Publication

Subspace Methods For Computing The Pseudospectral Abscissa And The Stability Radius

Daniel Kressner
2014
Journal Articles

Abstract

The pseudospectral abscissa and the stability radius are well-established tools for quantifying the stability of a matrix under unstructured perturbations. Based on first-order eigenvalue expansions, Guglielmi and Overton [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166-1192] recently proposed a linearly converging iterative method for computing the pseudospectral abscissa. In this paper, we propose to combine this method and its variants with subspace acceleration. Each extraction step computes the pseudospectral abscissa of a small rectangular matrix pencil, which is comparably cheap and guarantees monotonicity. We observe local quadratic convergence and prove local superlinear convergence of the resulting subspace methods. Moreover, these methods extend naturally to computing the stability radius. A number of numerical experiments demonstrate the robustness and efficiency of the subspace methods.

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https://infoscience.epfl.ch/entities/publication/c73cc723-bf31-4af8-8098-c72e04643e4b

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Daniel Kressner
2014
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/c73cc723-bf31-4af8-8098-c72e04643e4b

About this result

Related concepts (23)

Iterative method

In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method.

Newton's method

In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function f defined for a real variable x, the function's derivative f′, and an initial guess x0 for a root of f. If the function satisfies sufficient assumptions and the initial guess is close, then is a better approximation of the root than x0.

Jacobi method

In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization. The method is named after Carl Gustav Jacob Jacobi.

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