o) T

R '

G a) + hA 3X (0 + B 6X a) = 0

R ) + bXtOD + h6X )C = 0 .


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METHODS FOR THE SOLUTION OF

A X D - B X C = E . . .

345

Postm ult ipl ying (13) by C a nd D, respectively, and subs t i tu t ing (14) , then yie lds

for fiX l) and 61~ I),

(16)

A 5 X, t)D- B

jX(t)C = -

A R + G I)C

(17) A 5fifa)D - B 6~ C = (l/h) BR 0 - G D

both of which a r e iden t ica l in fo rm to (1) . No te tha t because bo th of the mat r ices

o)

O = [ei j ], C = - [ill)] may be singular , i t is generally necessary to solve both (16)

and (17). I f one of these is nonsi.ngular , i t is bett er to solve for jus t on e of 5X (1),

5Jf(~) and ob tai n the ot he r fr om (14) .

I n the special case tha t the m etho d is an impl ic i t Rung e-Ku t ta me thod , we have

e(o) =Stj (Kro neck er del ta) and only one sys tem n eeds to be solved. I t i s

ij

conventional to solve (17) . I f the

I R K

method i s a l so a co l loca t ion method , then

the matr ix penci l C 2 D

C - )~D = (o) T

U ~ i j ] - ~ [ 6 1 ~ 3 = - / ~ O ) T + ; , 1 )

can be r educed to Jo rdan canonica l f o rm by means o f a s imi la r ity t r ans format ion

that is expl ic i t ly computable . Thus , in this par t icular ins tance, the remark made

abov e advis ing agains t the use of Jor da n no rma l forms is i r re levant .

R E F E R E N C E S

1. F. R. Gantmacher ,

Theory o f Matrices,

Chelsea Pub lishin g Co. , New York, N.Y. 1977) .

2. R. H. Barte ls and G. W. Stewart , Algorithm 432,

Solution of the matrix equation A X X B = C,

AC M, 15 1972), 214-235.

3. W. H. Enright,

Improving the efficiency o f matrix operations in the numerical solution of sti ff ordinary

differential equations.

AC M Tran s. Mat h. Software, 4, No. 2 1978), 127-136.

4. C. B. Moler and G. W. Stewart ,

An algorithm for generalized matrix eigenvalue problems,

SlAM J .

Num . Anal. , 10, No. 2 1973), 241-256.

5. R. C. Ward,

The combination shift QZ algorithm,

SIA M J. Nu m. An al. , 12, No . 6 1975), 835-853.

6. L. Kaufmann,

The LZ algorithm to solve the generalized eigenvalue problem,

11, No . 5 1974), 99 7-

1024.

7. J. C. Butcher,

Implicit Runge-Kutta processes,

Mat h. Co mp , 18, 1964) , 50-64.

8. C. W. Gear,

Simultaneous numerical solutions of differential-algebraic equations,

IEEE Trans .

Circu it Theory , CT-18, No. 1 1971),

89-95.

9. T. A. Bickart and Z. Picel,

High o rder stiffly stable composite multistep methods for numerical

integration of stiff differential equations,

BIT 13, 1973), 272- 286.

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13 Methods for the solution of AXD − BXC = E