Solving Nonlinear Equations with Newton's Method
SIAM, 2003 - 104 strani
This brief book on Newton's method is a user-oriented guide to algorithms and implementation. In just over 100 pages, it shows, via algorithms in pseudocode, in MATLAB, and with several examples, how one can choose an appropriate Newton-type method for a given problem, diagnose problems, and write an efficient solver or apply one written by others. Solving Nonlinear Equations with Newton's Method contains trouble-shooting guides to the major algorithms, their most common failure modes, and the likely causes of failure. It also includes many worked-out examples (available on the SIAM website) in pseudocode and a collection of MATLAB codes, allowing readers to experiment with the algorithms easily and implement them in other languages.
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Algorithm Automatic differentiation BiCGSTAB Broyden's method brsola.m calls to F Chapter choice chord method columns convergence theory cost default difference increment difference Jacobian differential equation direct method discretization Evaluate F(x example factorization Figure forcing term forward difference forward difference approximation function evaluations Gaussian elimination GMRES GMRES(m ierr Imeth implementation inexact Newton methods initial iterate initial value problem iteration history iterative methods Jacobian-vector product Krylov methods Krylov subspace line search line search fails linear iteration linear solver Lipschitz continuous MATLAB code matrix maxit mesh Newton direction Newton step Newton-Armijo Newton-Krylov methods Newton's method newtsol.m nonlinear equation nonlinear iteration nonlinear residual nonlinear solver norm nsold.m nsolg nsoli parameter parms plot preconditioner q-linear q-order Right preconditioning secant method solution solve the equation sparse sparse matrix stagnation standard assumptions hold step length storage superlinearly termination criterion TFQMR Theorem updated vector