Nonlinear equations arise in essentially every branch of modern science, engineering, and mathematics. However, in only a very few special cases is it possible to obtain useful solutions to nonlinear equations via analytical calculations. As a result, many scientists resort to computational methods.

This book contains the proceedings of the Joint AMS-SIAM Summer Seminar, "Computational Solution of Nonlinear Systems of Equations," held in July 1988 at Colorado State University. The aim of the book is to give a wide-ranging survey of essentially all of the methods which comprise currently active areas of research in the computational solution of systems of nonlinear equations. A number of "entry-level" survey papers were solicited, and a series of test problems has been collected in an appendix. Most of the articles are accessible to students who have had a course in numerical analysis.

Readership

Table of Contents

• E. L. Allgower and K. Georg -- Numerically stable homotopy methods without an extra dimension
• G. Auchmuty -- Duality algorithms for smooth unconstrained optimization
• M. S. Berger -- Antidotes for nonintegrability of nonlinear systems: Quasi-periodic motions
• M. S. Berger -- Bifurcation into folds of infinite dimension
• S. L. Blish and J. H. Curry -- On the geometry of factorization algorithms
• K. Böhmer -- Defect corrections and mesh independence principle for operator equations and their discretizations
• K. Böhmer and M. Zhen -- On a numerical Lyapunov-Schmidt method
• A. Castelo, S. De Freitas, and G. Tavares -- PL approximation of manifolds and its application to implicit ODEs
• T. F. Coleman -- On characterizations of superlinear convergence for constrained optimization
• J. Duvallet -- Computation of solutions of two-point boundary value problems by a simplicial homotopy algorithm
• R. E. Ewing -- Computational methods for nonlinear systems of partial differential equations arising in contaminant transport in porous media
• R. Fletcher -- Low storage methods for unconstrained optimization
• A. Galantai -- Block ABS methods for nonlinear systems of algebraic equations
• S. Gélinas and R. Vaillancourt -- Application of Julia-Fatou iteration theory in dielectric spectroscopy
• K. Georg -- An introduction to PL algorithms
• K. Georg and D. Zachmann -- Nonlinear convection diffusion equations and Newton-like methods
• R. B. Guenther and J. W. Lee -- Convergence of the Newton-Raphson method for boundary value problems of ordinary differential equations
• P. T. Harker and J.-S. Pang -- A damped-Newton method for the linear complementarity problem
• A. Hoy and H. Schwetlick -- ome superlinearly convergent methods for solving singular nonlinear equations
• J. K. Hunter -- Numerical solutions of some nonlinear dispersive wave equations
• H. Th. Jongen -- Parametric optimization: Critical points and local minima
• R. B. Kearfott -- Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons
• C. T. Kelley -- Operator prolongation methods for nonlinear equations
• B. N. Lundberg, A. B. Poore, and B. Yang -- Smooth penalty functions and continuation methods for constrained optimization
• S. McKay and J. W. Thomas -- Application of the fast adaptive composite grid method to nonlinear partial differential equations
• R. Mejia -- Interactive program for continuation of solutions of large systems of nonlinear equations
• H. D. Mittelmann -- Nonlinear parametrized equations: New results for variational problems and inequalities
• A. P. Morgan and A. J. Sommese -- Generically nonsingular polynomial continuation
• A. P. Morgan, A. J. Sommese, and C. W. Wampler -- Polynomial continuation for mechanism design problems
• G. Russo -- A Lagrangian method for collisional kinetic equations
• R. Schaaf and K. Schmitt -- On the number of solutions of semilinear elliptic problems at resonance: Some numerical experiments
• J. Scheurle -- Splitting of separatrices and chaos
• P. H. Schmidt -- PL methods for constructing a numerical implicit function
• H. Schwetlick -- Nonstandard scaling matrices in trust region methods
• M. W. Smiley -- Numerical determination of breathers and forced oscillations of nonlinear wave equations
• M. Sun -- Numerical solutions of singular stochastic control problems in bounded intervals
• W. C. Thacker -- Large least-squares problems and the need for automating the generation of adjoint codes
• H. F. Walker -- Newton-like methods for underdetermined systems
• Y. Yomdin -- Sard's theorem and its improved versions in numerical analysis
• T. J. Ypma -- Finite difference approximation of sparse Jacobian matrices in Newton-like methods
• J. J. Moré -- A collection of nonlinear model problems