A brief introduction to numerical analysis:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Boston [u.a.]
Birkhäuser
1997
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XII, 202 S. graph. Darst. |
| ISBN: | 0817639160 3764339160 9781461264132 |
Staff View
MARC
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| 100 | 1 | |a Tyrtyšnikov, Evgenij E. |d 1955- |e Verfasser |0 (DE-588)115464891 |4 aut | |
| 245 | 1 | 0 | |a A brief introduction to numerical analysis |c Eugene E. Tyrtyshnikov |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 1997 | |
| 300 | |a XII, 202 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 0 | 7 | |a Numerische Mathematik |0 (DE-588)4042805-9 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804125951383044096 |
|---|---|
| adam_text | Contents
Lecture 1
1.1 Metric space 1
1.2 Some useful definitions 1
1.3 Nested balls 2
1.4 Normed space 2
1.5 Popular vector norms 3
1.6 Matrix norms 5
1.7 Equivalent norms 6
1.8 Operator norms 7
Lecture 2
2.1 Scalar product 11
2.2 Length of a vector 11
2.3 Isometric matrices 12
2.4 Preservation of length and unitary matrices 13
2.5 Schur theorem 13
2.6 Normal matrices 14
2.7 Positive definite matrices 14
2.8 The singular value decomposition 15
2.9 Unitarily invariant norms 16
2.10 A short way to the SVD 17
2.11 Approximations of a lower rank 17
2.12 Smoothness and ranks 18
Lecture 3
3.1 Perturbation theory 21
3.2 Condition of a matrix 21
3.3 Convergent matrices and series 22
3.4 The simplest iteration method 23
3.5 Inverses and series 23
3.6 Condition of a linear system 24
3.7 Consistency of matrix and right hand side 24
viii Contents
3.8 Eigenvalue perturbations 25
3.9 Continuity of polynomial roots 26
Lecture 4
4.1 Diagonal dominance 31
4.2 Gerschgorin disks 31
4.3 Small perturbations of eigenvalues and vectors 32
4.4 Condition of a simple eigenvalue 34
4.5 Analytic perturbations 35
Lecture 5
5.1 Spectral distances 39
5.2 Symmetric theorems 39
5.3 Hoffman Wielandt theorem 40
5.4 Permutational vector of a matrix 41
5.5 Unnormal extension 43
5.6 Eigenvalues of Hermitian matrices 44
5.7 Interlacing properties 44
5.8 What are clusters? 45
5.9 Singular value clusters 46
5.10 Eigenvalue clusters 47
Lecture 6
6.1 Floating Point numbers 49
6.2 Computer arithmetic axioms 49
6.3 Roundoff errors for the scalar product 50
6.4 Forward and backward analysis 51
6.5 Some philosophy 51
6.6 An example of bad operation 51
6.7 One more example 52
6.8 Ideal and machine tests 52
6.9 Up or down 53
6.10 Solving the triangular systems 54
Lecture 7
7.1 Direct methods for linear systems 57
7.2 Theory of the LU decomposition 57
7.3 Roundoff errors for the LU decomposition 59
7.4 Growth of matrix entries and pivoting 59
7.5 Complete pivoting 60
7.6 The Cholesky method 61
7.7 Triangular decompositions and linear systems solution .... 62
7.8 How to refine the solution 63
Contents ix
Lecture 8
8.1 The QR decomposition of a square matrix 65
8.2 The QR decomposition of a rectangular matrix 65
8.3 Householder matrices 66
8.4 Elimination of elements by reflections 66
8.5 Givens matrices 67
8.6 Elimination of elements by rotations 67
8.7 Computer realizations of reflections and rotations 68
8.8 Orthogonalization method 68
8.9 Loss of orthogonality 69
8.10 How to prevent the loss of orthogonality 70
8.11 Modified Gram Schmidt algorithm 70
8.12 Bidiagonalization 71
8.13 Unitary similarity reduction to the Hessenberg form 72
Lecture 9
9.1 The eigenvalue problem 75
9.2 The power method 75
9.3 Subspace iterations 76
9.4 Distances between subspaces 76
9.5 Subspaces and orthoprojectors 77
9.6 Distances and orthoprojectors 77
9.7 Subspaces of equal dimension 78
9.8 The CS decomposition 79
9.9 Convergence of subspace iterations for the block diagonal matrix 80
9.10 Convergence of subspace iterations in the general case .... 82
Lecture 10
10.1 The QR algorithm 85
10.2 Generalized QR algorithm 85
10.3 Basic formulas 86
10.4 The QR iteration lemma 86
10.5 Convergence of the QR iterations 88
10.6 Pessimistic and optimistic 89
10.7 Bruhat decomposition 90
10.8 What if the matrix X 1 is not strongly regular 91
10.9 The QR iterations and the subspace iterations 91
Lecture 11
11.1 Quadratic convergence 95
11.2 Cubic convergence 96
11.3 What makes the QR algorithm efficient 97
11.4 Implicit QR iterations 98
11.5 Arrangement of computations 99
11.6 How to find the singular value decomposition 100
x Contents
Lecture 12
12.1 Function approximation 103
12.2 Polynomial interpolation 103
12.3 Interpolating polynomial of Lagrange 104
12.4 Error of Lagrange interpolation 105
12.5 Divided differences 105
12.6 Newton formula 106
12.7 Divided differences with multiple nodes 107
12.8 Generalized interpolative conditions 108
12.9 Table of divided differences 109
Lecture 13
13.1 Convergence of the interpolation process 113
13.2 Convergence of the projectors 113
13.3 Sequences of linear continuous operators in a Banach space . 114
13.4 Algebraic and trigonometric polynomials 115
13.5 The Fourier series projectors 116
13.6 Pessimistic results 117
13.7 Why the uniform meshes are bad 117
13.8 Chebyshev meshes 118
13.9 Chebyshev polynomials 119
13.10 Bernstein s theorem 119
13.11 Optimistic results 120
Lecture 14
14.1 Splines 123
14.2 Natural splines 123
14.3 Variational property of natural splines 124
14.4 How to build natural splines 124
14.5 Approximation properties of natural splines 126
14.6 S splines 126
14.7 Quasi Local property and banded matrices 127
Lecture 15
15.1 Norm minimization 131
15.2 Uniform approximations 131
15.3 More on Chebyshev polynomials 132
15.4 Polynomials of the least deviation from zero 133
15.5 The Taylor series and its discrete counterpart 133
15.6 Least squares method 134
15.7 Orthogonal polynomials 134
15.8 Three term recurrence relationships 135
15.9 The roots of orthogonal polynomials 135
15.10 Three term relationships and tridiagonal matrices 136
15.11 Separation of the roots of orthogonal polynomials 137
15.12 Orthogonal polynomials and the Cholesky decomposition . . . 137
Contents xi
Lecture 16
16.1 Numerical integration 139
16.2 Interpolative quadrature formulas 139
16.3 Algebraic accuracy of a quadrature formula 140
16.4 Popular quadrature formulas 140
16.5 Gauss formulas 141
16.6 Compound quadrature formulas 142
16.7 Runge s rule for error estimation 142
16.8 How to integrate bad functions 143
Lecture 17
17.1 Nonlinear equations 145
17.2 When to quit? 145
17.3 Simple iteration method 146
17.4 Convergence and divergence of the simple iteration 146
17.5 Convergence and the Jacobi matrix 147
17.6 Optimization of the simple iteration 148
17.7 Method of Newton and Hermitian interpolation 148
17.8 Convergence of the Newton method 149
17.9 Newton everywhere 150
17.10 Generalization for n dimensions 150
17.11 Forward and backward interpolation 151
17.12 Secant method 152
17.13 Which is better, Newton or secant? 152
Lecture 18
18.1 Minimization methods 155
18.2 Newton again 155
18.3 Relaxation 156
18.4 Limiting the step size 156
18.5 Existence and uniqueness of the minimum point 157
18.6 Gradient method limiting the step size 158
18.7 Steepest descent method 159
18.8 Complexity of the simple computation 160
18.9 Quick computation of gradients 160
18.10 Useful ideas 162
Lecture 19
19.1 Quadratic functional and linear systems 165
19.2 Minimization over the subspace and projection methods ... 165
19.3 Krylov subspaces 166
19.4 Optimal subspaces 166
19.5 Optimality of the Krylov subspace 167
19.6 Method of minimal residuals 169
19.7 A norm and ^ orthogonality 170
19.8 Metod of conjugate gradients 170
xii Contents
19.9 Arnoldi method and Lanczos method 171
19.10 Arnoldi and Lanczos without Krylov 171
19.11 From matrix factorizations to iterative methods 172
19.12 Surrogate scalar product 173
19.13 Biorthogonalization approach 174
19.14 Breakdowns 174
19.15 Quasi Minimization idea 175
Lecture 20
20.1 Convergence rate of the conjugate gradient method 177
20.2 Chebyshev polynomials again 178
20.3 Classical estimate 178
20.4 Tighter estimates 179
20.5 Superlinear convergence and Vanishing eigenvalues . . . .180
20.6 Ritz values and Ritz vectors 180
20.7 Convergence of Ritz values 181
20.8 An important property 181
20.9 Theorem of van der Sluis and van der Vorst 182
20.10 Preconditioning 183
20.11 Preconditioning for Hermitian matrices 184
Lecture 21
21.1 Integral equations 187
21.2 Function spaces 187
21.3 Logarithmic kernel 188
21.4 Approximation, stability, convergence 189
21.5 Galerkin method 190
21.6 Strong ellipticity 190
21.7 Compact perturbation 190
21.8 Solution of integral equations 191
21.9 Splitting idea 191
21.10 Structured matrices 192
21.11 Circulant and Toeplitz matrices 192
21.12 Circulants and Fourier matrices 193
21.13 Fast Fourier transform 194
21.14 Circulant preconditioners 194
Bibliography 197
Index 199
|
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| author_facet | Tyrtyšnikov, Evgenij E. 1955- |
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| author_sort | Tyrtyšnikov, Evgenij E. 1955- |
| author_variant | e e t ee eet |
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| bvnumber | BV011435422 |
| classification_rvk | SK 900 |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:09:43Z |
| institution | BVB |
| isbn | 0817639160 3764339160 9781461264132 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007690390 |
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| physical | XII, 202 S. graph. Darst. |
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| publisher | Birkhäuser |
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| spelling | Tyrtyšnikov, Evgenij E. 1955- Verfasser (DE-588)115464891 aut A brief introduction to numerical analysis Eugene E. Tyrtyshnikov Boston [u.a.] Birkhäuser 1997 XII, 202 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s DE-604 Erscheint auch als Online-Ausgabe 978-0-8176-8136-4 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007690390&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tyrtyšnikov, Evgenij E. 1955- A brief introduction to numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 |
| title | A brief introduction to numerical analysis |
| title_auth | A brief introduction to numerical analysis |
| title_exact_search | A brief introduction to numerical analysis |
| title_full | A brief introduction to numerical analysis Eugene E. Tyrtyshnikov |
| title_fullStr | A brief introduction to numerical analysis Eugene E. Tyrtyshnikov |
| title_full_unstemmed | A brief introduction to numerical analysis Eugene E. Tyrtyshnikov |
| title_short | A brief introduction to numerical analysis |
| title_sort | a brief introduction to numerical analysis |
| topic | Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Numerische Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007690390&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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