Numerical Methods for Unconstrained Optimization and Nonlinear Equations

ISBN-10: 0898713641

ISBN-13: 9780898713640

Edition: N/A

List price: $48.00
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Book details

List price: $48.00
Publisher: Society for Industrial and Applied Mathematics
Binding: Paperback
Pages: 394
Size: 6.25" wide x 9.25" long x 0.75" tall
Weight: 1.188
Language: English

Preface
Introduction. Problems to be considered
Characteristics of 'real-world' problems
Finite-precision arithmetic and measurement of error
Exercises
Nonlinear Problems in One Variable
What is not possible
Newton's method for solving one equation in one unknown
Convergence of sequences of real numbers
Convergence of Newton's method
Globally convergent methods for solving one equation in one uknown
Methods when derivatives are unavailable
Minimization of a function of one variable
Exercises
Numerical Linear Algebra Background
Vector and matrix norms and orthogonality
Solving systems of linear equations'matrix factorizations
Errors in solving linear systems
Updating matrix factorizations
Eigenvalues and positive definiteness
Linear least squares
Exercises
Multivariable Calculus Background
Derivatives and multivariable models
Multivariable finite-difference derivatives
Necessary and sufficient conditions for unconstrained minimization
Exercises
Newton's Method for Nonlinear Equations and Unconstrained Minimization
Newton's method for systems of nonlinear equations
Local convergence of Newton's method
The Kantorovich and contractive mapping theorems
Finite-difference derivative methods for systems of nonlinear equations
Newton's method for unconstrained minimization
Finite difference derivative methods for unconstrained minimization
Exercises
Globally Convergent Modifications of Newton's Method
The quasi-Newton framework
Descent directions
Line searches
The model-trust region approach
Global methods for systems of nonlinear equations
Exercises
Stopping, Scaling, and Testing
Scaling
Stopping criteria
Testing
Exercises
Secant Methods for Systems of Nonlinear Equations
Broyden's method
Local convergence analysis of Broyden's method
Implementation of quasi-Newton algorithms using Broyden's update
Other secant updates for nonlinear equations
Exercises
Secant Methods for Unconstrained Minimization
The symmetric secant update of Powell
Symmetric positive definite secant updates
Local convergence of positive definite secant methods
Implementation of quasi-Newton algorithms using the positive definite secant update
Another convergence result for the positive definite secant method
Other secant updates for unconstrained minimization
Exercises
Nonlinear Least Squares
The nonlinear least-squares problem
Gauss-Newton-type methods
Full Newton-type methods
Other considerations in solving nonlinear least-squares problems
Exercises
Methods for Problems with Special Structure
The sparse finite-difference Newton method
Sparse secant methods
Deriving least-change secant updates
Analyzing least-change secant methods
Exercises
A Modular System of Algorithms for Unconstrained Minimization and Nonlinear Equations
Test Problems
References
Author Index
Subject Index
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