We consider Bernoulli's method for solving quadratic matrix equations (QMEs) having form Q(X) = AX^2 +BX+ C = 0 arising in hyperbolic quadratic eigenvalue problems (QEPs) and quasi-birth-death problems (QBDs) where A, B, C ∈ R^[m×m] satisfy Esenfeld's condition [8]. First, we analyze the exsistence of a solution and the convergence of the methods. Second, we sharpen bounds of the rates of convergence. Finally, in numerical experimentations, we show that the modified bounds give appropriate estimations of the numbers of iterations

KIM, HYUN-MIN

SEO, JONG-HYEON

English

Kyoto University Research Information Repository

RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:ESTIMATING THE CONVERGENCE RATE OFFUNCTIONAL ITERATIONS FOR SOLVINGQUADRATIC MATRIX EQUATIONS ARISING INHYPERBOLIC QUADRATIC EIGENVALUEPROBLEMS (Study on Nonlinear Analysisand Convex Analysis)SEO, JONG-HYEON; KIM, HYUN-MINSEO, JONG-HYEON ...[et al]. ESTIMATING THE CONVERGENCE RATE OF FUNCTIONAL ITERATIONS FOR SOLVINGQUADRATIC MATRIX EQUATIONS ARISING IN HYPERBOLIC QUADRATIC EIGENVALUE PROBLEMS (Study on NonlinearAnalysis and Convex Analysis). 数理解析研究所講究録 2021, 2 ...2021-07http://hdl.handle.net/2433/26565656ESTIMATING THE CONVERGENCE RATE OF FUNCTIONAL ITERATIONS FOR SOLVING QUADRATIC MATRIX EQUATIONS ARISING IN HYPERBOLIC QUADRATIC EIGENVALUE PROBLEMS JONG-HYEON SEO AND HYUN-MIN KIM ABSTRACT. We consider Bernoulli's method for solving quadratic matrix equa-tions (QMEs) having form Q(X) = AX2 +BX+ C = 0 arising in hyper-bolic quadratic eigenvalue problems (QEPs) and quasi-birth-death problems (QBDs) where A,B,C E ffi!.mxm satisfy Esenfeld's condition [8]. First, we an-alyze the exsistence of a solution and the convergence of the methods. Second, we sharpen bounds of the rates of convergence. Finally, in numerical experi-mentations, we show that the modified bounds give appropriate estimations of the numbers of iterations. 1. INTRODUCTION In this paper, we consider the quadratic matrix equation (QME) (1.1) Q(X) := AX2 +BX+ C = 0, where A, B, C E Rmxm satisfy Esenfeld's condition (1.2) [8,39]. If SE Rmxm satisfies Q(S) = 0 in (1.1) then we call that Sis a solvent [6]. Note that in [8] Esenfeld assumed that all coefficient matrices are nonsingular, but our condition needs that only B is nonsingular. From Bernoulli's iteration [7, 13, 21], we construct the functional iterative methods (1.3) and (1.4) 2000 Mathematics Subject Classification. 15A24, 65F10, 65H10. Key words and phrases. quadratic matrix eqaution (QME), Bernoulli 's method, functional iter-ation, hyperbolic quadratic eigenvalue problem (QEP), quasi-birth-death process (QBD), con-traction mapping theorem, iterated contraction mapping theorem, mass spring damping system, overdamped system, the rate of convergence, Esenfeld's condition, the number of iterations. 57JONG-HYEON SEO AND HYUN-MIN KIM Higham and Kim [21] showed that the iteration (1.4) converges to minmal solvents of the QMEs ( 1.1) if they have such solvents. Bai and Gao [3] modified the iteration by techniques of the Gauss-Seidal iteration. In stochastic areas, it is well-known that the iterations (1.3) and (1.4) converge to elementwise minimal nonnegative solvents ofQMEs (1.1) arising in QBDs (for more details, see Favati and Meini [11].). Definitions of minimal and elementwise minimal nonnegative solvents are found in [21] and [36] respectively. Those implies that the convergence of the iterations (1.3) and (1.4) are depend on the existence of such extreme solvents. There are well-known two conditions guar-anteeing the existence of such extreme sovlents. One is from quadratic eigenvalue problems (QEPs) and the other is from quasi-birth-death processes (QBDs). Let coefficient matrices A, B and C are Hermitian positive definite. Then QEPs satisfying (1.5) min ((x* Bx)2 - 4 (x* Ax) (x*Cx)) > 0, llxll=l are called hyperbolic [26, 37], and (1.5) guarantees the exisitence of minimal solvents of QMEs (1.1). Those problems have been widely studied together with elliptic problems [18, 23, 24, 34, 37]. Besides, let A, C, I+ B negative entries. Then QMEs satisfying N be real and have no (1.6) -Elm= (A+ C) lm and A+ N + C is irreducible have elementwise minimal nonnegative solvents and transition matrices from the associated QBDs have the unique stationary vectors where lm is the m-column vec-tor of which entries are 1. Those solvents of QMEs (1.1) play key roles in stochastic areas [5,9,17,27]. On the other hand, the existence of solvents of QMEs (1.1) in view point of func-tional analysis on Banach spaces were also discussed by some authors [2,8,21,25,30]. From the results, we found that Esenfeld's condition (1.2) is much involved in not only the existence of solvents but also the semi-local convergence of functional it-erative methods for special strarting matrices (note the relation to the condition that the discriminant of a scalar quadratic be positive [21]). Moreover, many exam-ples and applications of QMEs arising in hyperbolic QEPs including over-damped mass-spring systems referred in [3, 12, 16, 18, 21, 22, 24, 32, 34, 37-39] and stochastic problems including QBDs referred in [1, 4, 5, 9-11, 14, 15, 17, 19-21, 27-29, 31, 35] satisfy (1.2) as well as (1.5) and (1.6) respectively. 2. RELATED DEFINITIONS AND THEOREMS For SE JR.mxm and 8 > 0, B(S, 8) denotes an open ball centered by S with a radius 8 such as B(S, 8) := { X E JR.mxm : IIX - s11 < 8} ' 58Local Convergence of Iteration Methods for Solving QME and the spectral radius of A E JR.mxm is defined by p(A) := max{l>-1 : det(A - >.I) = O}. Since F 1 in (1.3) is well-defined on JR.mxm, from the properties of matrix norms we have Define (2.2) Since F 2 in (1.4) has the matrix inverse operations, we need the following lemmas. Lemma 2.1 (Neumann Lemma, see [33, 2.3.1 ]). For A, B E cmxm, if A be nonsingular and p(A-1B) < 1, then A-Bis also nonsingular and represented by (2.3) IIA-1 Ell < 1 instead of p (A-1 B) < 1 also leads to the same conclusion in Lemma 2.1. Lemma 2.2. (See [33, 2.3.1].) For L, M, N E cmxm, if L is nonsingular and IIL-1MII < 1 then L- Mis also nonsingular and II (L - M)-1 NII < IIL-1 NII - l - llL-1MII For X E JR.mxm if F 2(X) in (1.4) is well-defined, then from the Neumann lemma for a sufficiently small perturbation matrix H E JR.mxm we have F2(X +H) ((B - AX) - AH)- 1 C (B - AX)-1 C + (B - AX)-1 AH((B - AX) - AH)- 10 F2(X) + (B - AX)-1 AH(B - AX - AH)- 10. From the above expression H := Y - X yields and IIF2(Y) -F2(X)II II (B-AX)- 1 A(Y - X) (B -AY)-1 GIi :s: II (B - Ax)- 1 All II (B - AY)-1 CII IIY - x11. Define (2.4) r2(X, Y) := II (B - Ax)-1 All II (B -AY)-1 GIi-We show that the iterations (1.3) and (1.4) always converge to a solvent of which the existence is guaranteed. The followings are used in the analysis. 59JONG-HYEON SEO AND HYUN-MIN KIM One-step stationary iterations have the form (2.5) where F : D C ]Rmxm ---+ ]Rmxm_ This include Newton's method and some of minimization methods. Theorem 2.3 (Contraction Mapping Theorem, see [33, Thm. 12. 1. 2]). Let F : D c ]Rmxm ---+ ]Rmxm, and suppose that F maps a closed set Do C D into itself and that IIF(X) - F(Y)II::; allX - YII, VX, YE Do for some a < 1. Then, for any X 0 E D 0 , the sequence {Xk} generated by (2.5) converges to the unique fixed point S of F in D 0 and Definition 2.4. (See [33, Def. 12.3.1].) The matrix function F : D C JRmxm ---+ ]Rmxm is an iterated contraction on the set Do CD if there is an a< 1 such that IIF (F(X)) - F(X)II ::C: allF(X) - XII whenever X and F(X) are in D 0 . Theorem 2.5. (See [33, Thm. 12.3.2].) Suppose that F : D C ]Rmxm ---+ ]Rmxm is an iterated contraction on the closed set Do C D and that for some Xo E Do the sequence Xk+l = F(Xk), k = 0, 1,2, ... , remains in D 0 . Then, limk--+oo Xk :=SE D 0 , and the estimation (2.6) holds. Moreover, if F is continuous at S, then S = F(S). 3. ANALYSIS OF THE EXSISTENCE AND THE CONVERGENCE 3.1. Local Convergence of Xk+I = F 1 (Xk) for X 0 = 0. We show that F 1 in (1.3) is invariant and iterated contractive on some closed balls. Lemma 3.1. Suppose that B is nonsingular, where a and b are positive constants. If 'Y := 4ab < 1 then F1 in (1.3) is invariant and iterated contractive on B := B (0, 1/2a). Proof. Let X E B. Then from 'Y 1 1 b = - < - and IIXII::; -, 4a 4a 2a 60Local Convergence of Iteration Methods for Solving QME we have (3.1) Therefore F 1 is invariant on B. From (3.1) we get IIB-1AF1(X)II +allXII (3.2) 1+,, 1 3+,' < a---+a·-=--<l 4a 2a 4 where r 1 is in (2.2). From (3.2) we obtain (3.3) □ Since F 1 is Frechet differentiable and iterated contractive on B (0, 1/2a), from The-orem 2.5 the next theorem follows. Theorem 3.2. Under the assumption of Lemma 3.1, if I':= 4ab < l then the sequence {Xk} generated by (1.3) has a limit S in B. Moreover S is a solvent of QME (1.1) and the estimation holds. 3 + ,' 11xk - s11 s --11xk - xk-111, k = 1, 2, ... 1 - ,' Proof. From (3.3) in Lemma 3.3 the estimation is given by a 1-a □ 3.2. Local Convergence of Xk+l = F2 (Xk) for X 0 = 0. We show that F 2 in (1.4) is invariant and contractive on some closed balls. Lemma 3.3. Under the assumption of Lemma 3.1, If I':= 4ab < l then F 2 in (1.4) is invariant and contractive on B := B (0, 8) where (3.4) ():= v0_ 2a 61JONG-HYEON SEO AND HYUN-MIN KIM Proof. Since (3.5) ,,J1 = 2v'ah < 1, we have 1>1-v'ah>v'ah>O Let X E B. Then from IIXII :S ,n = v'ah 2a a we get IIB-1 AXIi :Sall XII < 1. Therefore from Lemma 2.2, B - AX is nonsingular and ::; ll(B -Ax)-1 ell b b :S -----,,-----,,- < -----== I - allXII I - v'ah (3.6) b v'ah ,/7 < -v'ali a 2a From (3.6), F 2 is invariant on B. Let Y, Z E B. Then from Lemma 2.2 and (3.6) we get ll(B -AY)-1 All ll(B -Az)-1C11 a b 1 - allYII . 1 - allZII a b 1 - v'ah . 1 - v'ah (3.7) a b < -·-=1 v'ah v'ah where r2 is in (2.4). From (3.7) we obtain (3.8) □ Since F 2 is Frechet differentiable and iterated contractive on B (0, 8), from the contraction mapping theorem we have the next theorem. Theorem 3.4. Under the assumption of Lemma 3.1, if 1 := 4ab < I then the sequence {Xk} generated by (1.4) has a limit S in B. Moreover S is the unique solvent of QME (I.I) in B and the estimation is given by r IIXk-SII:S ( )IIXk-Xk-1II, k=l,2, ... 4 1-,fi where 8 and I are in (3.4) and (3.5) in Lemma 3.3 respectively. 62Local Convergence of Iteration Methods for Solving QME Proof. From (3.3) in Lemma 3.3 the estimation is given by "I 1-o: ~ 1 -(2-✓-'f) ' ' 4- 4y'T 4. SHARPENING BOUNDS OF THE RATES OF CONVERGENCE 4.1. The Rate of Convergence of Xk+1 = .F1(Xk) for X 0 = 0. Theorem 4.1. Let A, B, C E JR.mxm and supoose that B is nonsingular and Then, for the sequence {Xk} generated by (1.3), we have ::~:+~ ~~~ :: ~ 1 - ~' k = l, 2, ... . Proof. Since we get (4.1) □ Since the sequence of the right side of (4.1) is increasing and bounded above by 1/2a, we have (4.2) \:/k EN. From (4.2), we obtain for all k EN IIxk+1 - Xkll ~ r1(Xk, xk-1)IIXk - xk-ill ~ IIB-1 All (IIXkll + IIXk-ill) IIXk - xk-ill -1 (1-~) < IIB All IIB-lAII IIXk - Xk-ill ~ (1-~) IIXk-Xk-ill-□ From Theorem 2.5 we have the next corollary. Corollary 4.2. Let A, B, C E JR.mxm and supoose that B is nonsingular and 63JONG-HYEON SEO AND HYUN-MIN KIM Then,, the sequence {Xk} generated by (1.3) has a limit S in B. Moreover S is a solvent of QME (1.1) and estimation 1-~ IIxk - sII :::; ~ IIxk - xk-ill, k = o, 1, 2, ... holds. 4.2. The Rate of Convergence of Xk+I = F2(Xk) for Xo = O. Theorem 4.3. Let A, B, C E ]Rmxm and supoose that B is nonsingular and Then, for the sequence {Xk} generated by (1.4), we have IIXk+l - Xkll 1 -~ II II < ' k = l, 2, .... xk - xk-1 - l + V 1 - 'Y Proof. Since we obtain By the mathematical induction, we easily prove that the sequence of the right side of the inequality (4.3) is increasing and bounded above by v"\/211B- 1 All- So we have From Lemma 2.2 and we get where r 2 is in (2.4). 1-~ IIXk II :::; 2IIB-1 All , k = o, I, 2, .... :::; r2 (F2(Xk), xk) IIXk - xk-1II IIB-1AII IIXk II 1-IIB-1AIIIIXkll IIXk - xk-ill < 1 - 1-y'I=y 2 From Theorem 2.3 we have the next corollary. Corollary 4.4. Let A, B, C E ]Rmxm and supoose that B is nonsingular and □ 64Local Convergence of Iteration Methods for Solving QME Then, the sequence { Xk} generated by (1.3) has the unique solvent solvent S in B and estimation holds and 1-~ 11s11 :::; 2IIB-1 All . In Corollary 4.4, since S is the unique solvent on neighborhood of the zero matrix, S must be the minimal solvent in view point of matrix norms and the spectral radius (i.e. p(S):::; p(S') and IISII :::; IIS'II if S' is arbitrary solvent of QME (1.1)). Therefore we have the following theorem. Theorem 4.5. Let A, B, C E JR.mxm and supoose that B is nonsingular and 0 <I:= 4IIB-l All llB-1011 < 1. Then, QME (1.1) has the minimal solvents S on B ( 0, ~) in view point of spectral radius. Furthermore, if S is nonsingular and (4.4) then the inverse of Sis a solvent ofQ2 (X) = CX2 +BX +A satisfying l..\min(S)I > 1. 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A structure-preserving doubling algorithm for quadratic matrix equa-tions arising form damped mass-spring system. Advan. Model. Optimiz, 12:85-100, 2010. DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY, BUSAN , REPUBLIC OF KOREA Email address: hyunmin©pusan. ac . kr 

ESTIMATING THE CONVERGENCE RATE OF FUNCTIONAL ITERATIONS FOR SOLVING QUADRATIC MATRIX EQUATIONS ARISING IN HYPERBOLIC QUADRATIC EIGENVALUE PROBLEMS (Study on Nonlinear Analysis and Convex Analysis)

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ESTIMATING THE CONVERGENCE RATE OF FUNCTIONAL ITERATIONS FOR SOLVING QUADRATIC MATRIX EQUATIONS ARISING IN HYPERBOLIC QUADRATIC EIGENVALUE PROBLEMS (Study on Nonlinear Analysis and Convex Analysis)

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