Abstract
An iterative algorithm is proposed for solving the least-squares problem of a general matrix equation∑i=1tMiZiNi=F, whereZi(i=1,2,…,t) are to be determined centro-symmetric matrices with given central principal submatrices. For any initial iterative matrices, we show that the least-squares solution can be derived by this method within finite iteration steps in the absence of roundoff errors. Meanwhile, the unique optimal approximation solution pair for given matricesZ~ican also be obtained by the least-norm least-squares solution of matrix equation∑i=1tMiZ-iNi=F-, in whichZ-i=Zi-Z~i, F-=F-∑i=1tMiZ~iNi. The given numerical examples illustrate the efficiency of this algorithm.
Highlights
Throughout this paper, we denote the set of all m × n real matrices by Rm×n
For any initial iterative matrices, we show that the least-squares solution can be derived by this method within finite iteration steps in the absence of roundoff errors
The symbol AT represents the transpose of matrix A. and identity
Summary
Throughout this paper, we denote the set of all m × n real matrices by Rm×n. The symbol AT represents the transpose of matrix A. and identity. Li et al [29] proposed an elegant algorithm for solving the generalized Sylvester (Lyapunov) matrix equation AXB + CYD = E with bisymmetric X and symmetric Y, the two unknown matrices include the given central principal submatrix and leading principal submatrix, respectively. This method shunned the difficulties in numerical instability and computational complexity, and solved the problem, completely.
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