This paper develops a unified theory for establishing the local and q-superlinear convergence of quasi-Newton methods from the convex class when part of the Hessian matrix is known. One first proves the bounded deterioration principle due to Dennis (and ...
It is shown that the effect of cancellation errors in a quasi-Newton method can be predicted with reasonable accuracy on the basis of simple formulae derived by using probabilistic arguments. Errors induced by cancellation are shown to have the ...
Based on eigenvalue analyses, well-structured upper bounds for the condition number of the scaled memoryless quasi-Newton updating formulae Broyden---Fletcher---Goldfarb---Shanno and Davidon---Fletcher---Powell are obtained. Then, it is shown that the ...
Florin Popentiu
Gurwitz describes a heuristic test for estimating the effect of the cancellation error on the approximating matrix in quasi-Newton methods. After a review of the theoretical results (which are largely due to R. Fletcher), the author suggests a test for stopping the update of the approximating matrix. This test can replace Fletcher's more complicated test based on the computation of the upper bound for the effect of cancellation errors and the scalar measure of the size of the correction. Tables I and II contain numerical trials on the Rosenbrock, Chebyquad, and trigonometric functions, which prove the correctness of the test. The proposed test can easily be encoded and saves computer resources. The text is concise, the references are good, and the terminology is suitable.
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