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.

Abstract. This paper presents a new context for using the sparse Broyden update method to solve systems of nonlinear equations. The setting for this work is that a Newton-like algorithm is assumed to be available which incorporates a workable strategy for improving poor initial guesses and providing a satisfactory Jacobian matrix approximation whenever required. The total cost of obtaining each Jacobian matrix, or the cost of factoring it to solve for the Newton step, is assumed to be sufficiently high to make it attractive to keep the same Jacobian approximation for several steps. This paper suggests the extremely convenient and apparently effective technique of applying the sparse Broyden update directly to the matrix factors in the iterations between réévaluations in the hope that fewer fresh factorizations will be required. The strategy is shown to be locally and ¡¡r-superlinearly convergent, and some encouraging numerical results are presented.

Abstract. This paper presents a new context for using the sparse Broyden update method to solve systems of nonlinear equations. The setting for this work is that a Newton-like algorithm is assumed to be available which incorporates a workable strategy for improving poor initial guesses and providing a satisfactory Jacobian matrix approximation whenever required. The total cost of obtaining each Jacobian matrix, or the cost of factoring it to solve for the Newton step, is assumed to be sufficiently high to make it attractive to keep the same Jacobian approximation for several steps. This paper suggests the extremely convenient and apparently effective technique of applying the sparse Broyden update directly to the matrix factors in the iterations between réévaluations in the hope that fewer fresh factorizations will be required. The strategy is shown to be locally and ¡¡r-superlinearly convergent, and some encouraging numerical results are presented.

A technique for dimensional reduction is combined with the ellipsoid algorithm and another acceleration technique lo solve linear programming problems. The new method yields exact rather than approximate solutions and provides a natural procedure for determining infeasibility.

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