Characterization and computation of singular points with maximum rank deficiency
SIAM Journal on Numerical Analysis
Bifurcations via singular value decompositions
Applied Mathematics and Computation
Fundamentals of matrix computations
Fundamentals of matrix computations
Matrix computations (3rd ed.)
Iterative solution of nonlinear equations in several variables
Iterative solution of nonlinear equations in several variables
Numerical methods for bifurcations of dynamical equilibria
Numerical methods for bifurcations of dynamical equilibria
SIAM Journal on Optimization
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
Solving rank-deficient separable nonlinear equations
Applied Numerical Mathematics
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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.