Numerical methods for unconstrained optimization and nonlinear equations / J. E. Dennis, Jr., Robert B. Schnabel.
Series Prentice-Hall series in computational mathematicsEditor: Englewood Cliffs, N.J. : Prentice-Hall, c1983Descripción: xiii, 378 p. : il. ; 24 cmISBN: 0136272169Tema(s): Mathematical optimization | Equations -- Numerical solutionsOtra clasificación: 65-01 (49D15 65H05 65K10 90C30) | 90-01 (65-01 65H05 65K05 90C30)PREFACE x i 1 INTRODUCTION [2] 1.1 Problems to be considered [2] 1.2 Characteristics of “real-world” problems [5] 1.3 Finite-precision arithmetic and measurement of error [70] 1.4 Exercises [13] 2 NONLINEAR PROBLEMS IN ONE VARIABLE [15] 2.1 What is not possible [15] 2.2 Newton’s method for solving one equation in one unknown [16] 2.3 Convergence of sequences of real numbers [19] 2.4 Convergence of Newton’s method [21] 2.5 Globally convergent methods for solving one equation in one unknown [24] 2.6 Methods when derivatives are unavailable [27] 2.7 Minimization of a function of one variable [32] 2.8 Exercises [36] 3 NUMERICAL LINEAR ALGEBRA BACKGROUND [40] 3.1 Vector and matrix norms and orthogonality [41] 3.2 Solving systems of linear equations—matrix factorizations [47] 3.3 Errors in solving linear systems [57] 3.4 Updating matrix factorizations [55] 3.5 Eigenvalues and positive definiteness [58] 3.6 Linear least squares [60] 3.7 Exercises [66] || 4 || MULTIVARIABLE CALCULUS BACKGROUND [69] 4.1 Derivatives and multivariable models [69] 4.2 Multivariable finite-difference derivatives [77] 4.3 Necessary and sufficient conditions for unconstrained minimization [80] 4.4 Exercises [83] || 5 || NEWTON’S METHOD FOR NONLINEAR EQUATIONS AND UNCONSTRAINED MINIMIZATION [86] 5.1 Newton’s method for systems of nonlinear equations [86] 5.2 Local convergence of Newton’s method [89] 5.3 The Kantorovich and contractive mapping theorems [92] 5.4 Finite-difference derivative methods for systems of nonlinear equations [94] 5.5 Newton’s method for unconstrained minimization [99] 5.6 Finite-difference derivative methods for unconstrained minimization [103] 5.7 Exercises [107] || 6 || GLOBALLY CONVERGENT MODIFICATIONS OF NEWTON’S METHOD [111] 6.1 The quasi-Newton framework [112] 6.2 Descent directions [113] 6.3 Line searches [116] 6.3.1 Convergence results for properly chosen steps [120] 6.3.2 Step selection by backtracking [126] 6.4 The model-trust region approach [129] 6.4.1 The locally constrained optimal (“hook”) step [134] 6.4.2 The double dogleg step [139] 6.4.3 Updating the trust region [143] 6.5 Global methods for systems of nonlinear equations [147] 6.6 Exercises [152] || 7 ||STOPPING, SCALING, AND TESTING [155] 7.1 Scaling [155] 7.2 Stopping criteria [159] 7.3 Testing [161] 7.4 Exercises [164] || 8 || SECANT METHODS FOR SYSTEMS OF NONLINEAR EQUATIONS [168] 8.1 Broyden’s method [169] 8.2 Local convergence analysis of Broyden’s method [174] 8.3 Implementation of quasi-Newton algorithms using Broyden’s update [186] 8.4 Other secant updates for nonlinear equations [189] 8.5 Exercises [190] || 9 || SECANT METHODS FOR UNCONSTRAINED MINIMIZATION [194] 9.1 The symmetric secant update of Powell [195] 9.2 Symmetric positive definite secant updates [198] 9.3 Local convergence of positive definite secant methods [203] 9.4 Implementation of quasi-Newton algorithms using the positive definite secant update [208] 9.5 Another convergence result for the positive definite secant method [210] 9.6 Other secant updates for unconstrained minimization [211] 9.7 Exercises [212] || 10 || NONLINEAR LEAST SQUARES [218] 10.1 The nonlinear least-squares problem [218] 10.2 Gauss-Newton-type methods [221] 10.3 Full Newton-type methods [228] 10.4 Other considerations in solving nonlinear least-squares problems [233] 10.5 Exercises [236] || 11 || METHODS FOR PROBLEMS WITH SPECIAL STRUCTURE [239] 11.1 The sparse finite-difference Newton method [240] 11.2 Sparse secant methods [242] 11.3 Deriving least-change secant updates [246] 11.4 Analyzing least-change secant methods [251] 11.5 Exercises [256] A APPENDIX: A MODULAR SYSTEM OF ALGORITHMS FOR UNCONSTRAINED MINIMIZATION AND NONLINEAR EQUATIONS (by Robert Schnabel) [259] ||B|| APPENDIX: TEST PROBLEMS [361] REFERENCES [364] (by Robert Schnabel) AUTHOR INDEX [371] SUBJECT INDEX [373]
| Item type | Home library | Call number | Materials specified | Copy number | Status | Date due | Barcode | Course reserves |
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Instituto de Matemática, CONICET-UNS | 90 D411 (Browse shelf) | Available | A-6567 | ||||
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Instituto de Matemática, CONICET-UNS | 90 D411 (Browse shelf) | Ej. 2 | Available | A-6867 |
Incluye referencias bibliográficas (p. 364-370) e índices.
PREFACE x i --
1 INTRODUCTION [2] --
1.1 Problems to be considered [2] --
1.2 Characteristics of “real-world” problems [5] --
1.3 Finite-precision arithmetic and measurement of error [70] --
1.4 Exercises [13] --
2 NONLINEAR PROBLEMS IN ONE VARIABLE [15] --
2.1 What is not possible [15] --
2.2 Newton’s method for solving one equation in one unknown [16] --
2.3 Convergence of sequences of real numbers [19] --
2.4 Convergence of Newton’s method [21] --
2.5 Globally convergent methods for solving one equation in one unknown [24] --
2.6 Methods when derivatives are unavailable [27] --
2.7 Minimization of a function of one variable [32] --
2.8 Exercises [36] --
3 NUMERICAL LINEAR ALGEBRA BACKGROUND [40] --
3.1 Vector and matrix norms and orthogonality [41] --
3.2 Solving systems of linear equations—matrix factorizations [47] --
3.3 Errors in solving linear systems [57] --
3.4 Updating matrix factorizations [55] --
3.5 Eigenvalues and positive definiteness [58] --
3.6 Linear least squares [60] --
3.7 Exercises [66] --
|| 4 || MULTIVARIABLE CALCULUS BACKGROUND [69] --
4.1 Derivatives and multivariable models [69] --
4.2 Multivariable finite-difference derivatives [77] --
4.3 Necessary and sufficient conditions for unconstrained minimization [80] --
4.4 Exercises [83] --
|| 5 || NEWTON’S METHOD --
FOR NONLINEAR EQUATIONS --
AND UNCONSTRAINED MINIMIZATION [86] --
5.1 Newton’s method for systems of nonlinear equations [86] --
5.2 Local convergence of Newton’s method [89] --
5.3 The Kantorovich and contractive mapping theorems [92] --
5.4 Finite-difference derivative methods for systems of nonlinear equations [94] --
5.5 Newton’s method for unconstrained minimization [99] --
5.6 Finite-difference derivative methods for unconstrained minimization [103] --
5.7 Exercises [107] --
|| 6 || GLOBALLY CONVERGENT MODIFICATIONS OF NEWTON’S METHOD [111] --
6.1 The quasi-Newton framework [112] --
6.2 Descent directions [113] --
6.3 Line searches [116] --
6.3.1 Convergence results for properly chosen steps [120] --
6.3.2 Step selection by backtracking [126] --
6.4 The model-trust region approach [129] --
6.4.1 The locally constrained optimal (“hook”) step [134] --
6.4.2 The double dogleg step [139] --
6.4.3 Updating the trust region [143] --
6.5 Global methods for systems of nonlinear equations [147] --
6.6 Exercises [152] --
|| 7 ||STOPPING, SCALING, AND TESTING [155] --
7.1 Scaling [155] --
7.2 Stopping criteria [159] --
7.3 Testing [161] --
7.4 Exercises [164] --
|| 8 || SECANT METHODS FOR SYSTEMS OF NONLINEAR EQUATIONS [168] --
8.1 Broyden’s method [169] --
8.2 Local convergence analysis of Broyden’s method [174] --
8.3 Implementation of quasi-Newton algorithms using Broyden’s update [186] --
8.4 Other secant updates for nonlinear equations [189] --
8.5 Exercises [190] --
|| 9 || SECANT METHODS --
FOR UNCONSTRAINED MINIMIZATION [194] --
9.1 The symmetric secant update of Powell [195] --
9.2 Symmetric positive definite secant updates [198] --
9.3 Local convergence of positive definite secant methods [203] --
9.4 Implementation of quasi-Newton algorithms using the positive definite secant update [208] --
9.5 Another convergence result for the positive definite secant method [210] --
9.6 Other secant updates for unconstrained minimization [211] --
9.7 Exercises [212] --
|| 10 || NONLINEAR LEAST SQUARES [218] --
10.1 The nonlinear least-squares problem [218] --
10.2 Gauss-Newton-type methods [221] --
10.3 Full Newton-type methods [228] --
10.4 Other considerations in solving nonlinear least-squares problems [233] --
10.5 Exercises [236] --
|| 11 || METHODS FOR PROBLEMS WITH SPECIAL STRUCTURE [239] --
11.1 The sparse finite-difference Newton method [240] --
11.2 Sparse secant methods [242] --
11.3 Deriving least-change secant updates [246] --
11.4 Analyzing least-change secant methods [251] --
11.5 Exercises [256] --
A APPENDIX: A MODULAR SYSTEM OF ALGORITHMS --
FOR UNCONSTRAINED MINIMIZATION AND NONLINEAR EQUATIONS (by Robert Schnabel) [259] --
||B|| APPENDIX: TEST PROBLEMS [361] --
REFERENCES [364] --
(by Robert Schnabel) --
AUTHOR INDEX [371] --
SUBJECT INDEX [373] --
MR, 85j:65001
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