An Iterative Method for Bisymmetric Solutions and Optimal Approximation Solution of AXB=C

2007 ◽  
Author(s):  
Ya-Xin Peng ◽  
Xi-Yan Hu ◽  
Lei Zhang
Keyword(s):  
Iterative Method ◽  
Optimal Approximation ◽  
Approximation Solution ◽  
Optimal Approximation Solution

scholarly journals A Study of Inverse Problems Based on Two Kinds of Special Matrix Equations in Euclidean Space

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Rui Huang ◽  
Xiaodong Wu ◽  
Ruihe Wang ◽  
Hui Li
Keyword(s):  
Inverse Problems ◽  
Euclidean Space ◽  
Optimal Approximation ◽  
Matrix Equations ◽  
Approximation Solution ◽  
Special Matrix ◽  
Optimal Approximation Solution ◽  
Special Classes

Two special classes of symmetric coefficient matrices were defined based on characteristics matrix; meanwhile, the expressions of the solution to inverse problems are given and the conditions for the solvability of these problems are studied relying on researching. Finally, the optimal approximation solution of these problems is provided.

scholarly journals A class of iteration methods for the matrix equation A×B = C

2007 ◽  
Vol 75 (2) ◽  
pp. 289-298 ◽  
Author(s):  
Konghua Guo ◽  
Xiyan Hu ◽  
Lei Zhang
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Iteration Method ◽  
Optimal Approximation ◽  
Approximation Solution ◽  
Numerical Examples ◽  
The Matrix ◽  
Iteration Methods ◽  
Optimal Approximation Solution

An iteration method for the matrix equation A×B = C is constructed. By this iteration method, the least-norm solution for the matrix equation can be obtained when the matrix equation is consistent and the least-norm least-squares solutions can be obtained when the matrix equation is not consistent. The related optimal approximation solution is obtained by this iteration method. A preconditioned method for improving the iteration rate is put forward. Finally, some numerical examples are given.

scholarly journals A class of constrained inverse eigenvalue problem and associated approximation problem for symmetrizable matrices

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1903-1909
Author(s):  
Xiangyang Peng ◽  
Wei Liu ◽  
Jinrong Shen
Keyword(s):  
Eigenvalue Problem ◽  
Symmetric Matrix ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Optimal Approximation ◽  
Original Matrix ◽  
Approximation Solution ◽  
Solvability Conditions ◽  
Inverse Eigenvalue ◽  
Symmetrizable Matrices

The real symmetric matrix is widely applied in various fields, transforming non-symmetric matrix to symmetric matrix becomes very important for solving the problems associated with the original matrix. In this paper, we consider the constrained inverse eigenvalue problem for symmetrizable matrices, and obtain the solvability conditions and the general expression of the solutions. Moreover, we consider the corresponding optimal approximation problem, obtain the explicit expressions of the optimal approximation solution and the minimum norm solution, and give the algorithm and corresponding computational example.

scholarly journals Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix

2020 ◽  
Vol 18 (1) ◽  
pp. 603-615
Author(s):  
Fan-Liang Li
Keyword(s):  
Eigenvalue Problem ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Optimal Approximation ◽  
Frobenius Norm ◽  
Approximation Solution ◽  
Inverse Eigenvalue ◽  
Left And Right ◽  
Centrosymmetric Matrix ◽  
Value Decomposition

Abstract Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix and its optimal approximation problem. Combining the special properties of left and right eigenpairs and the generalized singular value decomposition, we derive the solvability conditions of the problem and its general solutions. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. We present an algorithm and numerical experiment to give the optimal approximation solution. Our results extend and unify many results for left and right inverse eigenpairs problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint.

scholarly journals LSMR Iterative Method for General Coupled Matrix Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
F. Toutounian ◽  
D. Khojasteh Salkuyeh ◽  
M. Mojarrab
Keyword(s):  
Iterative Method ◽  
Symmetric Matrix ◽  
Matrix Group ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Approximation Solution ◽  
Matrix Groups ◽  
Special Cases ◽  
Least Frobenius Norm Solution ◽  
General Coupled Matrix Equations

By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations∑k=1qAikXkBik=Ci,i=1,2,…,p, (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups(X1,X2,…,Xq), such as symmetric, generalized bisymmetric, and(R,S)-symmetric matrix groups. By this iterative method, for any initial matrix group(X1(0),X2(0),…,Xq(0)), a solution group(X1*,X2*,…,Xq*)can be obtained within finite iteration steps in absence of round-off errors, and the minimum Frobenius norm solution or the minimum Frobenius norm least-squares solution group can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation solution group to a given matrix group(X¯1,X¯2,…,X¯q)in the Frobenius norm can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the presented method.

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