具体描述
《国外数学名著系列(续1)(影印版)42:无约束最优化与非线性方程的数值方法》is a standard for a complete description of the methods for unconstrained optimization and the solution ofnonlinear 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 satiated by looking only at this book. However, it contains much quite-well-presented material and I recommend reading it before going ,to other.publications.
这本书聚焦于无约束最优化与非线性方程求解领域，系统探讨了如何在实际工程应用中实现高效、精准的数值计算方法。内容涵盖了从理论到实践的全面介绍，详细描述了多种用于解决复杂优化问题的算法，包括但不限于遗传算法、粒子群优化和模拟退火等经典技术。书中深入剖析非线性方程的求解策略，论述了不同方法在精度与速度间的权衡，并结合具体案例展示其在工业与科研中的应用价值。 作者对数值方法的研究进行了系统化梳理，从经典微分方程求解到现代优化技术的演进，内容丰富且条理清晰。书中不仅注重理论框架，还注重实验验证和实际案例分析，为读者提供了强有力的工具箱，使他们能够灵活应对多变的工程需求。每一章节都经过细致设计，力求覆盖广泛的主题，确保读者在阅读过程中全面理解核心概念与应用技巧。 本文特别强调数值计算在复杂系统模拟中的重要性，详细介绍了如何通过优化算法提升求解效率，同时分析不同方法的适用场景。书中还结合最新的研究进展，探讨了当前技术在处理高维、非线性问题时的挑战与突破，帮助读者深刻理解领域的发展脉络。此外，为了增强实用性，内容特别注重操作步骤和参数调整建议，使其更易于实际应用。 通过整合经典理论与现代技术，作者为研究者、工程师以及相关领域的学习者提供了一本系统性的参考手册。这本书不仅帮助读者掌握解决复杂优化问题的科学方法，还强调了技术创新在提升计算效率和精确度方面的重要性。内容设计严谨、结构清晰，旨在满足不同层次用户的需求，为深入学习非线性方程求解与无约束优化的实践提供坚实基础。 总体而言，这本书以深刻分析和广泛覆盖的形式，为读者展现了当前数值计算领域的最新进展，同时也强调了理论与实践结合的重要性，成为一套兼具学术价值与应用指导意义的著作。
作者简介
目录信息
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 83
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
7.2 Stopping criteria
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 Exercises
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 SchnabeI)
REFERENCES
AUTHOR INDEX
SUBJECT INDEX
· · · · · · (收起)
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 83
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
7.2 Stopping criteria
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 Exercises
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 SchnabeI)
REFERENCES
AUTHOR INDEX
SUBJECT INDEX
· · · · · · (收起)
读后感
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very clear account of the knowledge in this area up to 1990's
评分very clear account of the knowledge in this area up to 1990's
评分very clear account of the knowledge in this area up to 1990's
评分very clear account of the knowledge in this area up to 1990's
评分very clear account of the knowledge in this area up to 1990's
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