Newton's method for singular nonlinear equations using approximate left and right nullspaces of the Jacobian

Authors:
Yun-Qiu Shen;Tjalling J. Ypma
Affiliations:
Department of Mathematics, Western Washington University, Bellingham, WA 98225-9063, USA;Department of Mathematics, Western Washington University, Bellingham, WA 98225-9063, USA
Venue:
Applied Numerical Mathematics - 6th IMACS International symposium on iterative methods in scientific computing
Year:
2005

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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 a simple regularity condition is satisfied at x^*, it is possible to define a bordered system for which Newton's method converges quadratically [Griewank, SIAM Rev. 27 (1985) 537]. In this paper we extend this technique to the case of higher rank deficiencies. We show that if a generalized regular singularity condition is satisfied then one 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. The theory and method are illustrated by several examples.