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

Abstract

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 Mor6 and Trangenstein is extended to show that a modified Schubert's method applied to linear equations is globally and q-superlinearly convergent.

A Hybrid Algorithm for Solving Sparse Nonlinear Systems of Equations

“…Let sk = xk+l -xk. The proof of this lemma is similar to that for Schubert's algorithm given by Reid [10] and Marwil [8].…”

Moving boundary transformation for American call options with transaction cost: finite difference methods and computing

“…Schubert's method [28] is widely used for sparse matrices because it preserves the sparsity of the Jacobian. Although it has good properties, it is sensitive to the problem under consideration and size of the matrix.…”

The column-updating method for solving nonlinear equations in Hilbert space

“…We wrote FORTRAN codes which implement the Column-Updating Method (CUM), as defined by Algorithm 4.1, Broyden's first method [2] using the idea of Matthies-Strang [20] and Schubert's method (see [3,19,24]). All the tests were run in a VAX11/785 at the State University of Campinas, using single précision, the FORTRAN 77 compiler and the VMS Operational System.…”

“…We observe that in Schubert's method, and other finite-dimensional methods, bounded détérioration principles are formulated in terms of the Frobenius norm (see [18,19]), whose natural generalization to Hilbert spaces is the norm of Hilbert-Schmidt. Therefore, we do not know if a)-b)-c) hold for the sparse Broyden (Schubert) method, which is also a very popular algorithm for nonlinear équations.…”

“…This condition is not necessary for many methods for solving nonlinear simultaneous équations. (See [2,3,4,6,7,9,18,19]). …”