On Modifying Singular Values to Solve Possible Singular Systems of Non-Linear Equations

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 maybe singular. The main idea is to modify small singular values of J(x) in such away that the modified Jacobian matrix J^(x) has a continuous pseudoinverse J^+(x)and that a solution x* of f(x) = 0 may be found by determining an asymptote of the solution to the initial value problem x(0) = x[sub0}, x’(t) = -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*) is non-singular, then x is unique), algorithms using(x) may enjoy a larger region of convergence than those that require(an approximation to) J[to the -1 power[(x).