PERGAMON

MATHEMATICAL AND COMPUTER MODELLING

Mathematical and Computer Modelling 35 (2002) 1061-1076 www.elsevier.com/locate/mcm

Chebyshev Rat ional Matr ix Approx imat ion with A Priori Error Bounds

for Linear and Riccati Matr ix Equat ions

J . C A M A C H O , J . C . CORTÉS A N D E . N A V A R R O Depar t amen to de Ma temä t i ca Apl icada

Universidad Poli técnica de Valencia Valencia, Spain 46071

<f camacho>< j c c o r t e s > < e n t o r r e s > ~ m a t , upv. es

A. E . P o s s o Depar t amen to de Matemät icas , Universidad Tecnolögica de Pereira

Pereira, Colombia possoa©utp, edu. co

(Received August 2001; accepted September 2001)

A b s t r a c t - - T h i s paper deals with initial value problems for Lipschitz continuous coefficient ma- trix Riccati equations. Using Chebyshev polynomial matrix approximations the coefficients of the Riccati equation are approximated by matrix polynomials in a constructive way. Then using the Fröbenius method developed in [1], given an admissible error e > 0 and the previously guaranteed existence domain, a rational matrix polynomial approximation is constructed so that the error is less than e in all the existence domain. The approach is Mso considered for the construction of matrix polynomial approximations of nonhomogeneous linear differential systems avoiding the integration of the transition matrix of the associated homogeneous problem. (~) 2002 Elsevier Science Ltd. All rights reserved.

K e y w o r d s - - R i c c a t i equation, Analytic numerical solution, A priori error bound, Fröbenius method, Chebyshev polynomial.

1. I N T R O D U C T I O N

In this paper, we consider rectangular nonsymmetric Riccati matrix differential equations of the type

W'(t ) = C(t) - D( t )W( t ) - W( t )A( t ) - W ( t ) B ( t ) W ( t ) , W(O) = Wo, (1.1)

where the unknown W(t) lies in C pxq, and coefficients A(t) E C q×q, B(t) E C q×p, C(t) E C pXq, D(t) E C p×p, are continuous matrix valued functions. If A T (t) denotes the transpose matr ix of A(t) and p = q, then equation (1.1) is said to be symmetric when the coefficients of (1.1) are real matrices and D(t) = AT(t).

This work has been partially supported by the Spanish D.G.I.C.Y.T. Grant BHA 2000-0206-C04-04, and the Spanish Office of International Cooperation.

0895-7177/02/$ - see front matter © 2002 Elsevier Science Ltd. All rights reserved. Typeset by .4¢k4S-TEX PII: S0895-7177(02)00070-5

1062 J. CAMACHO et al.

Initial value problems for Riccati differential systems arise frequently in important applications to classical control theory [2-5], also in decoupling techniques for both the analysis and numerical s tudy of boundary value problems [6,7], or in noncooperative control theory appearing in economic or military problems ([6,8-10] and the references therein). The case q ~ p appears, for instance, in differential garnes when Nash strategies in noncooperative control problems are tackled [11-13].

The Riccati equation (1.1) has been extensively studied in the autonomous case, but many real systems are nonautonomous, (see [6,14]), and this case has received few numerical treatments in the literature as rar as the study of accuracy and a priori error bounds in terms of the data for the proposed numerical solutions are concerned. Some exceptions can be found in [1,12,15-19].

In [17], the direct integration of (1.1) using well-known routines based on the modified Davidson-Maki algorithm is proposed, but the numerical solutions are tested on a set of con- crete examples without error bound information in terms of the data for the general case. In [18], the solution of symmetric Riccati equations is reconstructed starting from the previous deter- mination of the eigenvalues and eigenvectors of the solution matrix. In [20], an implicit Riccati matrix differential equation is considered, and a numerical method is proposed based on a contin- uous singular value decomposition of a matrix function and Kronecker canonical forms of matrix pencils.

The study of the Riccati equation (1.1) is closely related to the underlying linear system

X ' ( t ) = S ( t ) X ( t ) , X(O) = Wo '

X ( t ) = [U(t)] S(t) = lA(t) B( t ) ] [V(t)] ' [C(t) - D ( t ) '

(1.2)

where Iq denotes the identity matrix of C qxq. Specifying S(t) uniquely determines the related Riccati equation, but the converse is not true, see [19]. In fact, equation (1.1) is unaffected by the substitution

A(t ) -~ A( t ) + f ( t ) Iq , D(t) --* D( t ) + f ( t ) Ip , (1.3)

where f ( t ) is a scalar function. This can be written in the equivalent form

s ( t ) --, s(t) +/(t)I~+~. (lA)

The method proposed in [16] is based on the vectorization of problem (1.2) and uses the successive approximation method. In [15], an interesting numerical integration method is proposed and error bounds are given, but these are expressed in terms of the geometrical properties and involve dichotomy constants which are not know a priori in practice. In [19], a class of explicit one-step matrix methods for the numerical solution of problem (1.1) are given. These methods exploit the matrix group structure underlying the equation, and are simple and computationally cheap to implement. In [12,21], approximate solutions are constructed using matrix difference schemes. Finally in [12], the exact solution of (1.1) is approximated by a rational matrix function with a predetermined accuracy, for the case where the coefficients of (1.1) are analytic functions.

Note that if S(t) given by (1.2) is continuous in [0, T], equation (1.2) has a unique solution, that in general is not exactly computable in a closed form, [22, p. 99; 23]. This motivates the search of analytic approximate solutions with prefixed error with respect to the exact theoretical solution.

In [24], efficient numerical methods are developed preserving qualitative properties of the so- lution of problem

O'(t) = S( t )O(t) , ~(0) = I, (1.5)

where O(t) E C (p+q)x(p+q). In [25], error bounds for the approximations of problem (1.5) are given for the case where S(t) is analytic using the so-called methods of the iterated commutators

Chebyshev Rational Matrix Approximation 1063

and in [26] error bounds for the approximations using Fer's factorization are given for the case where S(t) is continuous.

The aim of this paper is the construction of analytic-numerical approximations with a priori error bounds for the solution of problem (1.1) where coefficients are r-times continuously differ- entiable. In this sense, the approach proposed hefe may be regarded a continuation of [1] where the coefficients are analytic matrix functions.

This paper is organized as follows. Section 2 deals with the study of the distance between the solution of the Riccati equation with continuous coefficients and its continuous perturbation. In Section 3, Chebyshev matrix series are introduced and explicit bounds for the distance between a continuous matrix valued function and its Chebyshev matrix polynomial t runcated series of certain degree are established. Section 4 may be regarded as a extension of Section 2 of [1] in two respects. First, it considers nonhomogeneous linear differential systems, and second, because the matr ix coefficients are assumed to be r-times continuously differentiable instead of analytic functions. A procedure for constructing matrix polynomial approximate solutions with a priori error bounds of nonhomogeneous linear differential systems with r-times continuously differen- tiable coefficients is given. The proposed procedure avoids the well-known variation of constants formula and the computation of the transition matrix of the associated homogeneous problem. Section 5 provides a procedure for constructing rational matrix polynomial approximations with a priori error bounds for r-times continuously differentiable coefficient Riccati matrix differential problems. The approach is based on the consideration of an appropriate polynomially per turbed Riccati equation and the analysis developed in [1] for the case where coefficients are analytic functions.

If M is a matrix in C pxq, we denote by ]] M ][ the square root of the maximum of the set {[ z I; z eigenvalue of M H M } where M H denotes the conjugate transpose of M. The set of all the eigenvalues of a square matrix P is denoted by a(P). If A is a matrix in C pxq, we have [27, p 57]

max I a~j ] - II A I1 -< v / ~ m a x I a~j I. (1.6)

In accordance with [22, p. 110], if D is a matrix in C rxr , the logarithmic norm It(D) defined by

[[ I + hD [[ - 1 It(D) = lim

h--*0, h>0 h

satisfies It(D) = max{z • a((D + DH)/2)} and

I~(D) I ~ II D I[, # ( aD) = alt(D), for a > O.

If f (n) , g(j) are scalar functions, then a change of variables gives

n--1 m

E E f ( n ) g ( J ) = E E f ( m + j ) g ( n ) + E E f ( m + j ) g ( m + n ) . n~_mq-1 j = 0 n=0 j_~l n_~l j~nq-1

(1.7)

2. O N P E R T U R B E D R I C C A T I E Q U A T I O N S

Let Ai(t) C C qxq, Bi(t) E C qxp, Ci(t) C C »×q, and Di(t) c C »×p be continuous functions for i -- 1,2. Consider the matrix valued functions

Fi(t, W) = Cdt ) - Di( t)W - W A d t ) - WBi(t)W, (2.1)

where W C C pXq, and consider the initial value problems

W'(t) = F~(t, W(t)), W(O) = Wo, 0 < t < T. (2.2)

1064 J . CAMACHO et al.

Let a~, bi, ci, and di defined by

ai = sup{ II Ai(t)Il , 0 < t < T},

ci = sup{ II c i ( t ) I I , o < t < T},

By defni t ion of Fi(t, W), if I[ W II -< M, one gets

II Fi(t, W) Il < ci + (di + ai)M + biM 2,

By (2.1), one obtains

bi = sup(ll Bi(t)II, 0 < t < T},

di = sup{ I[ Di(t)II, 0 < t < T}.

0 < t < T .

Il F~(t, W) - F2(t, W)II ~ II Cx(t) - C2(t)II + Il Bi(t) - B2(t) Il M~

+{[I Dl(t) - D2(t)[I + I[Al(t) - A2(t)Il}M, 0 < t < T.

Given e > 0, let us assume that for 0 < t < T,

Il c l ( t ) - c2(t)II < g, IF Bi(t) - B2(t)Il < 3M-------7,

If Al ( t ) - - A2(t)II < 6---~' II Dl ( t ) - D2(t)II < 6----M"

By (2.5),(2.6), one obtains

[[ FI(t, W) - F2(t, W) [[ < e , 0 < t < T .

By (2.1), if II w II -< M, II üz II --- M, and L = d2 -t- a2 -t- 2b2M, one gets

F 2 ( t , W ) - F 2 ( t , W ) < L W - V V , 0 < t < T . \ ]

(2.3)

(2.4)

(2.~)

(2.6)

(2.7)

(2.8)

In this section, matrix Chebyshev series are introduced and error bounds related to the prox- imity of a function to its t runcated Chebyshev series of an appropriate degree are extended to the matr ix framework. First, we recall some properties of scalar Chebyshev polynomials tha t may be found in [29]. Chebyshev polynomials {Tn(x)}n>0 are defined by the three terms recurrence formula

To(x) = 1, Tl(x) = x, Tn+l(x) = 2xTn(z) - Tn_l(X), (3.1)

n > l , - e ~ < x < oe.

3. O N C H E B Y S H E V M A T R I X S E R I E S E X P A N S I O N S

e ( e L o t _ l ) ' II x l ( t ) - x2( t ) II < Lö0

for all t with 0 < t < T and Lo = d e

(2.11) satisfy

By Theorem 3.1.1 of [28, p. 86], the solutions Wi, i = 1, 2, of problems (2.1),(2.2) satisfy

e (eLt 1), (2.9) II Wl(t) - w2(t)II < ~ -

for all values of t where both solutions W d t ) are defined. For the particular case where Ai(t) = Bi(t) = 0, the solutions Xl ( t ) and X2(t) of problems

X~ = Ci(t) - Di( t)Xi , Xi(O) = Wo, (2.10)

Chebyshev Ra t iona l Mat r ix Approx ima t ion 1065

Considering the change of variable x = cos 0, one gets Tn (x) = cos(n0) and normalizing we obtain the Chebyshev basis in the set E( [ -1 , 1]) of all continuous real valued functions defined on the interval [-1, 1] defined by

¢0(cos0) = 1, Cn(cos0) = x/2cos(n0), n _> 1.

For a function f in E( [ -1 , 1]), we denote

II f II~ = max{I f ( x ) I ; - 1 < x < 1}.

The Chebyshev series of f E E [ - 1 , 1] is defined by

f (x) = ao + v'~ E anTn(x), -1 < x < 1, n~_l

o r

f(cos O) = ao + v'2 E an cos(nO), 0 < 0 < 7r, n_>l

where 2 fo~ { x/-2, f o r n = O , an - f(cos O)Tn (cos 0) dO, e,~ =

7ren I, for n > O.

We denote by [[ f [[2 the norm defined by

[[ f [[2 = ~~_>0 [ an [2"

Let Pmf(X) be the t runcated Chebyshev series of degree m defined by

?n

Pmf(x) = ao + v / -2E anTn(x). n=l

Then by [29, p. 346] if f (x) is continuously differentiable, it follows that

~v~ PI ( I - P m ) f Iioo = V~n~-'>m+lanTn < IIf'll2, m »_ 1,

where I denotes the identity operator and

['f '[[2=/n~_>O a~n' 2,

where {a~n} is the sequence of the coefficients of Chebyshev series of f ' ( x ) ,

f ' (x) = aó + x/2 E a'nTn(x ). n>_l

If f , f ' , f(2), f(3) all lie in E [ -1 , 1], then [29, p. 347]

II (I - Pm)f I1, <- ~ I1(1 - P-,-1).f'll2

B 1 <- ~ m ( I - P,,_2)y ~2) = <_ - - B 1 1 _ f(3) 2"

B m m - 1

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

1066 J. CAMACHO et al.

In general, if f , f ' , f ( 2 ) , . . , f(~), all lie in E( [ -1 , 1]) then

Il f(r)112 I [ ( I - P m ) f l l ~ < ~ m ( m - 1 ) ( m - 2 ) . . . ( m - r + 2 ) ' m > _ r - 1 . (3.12)

Now we introduce the Chebyshev matrix series expansion of a continuous C pxq valued function f(x) = (fij(x)) with - 1 < x < 1. The Chebyshev series of f is defined as the matrix having in the (i, j ) -entry the Chebyshev series of fij for 1 < i < p, 1 _< j _< q,

f ( x ) = A o + V / 2 E A n T n ( x ) , - l < x < l , (3.13) n > l

f (cos O) = Ao + v~ ~_, An cos(n0), 0 < O < 7r, (3.14) n_>l

where for 1 < i < p, 1 _< j _< q, A,~ is the matrix in C pxq defined by

~ / 0 ~ An = (an(i,j)), an(i,j) = f«j(cosO)T~(cosO)dO. (3.15)

If we define by Pmf(X) the truncated Chebyshev series of degree m of f , by considering the previous results for each component fij of f , assuming that f , f ' , . . , f ( r ) are continuous in [-1, 1] and using (1.6) and (3.12), it follows that

H (Z - Pm)f Ha -< v ~ m a x { l l (I - Pm)f~j I1~, 1 < i <_ p, 1 <_ j < q}

~pmq max { f } f )~2~) l<_i<p, l<_j<_q} (3.16) <- m(m -. :-(~n---r~_-~5 , m _> r - 1.

Given f(x) we denote by Tm(f; x) = Pmf(X) the truncated matrix Chebyshev series written as an explicit polynomial in powers of x,

Tm(f; z) = k Ck(f; m)x k, (3.17) k=0

where matrix coefficients Ck(f; m) are computed from (3.1), (3.13), and (3.15).

4. O N N O N H O M O G E N E O U S L I N E A R D I F F E R E N T I A L S Y S T E M S

It is well known (see [28, p. 131; 30, p. 40]) that if (I)(t, 0) is the transition matrix of the homogeneous system

x ' ( t ) = s ( t ) x ( t ) , (4.1)

then the variation of constants formula gives the unique solution of

X'(t) = S(t)X(t) + B(t), X(O) = Xo, (4.2)

a s

{~o T } X(t) = ~(t,O) Xo + ~(s,O)-lB(s)ds . (4.3)

Hence, the exact solution of (4.2) involves the exact computation of (I)(t, 0) and its inverse, which in general is a difficult task. In the numerical implementation of (4.3), one approximates the integral by quadratures and (I)(., 0) taust be evaluated at several points in every step. This is very time-consuming and expensive. A modification which requires one evaluation of (I)(.,0)

Chebyshev Rational Matrix Approximation 1067

per step has been proposed in [25, p. 84]; however, the approximation of ~(t ,0) in terms of exponential transformations is a difficult task in spite of excellent recent methods for computing matrix exponential [9,11,15,21-23,25,27,28,31-34]. Another approach for computing the solution of (4.2) is to use one-step methods (see [12]). Thinking of the preservation of qualitative properties of the solution of (4.1), we must mention the method of iterated commutators [25,33] or Magnus methods [24], (see also [35, thesis]).

In this section, we address the construction of matrix polynomial approximations of the initial value problem (4.2) for the case where the coefficient S(t), B(t) are continuous in [0, T]. In the first stage, we assume that S(t) and B(t) are analytic functions and then we will use results of Sections 2 and 3 to construct approximate solutions of (1.1) for the case where the coefficients are continuous.

Let S(t) = E Smtm, B(t) = E Bmtm, O < t < T, (4.4)

n>_O m>O

where Sm(t) E C pxp, Bm C C pxq, and let us look for a series solution

X(t) = E Xmtm' 0 < t < T, (4.5) m>0

where the coefficients Xm are matrices in C p×p to be determined. in (4.5), one gets

Xt(t) = E ( n + 1)Xm+it m n>_O

and substituting (4.4)-(4.6) into (4.2), it follows that

m_>0 n_>0 k=0

Taking formal derivatives

(4.6)

(4.7)

By equating coefficients of t m in both members of (4.7), one gets

X n + l - n + 1 k=0

(4.8)

Let M1 and M2 be defined by

max II s(t)II : M1, 0<t<T

max II B(t)II = M2. 0<t<T

(4.9)

By Cauchy inequalities [32, p. 222], it follows that

M2 Il&Il < M----L~, Ilßmll < n>O, - - p m - - p-- '-n- ' - -

Taking norms in (4.8), by (4.10), one gets

0 < p < T . (4.10)

II Xm+~ II <-- M1 n

(n + 1)ph ~"~PÆII: Xk II + M2 (n + 1)p n'

n>_0. (4.11)

Let us consider the sequence of scalars {6n}n>_O defined by

6o = II Xo II, - 1 ( m-~ )

6n npm_ ~ M1 E 6kpk + M2 , 0 < p < T, k=0

n > l . (4.12)

1068 J. CAMACHO et al.

These satisfy 1[ Xn [1 _< 5~ and by (4.12), it follows that

1 M1 5kp k + M15np n + M2 • 5n+l = (n + 1)p n \ k=o (4.13)

By (4.12) and (4.13), one gets

(n + 1)pnSn+l -- rtpn-15n = M16np n. (4.14)

Hence, lim ~n+ltn+l lim ( u p - l + M, ~ ,t

ù-~oo 5ùt- - - - -~-~-*o~\ ~C-1- ) l t l = - f i - .

Thus, ~n_>0 5ntn is convergent for all 0 _< t < p < T and by the inequality Il Xn Il «- 5n, we deduce the absolute convergence of (4.5),(4.8) for 0 _ t < p < T. Let us now consider the truncated series

m

x(t,~n) = E x~t~, (4.15) n=O

and note that for 0 _< t < T, < T, one gets

Il x ( t ) - x ( t , m ) I I -< ~ II Xn Il It I n _< ~ II xn IIT2. (4.16) n>m+l n>m+l

Ler us consider the sequence of positive numbers {~,~},~>0 defined by

Bn = II x ~ II, n > o. (4.17)

By (4.11),(4.17), it follows that

Bn+l <: ~ II B~ II + II S~_~ I1~~ , n > O. (4.18) - n + l

j = 0

L e t 0 < t < T l < T , and

max Il S(t)II ~ M1, 0<t<T1

max }l B(t) ]I < M2. O<t<Tl

(4.19)

By Cauchy inequalities, it follows that

M1 /I/12 IlSnll <_ ~- , IIBùII <_ ~- , n_>0, 0 < t < T l . (4.20)

By (4.18) and (4.20), one gets

l(n ) Bn ~ n T ~ - i M 1 E ~JTJl + M2 •

j = o

(4.21)

By (4.16), (4.17), and (4.21) for 0 _< t < T. < Tl, it follows that

] ] X ( t ) - X ( t , m ) H «- E nTF-1 M I E ~ j T ~ + M 2 T. ~ n>m+l j = 0

= M1T1 E E ~jTf \ T l ] + M2T1 n>_m+l j=O n \ T l ] n>_m+l

(4.22)

Chebyshev Rational Matrix Approximation 1069

Considering (1.7) for f(n) = 1/n(T./T1) ~, g(j) = TJ~j, it follows that

n-1 . ( T , ~ n E E l(flJT~ ~TlB

n>m-F1 j=0 1 (T,~ m+j

= E m - F j \T1,] ~=o j>_l

1 (~.~~ [~ E I~*h' <-- - - \ T l ] \ T l ] m Ln=0 j_>l

<- L E :E,,~,,, - m k T ~ ) n = o j>_~

1 (T ,~ m [~-~~ [ (T , /T1 )1 = m \Tl ,] n=0 1 : (-~ß'T1)J T~q°n

( r , ~ ~ T, =-~ \TI) T~--T, ~zr~'n n>_O

m 1 (T~__..~_) m+.~ m~'~,~+Z ~ ~+s r?+n~om+ " n>_l j>m+l

T~~n + E E \Txl# T~n+n~m+~ n>l j>_n+l

Tr~°n-I- E E lkTll# n>_l j>l

[/~«1/1 ] +n~__l 1 : ~ ~ - ~ l ) J T~n+nq¢m+n

(4.23)

By [22, p. 114], one gets that theoretical solution X(t) of (4.2) in 0 _< t < T. < T1 < T2 < T, satisfies

IIX(t) II_ < IlXollexp(f)ó:~(S(v))dv (4.24)

~~ ~x~(i) ~~ + [~ Il B(v) Il ~ ( s (~) ) dA dv = M3. J0 V

For 0 < t < T2, Cauchy inequalities imply

Ma m m ~n = II x ù II -< T~ ~ > 0, (4.25)

and by (4.23),(4.25), it follows that

n-1

t, T1) n>m+l j=o (4.26)

<--1 ¢T,'~m T,__ M3[ 1 ] =--1 (T,) m T, T2M3 . - - m \ T 1 . ] T1 - T , 1 - ( : F I / T 2 ) m ~-1 ( T l - T , ) ( T 2 - Tl)

Since

E n ~~ <- E - - \ T 1 ] m = m ~~ Tl---T, n>m-'F1 n>_m...F1

by (4.22), (4.26), and (4.27), one gets

(4.27)

[[X( t )_X( t ,m)[[< l ( T . ~ m [ T.TIT2MIM3 (T,) Ma~e1 - -~ kT~] (Tl -T.)(T2 -T~) + ~ T, -T.J ( CT*'~ m [T.TI(T2MIM3 q-- M2(T2 - Tl)) 1 - \Tl) L ~ ---Z.~~--T11i J"

Hence, given e > 0, if

m > In [(2T.T1 (T2MxM3 + M2(T2 - T1)))/(e(T1 - T.)(T2 - Tl))] ln(T1/T.)

1070 J. CAMACHO et al.

one gets If x ( t ) - x ( t , m) Ir < «, O<_t < T , <TI < T 2 < T .

Take T1 = (T, + T) /2 and T2 = (Tl + T) /2 = (T, + 3T)/4, then if

m > In [(2T,(T, + T){(T, + 3T)M1M3 + M2(T - T , ) } ) / ( e ( T - T,)2)] In [(T, + T)/(2T,)] (4.28)

one gets E

Il x ( t ) - x ( t , m) II < ~, O < t < T , < T .

Suppose now that S(t) and B(t) are matrix polynomials of the form

(4.29)

no no

S(t) = Z Sntn' B(t) = E Bntn. n = 0 n = 0

(4.30)

With these data, the solution X( t ) of'problem (4.2) takes the form, see (4.7),(4.8),

n>O n>O j=~,~ (4.31)

where Bn = max{0, n - no}, an = 1 - min{l, fln}.

By equating coefficients of t n in both sides of (4.31), one gets

(4.32)

Xn+l = n + 1 j ~ n

(4.33)

Let us assume now that coefficients of problem (4.2) are r-times continuously differentiable.

Take no large enough such that

2v'~ (e M'T - 1) (M3 Il s<r> Il2 + Il B<r> Il2) (4.34) B n o ( n o - 1) . . . (n0 - r + 2 ) > cM1 '

where M1 satisfies (4.9) and M3 satisfies (4.24). Then, by (3.12), it follows that

11S(t) - Tno(S;t) [[M3 + II B(t) - Tno(B;t) [1 < cM1

2(e M1T -- 1)' (4.35)

By (2.10), (2.11), and (4.35), if Xno is the exact solution of problem

X' ( t ) = Tno (S; t )X( t ) + Tno (B; t), X(0) = Xo, (4.36)

and X( t ) is the exact theoretical solution of (4.2), one gets

é Il x ( t ) - Xno(t) II < 3' 0 < t < T. (4.37)

In accordance with (3.17), let us denote

no no

Tno(S;t) = E Ck(S;no)tk, Tno(B;t) = E Ck(B;no)tk. k = 0 k = 0

(4.38)

Chebyshev Rational Matr ix Approximat ion 1071

With the notation of (4.32), if we denote by Xno( t ,m) the truncation of order m of the exact series solution Xno(t) of problem (4.36), we obtain

m

Xno( t ,m) = E Xk(no)tk, Xno(0) = Xo, k=0

Xk+l(no) = k + 1 '

(4.39)

Taking m large enough to ensure (4.28), it follows that

C Il Xno(t) - X,~o(t,m) ll < -~, 0 < t < T . . (4.40)

By (4.37) and (4.40), one obtains

IIX(t)--Xno(t,m)ll<«, 0 < t < T . . (4.41)

Summarizing, the following resutt has been established.

THEOREM 1. Consider problem (4.2) where coefficients are r-times continuously differentiable and the previoas notation. Ler ¢ > 0 and ler no be the first positive integer satisfying (4.34). Ler T~o(S;t ), Ck(S;no), Tno(B;t), and Ck(B;no) be de/~ned by (4.38). Ler T. < T and ler M1-M3 be de/~ned by (4.9) and (4.24). Taking m /arge enough so that it satis/~es (4.28), the matrix polynomial Xno (t, m) de~ned by (4.39) is an approximate solution of problem (4.2) whose error wirb respect to the exact solution X (t ) satis/~es (4.41).

REMARK. Note that Theorem 1 provides the error bound in an interval [0, 21,] where T. < T and the solution of problem is guaranteed in [0, Tl. However, the approach may be modified in order to obtain error bounds in all the interval [0, T] where the exact solution X(t ) is defined. In fact, given problem (4.2) consider the problem

X'( t ) = S( t )X( t ) + b( t ) , X(O) = Xo, 0 < t < T + 5, (4.42)

where 5 > 0 and

.~(t) : { s(t), o < t < :r, ~( t ) = { B(t), o < t < :r, S(T), T < t < T + 5 , B(T) , T < t < T + 6 .

Now by application of Theorem 1 to problem (4.43) and replacing T by T + 6 and T. by T, one gets that error bounds are valid in all the existence interval. Note that since S(t) and B(t) take the constant values S(T) and/~(T), respectively, values M1 and M2 defined by (4.9) do not change, but M3 defined by (4.24) increases. Thus, the recommendation is taking 5 > 0 as small as possible in order to not increase the computational cost making m bigger through (4.28). Note that no does not change.

5. R A T I O N A L M A T R I X A P P R O X I M A T I O N S

In this section, we address the construction of a Chebyshev rational matrix approximation to the solution of problem (1.1), where coefficients are continuously differentiable. Let S(t) be defined by (1.2) and let T,~(S,t) be its Chebyshev matrix polynomial of order n. Taking into account the expression

Th(S, t) = PnS = S - (I - Ph)S,

it follows tha t I[ Th(S, t) H <- Il S II + H (I - Pù)S H. (5.1)

1072 J. CAMACHO et al.

According to the differentiability hypotheses of the coefficients of equations (1.1), the second term of the right-hand side of (5.1) can be bounded using (1.6) and (3.9) or (3.12). If S is only continuously differentiable, we have (see (1.13) of [29])

Il (z - Pn)S I1~ = max { l l ( I - P, JS(t)II, - 1 < t < 1 }

-< (P+ q)l/~-2 max {1[ S;~ I1~}. (5.2) V n ij

Let us denote

S1 ---- m .a~ {[I S:j I]2}, $2 ~-- n~.ax {H a:j ]12' H b:j 112}, (5.3) z3 3

then by (5.1)-(5.3), one gets ]1 ( I - Pn)S ]]oo <_" (p + q)Si ß and

max HT,~(S,t) ll=llTn(S,t)[l~ <llSll~+(p+q)S1V/~~=ko(n ). (5.4) --1<t<1

In an analogous way, we may introduce qo(n) by

max Il [Tn(A , t )Tn(B , t ) ] Il = II [Tn(A, t ) Tn(B, t )] II~ -l<_t<_l

(5.5) f ö <_ V / ~ + q)S2~/ n = qo(n).

Let X be the solution of problem

X' ( t ) = S ( t )X ( t ) , X(O) = Wo 0 < t < T, (5.6)

and taking into aceount the transformation x : [0, T] ~ [-1, 1] defined by x = x(t) = - 1 + 2 t /T , then t = t(x) = (x + 1)T/2, S(t) = S( t (x) ) = S ( ( x + 1)T/2) = ~(z) ,

X ( t ) = x ( ( X 2 1 ) T ) = f ( (x ) , X ' ( t ) = d X d l ( d X dx 2 - , - ~ = d--t- - ~xx ~ - ~ X (x), (5.7)

and problem (5.6) can be written in the form

Ä"'(x) = S (x ) f ( ( x ) , X ( - 1 ) = Wo ' - 1 < x < 1. (5.8)

Consider the truncation of order n of the Chebyshev series of the function S(x), denoted by Th(S; x) and )(n(x), the solution of problem

2 n Wo - 1 < x < 1. (5.9)

Thinking of the selection of the appropriate truncation index n, note that

T T

Thinking of the selection of the appropriate truncation index n, note that given e, if n is chosen so that

iM1 ( I - Pù):~ ~ < 2M3 (e (I/2)TM' - 1)' (5.11)

Chebyshev Rational Matrix Approximation 1073

then oo e (5.12) 2 - 2 , <3"

Assuming that coefficients of equation (1.1) are continuously differentiable, one satisfies

(I - P~)~ ~ < (p + q)S, V~ ~, (5.13)

and taking into account (3.12) if coefficients of (1.1) are r-times continuously differentiable, then

oo Vf~ Sr (5.14) ( I - P n ) S < ( ; ÷ q ) n ( n - 1 ) . - : - ( n - r + 2 ) '

where

Thus, if S(t) is once continuously differentiable, using (5.13), condition (5.12) is satisfied if

2x/~(p + q)S1M3 (e ('/2)TM1 - 1) vfn > cM1 (5.16)

If S(t) is r-times continuously differentiable, using (5.14), condition (5.12) is satisfied if

2v~(p + q)S,.M3 (e (1 /2)TM1 -- 1) (5.17) v ~ n ( n - 1 ) . . - ( n - r + 2) > eM1

Now we obtain explicit expression for constant M1 and M3 are given by [22, p. 114].

M1 = max [I S(t)[[ = [[ S [[o~, (5.18) 0<t<T

(; ) M3 = woexp ~ ( S ( v ) ) dv , wo'= (1 + II Wo 112) 1/2 (5.19)

Let no be the first positive integer satisfying (5.16) or (5.17) according with the differentiability hypothesis on the coeffieients of (1.1) and consider the solution Wno (t) of problem

W ( - 1 ) = Wo, - 1 < x < 1.

If )(no is the exact series solution of (5.9) for n = no and

Ü~o(X) = [Iq 012~o(X), ?no(t) = [0 xp])~~o(X), (5.21)

by the proof of Lemma I of [1] (in the interval [-1, 1] instead [0, Tl) and taking into account (5.4), (5.5), it foUows that the solution ITVno (x) of problem (5.20) is defined in the interval [ - 1 , - 1 + 5no] where 5no satisfies the inequality

ko(no)ó~o + ln(~~o) < - ln[qo(no)wo]. (5.22)

Furthermore, for x in [ - 1 , - 1 + 5no], one gets

/ \ ~ - 1

[Uno(X)) ~_ (1 - NnoSno) -1 , (5.23)

1074 J. CAMACHO et al.

where

Nno = qo(no)wo exp(ko(no)bno). (5.24)

In [ - 1 , - 1 + 5no], the solution l/Vno (x) of problem (5.21) takes the form

-- ~oo(X)[~oo(X)] -1 (5.25) Wno(X)

and ffVno(X) < M(n0), (5.26)

where M (no) = (1 - 5noqo(no)wo exp (bnoko(no) ) ) -1 wo exp (~noko(no) ) . (5.27)

Let Ä'no (x, 0 ) be the truncation of order m of )~no (x). Let

9no(X,O) = [0 I , ] 2 n o ( X , o ) , Ü~o(X,O) : [I« o]2no(X,O), (5.28)

and let I37n o (x, o ) be the rational matrix function defined by

W,o(X,O) = ~oo(~, o ) [Ono (X, o)] -1 (5.29)

By the choice of no, one gets (5.12) for n = no and by (2.9), it follows that

v¢(~) - V¢~o(~) < 2 ' - 1 < z < - 1 + 5no. (5.30)

Once no is chosen, the truncation index 0 may be found in the following way. Take ~ > 0 and 5,0 satisfying

* n 5no exp (bnoko( o)) = ( C o ( n o ) w o ) - 1 - - ~ . (5.31)

By [1, pp. 552-553], it follows that Üno(X) is invertible in the interval [ -1 , - 1 + 5*0] and

o(X < ~woqo(no), - 1 < x < -1 + 5* n O "

Furthermore, if 0 is chosen so that

2 n o ( x ) - X n o ( X , m ) <~woqo(no), - 1 < x < - 1 + 5 , o , (5 .32)

~ t X Xno( ) - X ' n o ( x , o ) <bnol{woqo(no), - l < x < - l + 5 ~ o ,

then Ü~o(X, 0 ) is invertible in [-1, - 1 + 5,0 ] and

- - 1

( ) -5,~ob~o)~qo(no)wo} -1 5" . (5.33) Üno(X,m) < {(1 • - 1 , - 1 < x < - l + no

Note that if l}V(x) is the exact solution of (1.1) written in the variable x,

T l~ ' (x) ---- ~ [Œ(x) - b(x)lTV - 17VÄ(x) - ffv/~(x)l/v], l /v(-1) = Wo, (5.34)

l~no (X) is the exact solution of (5.20) and IT¢'no(X, m) is the rational matrix polynomial defined by (4.28),(4.29), then it follows that

17V(t) - ITVno(X,m) < 17V(x) - ITVno(X) + I~Vno(X) - 17Vno(X,m) , (5.35)

Chebyshev Rational Matrix Approximation 1075

for x E [-1, - 1 + 5*0]. If 5,0 satisfies (5.31), 5,0 < ~1 < 5no and ~~" = (~1 + 5~o)/2, then if ml is the f rs t positive integer satisfying

In [min {(5~o15,o~qo(no)wo(~1 + 5no)(~~ - ~l))/(~31ko(no)M(no)), e/2Q}] (5.36) m l > in (5no/~1) '

where M(no) is defined by (5.27) and

1 + (~q0(no)) -1 (~qo(no) + exp (Snoko(no))) (1 -- 5*o/óno )

Q = ~~oqo(~o) one gets

, (5.37)

by (5.41) sa t ißes

II Wno(t, tal) - - W(t)II < «

where W(t) is the exact solution of problem (1.1).

EXAMPLE. Consider problem (lA) where

1 [ ö 0 ] C ( t ) - 2 + 2 t o s t 0.7 '

o~~], ~,,,:[o Oo, 1 -0.1

1 ~~o~_- [:~ o]

0 < t < 15~oT,

Taking into account the notation of Theorem 2, for the problem under consideration, the the- oretical solution W(t) is defined in [0, 0.5]. For different admissible values of ~, the following table provides the values of n(e) and ra(e) which according with (5.16) and (5.36) permit the construction of the rational matrix approximation given by (5.29).

10-1

10-2

10-3

10-4

Table 1.

n(~) m(~) 5.49 20.0603

5.49 x 102 28.0783

5.49 × 104 36.0771

5.49 × 106 44.0523

~Vn°(X)-~Vn°(X'?Ttl) < 2 ' - - l < x < - - l + 5 * n o . (5.38)

By (5.31), (5.36), and (5.38), it follows that

VV(x)-~Vno(X, ml) <e , - 1 < x < - 1 + 6 " o. (5.39)

Considering the transformation t = t(x) : [-1, 1] --~ [0, T] defined by t : t(x) = (1/2)(x + 1)T, the expressions appearing in (5.40) can be written in the form

[1W(t) - - Wno(t, rrtl)II < ~, 0 < t < ~5~oT, (5,40)

where Wno (., tal) is defined by

Wno(t, m l ) = l;Vn ° ( 2 ~ - l , m l ) = T~n o (~-~-~- 1,ml)[Üno (T-l'ml)] - t (5.41)

and l?Vno (., m) is defined by (5.29). Summarizing, the following result has been established.

THEOREM 2. Consider problem (1.1) where coetticients are p-times continuously differentiable and the previous notation. Ler e > 0 and/et no be the first positive integer satisfying (5.16) iE p = 1 or (5.17) i fp > 1, where M1 and M3 are given by (5.18) and (5.19). Ler 5~ o be a positive number satisfying (5.22), tet Xno (x) be the exact series solution of problem (5.9) and let Üno (x), ~/rno (X) be defined by (5.21) and 1et Xno (x, m), Üno (x, ?Tt), ~ß~no (X, ?TL) be defined by the trun«atlon of order m of Xno(X), Üno(x), and ~zno(x), respectively. I l ( > 0 and 5,o satisfy (5.31) and ml is the first positive integer satisfying (5.36), then the rational matrix function Wno (t, tal) defined

1076 J. CAMACHO et al.

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