Using the Singular Value Decomposition to Solve Nonlinear Equations with Rank Deficient Jacobians

Yun-Qiu Shen

and

Tjalling J. Ypma

Department of Mathematics, Western Washington University, Bellingham, Washington 98225, USA

Abstract

The convergence of Newton's method to a solution x* of f(x) = 0 may be unsatisfactory if the Jacobian matrix f '(x*) is singular. When the rank deficiency is one, and x* satisfies a simple regular singularity condition (Griewank [1]), it is possible to define a bordered system for which Newton's method converges quadratically. In this paper we extend this technique to the case of higher rank deficiencies. We show that if a corresponding regular singularity condition is satisfied then a singular value decomposition of f '(x^) for some point x^ near x* can be used to form a bordered system for which Newton's method converges quadratically. This method is illustrated by several numerical examples.

[1] A. Griewank, On solving nonlinear equations with simple singularities or nearly singular solutions, SIAM Review, 27(4)(1985), 537-563.