Sparse matrix technology:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London [u.a.]
Acad. Press
1984
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 321 S. graph. Darst. |
| ISBN: | 0125575807 |
Staff View
MARC
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| 100 | 1 | |a Pissanetzky, Sergio |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Sparse matrix technology |c Sergio Pissanetzky |
| 264 | 1 | |a London [u.a.] |b Acad. Press |c 1984 | |
| 300 | |a XIII, 321 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 4 | |a Matrices éparses - Informatique | |
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| 650 | 7 | |a élimination Gauss |2 inriac | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Sparse matrices |x Data processing | |
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Record in the Search Index
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| adam_text | Contents
Preface v
Introduction 1
Chapter 1 Fundamentals
1.1 Introduction .......... 4
1.2 Storage of arrays, lists, stacks and queues .... 5
1.3 Storage of lists of integers 8
1.4 Representation and storage of graphs 10
1.5 Diagonal storage of band matrices 13
1.6 Envelope storage of symmetric matrices 14
1.7 Linked sparse storage schemes 16
1.8 The sparse row wise format ....... 20
1.9 Ordered and unordered representations 22
1.10 Sherman s compression ........ 23
1.11 Storage of block partitioned matrices ..... 25
1.12 Symbolic processing and dynamic storage schemes ... 27
1.13 Merging sparse lists of integers ...... 30
1.14 The multiple switch technique 31
1.15 Addition of sparse vectors with the help of an expanded real
accumulator .......... 32
1.16 Addition of sparse vectors with the help of an expanded integer
array of pointers 35
1.17 Scalar product of two sparse vectors with the help of an array
of pointers .......... 36
Chapter 2 Linear Algebraic Equations
2.1 Introduction 38
2.2 Some definitions and properties 41
2.3 Elementary matrices and triangular matrices .... 43
2.4 Some properties of elementary matrices ..... 45
2.5 Some properties of triangular matrices 46
2.6 Permutation matrices 48
ix
X CONTENTS
2.7 Gauss elimination by columns 49
2.8 Gauss elimination by rows 53
2.9 Gauss Jordan elimination 55
2.10 Relation between the elimination form of the inverse and the
product form of the inverse 57
2.11 Cholesky factorization of a symmetric positive definite matrix 58
2.12 Practical implementation of Cholesky factorization. . . 60
2.13 Forward and backward substitution . . . . . 61
2.14 Cost considerations 62
2.15 Numerical examples 65
Chapter 3 Numerical Errors in Gausss Elimination
3.1 Introduction 69
3.2 Numerical errors in floating point operations . . . . 71
3.3 Numerical errors in sparse factorization 74
3.4 Numerical errors in sparse substitution 79
3.5 The control of numerical errors 84
3.6 Numerical stability and pivot selection 86
3.7 Monitoring or estimating element growth .... 90
3.8 Scaling 91
Chapter 4 Ordering for Gauss Elimination: Symmetric Matrices
4.1 Introduction: Statement of the problem 94
4.2 Basic notions of graph theory ....... 97
4.3 Breadth first search and adjacency level structures . . .103
4.4 Finding a pseudoperipheral vertex and a narrow level structure
of a graph 105
4.5 Reducing the bandwidth of a symmetric matrix . . .106
4.6 Reducing the profile of a symmetric matrix . . . .109
4.7 Graph theoretical background of symmetric Gauss elimination 112
4.8 The minimum degree algorithm 116
4.9 Tree partitioning of a symmetric sparse matrix . . .121
4.10 Nested dissection 125
4.11 Properties of nested dissection orderings . . . . 130
4.12 Generalized nested dissection 134
4.13 One way dissection for finite element problems . . .135
4.14 Orderings for the finite element method . . . . .141
4.15 Depth first search of an undirected graph . . . .146
4.16 Lexicographic search 152
4.17 Symmetric indefinite matrices 157
CONTENTS xi
Chapter 5 Ordering for Gauss Elimination: General Matrices
5.1 Introduction: Statement of the problem 159
5.2 Graph theory for unsymmetric matrices 162
5.3 The strong components of a digraph 165
5.4 Depth first search of a digraph 169
5.5 Breadth first search of a digraph and directed adjacency level
structures 172
5.6 Finding a maximal set of vertex disjoint paths in an acyclic
digraph 174
5.7 Finding a transversal: the algorithm of Hall . . . .176
5.8 Finding a transversal: the algorithm of Hopcroft and Karp . 179
5.9 The algorithm of Sargent and Westerberg for finding the strong
components of a digraph. . . . . . . .185
5.10 The algorithm of Tarjan for finding the strong components of a
digraph 187
5.11 Pivoting strategies for unsymmetric matrices . . . .191
5.12 Other methods and available software 194
Chapter 6 Sparse Eigenanalysis
6.1 Introduction 196
6.2 The Rayleigh quotient 200
6.3 Bounds for eigenvalues 202
6.4 The bisection method for eigenvalue calculations . . . 204
6.5 Reduction of a general matrix....... 205
6.6 Reduction of a symmetric band matrix to tridiagonal form . 207
6.7 Eigenanalysis of tridiagonal and Hessenberg matrices . . 209
6.8 Direct and inverse iteration 210
6.9 Subspaces and invariant subspaces 214
6.10 Simultaneous iteration 216
6.11 Lanczos algorithm . ¦ • . . 220
6.12 Lanczos algorithm in practice . ...... 225
6.13 Block Lanczos and band Lanczos algorithms .... 227
6.14 Trace minimization 230
6.15 Eigenanalysis of hermitian matrices 230
6.16 Unsymmetric eigenproblems 231
Chapter 7 Sparse Matrix Algebra
7.1 Introduction 234
7.2 Transposition of a sparse matrix 236
Xii CONTENTS
7.3 Algorithm for the transposition of a general sparse matrix . 238
7.4 Ordering a sparse representation ...... 239
7.5 Permutation of rows or columns of a sparse matrix: First
procedure .......... 240
7.6 Permutation of rows or columns of a sparse matrix: Second
procedure 241
7.7 Ordering of the upper representation of a sparse symmetric
matrix ........... 242
7.8 Addition of sparse matrices ....... 242
7.9 Example of addition of two sparse matrices .... 243
7.10 Algorithm for the symbolic addition of two sparse matrices with
N rows and M columns ........ 245
7.11 Algorithm for the numerical addition of two sparse matrices
with N rows 246
7.12 Product of a general sparse matrix by a column vector . 248
7.13 Algorithm for the product of a general sparse matrix by a full
column vector.......... 249
7.14 Product of a row vector by a general sparse matrix. . . 249
7.15 Example of product of a full row vector by a general sparse
matrix 250
7.16 Algorithm for the product of a full row vector by a general
sparse matrix . . . . . . . . . .251
7.17 Product of a symmetric sparse matrix by a column vector . 251
7.18 Algorithms for the product of a symmetric sparse matrix by a full
column vector.......... 252
7.19 Multiplication of sparse matrices ...... 253
7.20 Example of product of two matrices which are stored by rows 254
7.21 Algorithm for the symbolic multiplication of two sparse
matrices given in row wise format ...... 255
7.22 Algorithm for the numerial multiplication of two sparse
matrices given in row wise format ...... 256
7.23 Triangular factorization of a sparse symmetric matrix given in
row wise format ......... 258
7.24 Numerical triangular factorization of a sparse symmetric matrix
given in row wise format. . . . . . . .261
7.25 Algorithm for the symbolic triangular factorization of a sym¬
metric sparse matrix A 263
7.26 Algorithm for the numerical triangular factorization of a
symmetric positive definite sparse matrix A .... 265
7.27 Example of forward and backward substitution . . . 268
7.28 Algorithm for the solution of the system UTDUx = b . .269
contents xiii
Chapter 8 Connectivity and Nodal Assembly
8.1 Introduction 271
8.2 Boundary conditions for scalar problems. .... 274
8.3 Boundary conditions for vector problems .... 275
8.4 Example of a connectivity matrix ...... 279
8.5 Example of a nodal assembly matrix ..... 280
8.6 Algorithm for the symbolic assembly of a symmetric nodal
assembly matrix ... . . . . . . . 282
8.7 Algorithm for the numerical assembly of an element matrix and
vector into the nodal assembly matrix A and right hand vector
b: Symmetric case ......... 283
8.8 Algorithm for the numerical assembly of an element matrix and
vector into the nodal assembly matrix A and right hand vector
b: General case 286
Chapter 9 General Purpose Algorithms
9.1 Introduction 288
9.2 Multiplication of the inverse of a lower triangular matrix by a
general matrix 289
9.3 Algorithm for the symbolic multiplication of the inverse of a
lower triangular matrix U T by a general matrix B . . . 290
9.4 Algorithm for the numerical multiplication of the inverse of a
lower triangular matrix U~T by a general matrix B . . . 292
9.5 Algorithm for the multiplication of the inverse of an upper
triangular unit diagonal matrix U by a full vector x . . 293
9.6 Algorithm for the multiplication of the transpose inverse of an
upper triangular unit diagonal matrix U by a full vector . 294
9.7 Solution of linear equations by the Gauss Seidel iterative
method 295
9.8 Algorithm for the iterative solution of linear equations by the
Gauss Seidel method 295
9.9 Checking the representation of a sparse matrix . . . 297
9.10 Printing or displaying a sparse matrix 298
9.11 Algorithm for transforming a RR(C)U of a symmetric matrix
into a RR(U)U of the same matrix 298
9.12 Algorithm for the pre multiplication of a sparse matrix A by a
diagonal matrix D 299
9.13 Algorithm for copying a sparse matrix from I A, JA, AN to IB,
JB, BN 300
References 301
Subject Index 313
|
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| indexdate | 2024-07-09T15:43:39Z |
| institution | BVB |
| isbn | 0125575807 |
| language | English |
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| spelling | Pissanetzky, Sergio Verfasser aut Sparse matrix technology Sergio Pissanetzky London [u.a.] Acad. Press 1984 XIII, 321 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Matrices éparses - Informatique algèbre linéaire inriac algèbre matricielle inriac matrice creuse inriac élimination Gauss inriac Datenverarbeitung Sparse matrices Data processing Datenverarbeitung (DE-588)4011152-0 gnd rswk-swf Schwach besetzte Matrix (DE-588)4056053-3 gnd rswk-swf Schwach besetzte Matrix (DE-588)4056053-3 s DE-604 Datenverarbeitung (DE-588)4011152-0 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001512015&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pissanetzky, Sergio Sparse matrix technology Matrices éparses - Informatique algèbre linéaire inriac algèbre matricielle inriac matrice creuse inriac élimination Gauss inriac Datenverarbeitung Sparse matrices Data processing Datenverarbeitung (DE-588)4011152-0 gnd Schwach besetzte Matrix (DE-588)4056053-3 gnd |
| subject_GND | (DE-588)4011152-0 (DE-588)4056053-3 |
| title | Sparse matrix technology |
| title_auth | Sparse matrix technology |
| title_exact_search | Sparse matrix technology |
| title_full | Sparse matrix technology Sergio Pissanetzky |
| title_fullStr | Sparse matrix technology Sergio Pissanetzky |
| title_full_unstemmed | Sparse matrix technology Sergio Pissanetzky |
| title_short | Sparse matrix technology |
| title_sort | sparse matrix technology |
| topic | Matrices éparses - Informatique algèbre linéaire inriac algèbre matricielle inriac matrice creuse inriac élimination Gauss inriac Datenverarbeitung Sparse matrices Data processing Datenverarbeitung (DE-588)4011152-0 gnd Schwach besetzte Matrix (DE-588)4056053-3 gnd |
| topic_facet | Matrices éparses - Informatique algèbre linéaire algèbre matricielle matrice creuse élimination Gauss Datenverarbeitung Sparse matrices Data processing Schwach besetzte Matrix |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001512015&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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