An efficient algorithm based on Lanczos type of BCR to solve constrained quadratic inverse eigenvalue problems

Transactions of the Institute of Measurement and Control

2019

Self Cite

The study of linear matrix equations is extremely important in many scientific fields such as control systems and stability analysis. In this work, we aim to design the Hestenes-Stiefel (HS) version of biconjugate residual (Bi-CR) algorithm for computing the (least Frobenius norm) partially doubly symmetric solution [Formula: see text] of the general Sylvester matrix equations [Formula: see text] for [Formula: see text]. We show that the proposed algorithm converges in a finite number of iterations. Finally, numerical results compare the proposed algorithm to alternative algorithms.

Section: Preliminarymentioning

confidence: 99%

Section: Algorithm and Its Convergence Analysismentioning

confidence: 99%

Partially doubly symmetric solutions of general Sylvester matrix equations

Transactions of the Institute of Measurement and Control

2019

Self Cite

“…The inverse eigenvalue problem can appear in various forms depending on the applications 1‐7 . The most common type is the parameterized inverse eigenvalue problem 1,8 .…”

Section: Introductionmentioning

confidence: 99%

An extension of the Cayley transform method for a parameterized generalized inverse eigenvalue problem

Dalvand

Numerical Linear Algebra App

Román

2020

Self Cite

Summary Since recent studies have shown that the Cayley transform method can be an effective iterative method for solving the inverse eigenvalue problem, in this work, we consider using an extension of it for solving a type of parameterized generalized inverse eigenvalue problem and prove its locally quadratic convergence. This type of inverse eigenvalue problem, which includes multiplicative and additive inverse eigenvalue problems, appears in many applications. Also, we consider the case where the given eigenvalues are multiple. In this case, we describe a modified problem that is not overdetermined and discuss the extension of the Cayley transform method for this modified problem. Finally, to demonstrate the effectiveness of these algorithms, we present some numerical examples to show that the proposed methods are practical and efficient.

“…There are various forms of the inverse eigenvalue problem that appear depending on the applications; for several forms, one may refer to other studies 1‐9 . Generally, in all the inverse eigenvalue problems, the given information contains either all or part of the eigenvalues or eigenvectors, and the unknown parameters are the elements of a matrix or a matrix pencil 1 .…”

Section: Introductionmentioning

confidence: 99%

“…There are various forms of the inverse eigenvalue problem that appear depending on the applications; for several forms, one may refer to other studies. [1][2][3][4][5][6][7][8][9] Generally, in all the inverse eigenvalue problems, the given information contains either all or part of the eigenvalues or eigenvectors, and the unknown parameters are the elements of a matrix or a matrix pencil. 1 In this paper, we study a kind of inverse eigenvalue problem that appears in many practical applications in areas such as structural engineering, mechanics, and physics.…”

Section: Introductionmentioning

confidence: 99%

Newton‐like and inexact Newton‐like methods for a parameterized generalized inverse eigenvalue problem

Dalvand

Math Methods in App Sciences

2020

Self Cite

In this paper, we establish the Newton‐like and inexact Newton‐like based methods for solving a type of parameterized generalized inverse eigenvalue problem. This type of parameterized generalized inverse eigenvalue problem, including multiplicative and additive inverse eigenvalue problems, appears in many applications. We show that the direction produced by the Newton‐like method does not depend explicitly on the eigenvalues. Also, the inexact version can minimize the oversolving problem of Newton‐like methods and hence improve efficiency. We discuss the convergence properties of the presented methods. Finally, the performance and effectiveness of the algorithms are tested on three numerical examples and compared to the Newton algorithm.