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Mathematics of Computation

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On the sparse and symmetric least-change secant update


Author: Trond Steihaug
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
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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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DOI: https://doi.org/10.1090/S0025-5718-1984-0736450-2
Article copyright: © Copyright 1984 American Mathematical Society