Modifying singular values: existence of solutions to sytems of nonlinear equations having a possibly singular Jacobian matrix

Abstract

We show that if a certain nondegeneracy assumption holds, it is possible to guarantee the existence of a solution to a system of nonlinear equations f ( x ) = 0 f(x) = 0 whose Jacobian matrix J ( x ) J(x) exists but may be singular. The main idea is to modify small singular values of J ( x ) J(x) in such a way that the modified Jacobian matrix J ^ ( x ) \hat J(x) has a continuous pseudoinverse J ^ + ( x ) {\hat J^ + }(x) and that a solution x ∗ {x^\ast } of f ( x ) = 0 f(x) = 0 may be found by determining an asymptote of the solution to the initial value problem x ( 0 ) = x 0 , x ′ ( t ) = − J ^ + ( x ) f ( x ) x(0) = {x_0},x\prime (t) = - {\hat J^ + }(x)f(x) . We briefly discuss practical (algorithmic) implications of this result. Although the nondegeneracy assumption may fail for many systems of interest (indeed, if the assumption holds and J ( x ∗ ) J({x^\ast }) is nonsingular, then x ∗ {x^\ast } is unique), algorithms using J ^ + ( x ) {\hat J^ + }(x) may enjoy a larger region of convergence than those that require (an approximation to) J − 1 ( x ) {J^{ - 1}}(x) .