Functions of matrices: theory and computation
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Philadelphia
SIAM, Soc. for Industrial and Applied Math.
2008
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016430292&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Includes bibliographical references and index |
| Umfang: | XX, 425 S. Ill., graph. Darst. |
| ISBN: | 9780898716467 |
Internformat
MARC
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| 020 | |a 9780898716467 |9 978-0-898716-46-7 | ||
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| 100 | 1 | |a Higham, Nicholas J. |d 1961-2024 |e Verfasser |0 (DE-588)123564441 |4 aut | |
| 245 | 1 | 0 | |a Functions of matrices |b theory and computation |c Nicholas J. Higham |
| 264 | 1 | |a Philadelphia |b SIAM, Soc. for Industrial and Applied Math. |c 2008 | |
| 300 | |a XX, 425 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 7 | |a Matrices |2 gtt | |
| 650 | 7 | |a Matrixfuncties |2 gtt | |
| 650 | 7 | |a Numerieke wiskunde |2 gtt | |
| 650 | 4 | |a Matrices | |
| 650 | 4 | |a Functions | |
| 650 | 4 | |a Factorization (Mathematics) | |
| 650 | 0 | 7 | |a Funktion |g Mathematik |0 (DE-588)4071510-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Matrizenzerlegung |0 (DE-588)4376303-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Matrixfunktion |0 (DE-588)4169117-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Matrix |g Mathematik |0 (DE-588)4037968-1 |2 gnd |9 rswk-swf |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-016430292 | |
Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102 MAT 150f 2008 A 6668 0202 MAT 150f 2010 B 178 |
|---|---|
| DE-BY-TUM_katkey | 1627844 |
| DE-BY-TUM_location | 01 02 |
| DE-BY-TUM_media_number | 040010096310 040020765413 |
| _version_ | 1821933059400794112 |
| adam_text | Contents
List of Figures
xiii
List of Tables
xv
Preface
xvii
1
Theory of Matrix Functions
1
1.1
Introduction
................................ 1
1.2
Definitions of
f
(A)
............................ 2
1.2.1
Jordan Canonical Form
...................... 2
1.2.2
Polynomial Interpolation
..................... 4
1.2.3
Cauchy Integral Theorem
..................... 7
1.2.4
Equivalence of Definitions
.................... 8
1.2.5
Example: Function of Identity Plus Rank-
1
Matrix
...... 8
1.2.6
Example: Function of Discrete Fourier Transform Matrix
... 10
1.3
Properties
................................. 10
1.4
Nonprimary Matrix Functions
...................... 14
1.5
Existence of (Real) Matrix Square Roots and Logarithms
....... 16
1.6
Classification of Matrix Square Roots and Logarithms
......... 17
1.7
Principal Square Root and Logarithm
.................. 20
1.8
f
{AB)
and
f
(В А).............................
21
1.9
Miscellany
................................. 23
1.10
A Brief History of Matrix Functions
................... 26
1.11
Notes and References
........................... 27
Problems
.................................. 29
2
Applications
35
2.1
Differential Equations
........................... 35
2.1.1
Exponential Integrators
...................... 36
2.2
Nuclear Magnetic Resonance
....................... 37
2.3
Markov Models
.............................. 37
2.4
Control Theory
.............................. 39
2.5
The Nonsymmetric Eigenvalue Problem
................. 41
2.6
Orthogonalization and the Orthogonal Procrustes Problem
...... 42
2.7
Theoretical Particle Physics
....................... 43
2.8
Other Matrix Functions
.......................... 44
2.9
Nonlinear Matrix Equations
....................... 44
2.10
Geometric Mean
.............................. 46
2.11
Pseudospectra
............................... 47
2.12
Algebras
.................................. 47
vii
Contents
2.13
Sensitivity Analysis
............................ 48
2.14
Other Applications
............................ 48
2.14.1
Boundary Value Problems
.................... 48
2.14.2
Semidefinite
Programming
.................... 48
2.14.3
Matrix Sector Function
...................... 48
2.14.4
Matrix Disk Function
....................... 49
2.14.5
The Average Eye in Optics
.................... 50
2.14.6
Computer Graphics
........................ 50
2.14.7
Bregman Divergences
....................... 50
2.14.8
Structured Matrix Interpolation
................. 50
2.14.9
The Lambert
W
Function and Delay Differential Equations
. 51
2.15
Notes and References
........................... 51
Problems
.................................. 52
Conditioning
55
3.1
Condition Numbers
............................ 55
3.2
Properties of the
Frédiét
Derivative
................... 57
3.3
Bounding the Condition Number
.................... 63
3.4
Computing or Estimating the Condition Number
........... 64
3.5
Notes and References
........................... 69
Problems
.................................. 70
Techniques for General Functions
71
4.1
Matrix Powers
............................... 71
4.2
Polynomial Evaluation
.......................... 72
4.3
Taylor Series
................................ 76
4.4
Rational Approximation
......................... 78
4.4.1
Best
¿(χ
Approximation
..................... 79
4.4.2
Padé
Approximation
....................... 79
4.4.3
Evaluating Rational Functions
.................. 80
4.5
Diagoualization
.............................. 81
4.6 Schur
Decomposition and Triangular Matrices
............. 84
4.7
Block Diagonalization
........................... 89
4.8
Interpolating Polynomial and Characteristic Polynomial
....... 89
4.9
Matrix Iterations
............................. 91
4.9.1
Order of Convergence
....................... 91
4.9.2
Termination Criteria
....................... 92
4.9.3
Convergence
............................ 93
4.9.4
Numerical Stability
........................ 95
4.10
Preprocessing
............................... 99
4.11
Bounds for
lì{A)
............................ 102
4.12
Note« and References
........................... 104
Problems
.................................. 105
Matrix Sign Function
107
5.1
Sensitivity and Conditioning
....................... 109
5.2 Schur
Method
............................... 112
5.3
Newton s Method
............................. 113
5.4
The
Padé
Family of Iterations
...................... 115
5.5
Scaling the Newton Iteration
....................... 119
Contents
ix
5.6
Terminating the Iterations
........................ 121
5.7
Numerical Stability of Sign Iterations
.................. 123
5.8
Numerical Experiments and Algorithm
................. 125
5.9
Best Loo Approximation
......................... 128
5.10
Notes and References
........................... 129
Problems
.................................. 131
6
Matrix Square Root
133
6.1
Sensitivity and Conditioning
....................... 133
6.2 Schur
Method
............................... 135
6.3
Newton s Method and Its Variants
.................... 139
6.4
Stability and Limiting Accuracy
..................... 144
6.4.1
Newton Iteration
......................... 144
6.4.2
DB
Iterations
........................... 145
6.4.3
CR Iteration
............................ 146
6.4.4
IN Iteration
............................ 146
6.4.5
Summary
.............................. 147
6.5
Scaling the Newton Iteration
....................... 147
6.6
Numerical Experiments
.......................... 148
6.7
Iterations via the Matrix Sign Function
................. 152
6.8
Special Matrices
.............................. 154
6.8.1
Binomial Iteration
......................... 154
6.8.2
Modified Newton Iterations
................... 157
6.8.3
M-Matrices and ff-Matrices
................... 159
6.8.4
Hermitian Positive Definite Matrices
.............. 161
6.9
Computing Small-Normed Square Roots
................ 162
6.10
Comparison of Methods
.......................... 164
6.11
Involutory Matrices
............................ 165
6.12
Notes and References
........................... 166
Problems
.................................. 168
7
Matrix pth Root
173
7.1
Theory
................................... 173
7.2 Schur
Method
............................... 175
7.3
Newton s Method
............................. 177
7.4
Inverse Newton Method
.......................... 181
7.5 Schur-Newton
Algorithm
......................... 184
7.6
Matrix Sign Method
............................ 186
7.7
Notes and References
........................... 187
Problems
.................................. 189
8
The Polar Decomposition
193
8.1
Approximation Properties
........................ 197
8.2
Sensitivity and Conditioning
....................... 199
8.3
Newton s Method
............................. 202
8.4
Obtaining Iterations via the Matrix Sign Function
........... 202
8.5
The
Padé
Family of Methods
....................... 203
8.6
Scaling the Newton Iteration
....................... 205
8.7
Terminating the Iterations
........................ 207
8.8
Numerical Stability and Choice of
Я
.................. 209
Contents
8.9
Algorithm
.................................210
8.10
Notes and References
...........................213
Problems
..................................216
9
Schur-Parlett Algorithm
221
9.1
Evaluating Functions of the Atomic Blocks
............... 221
9.2
Evaluating the Upper Triangular Part of
ƒ
(T)
............. 225
9.3
Reordering and Blocking the
Schur Form................ 226
9.4
Schur-Parlett Algorithm for
f
(A)
.................... 228
9.5
Preprocessing
............................... 230
9.6
Notes and References
........................... 231
Problems
.................................. 231
10
Matrix Exponential
233
10.1
Basic Properties
.............................. 233
10.2
Conditioning
................................ 238
10.3
Scaling and Squaring Method
...................... 241
10.4 Schur
Algorithms
............................. 250
10.4.1
Newton Divided Difference Interpolation
............ 250
10.4.2
Schur-Fréchet
Algorithm
..................... 251
10.4.3 Schur
Parlett
Algorithm
..................... 251
10.5
Numerical Experiment
.......................... 252
10.6
Evaluating the
Frédiét
Derivative and Its Norm
............ 253
10.6.1
Quadrature
............................ 254
10.6.2
The
Kronecker
Formulae
..................... 256
10.6.3
Computing and Estimating the Norm
.............. 258
10.7
Miscellany
................................. 259
10.7.1
Hermitian Matrices and Best L^ Approximation
....... 259
10.7.2
Essentially
Nonnegative
Matrices
................ 260
10.7.3
Preprocessing
........................... 261
10.7.4
The V Functions
.......................... 261
10.8
Notes and References
........................... 262
Problems
.................................. 265
11
Matrix Logarithm
269
11.1
Basic Properties
.............................. 269
11.2
Conditioning
................................ 272
11.3
Series Expansions
............................. 273
11.4
Padé
Approximation
........................... 274
11.5
Inverse Scaling and Squaring Method
.................. 275
11.5.1 Schur
Decomposition: Triangular Matrices
........... 276
11.5.2
Full Matrices
........................... 278
11.6
Sdiur Algorithms
............................. 279
11.6.1 Schur
Frédiét
Algorithm
..................... 279
11.6.2
Schur-Parlett Algorithm
..................... 279
11.7
Numerical Experiment
.......................... 280
11.8
Evaluating the
Frechei
Derivative
.................... 281
11.9
Notes and References
........................... 283
Problems
.................................. 284
Contents
xi
12
Matrix Cosine and Sine
287
12.1
Basic Properties
.............................. 287
12.2
Conditioning
................................ 289
12.3
Padé
Approximation of Cosine
...................... 290
12.4
Double Angle Algorithm for Cosine
................... 290
12.5
Numerical Experiment
.......................... 295
12.6
Double Angle Algorithm for Sine and Cosine
.............. 296
12.6.1
Preprocessing
........................... 299
12.7
Notes and References
........................... 299
Problems
.................................. 300
13
Function of Matrix Times Vector:
f (A) b
301
13.1
Representation via Polynomial Interpolation
.............. 301
13.2
Krylov Subspace Methods
........................ 302
13.2.1
The
Arnoldi
Process
....................... 302
13.2.2
Arnoldi
Approximation of f(A)b
................. 304
13.2.3
Lanczos Biorthogonalization
................... 306
13.3
Quadrature
................................. 306
13.3.1
On the Real Line
......................... 306
13.3.2
Contour Integration
........................ 307
13.4
Differential Equations
........................... 308
13.5
Other Methods
.............................. 309
13.6
Notes and References
........................... 309
Problems
.................................. 310
14
Miscellany
313
14.1
Structured Matrices
............................ 313
14.1.1
Algebras and Groups
....................... 313
14.1.2
Monotone Functions
....................... 315
14.1.3
Other Structures
......................... 315
14.1.4
Data Sparse Representations
................... 316
14.1.5
Computing Structured /(A) for Structured A
......... 316
14.2
Exponential Decay of Functions of Banded Matrices
.......... 317
14.3
Approximating Entries of Matrix Functions
............... 318
A Notation
319
В
Background: Definitions and Useful Pacts
321
B.I Basic Notation
............................... 321
B.2 Eigenvalues and Jordan Canonical Form
................ 321
B.3 Invariant Subspaces
............................ 323
B.4 Special Classes of Matrices
........................ 323
B.5 Matrix Factorizations and Decompositions
............... 324
B.6
Pseudoinverse
and Orthogonality
.................... 325
B.6.1
Pseudoinverse
........................... 325
В.
6.2
Projector and Orthogonal Projector
............... 326
B.6.3 Partial Isometry
.......................... 326
B.7 Norms
................................... 326
B.8 Matrix Sequences and Series
....................... 328
B.9 Perturbation Expansions for Matrix Inverse
.............. 328
xii
Contents
8.
10
Sherman-Morrison-Woodbury Formula
................. 329
8.
11
Nonnegative
Matrices
........................... 329
В.
12
Positive (Senii)defmite Ordering
..................... 330
В.
13
Kroiiecker Product and Sum
....................... 331
B.14 Sylvester Equation
............................ 331
B.15 Floating Point Arithmetic
........................ 331
B.16 Divided Differences
............................ 332
Problems
..................................334
С
Operation Counts
335
D
Matrix Function Toolbox
339
Б
Solutions to Problems
343
Bibliography
379
Index
415
|
| any_adam_object | 1 |
| author | Higham, Nicholas J. 1961-2024 |
| author_GND | (DE-588)123564441 |
| author_facet | Higham, Nicholas J. 1961-2024 |
| author_role | aut |
| author_sort | Higham, Nicholas J. 1961-2024 |
| author_variant | n j h nj njh |
| building | Verbundindex |
| bvnumber | BV023244812 |
| callnumber-first | Q - Science |
| callnumber-label | QA188 |
| callnumber-raw | QA188 |
| callnumber-search | QA188 |
| callnumber-sort | QA 3188 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 780 SK 905 SK 220 SK 915 |
| classification_tum | MAT 150f |
| ctrlnum | (OCoLC)268627557 (DE-599)BVBBV023244812 |
| dewey-full | 512.9/434 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9/434 |
| dewey-search | 512.9/434 |
| dewey-sort | 3512.9 3434 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV023244812 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:11:31Z |
| institution | BVB |
| isbn | 9780898716467 |
| language | English |
| lccn | 2007061811 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016430292 |
| oclc_num | 268627557 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-20 DE-29T DE-706 DE-703 DE-384 DE-824 DE-83 DE-11 DE-M347 DE-188 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-739 DE-862 DE-BY-FWS |
| owner_facet | DE-91G DE-BY-TUM DE-20 DE-29T DE-706 DE-703 DE-384 DE-824 DE-83 DE-11 DE-M347 DE-188 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-739 DE-862 DE-BY-FWS |
| physical | XX, 425 S. Ill., graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | SIAM, Soc. for Industrial and Applied Math. |
| record_format | marc |
| spellingShingle | Higham, Nicholas J. 1961-2024 Functions of matrices theory and computation Matrices gtt Matrixfuncties gtt Numerieke wiskunde gtt Matrices Functions Factorization (Mathematics) Funktion Mathematik (DE-588)4071510-3 gnd Matrizenzerlegung (DE-588)4376303-0 gnd Matrixfunktion (DE-588)4169117-9 gnd Matrix Mathematik (DE-588)4037968-1 gnd |
| subject_GND | (DE-588)4071510-3 (DE-588)4376303-0 (DE-588)4169117-9 (DE-588)4037968-1 |
| title | Functions of matrices theory and computation |
| title_auth | Functions of matrices theory and computation |
| title_exact_search | Functions of matrices theory and computation |
| title_full | Functions of matrices theory and computation Nicholas J. Higham |
| title_fullStr | Functions of matrices theory and computation Nicholas J. Higham |
| title_full_unstemmed | Functions of matrices theory and computation Nicholas J. Higham |
| title_short | Functions of matrices |
| title_sort | functions of matrices theory and computation |
| title_sub | theory and computation |
| topic | Matrices gtt Matrixfuncties gtt Numerieke wiskunde gtt Matrices Functions Factorization (Mathematics) Funktion Mathematik (DE-588)4071510-3 gnd Matrizenzerlegung (DE-588)4376303-0 gnd Matrixfunktion (DE-588)4169117-9 gnd Matrix Mathematik (DE-588)4037968-1 gnd |
| topic_facet | Matrices Matrixfuncties Numerieke wiskunde Functions Factorization (Mathematics) Funktion Mathematik Matrizenzerlegung Matrixfunktion Matrix Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016430292&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT highamnicholasj functionsofmatricestheoryandcomputation |
Inhaltsverzeichnis
Paper/Kapitel scannen lassen
Paper/Kapitel scannen lassen
Teilbibliothek Mathematik & Informatik
| Signatur: |
0102 MAT 150f 2008 A 6668 Lageplan |
|---|---|
| Exemplar 1 | Ausleihbar Am Standort |
Teilbibliothek Physik
| Signatur: |
0202 MAT 150f 2010 B 178 Lageplan |
|---|---|
| Exemplar 1 | Ausleihbar Am Standort |