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Numerical Methods for Unconstrained Optimization and Nonlinear Equations

Numerical Methods for Unconstrained Optimization and Nonlinear Equations


Part of Classics in Applied Mathematics

  • Date Published: December 1996
  • availability: Available in limited markets only
  • format: Paperback
  • isbn: 9780898713640

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About the Authors
  • This book has become the standard for a complete, state-of-the-art description of the methods for unconstrained optimization and systems of nonlinear equations. Originally published in 1983, it provides information needed to understand both the theory and the practice of these methods and provides pseudocode for the problems. The algorithms covered are all based on Newton's method or 'quasi-Newton' methods, and the heart of the book is the material on computational methods for multidimensional unconstrained optimization and nonlinear equation problems. The republication of this book by SIAM is driven by a continuing demand for specific and sound advice on how to solve real problems.

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    Product details

    • Date Published: December 1996
    • format: Paperback
    • isbn: 9780898713640
    • length: 394 pages
    • dimensions: 233 x 150 x 23 mm
    • weight: 0.522kg
    • availability: Available in limited markets only
  • Table of Contents

    1. Introduction. Problems to be considered
    Characteristics of 'real-world' problems
    Finite-precision arithmetic and measurement of error
    2. 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
    3. 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
    4. Multivariable Calculus Background
    Derivatives and multivariable models
    Multivariable finite-difference derivatives
    Necessary and sufficient conditions for unconstrained minimization
    5. 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
    6. 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
    7. Stopping, Scaling, and Testing. Scaling
    Stopping criteria
    8. 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
    9. 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
    10. Nonlinear Least Squares. The nonlinear least-squares problem
    Gauss-Newton-type methods
    Full Newton-type methods
    Other considerations in solving nonlinear least-squares problems
    11. 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
    Appendix A. A Modular System of Algorithms for Unconstrained Minimization and Nonlinear Equations (by Robert Schnabel)
    Appendix B. Test Problems (by Robert Schnabel)
    Author Index
    Subject Index.

  • Authors

    J. E. Dennis

    Robert B. Schnabel

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