Implementation of different computational variations of biconjugate residual methods

2018

The inverse eigenvalue problem appears repeatedly in a variety of applications. The aim of this paper is to study a quadratic inverse eigenvalue problem of the form AXΛ2 + BXΛ + CX = 0 where A, B and C should be partially bisymmetric under a prescribed submatrix constraint. We derive an efficient matrix method based on the Hestenes‐Stiefel (HS) version of biconjugate residual (BCR) algorithm for solving this constrained quadratic inverse eigenvalue problem. The theoretical results demonstrate that the matrix method solves the constrained quadratic inverse eigenvalue problem within a finite number of iterations in the absence of round‐off errors. Finally we validate the accuracy and efficiency of the matrix method through the numerical results.

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

BCR Algorithm for Solving Quadratic Inverse Eigenvalue Problems for Partially Bisymmetric Matrices

2018

“…To describe our algorithm in this section, we first recall the BCR algorithm. In , Vespucci and Broyden described some different variants of BCR algorithm without any convergence analysis for solving nonsymmetric linear system of equations A x = b . One of the variants of BCR algorithm is as follows:…”

Section: Resultsmentioning

confidence: 99%

Matrix form of Biconjugate Residual Algorithm to Solve the Discrete‐Time Periodic Sylvester Matrix Equations

2017

There are important relationships between the discrete-time linear periodic descriptor systems and the discrete-time periodic matrix equations. In the present paper, we introduce the matrix form of the biconjugate residual (BCR) algorithm for solving the discrete-time periodic Sylvester matrix equations A i X i B i + C i X i+1 D i = E i , i = 1, 2, .... It is shown that the introduced algorithm converges to the solutions within a finite number of iterations in the absence of round-off errors. Finally, three numerical examples are given to demonstrate the efficiency and the performance of the algorithm.

“…In this section, we first present our algorithm in detail and then we describe the convergence analysis of the algorithm. In , Vespucci and Broyden described some different variants of BCR algorithm without any convergence analysis for solving nonsymmetric linear system of equations A x = b . Numerical results have shown that the variants of BCR algorithm described in are efficient to solve A x = b .…”

Section: Resultsmentioning

confidence: 99%

Convergence Results of the Biconjugate Residual Algorithm for Solving Generalized Sylvester Matrix Equation

2016

In this article, we investigate a variant of the biconjugate residual (BCR) algorithm to solve the generalized Sylvester matrix equation ∑i=1kAiXBi+∑j=1lCjYDj=E, which includes the well‐known Lyapunov, Stein and Sylvester matrix equations. We show that the BCR algorithm with any (special) initial matrix pair can smoothly compute the (least Frobenius norm) solution pair of the generalized Sylvester matrix equation within a finite number of iterations in the absence of round‐off errors. Finally the accuracy and effectiveness of the BCR algorithm in comparison to some existing algorithms are demonstrated by two numerical examples.