Convergence Results for Schubert’s Method for Solving Sparse Nonlinear Equations

Schubert’s method for solving sparse nonlinear equations is an extension of Broyden’s method. The zero-nonzero structure defined by the sparse Jacobian is preserved by updating the approximate Jacobian row by row. An estimate is presented which permits the extension of the convergence results for Broyden’s method to Schubert’s method. The analysis for local and q-superlinear convergence given here includes, as a special case, results in a recent paper by B. Lam; this generalization seems theoretically and computationally more satisfying. A Kantorovich analysis paralleling one for Broyden’s method is given. This leads to a convergence result for linear equations that includes another result by Lam. A result by Moré and Trangenstein is extended to show that a modified Schubert’s method applied to linear equations is globally and q-superlinearly convergent.