• 论文 •

### 计算矩阵函数双线性形式的Krylov子空间算法的误差分析

1. 清华大学数学科学系, 北京 100084
• 收稿日期:2018-10-05 出版日期:2020-02-15 发布日期:2020-02-15
• 基金资助:

国家自然科学基金资助（项目编号11771249）.

Jia Zhongxiao, Sun Xiaolin. THE ERROR ANALYSIS OF THE KRYLOV SUBSPACE METHODS FOR COMPUTING THE BILINEAR FORM OF MATRIX FUNCTIONS[J]. Mathematica Numerica Sinica, 2020, 42(1): 117-130.

### THE ERROR ANALYSIS OF THE KRYLOV SUBSPACE METHODS FOR COMPUTING THE BILINEAR FORM OF MATRIX FUNCTIONS

Jia Zhongxiao, Sun Xiaolin

1. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
• Received:2018-10-05 Online:2020-02-15 Published:2020-02-15

The bilinear form uTf(A)v of matrix functions is of wide interest in many applications, where u, v ∈ Rn, A ∈ Rn×n, f(z) is a given analytic function. In recent years, the efficient and reliable numerical algorithms for the bilinear form has been a research focus. Although there are numerous stopping criteria, they lack solid theoretical supports, and the reliability is unknown. In this paper, we consider the posteriori error estimates for the errors of approximate solutions of the matrix functions uTf(A)v. We derive an error expansion and prove that the first term of the error expansion can used as a reliable stopping criterion.

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 [1] 吕慧.计算大规模稀疏矩阵函数乘向量的Krylov子空间算法[D].清华大学, 2014.[2] Jia Z, Lv H. A posteriori error estimates of Krylov subspace approximations to matrix functions. Numerical Algorithms, 2015, 69(1):1-28.[3] Fika P, Mitrouli M, Roupa P. Estimates for the bilinear form xT A-1y with applications to linear algebra problems. Electronic Transactions on Numerical Analysis, 2014, 43:70-89.[4] Abramowitz M, Stegun I A. Handbook of mathematical functions:with formulas, graphs, and mathematical tables, volume 55. Courier Corporation, 1964.[5] Bai Z, Fahey G, Golub G. Some large-scale matrix computation problems. Journal of Computational and Applied Mathematics, 1996, 74(1-2):71-89.[6] Bai Z, Golub G. Computation of large-scale quadratic forms and transfer functions using the theory of moments, quadrature and Padé approximation. Proceedings of Modern Methods in Scientific Computing and Applications. Springer, 2002:1-30.[7] Calvetti D, Kim S M, Reichel L. Quadrature rules based on the Arnoldi process. SIAM Journal on Matrix Analysis and Applications, 2005, 26(3):765-781.[8] Golub G. Matrix Computation and the Theory of Moments. Birkhäuser, Basel, 1995:1440-1448.[9] Golub G H, Meurant G. Matrices, moments and quadrature. Pitman Research Notes In Mathematics Series, 1994. 105-105.[10] Golub G H, Meurant G. Matrices, moments and quadrature II; how to compute the norm of the error in iterative methods. BIT Numerical Mathematics, 1997, 37(3):687-705.[11] Golub G H, Strakošs Z. Estimates in quadratic formulas. Numerical Algorithms, 1994, 8(2):241-268.[12] Guo H, Renaut R A. Estimation of uTf(A)v for large scale unsymmetric matrices. Numerical Linear Algebra with Applications, 2004, 11(1):75-89.[13] 郭洪斌.矩阵函数双线性形式的计算及其应用[D].复旦大学, 1999.[14] Strakoš Z. Model reduction using the Vorobyev moment problem. Numerical Algorithms, 2009, 51(3):363-379.
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