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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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