Solving linear systems: an analysis of matrix prefactorization iterative methods
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Ithaca, NY
Matrix Editions
[2009]
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | xiv, 554 Seiten Illustrationen |
| ISBN: | 9780971576667 |
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MARC
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| 100 | 1 | |a Woźnicki, Zbigniew Ignacy |d 1937-2008 |0 (DE-588)139462465 |4 aut | |
| 245 | 1 | 0 | |a Solving linear systems |b an analysis of matrix prefactorization iterative methods |c Zbigniew Ignacy Woźnicki |
| 264 | 1 | |a Ithaca, NY |b Matrix Editions |c [2009] | |
| 264 | 4 | |c © 2009 | |
| 300 | |a xiv, 554 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Iterative methods (Mathematics) | |
| 650 | 4 | |a Factorization (Mathematics) | |
| 650 | 4 | |a Matrices | |
| 650 | 4 | |a Numerical analysis | |
| 650 | 4 | |a Differential equations, Linear |x Numerical solutions | |
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Record in the Search Index
| _version_ | 1804140736641236992 |
|---|---|
| adam_text | Contents
Foreword by Richard
Varga
xi
Preface
xiii
0
Introduction to iterative methods and preconditioning
1
1
Numerical linear algebra: background
5
1.1
Review of matrix theory
........................... 5
1.1.1
Basic matrix operations
....................... 7
1.1.2
Matrix partitionings
......................... 9
1.1.3
Basic concepts of matrix analysis
.................. 10
1.1.4
Vector norms and matrix norms
.................. 12
1.1.5
Computational work
........................ 15
1.2
Eigenvalues and eigenvectors
........................ 15
1.2.1
Relating norms and eigenvalues
................... 18
1.2.2
Convergence of vector and matrix sequences
............ 18
1.2.3
Perron-Frobenius theory of
nonnegative
matrices
......... 24
1.2.4
Diagonally dominant matrices
.................... 28
1.2.5
Power method
............................ 29
1.3
General linear systems
............................ 32
1.3.1
Direct methods
............................ 36
1.3.2
Iterative refinement
......................... 40
2
The theory of matrix splitting
41
2.1
General properties of matrix splittings
................... 41
2.2
Regular splittings
............................... 46
2.3
Nonnegative
and weak
nonnegative
splittings
............... 52
2.4
Weak and weaker splittings
......................... 61
2.5
Summary
................................... 68
3
Discretization of partial differential equations
70
3.1
Finite-difference approximations
...................... 71
3.2
One-dimensional problems
.......................... 73
3.2.1
Forward elimination
-
backward substitution
........... 75
3.2.2
Backward elimination
-
forward substitution
........... 76
3.3
Band matrices
................................ 78
3.3.1
Pentadiagonal matrices
....................... 78
viii
Contents
3.3.2 General band
systems
........................ 80
3.4
Two-dimensional problems
.......................... 83
3.4.1
Rectangular geometry
........................ 83
3.4.2
Triangular geometry
......................... 94
3.4.3
Hexagonal geometry
......................... 96
3.4.4
Reduced systems
........................... 101
3.4.5
Line
orderings
............................ 114
3.4.6
Computational molecules
...................... 118
3.4.7
Irregular mesh structures
...................... 121
3.5
Test problems
................................. 122
Standard iterative methods
129
4.1
General theory of iterative methods
.................... 129
4.1.1
Stopping criteria
........................... 133
4.1.2
Starting vectors
............................ 134
4.2
Point iterative methods
........................... 135
4.2.1
Basic algorithms
........................... 135
4.2.2
Consistent
orderings
......................... 138
4.2.3
The successive overrelaxation method
(SOR)
........... 142
4.2.4
Determining the optimum relaxation parameter
.......... 148
4.2.5
Experimental examination of
SOR
convergence
.......... 158
4.2.6
Computational aspects
........................ 171
4.3
Line iterative methods
............................ 176
4.3.1
1-line algorithms
........................... 178
4.3.2
2-line algorithms
........................... 189
4.3.3
3-line algorithms
........................... 193
4.4
Results of numerical experiments
...................... 195
Explicit prefactorization methods
(AGA)
199
5.1
Matrix notation
................................ 200
5.1.1
Basic algorithms
........................... 200
5.1.2
AGA
algorithms with point modification
.............. 208
5.1.3
AGA
algorithms with line modification
.............. 208
5.1.4
Techniques for accelerating convergence
.............. 209
5.2
Implementing prefactorization algorithms in mesh structures
...... 213
5.2.1
Rectangular geometry
........................ 213
5.2.2
Triangular geometry
......................... 234
5.2.3
Hexagonal geometry
......................... 240
5.3
Results of numerical experiments
...................... 243
Semi-explicit prefactorization with implicit backward sweep
248
6.1
Matrix notation
................................ 248
6.1.1
Modified line methods
........................ 250
6.2
Implementation in mesh structures
..................... 253
6.2.1
Rectangular geometry
........................ 253
6.2.2
Triangular geometry
......................... 260
6.2.3
Hexagonal geometry
......................... 262
6.3
Numerical experiments
............................ 265
x
Contents
7
Semi-explicit prefactorization with implicit forward sweep
267
7.1
Implementing OLA methods in mesh structures
.............. 268
7.1.1
Rectangular geometry
........................ 268
7.1.2
Triangular geometry
......................... 285
7.1.3
Hexagonal geometry
......................... 291
7.1.4
Successive overrelaxation
...................... 294
7.2
Matrix notation
................................ 295
7.3
Numerical experiments
............................ 302
7.4
Final discussion
................................ 304
8
Advances in solving linear control systems
308
8.1
Sylvester equations
.............................. 308
8.1.1
The SOR-like method
........................ 309
8.1.2
Numerical experiments
........................ 312
8.1.3
Sylvester equations: conclusion
................... 319
8.2
Continuous-time algebraic Riccati equations
................ 320
8.2.1
The SOR-like method
........................ 320
8.2.2
Numerical experiments
........................ 321
8.2.3
Concluding remarks
......................... 323
A Numerical experiments for chapter
4 325
A.I Standard iterative methods: self-adjoint problems
............. 325
A.
2
Standard iterative methods: non-self-adjoint problems
.......... 351
В
Numerical experiments for chapter
5 375
B.I
AGA
algorithms: self-adjoint problems
................... 375
B.2
AGA
algorithms: non-self-
adj
oint
problems
................ 455
С
Numerical experiments for chapter
6 473
(Semi-explicit prefactorization with implicit backward sweep)
C.I Self-adjoint problems
............................. 473
C.2 Non-self-adjoint problems
.......................... 487
D
Numerical experiments for chapter
7 496
(Semi-explicit prefactorization with implicit forward sweep)
D.I Self-adjoint problems
............................. 496
D.2 Non-self-adjoint problems
.......................... 514
Bibliography
529
Index
535
|
| any_adam_object | 1 |
| author | Woźnicki, Zbigniew Ignacy 1937-2008 |
| author_GND | (DE-588)139462465 |
| author_facet | Woźnicki, Zbigniew Ignacy 1937-2008 |
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| classification_rvk | SK 920 |
| ctrlnum | (OCoLC)320524435 (DE-599)BVBBV035794096 |
| dewey-full | 518/.26 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518/.26 |
| dewey-search | 518/.26 |
| dewey-sort | 3518 226 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T22:04:43Z |
| institution | BVB |
| isbn | 9780971576667 |
| language | English |
| lccn | 2009017867 |
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| physical | xiv, 554 Seiten Illustrationen |
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| publisher | Matrix Editions |
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| spelling | Woźnicki, Zbigniew Ignacy 1937-2008 (DE-588)139462465 aut Solving linear systems an analysis of matrix prefactorization iterative methods Zbigniew Ignacy Woźnicki Ithaca, NY Matrix Editions [2009] © 2009 xiv, 554 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Iterative methods (Mathematics) Factorization (Mathematics) Matrices Numerical analysis Differential equations, Linear Numerical solutions Elliptisches Randwertproblem (DE-588)4193399-0 gnd rswk-swf Iteration (DE-588)4123457-1 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 s Elliptisches Randwertproblem (DE-588)4193399-0 s Iteration (DE-588)4123457-1 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018653363&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Woźnicki, Zbigniew Ignacy 1937-2008 Solving linear systems an analysis of matrix prefactorization iterative methods Iterative methods (Mathematics) Factorization (Mathematics) Matrices Numerical analysis Differential equations, Linear Numerical solutions Elliptisches Randwertproblem (DE-588)4193399-0 gnd Iteration (DE-588)4123457-1 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4193399-0 (DE-588)4123457-1 (DE-588)4035811-2 |
| title | Solving linear systems an analysis of matrix prefactorization iterative methods |
| title_auth | Solving linear systems an analysis of matrix prefactorization iterative methods |
| title_exact_search | Solving linear systems an analysis of matrix prefactorization iterative methods |
| title_full | Solving linear systems an analysis of matrix prefactorization iterative methods Zbigniew Ignacy Woźnicki |
| title_fullStr | Solving linear systems an analysis of matrix prefactorization iterative methods Zbigniew Ignacy Woźnicki |
| title_full_unstemmed | Solving linear systems an analysis of matrix prefactorization iterative methods Zbigniew Ignacy Woźnicki |
| title_short | Solving linear systems |
| title_sort | solving linear systems an analysis of matrix prefactorization iterative methods |
| title_sub | an analysis of matrix prefactorization iterative methods |
| topic | Iterative methods (Mathematics) Factorization (Mathematics) Matrices Numerical analysis Differential equations, Linear Numerical solutions Elliptisches Randwertproblem (DE-588)4193399-0 gnd Iteration (DE-588)4123457-1 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Iterative methods (Mathematics) Factorization (Mathematics) Matrices Numerical analysis Differential equations, Linear Numerical solutions Elliptisches Randwertproblem Iteration Lineare Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018653363&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT woznickizbigniewignacy solvinglinearsystemsananalysisofmatrixprefactorizationiterativemethods |