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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Direct secant updates of matrix factorizations
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by J. E. Dennis and Earl S. Marwil PDF
Math. Comp. 38 (1982), 459-474 Request permission

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 reevaluations in the hope that fewer fresh factorizations will be required. The strategy is shown to be locally and q-superlinearly convergent, and some encouraging numerical results are presented.
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Math. Comp. 38 (1982), 459-474
  • MSC: Primary 65H10
  • DOI: https://doi.org/10.1090/S0025-5718-1982-0645663-8
  • MathSciNet review: 645663