Some applications of the Hermite matrix polynomials series expansions 1

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1 Journal of Computational and Applied Mathematics 99 (1998) Some applications of the Hermite matrix polynomials series expansions 1 Emilio Defez, Lucas Jodar Departamento de Matematica Aplicada, Universidad Politecnica de Valencia, Spain Received 0 October 1997; received in revised form 5 May 1998 Abstract This paper deals with Hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of dierential systems. Properties of Hermite matrix polynomials such as the three terms recurrence formula permit an ecient computation of matrix functions avoiding important computational drawbacks of other well-known methods. Results are applied to compute accurate approximations of certain dierential systems in terms of Hermite matrix polynomials. c 1998 Elsevier Science B.V. All rights reserved. AMS classication: 15A60; 33C5; 34A50; 41A10 Keywords: Hermite matrix polynomials; Matrix functions; Dierential systems; Approximate solutions; Error bounds; Sylvester equation 1. Introduction and preliminaries The evaluation of matrix functions is frequent in the solution of dierential systems. So, the system Y = AY; Y (0) = y 0 ; (1) where A is matrix and y 0 is a vector, arises in the semidiscretization of the heat equation [17]. The matrix dierential problem Y + A Y =0; Y(0) = P; Y (0) = Q; () Corresponding author. edefez@mat.upv.es. 1 This work has been partially supported by the Generalitat Valenciana grant GV-C-CN and the Spanish D.G.I.C.Y.T grant PB CO /98/$ see front matter c 1998 Elsevier Science B.V. All rights reserved. PII: S (98)

2 106 E. Defez, L. Jodar / Journal of Computational and Applied Mathematics 99 (1998) where A is a matrix and P and Q are vectors, arises from semidiscretization of the wave equation [15]. The Sylvester matrix dierential equation X = AX + XB; X(0) = C; (3) where A, B and C are matrices, appears in systems stability and control [4, p. 6]. The solutions of problems (1) (3) can be expressed in terms of exp(at); cos(at); sin(at) and exp(bt) and the computation of these matrix functions has motivated many and varied approaches. An excelent survey about the matrix exponential is [11] and the study of cos(at) is treated in [1, 14, 15]. Some of the drawbacks of the existing methods are (i) The computation of eigenvalues or eigenvectors [11, 16]; (ii) The computation of the inverses of matrices (see Pade methods, [5, p. 557]) [11, 14]; (iii) Storage problems and expensive computational time [5, Chap. 1]; (iv) Round-o accumulation errors [5, p. 551]; (v) Diculties for computing approximations of matrix functions with a prexed accuracy [15, 16]. In this paper we propose a new method for computing the above matrix functions using Hermite matrix polynomials which avoids the quoted computational diculties. Results are applied to construct approximations of problems (1) (3) with a prexed accuracy in a bounded domain. If D 0 is the complex plane cut along the negative real axis and log(z) denotes the principal logarithm of z, [13, p. 7], then z 1= represents exp( 1log(z)). If A is a matrix in Cr r ; its -norm denoted A is dened by A = Ax = x, where for a vector y in C r, y denotes the usual euclidean norm of y, y =(y T y) 1=. The set of all the eigenvalues of A is denoted by (A). If f(z) and g(z) are holomorphic functions of the complex variable z, which are dened in an open set of the complex plane, and A is a matrix in C r r such that (A), then from the properties of the matrix functional calculus, [3, p. 558], it follows that f(a)g(a)=g(a)f(a). If A is a matrix with (A) D 0, then A 1= = A denotes the image by z 1= of the matrix functional calculus acting on the matrix A. We say that A is a positive stable matrix if Re(z) 0 for all z (A). For the sake of clarity in the presentation we recall some properties of the Hermite matrix polynomials which will be used below and that have been established in [7], see also [8]. If A is a positive stable matrix in C r r, the nth Hermite matrix polynomial is dened by [n=] H n (x; A)=n! ( 1) k ( A) n k x n k ; (4) k!(n k)! and satisfy the three terms recurrence relationship H n (x; A)=Ix (A)H n 1 (x; A) (n 1)H n (x; A); n 1 H 1 (x; A)=0; H 0 (x; A)=I; (5) where I is the identity matrix in C r r. By [7] we also have e xt A t I = 1 n! H n(x; A)t n ; t + ; (6)

3 E. Defez, L. Jodar / Journal of Computational and Applied Mathematics 99 (1998) and if H n (x) denotes the classical nth scalar Hermite polynomial, then one gets H n (x; A) 6 H n( ia A ) ; x a; n N; (7) i n H n ( A ia ) n!i n t n = exp( t a A + t ); t : (8) If A(k; n) and B(k; n) are matrices on C r r Lemma 11 of [1, p. 57] it follows that for ; k 0, in an analogous way to the proof of [n=] A(k; n k); k 0 A(k; n)= B(k; n)= k 0 B(k; n k): (9) If B is a matrix in C r r and n 0 is a positive integer we denote by A(B; n 0 ), D(B; n 0 ) and E(B; n 0 ) the real numbers: n 0 [n=] A(B; n 0 )= E(B; n 0 )= n=0 n 0 n=0 ( B ) n k k!(n k)! ; D(B; n 0)= n 0 n=0 ( B ) (n k) k!((n k))! ; (10) ( B ) (n k)+1 k!((n k) + 1)! : (11) This paper is organized as follows. In Section some new properties of Hermite matrix polynomials are established. Section 3 deals with the Hermite matrix polynomial series expansions of e At, sin(at) and cos(at) of an arbitrary matrix as well as with theirs nite series truncation with a prexed accuracy in a bounded domain. Finally, in Section 4 analytic-numerical approximations of problems (1) (3) are contructed in terms of Hermite matrix polynomial series expansions. Given an admissible error 0 and a bounded domain D, an approximation in terms of Hermite matrix polynomials is contructed so that the error with respect to the exact solution is uniformly upper bounded by in D.. On Hermite matrix polynomials Let A be a positive stable matrix. By (6) it follows that e xt A+t I = H n (x; A) t n : n! Hence ( ( Ax) n t n = n! )( t n n! ) H n (x; A) t n n! (1)

4 108 E. Defez, L. Jodar / Journal of Computational and Applied Mathematics 99 (1998) and by (9) one gets ( )( ) t n H n (x; A) t n = [n=] H n k (x; A) n! n! k!(n k)! t n : (13) By identication of the coecient of t n in (1) and (13) one gets x n I =( [n=] A) n n! k!(n k)! H n k(x; A); x + : (14) Lemma.1. Let be A a positive stable matrix; K and integer. Then H n (x; A) 6 n! n K n e x ; x K A : (15) Proof. It is clear that for n = 0 the inequality (15) holds true. Suppose that (15) is true for k =0; 1;:::;n. Taking norms in the three terms formula (5) and using the induction hypothesis one gets H n+1 (x; A) 6 x (A) H n (x; A) +n H n 1 (x; A) 6 K n! n K n e x +n (n 1)! n 1 K n 1 e x = n! n K n 1 e x (K + n ) = (n + 1)! n K n 1 e x K n + : n +1 n +1 (16) Since K ( n + K 6 + ) = K + 6 K ; n +1 n +1 by (16) one gets (15) for n +1. The following result that may be regarded as a matrix version of the algorithm given in [10], provides an ecient procedure for computing nite matrix polynomial series expansions in terms of matrix polynomials satisfying a three terms formula. In particular, it is applicable to the Hermite matrix polynomials. Note the remarkable fact from a computational point of view that the evaluation of a matrix polynomial at x is only expressed in terms of P 0 (x). Theorem.1. Let {P n (x)} be a sequence of matrix polynomials such that A n P n (x)=(xi B n )P n 1 (x) C n P n (x); n 1;

5 E. Defez, L. Jodar / Journal of Computational and Applied Mathematics 99 (1998) where A n is an invertible matrix in C r r and the degree of P n (x) is n. Let Q(x) be a matrix polynomial dened by Q(x)= E j P j (x); j=0 where E j is a matrix in C r r. Let x be any real number and consider the sequence of matrices dened by D n = E n ; D n 1 = E n 1 + D n A 1 n (xi B n ); and for j = n ;:::;0; D j = E j + D j+1 A 1 j+1(xi B j+1 ) D j+ A 1 j+c j+ : Then Q(x)=D 0 P 0 (x). Proof. See [10]. 3. Hermite matrix polynomials series expansions We begin this section with Hermite matrix polynomial series expansion of exp(bt); sin(bt) and cos(bt) for matrices satisfying the spectral property Re(z) Im(z) for all z (B): (17) Theorem 3.1. Let B be a matrix in C r r satisfying (17). Then e Bx = e cos(bx)= 1 e sin(bx)= 1 e H n (x; 1 B ) ; x + ; (18) n! ( 1) n (n)! H n ( 1) n (n + 1)! H n+1 (x; 1 B ) ; x + ; (19) (x; 1 B ) ; x + : (0) Let E(B; x; n); C(B; x; n) and S(B; x; n) be respectively the nth partial sum of series (18) (0) respectively. Given 0 and c 0; let a = c + and let n 0 ;n 1 and n be the rst positive integers satisfying A(Ba; n 0 ) exp(a B +1) e ; D(Ba; n 1 ) e(cosh(a B ) ); E(Ba; n ) e(sinh(a B ) ); (1) () (3)

6 110 E. Defez, L. Jodar / Journal of Computational and Applied Mathematics 99 (1998) where A; D and E are dened by (10) and (11). Then for x c and n max{n 0 ;n 1 ;n } one gets e Bx E(B; x; n) cos(bx) C(B; x; n) ; sin(bx) S(B; x; n) : (4) (5) (6) Proof. Let A = 1 B. By the spectral mapping theorem [3, p. 569] and (17) it follows that { } ( ) 1 1 (A)= b ; b (B) ; Re b = 1 { (Re(b)) (Im(b)) } 0; b (B): Thus A is a positive stable matrix and taking t = 1 in (6), B = A, one gets e Bx = e n! H n B ; x + : If n 0 1 one gets n 0 ( e ebx k H k! x; 1 ) B 6 ( e k n 0 k! H k x; 1 ) B = e H k (x; 1 B ) : (7) k n 0 k! By (7) for 0, c 0, a = c + and x a, it follows that e H k (x; 1 B ) 6e H k ( B ai ) n 0 H k ( B ai ) e : (8) k n 0 k! k!i k k!i k k 0 By (8), (10), (7) and (8) one gets n 0 ( e ebx k! H k x; 1 ) n 0 H k ( B ai ) B 6 exp(a B +1) e k!i k By [1, p. 57] one gets lim n [k=] l=0 ( B a) k l l!(k l)! = exp(a B +1): = exp(a B +1) A(Ba; n 0 ): (9) Hence, taking the rst positive integer n 0 satisfying (1), then by (9) one gets n 0 ( e ebx k! H k x; 1 ) B 6; x c; n n 0 : Considering (14) for the positive stable matrix A = 1 B, it follows that x n I = B n (n)! k!((n k))! H (n k) B :

7 E. Defez, L. Jodar / Journal of Computational and Applied Mathematics 99 (1998) Taking into account the series expansion of cos(bx) and (9) we can write cos(bx)= ( 1) n B n x n = ( 1) n (n)! k! ((n k))! H (n k) B = ( 1) n+k k! (n)! H n B = ( ) ( 1) k ( 1) n k! (n)! H n B : k 0 Hence cos(bx)= 1 e By [1, p. 57] one gets lim n k l=0 k 0 ( 1) n (n)! H n B ; x + : (30) ( B a) (k l) l!((k l))! = e cosh(a B ): Let us consider the n 1 th partial sum of the series (30). Then in an analogous way to the previous computations, for x c one gets cos(bx) C(B; x; n 1 ) 6 cosh(a B ) 1 e D(aB; n 1); a= c + ; where D(aB; n 1 ) is given by (10). Taking the rst integer n 1 1 satisfying () one gets (5). By (14) we also have x n+1 I = B (n+1) (n + 1)! k!((n k) + 1)! H (n k)+1 B : and in an analogous way to the series expansion of cos(bx), one gets (0). By [1, p. 57] one gets lim n k l=0 ( B a) (k l)+1 l!((k l) + 1)! = e sinh(a B ): Then in an analogous way to the previous computations, one gets (6). Hence the result is established. Now we show that condition (17) can be removed in the construction of accurate Hermite matrix polynomial expansions of exp(at); cos(at) and sin(at) of an arbitrary matrix A in C r r. Lemma 3.1. Let A be a matrix in C r r and let any positive number such that max{ Re(z) + Im(z) ; z (A)}: Then the matrix B = A + I satises (17). Proof. By the spectral mapping theorem (A + I)= {z + ; z (A)} and for z (A) one gets Re(z + ) = Re(z) + Re(z) Im(z) = Im(z + ). Hence the result is established.

8 11 E. Defez, L. Jodar / Journal of Computational and Applied Mathematics 99 (1998) Corollary 3.1. Let A be a matrix in C r r and let be a number satisfying the condition of Lemma 3.1. Let 0; c 0; a= c +. With the notacion of Theorem 3.1 it follows that (i) If n 0 satises (1) for B = A + I; then e Ax e x E(A + I;x;n) ; n n 0 ; x c: (31) (ii) Let n 1 and n be positive integers satisfying () and (3) respectively for == ; and B = A + I; then for x c and n max{n 1 ;n }; sin(ax) {cos(x)s(a + I;x;n) sin(x)c(a + I;x;n)} ; (3) cos(ax) {cos(x)c(a + I;x;n)+ sin(x)s(a + I;x;n)} : (33) Proof. (i) The result is a consequence of Theorem 3.1 and the formula e Ax =e x e (A+I)x. The proof of part (ii) follows from the expressions sin(ax) = sin[(a + I)x] cos(x) cos[(a + I)x] sin(x); cos(ax) = cos[(a + I)x] cos(x)+ sin[(a + I)x] sin(x); and Theorem 3.1. In the following example we illustrate the use of Hermite matrix polynomials for computing e A, sin A and cos A for a matrix A satisfying (17). As it has been proved in Corollary 3.1, condition (17) is not necessary taking an appropiate value of in accordance with Lemma 3.1. It is interesting to point out that algorithm of Theorem.1 has been adapted for the Hermite matrix polynomials using (5). Computations have been perfomed using Mathematica version ::1. Example 3.1. Consider the matrix A = 0 1 ; 1 1 where (A)={1; ; }. Using the minimal theorem [3, p. 571] one gets the exact value of e A : e e e e A = e +e e e e e + e e e e 14: : : = 1: : : : 4: : :

9 E. Defez, L. Jodar / Journal of Computational and Applied Mathematics 99 (1998) Given an admissible error =10 5, from (1), taking n 0 = 30 the approximation E(A; 1; 30) provides the required accuracy 30 1 E (A; 1; 30) = e n! H n (1; 1 ) A n=0 14: : : = 1: : : ; 4: : : e A E (A; 1; 30) =3: : Of course, in practice the number of terms required to obtain a prexed accuracy uses to be smaller than the one provided by (1), because the error bounds given by Theorem 3.1 is valid for any matrix satisfying (17). So for instance taking n 0 = 19 one gets 14: : : E (A; 1; 19) = 1: : : ; 4: : : e A E (A; 1; 19) =6: : In an analogous way we have sin() + cos() cos() cos() sin A = sin(1) + sin() + cos() sin(1) cos() cos() sin(1) + sin() sin(1) sin() sin() 0: : : = 0: : : : 0: : : Taking n 0 = 8 one gets the Hermite matrix polynomial approximation with S (A; 1; 8) = 1 8 ( 1) n e (n + 1)! H n+1 (1; 1 ) A n=0 0: : : = 0: : : ; 0: : : sin A S(A; 1; 8) =4: :

10 114 E. Defez, L. Jodar / Journal of Computational and Applied Mathematics 99 (1998) The number of terms provided by (3) for the accuracy =10 5 is n 0 = 19. Finally approximating cos A by Hermite matrix polynomial series we have cos() sin() sin() sin() cos A = cos(1) + cos() sin() cos(1) + sin() sin() cos(1) + cos() cos(1) cos() cos() 1: : : = 1: : : : 0: : : Taking n 0 = 8 one gets the approximation C (A; 1; 8) = 1 8 ( 1) n e (n)! H n (1; 1 ) A n=0 1: : : = 1: : : ; 0: : : with error cos A C (A; 1; 8) =9: : For a prexed accuracy =10 5, the expression () gives n 0 = Applications In this section we construct matrix polynomial approximations of problems (1) (3) expressed in terms of Hermite matrix polynomials. It is well known that the solution of problem (1) is Y (x)=e Ax y 0 : Let be as in Lemma 3.1, 0, c 0, a = c +. Let n 0 be the rst positive integer such that A((A + I)a; n 0 ) exp(a A + I +1) e(1 + y 0 ) : Then by Corollary 3.1 it follows that Thus e Ax y 0 e x E(A + I;x;n)y 0 ; x c; n n 0 : Ỹ n (x)=e (1 x) [ H k (x; 1(A + ] I) ) y 0 ; (34) k!

11 E. Defez, L. Jodar / Journal of Computational and Applied Mathematics 99 (1998) is an approximate solution of problem (1) such that if Y (x) is the exact solution one gets Y (x) Ỹn(x) ; x c; n n 0 : (35) Problem () can be solved considering the extended system [ ] [ ] [ ] Y 0 I P Z = Y ; Z = A Z; Z(0) = ; 0 Q but such an approach increases the computational cost [1], and involves a lack of explicitness in terms of two vector parameters that is very interesting to study boundary value problems associated to () using the shooting method [9]. Consider problem () where A is an invertible matrix in C r r. By [6] the pair {cos(ax); sin(ax)} is a fundamental set of solutions of the equation Y +A Y =0; x + ; because Y 1 (x) = cos(ax), Y (x) = sin(ax) satisfy [ ] [ ] Y1 (0) Y W (0) = (0) I 0 Y 1(0) Y (0) = is invertible in C 0 A r r : Hence, the unique solution of problem () is given by Y (x) = cos(ax)p+ sin(ax)a 1 Q; x + : (36) If is given by Lemma 3.1, the expresion (36) and Corollary 3.1 suggest the approximation Ỹ n (x)={cos(x)c (A + I;x;n) + sin(x) (A + I;x;n)} P + {cos(x)s (A + I;x;n) sin(x)c (A + I;x;n)} A 1 Q = C (A + I;x;n) ( cos(x)p sin(x)a 1 Q ) +S (A + I;x;n) ( sin(x)p+ cos(x)a 1 Q ) ; [ ]( ( 1) k H k (x; 1 Ỹ n (x)= (A+I) ) cos(x) P ) (k)! e Q sin(x)a 1 e [ ]( ( 1) k H k+1 (x; 1 + (A+I) ) sin(x) P ) (k + 1)! e + Q cos(x)a 1 : (37) e By Corollary 3.1, for n max{n 1 ;n }, where n 1 and n are given by the Corollary 3.1, one gets Y (x) Ỹn(x) 6( P + A 1 Q ); x c: (38) We conclude this section with the construction of Hermite matrix polynomials approximations of the solution of problem (3). By [, p. 195] the solution of (3) is given by X (t)=e At Ce Bt : (39)

12 116 E. Defez, L. Jodar / Journal of Computational and Applied Mathematics 99 (1998) Let max{ A ; B } where and A max{ Re(z)+ Im(z) ; z (A)}; B max{ Re(z)+ Im(z) ; z (B)}: Corollary 3.1 suggests the approximation X n (t)=e t E(A + I;t;n)Ce t E(B + I;t;n); (40) and note that X (t) X n (t) =(e At e t E(A + I;t;n))Ce Bt +e t E(B + I;t;n)C(e Bt e t E(B + I;t;n)): (41) Consider the domain t c and take K = max{c( B +); }+1: By (15) and (18) it follows that t K=( B +) and E(B + I;t;n) 6e c +1 (K) j = L: (43) j! j 0 Taking norms in (41) and using (43) one gets X (t) X n (t) 6 e At e t E(A + I;t;n) C e c B + L C e Bt Taking the rst positive integer n 0 such that (4) e t E(B + I;t;n) : (44) e c B A((A+I)a; n 0) exp(w A+I a+1) e( C +1) ; and the rst positive integer n 1 such that A((B + I)a; n 1) exp( B + I a +1) L( C +1)e ; then for n max{n 0;n 1} and t c, by Corollary 3.1 and (39), (40), (43) it follows that X (t) X n (t) ; t c: References [1] T. Apostol, Explicit formulas for solutions of the second-order matrix dierential equation Y = AY, Amer. Math. Monthly 8 (1975) [] R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York, 1965.

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