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 $f(x) = 0$ whose Jacobian matrix $J(x)$ exists but may be singular. The main idea is to modify small singular values of $J(x)$ in such a way that the modified Jacobian matrix $\hat J(x)$ has a continuous pseudoinverse ${\hat J^ + }(x)$ and that a solution ${x^\ast }$ of $f(x) = 0$ may be found by determining an asymptote of the solution to the initial value problem $x(0) = {x_0},x\prime (t) = - {\hat J^ + }(x)f(x)$. 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 $J({x^\ast })$ is nonsingular, then ${x^\ast }$ is unique), algorithms using ${\hat J^ + }(x)$ may enjoy a larger region of convergence than those that require (an approximation to) ${J^{ - 1}}(x)$.