Solving a Quadratic Matrix Equation by Newton's Method with Exact Line Searches
Nicholas J. Higham,Hyun-Min Kim +1 more
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This work shows how to incorporate exact line searches into Newton's method for solving the quadratic matrix equation AX2 + BX + C = 0, where A, B and C are square matrices.Abstract:
We show how to incorporate exact line searches into Newton's method for solving the quadratic matrix equation AX2 + BX + C = 0, where A, B and C are square matrices. The line searches are relatively inexpensive and improve the global convergence properties of Newton's method in theory and in practice. We also derive a condition number for the problem and show how to compute the backward error of an approximate solution.read more
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Journal ArticleDOI
The Quadratic Eigenvalue Problem
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Structured Pseudospectra for Polynomial Eigenvalue Problems, with Applications
TL;DR: The usual definitions of pseudospectra are extended in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory.
Journal ArticleDOI
Numerical analysis of a quadratic matrix equation
Nicholas J. Higham,Hyun-Min Kim +1 more
TL;DR: In this article, the quadratic matrix equation AX2+BX + C = 0 in n x n matrices arises in applications and is of intrinsic interest as one of the simplest nonlinear matrix equations.
Universidad central de venezuela
TL;DR: This paper proposes a specialized secant method for the special problem of computing the inverse or the pseudoinverse of a given matrix, for which stability and q-superlinear convergence are established, and for which some numerical results are presented.
References
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Numerical Optimization
Jorge Nocedal,Stephen J. Wright +1 more
TL;DR: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization, responding to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems.
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TL;DR: The aim of this book is to provide a Discussion of Constrained Optimization and its Applications to Linear Programming and Other Optimization Problems.
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TL;DR: The field of values as discussed by the authors is a generalization of the field of value of matrices and functions, and it includes singular value inequalities, matrix equations and Kronecker products, and Hadamard products.