Compact numerical methods for computers: linear algebra and function minimisation
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Bristol [u.a.]
Hilger
1990
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| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 278 S. graph. Darst. |
| ISBN: | 0852743181 085274319X 0750300361 0750300434 |
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Record in the Search Index
| _version_ | 1804120852319436800 |
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| adam_text | Titel: Compact numerical methods for computers
Autor: Nash, John C
Jahr: 1990
CONTENTS Preface to the Second Edition ix Preface to the First Edition xi 1. A STARTING POINT 1 1.1. Purpose and scope 1 1.2. Machine characteristics 3 1.3. Sources of programs 9 1.4. Programming languages used and structured programming 11 1.5. Choice of algorithms 13 1.6. A method for expressing algorithms 15 1.7. General notation 17 1.8. Software engineering issues 17 2. FORMAL PROBLEMS IN LINEAR ALGEBRA 19 2.1. Introduction 19 2.2. Simultaneous linear equations 19 2.3. The linear least-squares problem 21 2.4. The inverse and generalised inverse of a matrix 24 2.5. Decompositions of a matrix 26 2.6. The matrix eigenvalue problem 28 3. THE SINGULAR-VALUE DECOMPOSITION AND ITf USE TO SOLVE LEAST-SQUARES PROBLEMS ;1 v .V 30 3.1. Introduction ?* 30 3.2. A singular-value decomposition algorithm , 31 3.3. Orthogonalisation by plane rotations . ? 32 3.4. A fine point y. : - / 35 3.5. An alternative implementation of the singular-yalye decojnpdsi- tion - V • 3 g 3.6. Using the singular-value decomposition to solve least-squares problems 40 4. HANDLING LARGER PROBLEMS 49 4.1. Introduction 49 4.2. The Givens’ reduction 49 4.3. Extension to a singular-value decomposition 54 4.4. Some labour-saving devices 54 4.5. Related calculations 63 5. SOME COMMENTS ON THE FORMATION OF THE CROSS- PRODUCTS MATRIX A T A 66 v
vi Compact numerical methods for computers 6. LINEAR EQUATIONS—A DIRECT APPROACH 72 6.1. Introduction 72 6.2. Gauss elimination 72 6.3. Variations on the theme of Gauss elimination 80 6.4. Complex systems of equations 82 6.5. Methods, for special matrices 83 7. THE CHOLESKI DECOMPOSITION 84 7.1. The Choleski decomposition 84 7.2. Extension of the Choleski decomposition to non-negative definite matrices 86 7.3. Some organisational details 90 8. THE SYMMETRIC POSITIVE DEFINITE MATRIX AGAIN 94 8.1. The Gauss-Jordan reduction 94 8.2. The Gauss-Jordan algorithm for the inverse of a symmetric positive definite matrix 97 9. THE ALGEBRAIC EIGENVALUE PROBLEM 102 9.1. Introduction 102 9.2. The power method and inverse iteration 102 9.3. Some notes on the behaviour of inverse iteration 108 9.4. Eigensolutions of non-symmetric and complex matrices 110 10. REAL SYMMETRIC MATRICES 119 10.1. The eigensolutions of a real symmetric matrix 119 10.2. Extension to matrices which are not positive definite 121 10.3. The Jacobi algorithm for the eigensolutions of a real symmetric matrix 126 10.4. Organisation of the Jacobi algorithm 128 10.5. A brief comparison of methods for the eigenproblem of a real symmetric matrix 133 11. THE GENERALISED SYMMETRIC MATRIX EIGENVALUE PROBLEM 135 12. OPTIMISATION AND NONLINEAR EQUATIONS 142 12.1. Formal problems in unconstrained optimisation and nonlinear equations 142 12.2. Difficulties encountered in the solution of optimisation and nonlinear-equation problems 146 13. ONE-DIMENSIONAL PROBLEMS 148 13.1. Introduction 148 13.2. The linear search problem 148 13.3. Real roots of functions of one variable 160
Contents vii 14. DIRECT SEARCH METHODS 168 14.1. The Nelder-Mead simplex search for the minimum of a function of several parameters 168 14.2. Possible modifications of the Nelder-Mead algorithm 172 14.3. An axial search procedure 178 14.4. Other direct search methods 182 15. DESCENT TO A MINIMUM I: VARIABLE METRIC ALGORITHMS 186 15.1. Descent methods for minimisation 186 15.2. Variable metric algorithms 187 15.3. A choice of strategies 190 16. DESCENT TO A MINIMUM II: CONJUGATE GRADIENTS 197 16.1. Conjugate gradients methods 197 16.2. A particular conjugate gradients algorithm 198 17. MINIMISING A NONLINEAR SUM OF SQUARES 207 17.1. Introduction 207 17.2. Two methods 208 17.3. Hartley’s modification 210 17.4. Marquardt’s method 211 17.5. Critique and evaluation 212 17.6. Related methods 215 18. LEFT-OVERS 218 18.1. Introduction 218 18.2. Numerical approximation of derivatives 218 18.3. Constrained optimisation 221 18.4. A comparison of function minimisation and nonlinear least- squares methods 226 19. THE CONJUGATE GRADIENTS METHOD APPLIED TO PROBLEMS IN LINEAR ALGEBRA 234 19.1. Introduction 234 19.2. Solution of linear equations and least-squares problems by conjugate gradients 235 19.3. Inverse iteration by algorithm 24 241 19.4. Eigensolutions by minimising the Rayleigh quotient 243 APPENDICES 253 1. Nine test matrices 253 2. List of algorithms 255 3. List of examples 256 4. Files on the software diskette 258 BIBLIOGRAPHY 263 INDEX 271
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| author | Nash, John C. 1947- |
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| author_sort | Nash, John C. 1947- |
| author_variant | j c n jc jcn |
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| callnumber-first | Q - Science |
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| dewey-full | 519.4 |
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| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| spelling | Nash, John C. 1947- Verfasser (DE-588)1045402818 aut Compact numerical methods for computers linear algebra and function minimisation J. C. Nash 2. ed. Bristol [u.a.] Hilger 1990 XII, 278 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algèbre linéaire - Informatique Maximums et minimums - Informatique Optimisation mathématique - Informatique Datenverarbeitung Algebras, Linear Data processing Functions Data processing Maxima and minima Data processing Numerical analysis Data processing Algorithmus (DE-588)4001183-5 gnd rswk-swf TURBO-PASCAL (DE-588)4117264-4 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Datenverarbeitung (DE-588)4011152-0 gnd rswk-swf Minimierung (DE-588)4251074-0 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf 1\p (DE-588)4055382-6 Software gnd-content Numerische Mathematik (DE-588)4042805-9 s Algorithmus (DE-588)4001183-5 s Datenverarbeitung (DE-588)4011152-0 s DE-604 Funktion Mathematik (DE-588)4071510-3 s Minimierung (DE-588)4251074-0 s TURBO-PASCAL (DE-588)4117264-4 s 2\p DE-604 Lineare Algebra (DE-588)4035811-2 s 3\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004200181&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Nash, John C. 1947- Compact numerical methods for computers linear algebra and function minimisation Algèbre linéaire - Informatique Maximums et minimums - Informatique Optimisation mathématique - Informatique Datenverarbeitung Algebras, Linear Data processing Functions Data processing Maxima and minima Data processing Numerical analysis Data processing Algorithmus (DE-588)4001183-5 gnd TURBO-PASCAL (DE-588)4117264-4 gnd Funktion Mathematik (DE-588)4071510-3 gnd Lineare Algebra (DE-588)4035811-2 gnd Datenverarbeitung (DE-588)4011152-0 gnd Minimierung (DE-588)4251074-0 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
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| title | Compact numerical methods for computers linear algebra and function minimisation |
| title_auth | Compact numerical methods for computers linear algebra and function minimisation |
| title_exact_search | Compact numerical methods for computers linear algebra and function minimisation |
| title_full | Compact numerical methods for computers linear algebra and function minimisation J. C. Nash |
| title_fullStr | Compact numerical methods for computers linear algebra and function minimisation J. C. Nash |
| title_full_unstemmed | Compact numerical methods for computers linear algebra and function minimisation J. C. Nash |
| title_short | Compact numerical methods for computers |
| title_sort | compact numerical methods for computers linear algebra and function minimisation |
| title_sub | linear algebra and function minimisation |
| topic | Algèbre linéaire - Informatique Maximums et minimums - Informatique Optimisation mathématique - Informatique Datenverarbeitung Algebras, Linear Data processing Functions Data processing Maxima and minima Data processing Numerical analysis Data processing Algorithmus (DE-588)4001183-5 gnd TURBO-PASCAL (DE-588)4117264-4 gnd Funktion Mathematik (DE-588)4071510-3 gnd Lineare Algebra (DE-588)4035811-2 gnd Datenverarbeitung (DE-588)4011152-0 gnd Minimierung (DE-588)4251074-0 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Algèbre linéaire - Informatique Maximums et minimums - Informatique Optimisation mathématique - Informatique Datenverarbeitung Algebras, Linear Data processing Functions Data processing Maxima and minima Data processing Numerical analysis Data processing Algorithmus TURBO-PASCAL Funktion Mathematik Lineare Algebra Minimierung Numerische Mathematik Software |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004200181&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT nashjohnc compactnumericalmethodsforcomputerslinearalgebraandfunctionminimisation |