Abstract
We propose an algorithm for implementing Newton’s method for a general nonlinear system $f(x) = 0$ where the linear systems that arise at each step of Newton’s method are solved by a preconditioned Krylov subspace iterative method. The algorithm requires only function evaluations and does not require the evaluation or storage of the Jacobian matrix. Matrix-vector products involving the Jacobian matrix are approximated by directional differences. We develop a framework for constructing preconditionings for this inner iterative method which do not reference the Jacobian matrix explicitly. We derive a nonlinear SSOR type preconditioning which numerical experiments show to be as effective as the linear SSOR preconditioning that uses the Jacobian explicitly.
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More From: SIAM Journal on Scientific and Statistical Computing
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