COMPARISON OF PERTURBATION BOUNDS FOR THE MATRIX. Abstract. The paper deals with the perturbation estimates proposed by Xu[4],Sun,

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1 COMPARISON OF PERTURBATION BOUNDS FOR THE MATRIX EQUATION X = A 1 + A H X;1 A M.M. Konstantinov V.A. Angelova y P.H. Petkov z I.P. Popchev x Abstract The paper deals with the perturbation estimates proposed by Xu[4],Sun, Xu [3] and Konstantinov et al [] for evaluating the sensitivity of the solution to the complex fractional-ane matrix equation X = A 1 + A H X ;1 A relative to rounding and parameter errors. The eectiveness and reliability of the dierent methods are analyzed by experiments with numerical examples. Key Words: Perturbations, complex fractional-ane matrix equation, sensitivity. MSC 000: 15A4. 1 Introduction Shu-Fang Xu in [4], Ji-gunag Sun in [3] and Konstantinov et al in [] consider the sensitivity of the complex fractional-ane matrix equation X = A 1 + A H X ;1 A (1) where A 1 C nn and the solution X C nn are Hermitian positive denite matrices. Dierent schemes for estimating the sensitivity of the solution relative University ofarchitecture and Civil Engineering, 1 Hr. Smirnenski Blvd., 1046 Soa, Bulgaria, mmk ;fte@uacg.bg y Institute for Information Technologies, Akad. G. Bonchev Str., Bl., 1113 Soa, Bulgaria, verandi@abv.bg z Department of Automatics, Technical University of Soa, 1756 Soa, Bulgaria, php@tusoa.acad.bg x Institute for Information Technologies, Akad. G. Bonchev Str., Bl., 1113 Soa, Bulgaria, ipopchev@iit.bas.bg

2 to rounding and parameter errors are presented in the above papers. Therefore, it is interesting to compare the eectivness and the eld of application of these dierent perturbation results. Perturbation analysis of real equations of type (1) is done in [1]. In this paper, by means of numerical experiments, we present a comparison analysis of the eectiveness and applicability of the perturbation bounds, proposed in [4, 3, ]. Throughout the paper we use the following notation: C nn { the space of n n complex matrices R + =[0 1) H nn { the space of Hermitian matrices A > { the transpose of the matrix A A { the complex conjugate of A A H = A > vec(a) C n { the column-wise vector representation of the matrix A C nn Mat(L) C n n { the matrix representation of the linear matrix operator L : C nn! C nn I n { the identity n n matrix A B =[a pqb] { the Kronecker product of the matrices A =[a pq] and B kk { the Euclidean norm in C n or the spectral (or -) norm in C nn kk F {thefrobenius (or F-) norm in C nn kk { a replacement of either kk or kk F z R R n { the real version of the vector z C n ; R R nn { the real version of the matrix ; C nn (; ) { the matrix of the real version of the operator z! ;z +z n C n n {thevec-permutation matrix such that vec(y > )= n vec(y ) for each Y C nn. The notation `:=' stands for `equal by denition'. The sub-indices k l take values 1,. Statement of the problem Consider the matrix equation (1). The round{o and parameter errors, accompanying the numerical solution of the equation, are represented by equivalent perturbations in the matrix coecients, A k! A ~ k := A k + A k. They lead to a perturbation in the solution X! X ~ = X + X, where A1 X H nn and A C nn and for some k 0wehave ka kk F k. The perturbed equation (1) is ~X = ~ A 1 + ~ A H ~ X ;1 ~ A: () The purpose of the norm-wise perturbation analysis of equation (1) is to derive bounds for X := kxk F as a linear or non-linear function of the perturbations ka kk F in the data. 3 Sensitivity estimates 3.1 The estimate of Xu [4] In [4] Xu proves the following theorem.

3 Theorem 1. Let A ~ A A 1 ~ A 1 C nn with A 1 and ~ A 1 Hermitian positive denite. If ka k ka ;1 1 k < 1 k A ~ ; A k < 1 1 k ~ A 1 ; A 1k ;kakka;1 1 k ka ;1 1 k ;1 (3) 1 ;kakka;1 1 k ka ;1 1 k ;1 then the solutions X and ~ X of the matrix equations (1) and () exist and satisfy that k ~ X ; Xk kxk 1 := 1 1 ;kakka;1 1 k k A ~ ; A k + k ~ A1 ; A 1k : (4) ka k ka 1k 3. The estimate of Sun and Xu [3] Denote Let, in the real case, B := X ;1 A := ka k := kbk := kx ;1 k : L := I ; B > B H = I ; ; X ;1 > ; A X ;1 H A Q := L I ;1 B H +(B > I n) n = L ;1 (I n (X ;1 A) H )+L ;1 (X ;1 A ) > I n n and q := kqk F, l := kl ;1 k ;1 F. Similarly let, in the complex case, L ;1 =: S + i L ;1 I n B H = =: U 1 + i 1 L ;1 B > I n n = =: U + i S ; S c := S U1 + U U c := ; 1 ; 1 + U 1 ; U and q := ku ck F, l := ks ck ;1. Denote F = 1 l ka1kf + qkakf + l kak F = (( + kakf) + ) kakf: l In [3] SunandXuprove the following theorem. Theorem. If (1 ; )( + ) <min 1 l and

4 ( l(1 ; ) <min (l + + l + p (l + )(l + )) (l ; )((1 ; )( + ) ; l) ((1 + )( + )+l) then the perturbed matrix equation () has a solution ~ X, and moreover, k ~ X ; XkF := l l(1 + ; )+ p l (1 + ; ) ; 4l(l + ) : (5) 3.3 The estimate of Konstantinov et al [] Denote := kxk F and := [ 1 ] > = ka 1k F ka k F > R +: In [] the following local bound is derived for the perturbation X in the solution X of equation (1) g() +O(kk )! 0 (6) where n g() := min k[m 0 1 M 0 ]k p o kk > M 0 (7) L := I n ; (X ;1 A ) > (A H X ;1 ) M 1 := ;L ;1 =: ; + i M 1 := +L I ;1 n (A H X ;1 ) =: ; 1 + i 1 M := +L ;1 ((X ;1 A ) > I n) n =: ; + i ; ; M 0 1 := M 0 ;1 +; := ; 1 ; 1 + ; 1 ; ; and M 0 =[m 0 kl] R + Let := kx ;1 k and is a symmetric matrix with elements m 0 kl = Mk 0 > M 0 l : a 0() := g()+c 1 a 1() := a 11 + a 1 a () := a 0 + a 1 + a where a 11 := L ;1 I n A H X ;1 ;X + L ;1 ;1 > A I n P n a 1 := c 1 a 0 := 3 L ;1 A > A H a 1 := 3 L ;1 a := c 1 3 : A > I n n + L ;1 I n A H

5 The following theorem is proved in [] Theorem 3. Let n := R + : a 1 ; a 0 + p o a 0(a + (1 ; a 1)) 1 : (8) Then the non-local perturbation bound kxk F f() := is valid for equation (1). a 0() 1 ; a 1 + a 0 + p d() (9) d() := (1 ; a 1()+a 0()) ; 4a 0()(a ()+(1 ; a 1())) 4 Experimental results Example 1. The model from Example 5.1 in [3] is used. Consider equation (1) 0 a with matrix coecients A 1 = I, A =,wherea =0:5 ; ;k. The solution of the equation is X = diag(1 1 ; a ). The perturbations in the data are taken as: ;0:436 ;0:7701 0:9501 0:6068 A 1 = 10 ;8 A 0:7701 0:877 = 10 ;9 0:311 0:4860 for k = : In Table 1 we give the numerical results for the relative perturbations bounds 1 (4) of Xu [4] =kxk (5) of Sun and Xu the local bound g()=kxk (6) and the non-local bound f()=kxk (9) of Konstantinov et al []. Table 1. k ; ; ; ; =kxk ; ; ; ; ;8 g()=kxk ; ; ; ; ;8 f()=kxk ; ; ; ; ;8. Let now a =0:99 and the perturbations in the matrix coecients be ;0:436 ;0:7701 A 1 = 10 ;k 0:9501 0:6068 A 0:7701 0:877 = 10 ;k 0:311 0:4860 for k =6 ::: 10. For this example the estimate 1 from Xu [4] is not valid, since the conditions (3) from Theorem 1 are not satised. The results for (5) of Sun and Xu, the

6 local bound g() (6) and the non-local bound f() (9) of Konstantinov et al. are listed in Table. The case when the non-local estimate f() is not valid, since the existence condition is violated, is denoted by asterisk. Table. k ; ; ; ; ;10 g() ; ; ; ; ;10 f() * ; ; ; ;10 Example. For this example we use the model from []. Consider the complex fractional-ane matrix equation (1) with matrices 0: :0535i 0: :0077i 0: :4175i 3 A = 4 0: :597i 0: :3834i 0: :6868i 5 0: :6711i 0: :0668i 0: :5890i A 1 = I 3 + A H A : The perturbations in the data are taken as 3 1+i 1+i 1+i A = 10 (;k) 4 1+i 1+i 1+i 5 1+i 1+i 1+i A 1 = A +(A + A ) H (I 3 + A ) ;1 (A + A ) ; A H A for k =10 9 ::: : This problem was designed so as to have solutions X = I 3 and X + X = I 3 + A of the unperturbed and perturbed equations respectively. The perturbation kxk F in the solution is estimated by the bound (5) of Sun and Xu [3], the local bound g() (6) and the non-local bound () (9) of Konstantinov et al []. For this example the bound 1 (see (4)) of Xu [4] is not valid, since the conditions (3) are violeted. The results obtained for dierent values of k are shown at Table 3. When k decreases from 10 to the non-local estimates (5) and () (9) are slightly more pessimistic than the local bound g(). The cases when the non-local estimates are not valid, since the existence conditions are violated, are denoted by asterisk.

7 Table 3. k kxk F g() () ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;3 * ; ; ; * ;1 * 5 Concluding remarks The results of the experimental analysis show that in the given cases the non{ linear method of Konstantinov et al. [] for estimating the sensitivity ofthe solution to equation(1) is superior to the methods of Xu [4] and Sun and Xu [3] with respect of closeness to the estimated quantity. References [1] M. Konstantinov. Perturbation analysis of a class of real fractional{ane matrix equations. In Proc. Jub. Sci. Conf. Univ. Arch. Civil Eng. Geod., volume 8, pages 489{494, Soa, 00. [] M. Konstantinov, P. Petkov, V. Angelova, and I. Popchev. Sensitivity ofa complex fractional{ane matrix equation. In Proc. Jub. Sci. Conf. Univ. Arch. Civil Eng. Geod., volume 8, pages 495{504, Soa, 00. [3] J.-G. Sun and S. Xu. Perturbation analysis of the maximal solution of the matrix equation X + A X ;1 A = P.II. Linear Algebra Appl., 36:11{8, 003. [4] S. Xu. Perturbation analysis of the maximal solution of the matrix equation X + A X ;1 A = P. Linear Algebra Appl., 336:61{70, 001. Submitted: January 003

8 X = A 1 + A H X ;1 A..,..,..,.. [4], [3].[] - X = A 1 + A H X ;1 A..

9 I X = A 1 + A H X ;1 A..,..,..,.. [4], [3].[] - X = A 1 + A H X ;1 A..