Two Iterative Algorithms to Compute the Bisymmetric Solution of the Matrix Equation A1X1B1 + A2X2B2 + ... + AlXlBl = C

Authors: A.Tajaddini

Abstract:

In this paper, two matrix iterative methods are presented to solve the matrix equation A1X1B1 + A2X2B2 + ... + AlXlBl = C the minimum residual problem l i=1 AiXiBi−CF = minXi∈BRni×ni l i=1 AiXiBi−CF and the matrix nearness problem [X1, X2, ..., Xl] = min[X1,X2,...,Xl]∈SE [X1,X2, ...,Xl] − [X1, X2, ..., Xl]F , where BRni×ni is the set of bisymmetric matrices, and SE is the solution set of above matrix equation or minimum residual problem. These matrix iterative methods have faster convergence rate and higher accuracy than former methods. Paige’s algorithms are used as the frame method for deriving these matrix iterative methods. The numerical example is used to illustrate the efficiency of these new methods.

Keywords: Bisymmetric matrices, Paige’s algorithms, Least square.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1087854

References:

[1] K. E. Chu, Singular value and generalized value decomposition and the solution of linear matrix equations, Linear Algebra Appl. , 87 (1987) 83-98.
[2] H. Dai, On the symmetric solutions of linear matrix equations, Linear Algebra Appl. , 131 (1990) 1-7.
[3] GH. Golub, W. Kahan, Calculating the singular values and pseudoinverse of a matrix, SIAM Journal on Numerical Analysis, (1965) 197-209.
[4] G. X. Huang, F. Yin, K. Guo., An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB = C, J. Comput. Appl. Math. , 212 (2008) 231-244.
[5] C.C.paige, Bidiagonalization of matrices and solution of linear equation, SIAM. J. Numer. Anal. , 11 (1974) 197-209.
[6] Z.-Y. Peng, The inverse problem of bisymmetric matrices, Numer. Linear Algebra Appl. , 11 (2004) 59-73.
[7] Z.-Y. Peng, The solutions of matrix AXC +BY D = E and its optimal approximation, Math. Theory Appl. , 22 (2) (2002) 99-103.
[8] Z.-Y. Peng, Solutions of symmetry-constrained least-squares problems, Numer. Linear Algebra Appl. , 15 (2008) 373-389.
[9] Z.-Y. Peng, New matrix iterative methods for constrained solutions of the matrix equation AXB = C, J. Comput. Appl. Math. 235 (2010) 726-735.
[10] Z.-Y. Peng, The nearest bisymmetric solutions of linear matrix equations, J. Comput. Math., 22 (6) (2004) 873-880.
[11] Z.-H. Peng, X.-Y. Hu, L. Zhang, The bisymmetric solutions of the matrix equation A1X1B1 + A2X2B2 + ... + AlXlBl = C and its optimal approximation, Linear Algebra Appl. , 426 (2007) 583-595.
[12] S.- Y. Shim, Y. chen, Least squares solution of matrix equation AXB∗+ CY D∗ = E, SIAM J. Matrix Anal.Appl., 3 (2003) 802-808.
[13] Q.W.Wang, H. Sun, S. Z.Li, Consistency for bi(skew) symmetric solutions to systems of generalized Sylvester equations over a finite central algebra, Linear Algebra Appl. 353 (2002) 169-182.