Browsing by Author "Walker, Homer F."

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    Inaccuracy in Quasi-Newton Methods: Local Improvement Theorems
    (1983-03) Dennis, J.E. Jr.; Walker, Homer F.
    In this paper, we consider the use of bounded-deterioration quasi-Newton methods implemented in floating-point arithmetic to find solutions to F(x)=0 where only inaccurate F-values are available. Our analysis is for the case where the relative error in F is less than one. We obtain theorems specifying local rates of improvement and limiting accuracies depending on the nearness to Newton's method of the basic algorithm, the accuracy of its implementation, the relative errors in the function values, the accuracy of the solutions of the linear systems for the Newton steps, and the unit-rounding errors in the addition of the Newton steps.
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    Least-Change Secant Update Methods with Inaccurate Secant Conditions
    (1983-11) Dennis, J.E. Jr.; Walker, Homer F.
    In this paper, we investigate the role of the secant or quasi-Newton condition in the sparse Broyden or Schubert update method for solving systems of nonlinear equations whose Jacobians are either sparse, or can be approximated acceptably by conveniently sparse matrices. We develop a general theory on perturbations to the secant equation that will still allow a proof of local q-linear convergence. To illustrate the theory, we show how to generalize the standard secant condition to the case when the function difference is contaminated by noise.