Sparse Matrix Technology - electronic edition (Google eBook)

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Contents

511 Pivoting strategies for unsymmetric matrices
173
512 Other methods and available software
175
Sparse Eigenanalysis
177
62 The Rayleigh quotient
180
63 Bounds for eigenvalues
182
64 The bisection method for eigenvalue calculations
184
65 Reduction of a general matrix
185
66 Reduction of a symmetric band matrix to tridiag onal form
188

110 Shermans compression
22
111 Storage of blockpartitioned matrices
24
112 Symbolic processing and dynamic storage schemes
26
113 Merging sparse lists of integers
28
114 The multiple switch technique
29
115 Addition of sparse vectors with the help of an expanded real accumulator
30
116 Addition of sparse vectors with the help of an expanded integer array of pointers
32
117 Scalar product of two sparse vectors with the help of an array of pointers
33
Linear Algebraic Equations
35
22 Some definitions and properties
37
23 Elementary matrices and triangular matrices
40
24 Some properties of elementary matrices
41
25 Some properties of triangular matrices
42
26 Permutation matrices
44
27 Gauss elimination by columns
45
28 Gauss elimination by rows
49
29 GaussJordan elimination
50
210 Relation between the elimination form of the inverse and the product form of the inverse
52
211 Cholesky factorization of a symmetric positive definite matrix
53
212 Practical implementation of Cholesky factorization
55
213 Forward and backward substitution
56
214 Cost considerations
57
215 Numerical examples
59
Numerical Errors in Gauss Elimination
63
32 Numerical errors in floating point operations
65
33 Numerical errors in sparse factorization
68
34 Numerical errors in sparse substitution
73
35 The control of numerical errors
77
36 Numerical stability and pivot selection
78
37 Monitoring or estimating element growth
82
38 Scaling
83
Ordering for Gauss Elimination Symmetric Matrices
85
42 Basic notions of graph theory
87
43 Breadthfirst search and adjacency level structures
93
44 Finding a pseudoperipheral vertex and a narrow level structure of a graph
95
45 Reducing the bandwidth of a symmetric matrix
96
46 Reducing the profile of a symmetric matrix
98
47 Graphtheoretical background of symmetric Gauss elimination
101
48 The minimum degree algorithm
104
49 Tree partitioning of a symmetric sparse matrix
109
410 Nested dissection
113
411 Properties of nested dissection orderings
118
412 Generalized nested dissection
121
413 Oneway dissection of finite element problems
122
414 Orderings for the finite element method
127
415 Depthfirst search of an undirected graph
132
416 Lexicographic search
136
417 Symmetric indefinite matrices
140
Ordering for Gauss Elimination General Matrices
143
52 Graph theory for unsymmetric matrices
146
53 The strong components of a digraph
148
54 Depthfirst search of a digraph
151
55 Breadthfirst search of a digraph and directed adjacency level structures
155
56 Finding a maximal set of vertex disjoint paths in an acyclic digraph
157
the algorithm of Hall
158
the algorithm of Hopcroft and Karp
161
59 The algorithm of Sargent and Westerberg for finding the strong components of a digraph
167
510 The algorithm of Tarjan for finding the strong components of a digraph
168
67 Eigenanalysis of tridiagonal and Hessenberg matrices
189
69 Subspaces and invariant subspaces
193
610 Simultaneous iteration
195
611 Lanczos algorithm
199
612 Lanczos algorithm in practice
203
613 Block Lanczos and band Lanczos algorithms
206
614 Trace minimization
208
615 Eigenanalysis of hermitian matrices
209
616 Unsymmetric eigenproblems
210
Sparse Matrix Algebra
211
72 Transposition of a sparse matrix
213
73 Algorithm for the transposition of a general sparse matrix
215
74 Ordering a sparse representation
216
First procedure
217
Second procedure
218
78 Addition of sparse matrices
219
79 Example of addition of two sparse matrices
220
710 Algorithm for the symbolic addition of two sparse matrices with N rows and M columns
221
711 Algorithm for the numerical addition of two sparse matrices with N rows
223
712 Product of a general sparse matrix by a column vector
224
713 Algorithm for the product of a general sparse matrix by a full column vector
225
714 Product of a row vector by a general sparse matrix
226
716 Algorithm for the product of a full row vector by a general sparse matrix
227
717 Product of a symmetric sparse matrix by a column vector
228
718 Algorithm for the product of a symmetric sparse matrix by a full column vector
229
719 Multiplication of sparse matrices
230
720 Example of product of two matrices which are stored by rows
231
721 Algorithm for the symbolic multiplication of two sparse matrices given in rowwise format
233
722 Algorithm for the numerical multiplication of two sparse matrices given in rowwise format
234
723 Triangular factorization of a sparse symmetric matrix given in rowwise format
235
724 Numerical triangular factorization of a sparse symmetric matrix given in rowwise format
238
725 Algorithm for the symbolic triangular factorization of a symmetric sparse matrix A
240
726 Algorithm for the numerical triangular factorization of a symmetric positive definite sparse matrix A
242
727 Example of forward and backward substitution
245
728 Algorithm for the solution of the system UTDUx b
246
Connectivity and Nodal Assembly
249
82 Boundary conditions for scalar problems
251
83 Boundary conditions for vector problems
252
84 Example of a connectivity matrix
256
85 Example of a nodal assembly matrix
257
86 Algorithm for the symbolic assembly of a symmetric nodal assembly matrix
259
Symmetric case
261
General case
264
General Purpose Algorithms
267
92 Multiplication of the inverse of a lower triangular matrix by a general matrix
268
93 Algorithm for the symbolic multiplication of the inverse of a lower triangular matrix U T by a general matrix B
269
94 Algorithm for the numerical multiplication of the inverse of a lower triangular matrix U T by a general matrix B
270
95 Algorithm for the multiplication of the inverse of an upper triangular unit diagonal matrix U by a full vector x
272
96 Algorithm for the multiplication of the transpose inverse of an upper triangular unit diagonal matrix U by a full vector
273
97 Solution of linear equations by the GaussSeidel iterative method
274
98 Algorithm for the iterative solution of linear equations by the GaussSeidel method
275
99 Checking the representation of a sparse matrix
276
910 Printing and displaying a sparse matrix
277
911 Algorithm for transforming a RRCU of a symmetric matrix into a RRUU of the same matrix
278
912 Algorithm for the premultiplication of a sparse matrix A by a diagonal matrix D
279
References
281
Index
295
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