“...For anyone who needs to go beyond the basic treatment of numerical methods for nonlinear equations given in any number of standard numerical texts this book is ideal.” —Duncan Lawson, Mathematics Today, August 1998. “With 206 exercises aiming to illustrate and develop the ideas in the text and 134 bibliographical references, this very well written and organized monograph provides the basic infiormation needed to understand both the theory and the practice of the methods for solving problems related to unconstrained optimization and systems of nonlinear equations.” —Alfred Braier, Buletinul Institutului Politehnic Din Iasi, Tomul XLII(XLVI), Fasc. 3-4, 1996. “This book is a standard for a complete description of the methods for unconstrained optimization and the solution of nonlinear equations. ...this republication is most welcome and this volume should be in every library. Of course, there exist more recent books on the topics and somebody interested in the subject cannot be sati~sfied by looking only at this book. However, it contains much quite-well-presented material and I recommend reading it before going to other publications.” —Claude Brejinski, Numerical Algorithms, 13, October 1996.
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目录
PREFACE TO THE CLASSICS EDITION PREFACE 1 INTRODUCTION 1.1 Problems to be considered 1.2 Characteristics of "real-world" problems 1.3 Finite-precision arithmetic and measurement of error 1.4 Exercises 2 NONLINEAR PROBLEMS IN ONE VARIABLE 2.1 What is not possible 2.2 Newton's method for solving one equation in one unknown 2.3 Convergence of sequences of real numbers 2.4 Convergence of Newton's method 2.5 Globally convergent methods for solving one equation in one unknown 2.6 Methods when derivatives are unavailable 2.7 Minimization of a function of one variable 2.8 Exercises 3 NUMERICAL LINEAR ALGEBRA BACKGROUND 3.1 Vector and matrix norms and orthogonality 3.2 Solving systems of linear equations-matrix factorizations 3.3 Errors in solving linear systems 3.4 Updating matrix factorizations 3.5 Eigenvalues and positive definiteness 3.6 Linear least squares 3.7 Exercises 4 MULTIVARIABLE CALCULUS BACKGROUND 4.1 Derivatives and multivariable models 4.2 Multivariable finite-difference derivatives 4.3 Necessary and sufficient conditions for unconstrained minimization 4.4 Exercises 5 NEWTON'S METHOD FOR NONLINEAR EQUATIONS AND UNCONSTRAINED MINIMIZATION 5.1 Newton's method for systems of nonlinear equations 5.2 Local convergence of Newton's method 5.3 The Kantorovich and contractive mapping theorems 5.4 Finite-difference derivative methods for systems of nonlinear equations 5.5 Newton's method for unconstrained minimization 5.6 Finite-difference derivative methods for unconstrained minimization 5.7 Exercises 6 GLOBALLY CONVERGENT MODIFICATIONS OF NEWTON'S METHOD 6.1 The quasi-Newton framework 6.2 Descent directions 6.3 Line searches 6.3.1 Convergence results for properly chosen steps 6.3.2 Step selection by backtracking 6.4 The model-trust region approach 6.4.1 The locally constrained optimal ("hook") step 6.4.2 The double dogleg step 6.4.3 Updating the trust region 6.5 Global methods for systems of nonlinear equations 6.6 Exercises 7 STOPPING, SCALING, AND TESTING 7.1 Scaling 7.1 Scaling 7.3 Testing 7.4 Exercises 8 SECANT METHODS FOR SYSTEMS OF NONLINEAR EQUATIONS 8.1 Broyden's method 8.2 Local convergence analysis of Broyden's method 8.3 Implementation of quasi-Newton algorithms using Broyden's update 8.4 Other secant updates for nonlinear equations 8.5 Exercises 9 SECANT METHODS FOR UNCONSTRAINED MINIMIZATION 9.1 The symmetric secant update of Powell 9.2 Symmetric positive definite secant updates 9.3 Local convergence of positive definite secant methods 9.4 Implementation of quasi-Newton algorithms using the positive definite secant update 9.5 Another convergence result for the positive definite secant method 9.6 Other secant updates for unconstrained minimization 9.7 Exercises 10 NONLINEAR LEAST SQUARES 10.1 The nonlinear least-squares problem 10.2 Gauss-Newton-type methods 10.3 Full Newton-type methods 10.4 Other considerations in solving nonlinear least-squares problems 10.5 Exercise 11 METHODS FOR PROBLEMS WITH SPECIAL STRUCTURE 11.1 The sparse finite-difference Newton method 11.2 Sparse secant methods 11.3 Deriving least-change secant updates 11.4 Analyzing least-change secant methods 11.5 Exercises A APPENDIX: A MODULAR SYSTEM OF ALGORITHMS FOR UNCONSTRAINED MINIMIZATION AND NONLINEAR EQUATIONS(by Robert Schnabel) B APPENDIX: TEST PROBLEMS (by Robert Schnabel) REFERENCES AUTHOR INDEX SUBJECT INDEX
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