LEAST-CHANGE SECANT UPDATE METHODS FOR UNDERDETERMINED SYSTEMS*

Abstract. Least-change secant updates for nonsquare matrices have been addressed recently in [6]. Here the use of these updates in iterative procedures for the numerical solution of underde-retrained systems is considered. The 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 al-gorithm. 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. In conclusion, the results of some numerical experiments are given. Key words, underdetermined systems, least-change secant update methods, quasi-Newton methods, normal flow algorithm, augmented Jacobian matrix algorithm, continuation methods, ho-motopy methods, curve-tracking algorithms, parameter-dependent systems AMS(MOS) subject classification. 65H1