TABLE OF CONTENTS Nonlinear equations are solved as part of almost all simulations of physical processes. Physical models that are expressed as nonlinear partial differential equations, for example, become large systems of nonlinear equations when discretized. Authors of simulation codes must either use a nonlinear solver as a tool or write one from scratch. The purpose of this book is to show these authors what technology is available, sketch the implementation, and warn of the problems. We do this via algorithmic outlines, examples in MATLAB, nonlinear solvers in MATLAB that can be used for production work, and chapter-ending projects.
We use the standard notation
for systems of N equations in N unknowns. Here F: R N ? R N. We will call F the nonlinear residual or simply the residual. Rarely can the solution of a nonlinear equation be given by a closed-form expression, so iterative methods must be used to approximate the solution numerically. The output of an iterative method is a sequence of approximations to a solution.
In this book, following the convention in [42, 43], vectors are to be understood as column vectors. The vector x* will denote a solution, x a potential solution, and { x n} n ? 0 the sequence of iterates. We will refer to x 0 as the initial iterate (not guess!). We will denote the ith component of a vector x
