求解Stein张量方程的张量格式BCGSTAB算法

Abstract

Abstract:

The biconjugate gradient stabilized (BCGSTAB) method is a fast and smoothly converging variant of biconjugate gradient (BiCG) method. In this paper, we generalize BCGSTAB method to solve the Stein tensor equation. We raise the algorithm of tensor format and specific proof process of existence of a solution. And we present the convergence theorem of this algorithm. Numerical experiments demonstrate this algorithm is effective and feasible for solving the Stein tensor equation.

Key words: tein tensor equation, BCGSTAB method, Convergence analysis, Numerical experiment

CLC Number:

O241.1

Ma Changfeng, Xie Yajun, Bu Fan. The Tensor Scheme BCGSTAB Algorithm for Solving Stein Tensor Equations[J].Acta mathematica scientia,Series A, 2024, 44(6): 1652-1664.

Trendmd

References

[1]	Hestenes M R, Stiefel E. Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards (United States), 1952, 49(6): 409-436
[2]	Paige C C, Saunders M A. Solution of sparse indefinite systems of linear equations. SIAM Journal on Numerical Analysis, 1975, 12(4): 617-629
[3]	Saad Y. Iterative Methods for Sparse Linear Systems. SIAM: Philadelphia PA USA, 2003
[4]	Lanczos C. Solution of systems of linear equations by minimized iterations. Journal of Research of the National Bureau of Standards, 1952, 49: 33-53
[5]	Fletcher R. Conjugate gradient methods for indefinite systems. Springer, 1976, 506: 73-89
[6]	Saad Y, Shultz M H. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 1986, 7(3): 856-869
[7]	Eisenstat S C, Elman M H, Schultz M H. Variational iterative methods for nonsymmetric systems of linear equations. SIAM Journal on Numerical Analysis, 1983, 20(2): 345-357
[8]	Sogabe T, Sugihara M, Zhang S L. An extension of the conjugate residual method to nonsymmetric linear systems. Journal of Computational and Applied Mathematics, 2009, 226(1): 103-113
[9]	Kressner D, Tobler C. Krylov subspace methods for linear systems with tensor product structure. SIAM Journal on Matrix Analysis and Applications, 2010, 31(4): 1688-1714
[10]	Van der Vorst H A. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 1992, 13(2): 631-644
[11]	Gutknecht M H. Variants of BICGSTAB for matrices with complex spectrum. SIAM Journal on Scientific Computing, 1993, 14(5): 1020-1033
[12]	Zhang S L. GPBi-CG: Generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems. SIAM Journal Scientific Computing, 1997, 18(3): 537-551
[13]	Zhou B, Lam J, Duan G R. On smith-type iterative algorithms for the stein matrix equation. Applied Mathematics Letters, 2009, 22(7): 1038-1044
[14]	DeAlba L M, Johnson C R. Possible inertia combinations in the stein and lyapunov euations. Linear Algebra and its Applications, 1995, 222: 227-240
[15]	Fuhrmann P A. A functional approach to the stein equation. Linear Algebra and its Applications, 2010, 432(12): 3031-3071
[16]	Gouveia M C. On the solution of sylvester, lyapunov and stein equations over arbitrary rings. International Journal of Pure and Applied Mathematics, 2005, 24(1): 135-141
[17]	Betser A, Cohen N, Zeheb E. On solving the lyapunov and stein equations for a companion matrix. Systems and Control Letters, 1995, 25(3): 211-218
[18]	Chiang C Y. A note on the $\top$-Stein matrix equation. Abstract and Applied Analysis. Hindawi Publishing Corporation, 2013, 2013(1): 824641
[19]	Xu X J, Wang Q W. Extending BiCG and BiCR methods to solve the Stein tensor equation. Computers & Mathematics with Applications, 2019, 77(12): 3117-3127
[20]	Zhang X F, Wang Q W. On RGI algorithms for solving sylvester tensor equations. Taiwanese Journal of Mathematics, 2022, 26(3): 501-519
[21]	Chen Y, Li C. Preconditioned tensor format conjugate gradient squared and biconjugate gradient stabilized methods for solving stein tensor equations. Numerical Linear Algebra with Applications, 2023, 30(5): e2502

The Tensor Scheme BCGSTAB Algorithm for Solving Stein Tensor Equations

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 9

Metrics

Comments

Recommended 0

[1]	Yu Dongmei, Liu Dayi. The Levenberg-Marquardt Algorithm for Solving the Generalized Complementarity Problems [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1311-1326.
[2]	Yu Yikang,Niu Jing. A Kind of Numerical Algorithm for Elliptic Interface Problem [J]. Acta mathematica scientia,Series A, 2023, 43(3): 883-895.
[3]	Yajun Xie. A Class of Efficient Modified Algorithms Based onHalley-Newton Methods [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1066-1078.
[4]	Zhihao Ge,Ruihua Li. Numerical Methods for the Critical Temperature and Gap Solution of Bogoliubov-Tolmachev-Shirkov Model [J]. Acta mathematica scientia,Series A, 2020, 40(6): 1699-1711.
[5]	Linjuan Pu,Xiaozhong Yang,Shuzhen Sun. Numerical Analysis of a Class of Fractional Langevin Equation by Predictor-Corrector Method [J]. Acta mathematica scientia,Series A, 2020, 40(4): 1018-1028.
[6]	Lixin Feng,Xiaoxu Yang. Spectral Regularization Method for Volterra Integral Equation of the First Kind with Noise Data [J]. Acta mathematica scientia,Series A, 2020, 40(3): 650-661.
[7]	Yadi Zhao, Lifei Wu, Xiaozhong Yang, Shuzhen Sun. A Kind of Efficient Difference Method for the Time Fractional Sub-Diffusion Equation [J]. Acta mathematica scientia,Series A, 2018, 38(6): 1122-1134.
[8]	SUN Qing-Ying, DONG Jie-Hong, SANG Zhao-Yang. A New Trust Region Algorithm with Simple Quadratic Models and Line Search [J]. Acta mathematica scientia,Series A, 2010, 30(6): 1562-1574.
[9]	SHI Zhen-Jun. A Nonlinear Conjugate Gradient Method under Exact Line Search [J]. Acta mathematica scientia,Series A, 2004, 4(6): 675-682.