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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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On the sparse and symmetric least-change secant update
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by Trond Steihaug PDF
Math. Comp. 42 (1984), 521-533 Request permission

Abstract:

To find the sparse and symmetric n by n least-change secant update we have to solve a consistent linear system of n equations in n unknowns, where the coefficient matrix is symmetric and positive semidefinite. We give bounds on the eigenvalues of the coefficient matrix and show that the preconditioned conjugate gradient method is a very efficient method for solving the linear equation. By solving the linear system only approximately, we generate a family of sparse and symmetric updates with a residual in the secant equation. We address the question of how accurate a solution is needed not to impede the convergence of quasi-Newton methods using the approximate least-change update. We show that the quasi-Newton methods are locally and superlinearly convergent after one or more preconditioned conjugate gradient iterations.
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Math. Comp. 42 (1984), 521-533
  • MSC: Primary 65H05; Secondary 65F50
  • DOI: https://doi.org/10.1090/S0025-5718-1984-0736450-2
  • MathSciNet review: 736450