具體描述
《國外數學名著係列(續1)(影印版)42:無約束最優化與非綫性方程的數值方法》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.
這本書聚焦於無約束最優化與非綫性方程求解領域，係統探討瞭如何在實際工程應用中實現高效、精準的數值計算方法。內容涵蓋瞭從理論到實踐的全麵介紹，詳細描述瞭多種用於解決復雜優化問題的算法，包括但不限於遺傳算法、粒子群優化和模擬退火等經典技術。書中深入剖析非綫性方程的求解策略，論述瞭不同方法在精度與速度間的權衡，並結閤具體案例展示其在工業與科研中的應用價值。 作者對數值方法的研究進行瞭係統化梳理，從經典微分方程求解到現代優化技術的演進，內容豐富且條理清晰。書中不僅注重理論框架，還注重實驗驗證和實際案例分析，為讀者提供瞭強有力的工具箱，使他們能夠靈活應對多變的工程需求。每一章節都經過細緻設計，力求覆蓋廣泛的主題，確保讀者在閱讀過程中全麵理解核心概念與應用技巧。 本文特彆強調數值計算在復雜係統模擬中的重要性，詳細介紹瞭如何通過優化算法提升求解效率，同時分析不同方法的適用場景。書中還結閤最新的研究進展，探討瞭當前技術在處理高維、非綫性問題時的挑戰與突破，幫助讀者深刻理解領域的發展脈絡。此外，為瞭增強實用性，內容特彆注重操作步驟和參數調整建議，使其更易於實際應用。 通過整閤經典理論與現代技術，作者為研究者、工程師以及相關領域的學習者提供瞭一本係統性的參考手冊。這本書不僅幫助讀者掌握解決復雜優化問題的科學方法，還強調瞭技術創新在提升計算效率和精確度方麵的重要性。內容設計嚴謹、結構清晰，旨在滿足不同層次用戶的需求，為深入學習非綫性方程求解與無約束優化的實踐提供堅實基礎。 總體而言，這本書以深刻分析和廣泛覆蓋的形式，為讀者展現瞭當前數值計算領域的最新進展，同時也強調瞭理論與實踐結閤的重要性，成為一套兼具學術價值與應用指導意義的著作。
作者簡介
目錄資訊
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)
讀後感
評分
very clear account of the knowledge in this area up to 1990's
評分very clear account of the knowledge in this area up to 1990's
評分very clear account of the knowledge in this area up to 1990's
評分very clear account of the knowledge in this area up to 1990's
評分very clear account of the knowledge in this area up to 1990's
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