Solving a Quadratic Matrix Equation by Newton's Method with Exact Line Searches

Abstract

Abstract. We show how to incorporate exact line searches into Newton's method for solving the quadratic matrix equation AX 2 + 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.

A contour integral approach to the computation of invariant pairs

“…Several works address the problem of computing a numerical approximation for the solution of the quadratic matrix equation: an approach to compute, when possible, the dominant solvent is proposed in [12]. Newton's method and some variations are also used to approximate solvents numerically: see for example [11], [22], [21], [27]. The work in [19] uses interval arithmetic to compute an interval matrix containing the exact solution to the quadratic matrix equation.…”

“…We follow the ideas presented in the articles [36] and [21], which give expressions for backward errors and condition numbers for the polynomial eigenvalue problem and for a solvent of the quadratic matrix equation.…”

“…For instance, in [20] and [21] the authors find formulations for the condition number and the backward error. They also propose functional iteration approaches based on Bernoulli's method and Newton's method with line search to compute the solution numerically.…”

“…An analysis and a computable formulation for the condition number and backward error of the quadratic matrix equation Q(S) = 0 can be found in [20] and [21].…”

The Condition Numbers of a Quadratic Matrix Equation

“…Also, the mixed perturbation analysis was suggested by Skeel [6] and he obtained the mixed analysis for Gaussian elimination. Higham and Kim [3] considered the componentwise perturbation theory for the equation Q(X). Gohberg and Koltracht [2] obtained explicit expressions for both mixed and componentwise condition numbers.…”