Linear numerical analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Paris
Hermann
1970
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 341 S. graph. Darst. |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804118058680188928 |
|---|---|
| adam_text | TABLE OF CONTENTS
Preface xi
Chapter 1 Elementary properties of matrices 1
1. Introduction 1
2. Addition of vectors 1
3. Multiplication of a vector by a scalar 2
4. Linear mappings 3
5. Operations on matrices: addition 6
6. Multiplication of a matrix by a scalar 9
7. Product of two matrices 10
8. Particular matrix products 14
9. Basis matrices 15
10. Products A . Ew and Eu . A 16
11. Numerical calculation of matrix products 18
12. Transposition 19
13. Conjugate and associate 21
Chapter 2 Vector and matrix norms 23
1. Elementary properties 23
2. Examples of vector norms 23
3. Matrix norms 26
4. Comparison of Holder norms 27
5. Definition of geometric norms 29
6. Geometric norms of matrices 31
7. Norms of square matrices 35
8. R as a Hilbert space 37
Chapter 3 Inversion of matrices—Theory 43
1. Linear independence of vectors 43
2. Systems of generators 43
3. Definition of basis 44
4. Fundamental theorem on the existence of a solution of a homogeneous system
with more unknowns than equations 44
5. Dimension 46
v
Table of Contents
6. Isomorphism of R or C with every vector space over R or C with finite
dimension 48
7. Inverse of a linear transformation 48
8. Linearity of the inverse transformation 50
9. Indicator of linear independence 51
10. Properties of determinants 56
11. Existence and construction of determinants 57
12. Formulae and definitions 58
13. Necessary and sufficient conditions for the existence of the inverse matrix 59
14. Existence of the inverse in a normed space 60
15. Theoretical solution of a system of linear equations 62
Chapter 4 Direct methods for the solution of a system of linear equations 63
1. Systems with a diagonal matrix 63
2. Systems with a triangular matrix 63
3. Inversion of a triangular matrix 66
4. General case: Gauss method of simple elimination 68
5. Decomposition of a matrix into the form T. A 72
6. An expression for the elements arising in the normal elimination process 74
7. Gauss method: formulae for computation 76
8. Crout s method 81
9. Jordan s method of complete elimination 84
10. Orthogonalization methods: the Schmidt process 87
11. Conservation of scalar products 88
12. Schmidt s algorithm 89
13. Solution of a system of equations by row orthogonalization 94
14. Method of rotations 97
15. Rotation algorithm 98
16. General direct methods for the inverse of a matrix 99
17. Evaluation of a determinant 104
18. Systems with a symmetric matrix 105
19. Jordan s method 106
20. Choleski s method 106
21. Use of partitioned matrices 112
22. Extension method 113
Problems on Chapters A 115
Chapter 5 Indirect methods 119
1. Introduction 119
2. Iteration and relaxation 119
3. Relaxation . 121
4. Southwell s method 125
5. Convergence of Southwell s method 127
6. The Gauss Seidel method 131
7. The successive over relaxation method 134
8. Linear iterative methods 138
9. Convergence of linear iterative methods 141 •
10. Application to the classical iterative methods 145 J
Table of Contents
11. Sufficient conditions derived from Hadamard s theorem 146
12. Frobenius Mises theorem 147
13. Successive over relaxation for Hermitian (or real symmetric) matrices 149
14. Other theorems on the location of eigenvalues 151
15. Projection methods 152
16. Decomposition of a norm 153
17. Projection on a plane perpendicular to ATzr 155
18. Projection on the plane corresponding to the largest residual 156
19. Kacmarz s method 157
20. Projections corresponding to decompositions of the norm q v 159
21. Process corresponding to the decomposition of the norm fi 160
22. Cimmino s process 160
23. Iterative methods for systems with a symmetric matrix 162
24. The gradient method 169
25. Stiefel Hestenes conjugate gradient method 170
26. Note on non symmetric systems 174
27. Note on convergence and its improvement 175
28. Iterative methods for the inversion of a matrix (Hotelling Bodewig) 176
Problems on Chapter 5 180
Chapter 6 Invariant subspaces 183
1. Introduction 183
2. Invariant subspaces 185
3. Polynomial transformations 186
4. Invariant subspaces and polynomial transformations 188
5. Diagonal form 195
6. Characteristic polynomial 197
7. Elementary divisors of a polynomial matrix 199
8. Normal forms 204
9. Functions of a linear transformation 215
Chapter 7 Some applications of the properties of invariant subspaces 227
1. Schur s theorem 227
2. Polar decomposition 231
3. Matrices with non negative elements 234
4. Graph theory and matrices with positive elements 244
5. Comparison of the classical iterative methods 250
6. Young and Frankel s theory of successive over relaxation 254
7. Polynomial iteration and the Peaceman Rachford method 261
8. Approximation of the spectral radius of a matrix by a norm 265
Chapter 8 Numerical methods for the calculation of eigenvalues and eigenvectors.. 271
1. Direct calculation of Det(A XI) = F(A) 271
2. Direct use of the Cayley Hamilton theorem 272
3. Leverrier s method 274
4. Souriau s method 277
5. Samuelson s method 278
vii
Table of Contents
6. Partitioning methods 281
7. Transformation methods: non symmetric matrices 282
8. Reduction to Frobenius form 283
9. First transformation 285
10. General transformation 287
11. Transformation to triple diagonal form 294
12. Lanczos method 298
13. Symmetric matrices: Given s method for real matrices 306
14. Characteristic polynomial of a symmetric tridiagonal matrix 309
15. Iterative methods for eigenvalues and eigenvectors—non symmetric matrices.. 314
16. Deflation 318
17. Jacobi s method for Hermitian or symmetric matrices 321
18. Rutishauser s LR method 328
Problems on Chapters 6, 7 and 8 336
Short Bibliography 341
viii
i
|
| any_adam_object | 1 |
| author | Gastinel, Noël 1925-1984 |
| author_GND | (DE-588)107915448 |
| author_facet | Gastinel, Noël 1925-1984 |
| author_role | aut |
| author_sort | Gastinel, Noël 1925-1984 |
| author_variant | n g ng |
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| bvnumber | BV003713889 |
| callnumber-first | Q - Science |
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| callnumber-raw | QA297 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 910 |
| ctrlnum | (OCoLC)134987 (DE-599)BVBBV003713889 |
| dewey-full | 515/.62/0285 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.62/0285 |
| dewey-search | 515/.62/0285 |
| dewey-sort | 3515 262 3285 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T16:04:16Z |
| institution | BVB |
| language | English French |
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| physical | IX, 341 S. graph. Darst. |
| publishDate | 1970 |
| publishDateSearch | 1970 |
| publishDateSort | 1970 |
| publisher | Hermann |
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| spelling | Gastinel, Noël 1925-1984 Verfasser (DE-588)107915448 aut Analyse numérique linéaire Linear numerical analysis Noel Gastinel Paris Hermann 1970 IX, 341 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier ALGOL (Langage de programmation) Algol larpcal Analyse numérique - Informatique Análise numérica larpcal Métodos numéricos de álgebra linear larpcal Datenverarbeitung ALGOL (Computer program language) Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s Lineare Algebra (DE-588)4035811-2 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002363967&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 |
| spellingShingle | Gastinel, Noël 1925-1984 Linear numerical analysis ALGOL (Langage de programmation) Algol larpcal Analyse numérique - Informatique Análise numérica larpcal Métodos numéricos de álgebra linear larpcal Datenverarbeitung ALGOL (Computer program language) Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4035811-2 |
| title | Linear numerical analysis |
| title_alt | Analyse numérique linéaire |
| title_auth | Linear numerical analysis |
| title_exact_search | Linear numerical analysis |
| title_full | Linear numerical analysis Noel Gastinel |
| title_fullStr | Linear numerical analysis Noel Gastinel |
| title_full_unstemmed | Linear numerical analysis Noel Gastinel |
| title_short | Linear numerical analysis |
| title_sort | linear numerical analysis |
| topic | ALGOL (Langage de programmation) Algol larpcal Analyse numérique - Informatique Análise numérica larpcal Métodos numéricos de álgebra linear larpcal Datenverarbeitung ALGOL (Computer program language) Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | ALGOL (Langage de programmation) Algol Analyse numérique - Informatique Análise numérica Métodos numéricos de álgebra linear Datenverarbeitung ALGOL (Computer program language) Numerical analysis Data processing Numerische Mathematik Lineare Algebra |
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