Least Change Secant Update Methods for Undetermined Systems

Abstract

Least-change secant updates for nonsquare matrices have been addressed recently in [6]. Here we consider the use of these updates in iterative procedures for the numerical solution of underdetermined systems. Our model method is the normal flow algorithm used in homotopy or continuation methods for determining points on an implicitly defined curve. A Kantorovich-type local convergence analysis is given which supports the use of least-change secant updates in this algorithm. This analysis also provides a Kantorovich-type local convergence analysis for least-change secant update methods in the usual case of an equal number of equations and unknowns. This in turn gives a local convergence analysis for augmented Jacobian algorithms which use least-change secant updates. We conclude with the results of some numerical experiments. Key words. underdetermined systems, least-change secant update methods, quasi-Newton methods, normal flow algorithm, augmented Jacobian matrix algorithm, continuation methods, homotopy methods, curve-tracking algorithms, parameter-dependent systems