Iterative method for mirror-symmetric solution of matrix equation \(AXB+CYD=E\)
An iterative algorithm for the mirror-symmetric solution of the matrix equation \(AXB+CYD=E\) is proposed. It is an extension of the classical conjugate gradient method. Using this algorithm, the solvability of the matrix equation can be determined automatically. If the matrix equation is consistent, it is shown that for a special initial iteration the least norm mirror-symmetric solution can be computed by finite iterations in the absence of round-off errors. Numerical experiments illustrate the efficiency of the algorithm. The paper deals with a special problem of the linear algebra, however, the authors claim that it has important applications, e.g., in pattern recognition, antenna theory, quantum physic, etc.. The mirror-symmetric matrices are defined as the generalization of the centrosymmetric matrices. After proposing the variant of the conjugate gradient method, its properties in a numerous of technical lemmas and theorems are studied. This extends results known from the standard conjugate gradient method. The least norm solution is studied using the Kronecker product. The method is tested on one academic example in the Matlab environment. The precision of computed results is comparable with the computer epsilon.
scientific article; zbMATH DE number 5932929
scientific article