Mercurial > octave-nkf
annotate doc/interpreter/sparse.txi @ 19630:0e1f5a750d00
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| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Tue, 20 Jan 2015 10:24:46 -0500 |
| parents | 861a46deae1f 446c46af4b42 |
| children | 4197fc428c7d |
| rev | line source |
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1 @c Copyright (C) 2004-2013 David Bateman |
| 7018 | 2 @c |
| 3 @c This file is part of Octave. | |
| 4 @c | |
| 5 @c Octave is free software; you can redistribute it and/or modify it | |
| 6 @c under the terms of the GNU General Public License as published by the | |
| 7 @c Free Software Foundation; either version 3 of the License, or (at | |
| 8 @c your option) any later version. | |
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9 @c |
| 7018 | 10 @c Octave is distributed in the hope that it will be useful, but WITHOUT |
| 11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 13 @c for more details. | |
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14 @c |
| 7018 | 15 @c You should have received a copy of the GNU General Public License |
| 16 @c along with Octave; see the file COPYING. If not, see | |
| 17 @c <http://www.gnu.org/licenses/>. | |
| 5164 | 18 |
| 5648 | 19 @ifhtml |
| 20 @set htmltex | |
| 21 @end ifhtml | |
| 22 @iftex | |
| 23 @set htmltex | |
| 24 @end iftex | |
| 25 | |
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26 @node Sparse Matrices |
| 5164 | 27 @chapter Sparse Matrices |
| 28 | |
| 29 @menu | |
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30 * Basics:: Creation and Manipulation of Sparse Matrices |
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31 * Sparse Linear Algebra:: Linear Algebra on Sparse Matrices |
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32 * Iterative Techniques:: Iterative Techniques |
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33 * Real Life Example:: Using Sparse Matrices |
| 5164 | 34 @end menu |
| 35 | |
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36 @node Basics |
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37 @section Creation and Manipulation of Sparse Matrices |
| 5164 | 38 |
| 39 The size of mathematical problems that can be treated at any particular | |
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40 time is generally limited by the available computing resources. Both, |
| 5164 | 41 the speed of the computer and its available memory place limitation on |
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42 the problem size. |
| 5164 | 43 |
| 5506 | 44 There are many classes of mathematical problems which give rise to |
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45 matrices, where a large number of the elements are zero. In this case |
| 5164 | 46 it makes sense to have a special matrix type to handle this class of |
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47 problems where only the nonzero elements of the matrix are |
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48 stored. Not only does this reduce the amount of memory to store the |
| 5164 | 49 matrix, but it also means that operations on this type of matrix can |
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50 take advantage of the a priori knowledge of the positions of the |
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51 nonzero elements to accelerate their calculations. |
| 5164 | 52 |
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53 A matrix type that stores only the nonzero elements is generally called |
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54 sparse. It is the purpose of this document to discuss the basics of the |
| 5164 | 55 storage and creation of sparse matrices and the fundamental operations |
| 56 on them. | |
| 57 | |
| 58 @menu | |
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59 * Storage of Sparse Matrices:: |
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60 * Creating Sparse Matrices:: |
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61 * Information:: |
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62 * Operators and Functions:: |
| 5164 | 63 @end menu |
| 64 | |
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65 @node Storage of Sparse Matrices |
| 5164 | 66 @subsection Storage of Sparse Matrices |
| 67 | |
| 68 It is not strictly speaking necessary for the user to understand how | |
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69 sparse matrices are stored. However, such an understanding will help |
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70 to get an understanding of the size of sparse matrices. Understanding |
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71 the storage technique is also necessary for those users wishing to |
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72 create their own oct-files. |
| 5164 | 73 |
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74 There are many different means of storing sparse matrix data. What all |
| 5648 | 75 of the methods have in common is that they attempt to reduce the complexity |
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76 and storage given a priori knowledge of the particular class of problems |
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77 that will be solved. A good summary of the available techniques for storing |
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78 sparse matrix is given by @nospell{Saad} @footnote{Y. Saad "SPARSKIT: A basic |
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79 toolkit for sparse matrix computation", 1994, |
| 6620 | 80 @url{http://www-users.cs.umn.edu/~saad/software/SPARSKIT/paper.ps}}. |
| 5164 | 81 With full matrices, knowledge of the point of an element of the matrix |
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82 within the matrix is implied by its position in the computers memory. |
| 5164 | 83 However, this is not the case for sparse matrices, and so the positions |
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84 of the nonzero elements of the matrix must equally be stored. |
| 5164 | 85 |
| 86 An obvious way to do this is by storing the elements of the matrix as | |
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87 triplets, with two elements being their position in the array |
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88 (rows and column) and the third being the data itself. This is conceptually |
| 5164 | 89 easy to grasp, but requires more storage than is strictly needed. |
| 90 | |
| 5648 | 91 The storage technique used within Octave is the compressed column |
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92 format. It is similar to the Yale format. |
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93 @footnote{@url{http://en.wikipedia.org/wiki/Sparse_matrix#Yale_format}} |
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94 In this format the position of each element in a row and the data are |
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95 stored as previously. However, if we assume that all elements in the |
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96 same column are stored adjacent in the computers memory, then we only |
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97 need to store information on the number of nonzero elements in each |
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98 column, rather than their positions. Thus assuming that the matrix has |
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99 more nonzero elements than there are columns in the matrix, we win in |
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100 terms of the amount of memory used. |
| 5164 | 101 |
| 102 In fact, the column index contains one more element than the number of | |
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103 columns, with the first element always being zero. The advantage of |
| 7001 | 104 this is a simplification in the code, in that there is no special case |
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105 for the first or last columns. A short example, demonstrating this in |
| 5164 | 106 C is. |
| 107 | |
| 108 @example | |
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109 @group |
| 5164 | 110 for (j = 0; j < nc; j++) |
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111 for (i = cidx(j); i < cidx(j+1); i++) |
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112 printf ("nonzero element (%i,%i) is %d\n", |
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113 ridx(i), j, data(i)); |
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114 @end group |
| 5164 | 115 @end example |
| 116 | |
| 117 A clear understanding might be had by considering an example of how the | |
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118 above applies to an example matrix. Consider the matrix |
| 5164 | 119 |
| 120 @example | |
| 121 @group | |
| 122 1 2 0 0 | |
| 123 0 0 0 3 | |
| 124 0 0 0 4 | |
| 125 @end group | |
| 126 @end example | |
| 127 | |
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128 The nonzero elements of this matrix are |
| 5164 | 129 |
| 130 @example | |
| 131 @group | |
| 132 (1, 1) @result{} 1 | |
| 133 (1, 2) @result{} 2 | |
| 134 (2, 4) @result{} 3 | |
| 135 (3, 4) @result{} 4 | |
| 136 @end group | |
| 137 @end example | |
| 138 | |
| 139 This will be stored as three vectors @var{cidx}, @var{ridx} and @var{data}, | |
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140 representing the column indexing, row indexing and data respectively. The |
| 5164 | 141 contents of these three vectors for the above matrix will be |
| 142 | |
| 143 @example | |
| 144 @group | |
| 5506 | 145 @var{cidx} = [0, 1, 2, 2, 4] |
| 5164 | 146 @var{ridx} = [0, 0, 1, 2] |
| 147 @var{data} = [1, 2, 3, 4] | |
| 148 @end group | |
| 149 @end example | |
| 150 | |
| 151 Note that this is the representation of these elements with the first row | |
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152 and column assumed to start at zero, while in Octave itself the row and |
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153 column indexing starts at one. Thus the number of elements in the |
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154 @var{i}-th column is given by @code{@var{cidx} (@var{i} + 1) - |
| 5164 | 155 @var{cidx} (@var{i})}. |
| 156 | |
| 5648 | 157 Although Octave uses a compressed column format, it should be noted |
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158 that compressed row formats are equally possible. However, in the |
| 5648 | 159 context of mixed operations between mixed sparse and dense matrices, |
| 160 it makes sense that the elements of the sparse matrices are in the | |
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161 same order as the dense matrices. Octave stores dense matrices in |
| 5648 | 162 column major ordering, and so sparse matrices are equally stored in |
| 163 this manner. | |
| 5164 | 164 |
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165 A further constraint on the sparse matrix storage used by Octave is that |
| 5164 | 166 all elements in the rows are stored in increasing order of their row |
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167 index, which makes certain operations faster. However, it imposes |
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168 the need to sort the elements on the creation of sparse matrices. Having |
| 7001 | 169 disordered elements is potentially an advantage in that it makes operations |
| 5164 | 170 such as concatenating two sparse matrices together easier and faster, however |
| 171 it adds complexity and speed problems elsewhere. | |
| 172 | |
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173 @node Creating Sparse Matrices |
| 5164 | 174 @subsection Creating Sparse Matrices |
| 175 | |
| 176 There are several means to create sparse matrix. | |
| 177 | |
| 178 @table @asis | |
| 179 @item Returned from a function | |
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180 There are many functions that directly return sparse matrices. These include |
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181 @dfn{speye}, @dfn{sprand}, @dfn{diag}, etc. |
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182 |
| 5164 | 183 @item Constructed from matrices or vectors |
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184 The function @dfn{sparse} allows a sparse matrix to be constructed from |
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185 three vectors representing the row, column and data. Alternatively, the |
| 5164 | 186 function @dfn{spconvert} uses a three column matrix format to allow easy |
| 187 importation of data from elsewhere. | |
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188 |
| 5164 | 189 @item Created and then filled |
| 190 The function @dfn{sparse} or @dfn{spalloc} can be used to create an empty | |
| 191 matrix that is then filled by the user | |
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192 |
| 5164 | 193 @item From a user binary program |
| 194 The user can directly create the sparse matrix within an oct-file. | |
| 195 @end table | |
| 196 | |
| 197 There are several basic functions to return specific sparse | |
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198 matrices. For example the sparse identity matrix, is a matrix that is |
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199 often needed. It therefore has its own function to create it as |
| 5164 | 200 @code{speye (@var{n})} or @code{speye (@var{r}, @var{c})}, which |
| 201 creates an @var{n}-by-@var{n} or @var{r}-by-@var{c} sparse identity | |
| 202 matrix. | |
| 203 | |
| 204 Another typical sparse matrix that is often needed is a random distribution | |
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205 of random elements. The functions @dfn{sprand} and @dfn{sprandn} perform |
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206 this for uniform and normal random distributions of elements. They have exactly |
| 5506 | 207 the same calling convention, where @code{sprand (@var{r}, @var{c}, @var{d})}, |
| 208 creates an @var{r}-by-@var{c} sparse matrix with a density of filled | |
| 5164 | 209 elements of @var{d}. |
| 210 | |
| 7001 | 211 Other functions of interest that directly create sparse matrices, are |
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212 @dfn{diag} or its generalization @dfn{spdiags}, that can take the |
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213 definition of the diagonals of the matrix and create the sparse matrix |
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214 that corresponds to this. For example, |
| 5164 | 215 |
| 216 @example | |
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217 s = diag (sparse (randn (1,n)), -1); |
| 5164 | 218 @end example |
| 219 | |
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220 @noindent |
| 5164 | 221 creates a sparse (@var{n}+1)-by-(@var{n}+1) sparse matrix with a single |
| 222 diagonal defined. | |
| 223 | |
| 6620 | 224 @DOCSTRING(spdiags) |
| 225 | |
| 226 @DOCSTRING(speye) | |
| 227 | |
| 228 @DOCSTRING(spones) | |
| 229 | |
| 230 @DOCSTRING(sprand) | |
| 231 | |
| 232 @DOCSTRING(sprandn) | |
| 233 | |
| 234 @DOCSTRING(sprandsym) | |
| 235 | |
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236 The recommended way for the user to create a sparse matrix, is to create |
| 5648 | 237 two vectors containing the row and column index of the data and a third |
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238 vector of the same size containing the data to be stored. For example, |
| 5164 | 239 |
| 240 @example | |
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241 @group |
| 6421 | 242 ri = ci = d = []; |
| 243 for j = 1:c | |
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244 ri = [ri; randperm(r,n)']; |
| 6421 | 245 ci = [ci; j*ones(n,1)]; |
| 246 d = [d; rand(n,1)]; | |
| 247 endfor | |
| 248 s = sparse (ri, ci, d, r, c); | |
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249 @end group |
| 5164 | 250 @end example |
| 251 | |
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252 @noindent |
| 6421 | 253 creates an @var{r}-by-@var{c} sparse matrix with a random distribution |
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254 of @var{n} (<@var{r}) elements per column. The elements of the vectors |
| 6421 | 255 do not need to be sorted in any particular order as Octave will sort |
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256 them prior to storing the data. However, pre-sorting the data will |
| 6421 | 257 make the creation of the sparse matrix faster. |
| 5164 | 258 |
| 259 The function @dfn{spconvert} takes a three or four column real matrix. | |
| 260 The first two columns represent the row and column index respectively and | |
| 261 the third and four columns, the real and imaginary parts of the sparse | |
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262 matrix. The matrix can contain zero elements and the elements can be |
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263 sorted in any order. Adding zero elements is a convenient way to define |
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264 the size of the sparse matrix. For example: |
| 5164 | 265 |
| 266 @example | |
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267 @group |
| 5164 | 268 s = spconvert ([1 2 3 4; 1 3 4 4; 1 2 3 0]') |
| 269 @result{} Compressed Column Sparse (rows=4, cols=4, nnz=3) | |
| 270 (1 , 1) -> 1 | |
| 271 (2 , 3) -> 2 | |
| 272 (3 , 4) -> 3 | |
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273 @end group |
| 5164 | 274 @end example |
| 275 | |
| 276 An example of creating and filling a matrix might be | |
| 277 | |
| 278 @example | |
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279 @group |
| 5164 | 280 k = 5; |
| 281 nz = r * k; | |
| 282 s = spalloc (r, c, nz) | |
| 283 for j = 1:c | |
| 284 idx = randperm (r); | |
| 5648 | 285 s (:, j) = [zeros(r - k, 1); ... |
| 286 rand(k, 1)] (idx); | |
| 5164 | 287 endfor |
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288 @end group |
| 5164 | 289 @end example |
| 290 | |
| 5324 | 291 It should be noted, that due to the way that the Octave |
| 5164 | 292 assignment functions are written that the assignment will reallocate |
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293 the memory used by the sparse matrix at each iteration of the above loop. |
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294 Therefore the @dfn{spalloc} function ignores the @var{nz} argument and |
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295 does not pre-assign the memory for the matrix. Therefore, it is vitally |
| 5648 | 296 important that code using to above structure should be vectorized |
| 297 as much as possible to minimize the number of assignments and reduce the | |
| 5164 | 298 number of memory allocations. |
| 299 | |
| 6620 | 300 @DOCSTRING(full) |
| 301 | |
| 302 @DOCSTRING(spalloc) | |
| 303 | |
| 304 @DOCSTRING(sparse) | |
| 305 | |
| 306 @DOCSTRING(spconvert) | |
| 307 | |
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308 The above problem of memory reallocation can be avoided in |
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309 oct-files. However, the construction of a sparse matrix from an oct-file |
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310 is more complex than can be discussed here. @xref{External Code Interface}, |
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311 for a full description of the techniques involved. |
| 5164 | 312 |
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313 @node Information |
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314 @subsection Finding Information about Sparse Matrices |
| 5648 | 315 |
| 316 There are a number of functions that allow information concerning | |
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317 sparse matrices to be obtained. The most basic of these is |
| 5648 | 318 @dfn{issparse} that identifies whether a particular Octave object is |
| 319 in fact a sparse matrix. | |
| 320 | |
| 321 Another very basic function is @dfn{nnz} that returns the number of | |
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322 nonzero entries there are in a sparse matrix, while the function |
| 5648 | 323 @dfn{nzmax} returns the amount of storage allocated to the sparse |
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324 matrix. Note that Octave tends to crop unused memory at the first |
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325 opportunity for sparse objects. There are some cases of user created |
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326 sparse objects where the value returned by @dfn{nzmax} will not be |
| 5648 | 327 the same as @dfn{nnz}, but in general they will give the same |
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328 result. The function @dfn{spstats} returns some basic statistics on |
| 5648 | 329 the columns of a sparse matrix including the number of elements, the |
| 330 mean and the variance of each column. | |
| 331 | |
| 6620 | 332 @DOCSTRING(issparse) |
| 333 | |
| 334 @DOCSTRING(nnz) | |
| 335 | |
| 336 @DOCSTRING(nonzeros) | |
| 337 | |
| 338 @DOCSTRING(nzmax) | |
| 339 | |
| 340 @DOCSTRING(spstats) | |
| 341 | |
| 5648 | 342 When solving linear equations involving sparse matrices Octave |
| 343 determines the means to solve the equation based on the type of the | |
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344 matrix (@pxref{Sparse Linear Algebra}). Octave probes the |
| 5648 | 345 matrix type when the div (/) or ldiv (\) operator is first used with |
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346 the matrix and then caches the type. However the @dfn{matrix_type} |
| 5648 | 347 function can be used to determine the type of the sparse matrix prior |
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348 to use of the div or ldiv operators. For example, |
| 5648 | 349 |
| 350 @example | |
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351 @group |
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352 a = tril (sprandn (1024, 1024, 0.02), -1) ... |
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353 + speye (1024); |
| 5648 | 354 matrix_type (a); |
| 355 ans = Lower | |
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356 @end group |
| 5648 | 357 @end example |
| 358 | |
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359 @noindent |
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360 shows that Octave correctly determines the matrix type for lower |
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361 triangular matrices. @dfn{matrix_type} can also be used to force |
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362 the type of a matrix to be a particular type. For example: |
| 5648 | 363 |
| 364 @example | |
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365 @group |
| 5648 | 366 a = matrix_type (tril (sprandn (1024, ... |
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367 1024, 0.02), -1) + speye (1024), "Lower"); |
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368 @end group |
| 5648 | 369 @end example |
| 370 | |
| 371 This allows the cost of determining the matrix type to be | |
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372 avoided. However, incorrectly defining the matrix type will result in |
| 5648 | 373 incorrect results from solutions of linear equations, and so it is |
| 374 entirely the responsibility of the user to correctly identify the | |
| 375 matrix type | |
| 376 | |
| 377 There are several graphical means of finding out information about | |
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378 sparse matrices. The first is the @dfn{spy} command, which displays |
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379 the structure of the nonzero elements of the |
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380 matrix. @xref{fig:spmatrix}, for an example of the use of |
| 5704 | 381 @dfn{spy}. More advanced graphical information can be obtained with the |
| 5648 | 382 @dfn{treeplot}, @dfn{etreeplot} and @dfn{gplot} commands. |
| 383 | |
| 384 @float Figure,fig:spmatrix | |
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385 @center @image{spmatrix,4in} |
| 5648 | 386 @caption{Structure of simple sparse matrix.} |
| 387 @end float | |
| 388 | |
| 389 One use of sparse matrices is in graph theory, where the | |
| 7001 | 390 interconnections between nodes are represented as an adjacency |
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391 matrix. That is, if the i-th node in a graph is connected to the j-th |
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392 node. Then the ij-th node (and in the case of undirected graphs the |
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393 @nospell{ji-th} node) of the sparse adjacency matrix is nonzero. If each node |
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394 is then associated with a set of coordinates, then the @dfn{gplot} |
| 5648 | 395 command can be used to graphically display the interconnections |
| 396 between nodes. | |
| 397 | |
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398 As a trivial example of the use of @dfn{gplot} consider the example, |
| 5648 | 399 |
| 400 @example | |
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401 @group |
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402 A = sparse ([2,6,1,3,2,4,3,5,4,6,1,5], |
| 5648 | 403 [1,1,2,2,3,3,4,4,5,5,6,6],1,6,6); |
| 404 xy = [0,4,8,6,4,2;5,0,5,7,5,7]'; | |
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405 gplot (A,xy) |
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406 @end group |
| 5648 | 407 @end example |
| 408 | |
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409 @noindent |
| 5648 | 410 which creates an adjacency matrix @code{A} where node 1 is connected |
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411 to nodes 2 and 6, node 2 with nodes 1 and 3, etc. The coordinates of |
| 5648 | 412 the nodes are given in the n-by-2 matrix @code{xy}. |
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413 @ifset htmltex |
| 5648 | 414 @xref{fig:gplot}. |
| 415 | |
| 416 @float Figure,fig:gplot | |
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417 @center @image{gplot,4in} |
| 5648 | 418 @caption{Simple use of the @dfn{gplot} command.} |
| 419 @end float | |
| 420 @end ifset | |
| 421 | |
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422 The dependencies between the nodes of a Cholesky@tie{}factorization can be |
| 5648 | 423 calculated in linear time without explicitly needing to calculate the |
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424 Cholesky@tie{}factorization by the @code{etree} command. This command |
| 5648 | 425 returns the elimination tree of the matrix and can be displayed |
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426 graphically by the command @code{treeplot (etree (A))} if @code{A} is |
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427 symmetric or @code{treeplot (etree (A+A'))} otherwise. |
| 5648 | 428 |
| 6620 | 429 @DOCSTRING(spy) |
| 430 | |
| 431 @DOCSTRING(etree) | |
| 432 | |
| 433 @DOCSTRING(etreeplot) | |
| 434 | |
| 435 @DOCSTRING(gplot) | |
| 436 | |
| 437 @DOCSTRING(treeplot) | |
| 438 | |
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439 @DOCSTRING(treelayout) |
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440 |
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441 @node Operators and Functions |
| 5164 | 442 @subsection Basic Operators and Functions on Sparse Matrices |
| 443 | |
| 444 @menu | |
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445 * Sparse Functions:: |
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446 * Return Types of Operators and Functions:: |
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447 * Mathematical Considerations:: |
| 5164 | 448 @end menu |
| 449 | |
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450 @node Sparse Functions |
| 5648 | 451 @subsubsection Sparse Functions |
| 452 | |
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453 Many Octave functions have been overloaded to work with either sparse or full |
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454 matrices. There is no difference in calling convention when using an |
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455 overloaded function with a sparse matrix, however, there is also no access to |
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456 potentially sparse-specific features. At any time the sparse matrix specific |
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457 version of a function can be used by explicitly calling its function name. |
| 5648 | 458 |
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459 The table below lists all of the sparse functions of Octave. Note that the |
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460 names of the specific sparse forms of the functions are typically the same as |
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461 the general versions with a @dfn{sp} prefix. In the table below, and in the |
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462 rest of this article, the specific sparse versions of functions are used. |
| 6620 | 463 |
| 464 @c Table includes in comments the missing sparse functions | |
| 5648 | 465 |
| 466 @table @asis | |
| 467 @item Generate sparse matrices: | |
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468 @dfn{spalloc}, @dfn{spdiags}, @dfn{speye}, @dfn{sprand}, |
| 5648 | 469 @dfn{sprandn}, @dfn{sprandsym} |
| 470 | |
| 471 @item Sparse matrix conversion: | |
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472 @dfn{full}, @dfn{sparse}, @dfn{spconvert} |
| 5648 | 473 |
| 474 @item Manipulate sparse matrices | |
| 475 @dfn{issparse}, @dfn{nnz}, @dfn{nonzeros}, @dfn{nzmax}, | |
| 6620 | 476 @dfn{spfun}, @dfn{spones}, @dfn{spy} |
| 5164 | 477 |
| 5648 | 478 @item Graph Theory: |
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479 @dfn{etree}, @dfn{etreeplot}, @dfn{gplot}, |
| 6620 | 480 @dfn{treeplot} |
| 481 @c @dfn{treelayout} | |
| 5648 | 482 |
| 483 @item Sparse matrix reordering: | |
| 7619 | 484 @dfn{amd}, @dfn{ccolamd}, @dfn{colamd}, @dfn{colperm}, @dfn{csymamd}, |
| 6620 | 485 @dfn{dmperm}, @dfn{symamd}, @dfn{randperm}, @dfn{symrcm} |
| 5648 | 486 |
| 487 @item Linear algebra: | |
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488 @dfn{condest}, @dfn{eigs}, @dfn{matrix_type}, |
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489 @dfn{normest}, @dfn{sprank}, @dfn{spaugment}, @dfn{svds} |
| 5648 | 490 |
| 491 @item Iterative techniques: | |
| 19182 | 492 @dfn{ichol}, @dfn{ilu}, @dfn{pcg}, @dfn{pcr} |
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493 @c @dfn{bicg}, @dfn{bicgstab}, @dfn{cholinc}, @dfn{cgs}, @dfn{gmres}, |
| 6620 | 494 @c @dfn{lsqr}, @dfn{minres}, @dfn{qmr}, @dfn{symmlq} |
| 5648 | 495 |
| 496 @item Miscellaneous: | |
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497 @dfn{spparms}, @dfn{symbfact}, @dfn{spstats} |
| 5648 | 498 @end table |
| 499 | |
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500 In addition all of the standard Octave mapper functions (i.e., basic |
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501 math functions that take a single argument) such as @dfn{abs}, etc. |
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502 can accept sparse matrices. The reader is referred to the documentation |
| 5648 | 503 supplied with these functions within Octave itself for further |
| 504 details. | |
| 5164 | 505 |
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506 @node Return Types of Operators and Functions |
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507 @subsubsection Return Types of Operators and Functions |
| 5164 | 508 |
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509 The two basic reasons to use sparse matrices are to reduce the memory |
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510 usage and to not have to do calculations on zero elements. The two are |
| 5164 | 511 closely related in that the computation time on a sparse matrix operator |
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512 or function is roughly linear with the number of nonzero elements. |
| 5164 | 513 |
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514 Therefore, there is a certain density of nonzero elements of a matrix |
| 5164 | 515 where it no longer makes sense to store it as a sparse matrix, but rather |
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516 as a full matrix. For this reason operators and functions that have a |
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517 high probability of returning a full matrix will always return one. For |
| 5164 | 518 example adding a scalar constant to a sparse matrix will almost always |
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519 make it a full matrix, and so the example, |
| 5164 | 520 |
| 521 @example | |
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522 @group |
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523 speye (3) + 0 |
| 5164 | 524 @result{} 1 0 0 |
| 525 0 1 0 | |
| 526 0 0 1 | |
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527 @end group |
| 5164 | 528 @end example |
| 529 | |
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530 @noindent |
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531 returns a full matrix as can be seen. |
| 7330 | 532 |
| 533 | |
| 534 Additionally, if @code{sparse_auto_mutate} is true, all sparse functions | |
| 535 test the amount of memory occupied by the sparse matrix to see if the | |
| 536 amount of storage used is larger than the amount used by the full | |
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537 equivalent. Therefore @code{speye (2) * 1} will return a full matrix as |
| 5164 | 538 the memory used is smaller for the full version than the sparse version. |
| 539 | |
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540 As all of the mixed operators and functions between full and sparse |
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541 matrices exist, in general this does not cause any problems. However, |
| 5164 | 542 one area where it does cause a problem is where a sparse matrix is |
| 543 promoted to a full matrix, where subsequent operations would resparsify | |
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544 the matrix. Such cases are rare, but can be artificially created, for |
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545 example @code{(fliplr (speye (3)) + speye (3)) - speye (3)} gives a full |
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546 matrix when it should give a sparse one. In general, where such cases |
| 5164 | 547 occur, they impose only a small memory penalty. |
| 548 | |
| 5648 | 549 There is however one known case where this behavior of Octave's |
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550 sparse matrices will cause a problem. That is in the handling of the |
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551 @dfn{diag} function. Whether @dfn{diag} returns a sparse or full matrix |
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552 depending on the type of its input arguments. So |
| 5164 | 553 |
| 554 @example | |
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555 a = diag (sparse ([1,2,3]), -1); |
| 5164 | 556 @end example |
| 557 | |
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558 @noindent |
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559 should return a sparse matrix. To ensure this actually happens, the |
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560 @dfn{sparse} function, and other functions based on it like @dfn{speye}, |
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561 always returns a sparse matrix, even if the memory used will be larger |
| 5164 | 562 than its full representation. |
| 563 | |
| 7330 | 564 @DOCSTRING(sparse_auto_mutate) |
| 565 | |
| 566 Note that the @code{sparse_auto_mutate} option is incompatible with | |
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567 @sc{matlab}, and so it is off by default. |
| 7330 | 568 |
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569 @node Mathematical Considerations |
| 5164 | 570 @subsubsection Mathematical Considerations |
| 571 | |
| 572 The attempt has been made to make sparse matrices behave in exactly the | |
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573 same manner as there full counterparts. However, there are certain differences |
| 5164 | 574 and especially differences with other products sparse implementations. |
| 575 | |
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576 First, the @qcode{"./"} and @qcode{".^"} operators must be used with care. |
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577 Consider what the examples |
| 5164 | 578 |
| 579 @example | |
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580 @group |
| 5164 | 581 s = speye (4); |
| 582 a1 = s .^ 2; | |
| 583 a2 = s .^ s; | |
| 584 a3 = s .^ -2; | |
| 585 a4 = s ./ 2; | |
| 586 a5 = 2 ./ s; | |
| 587 a6 = s ./ s; | |
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588 @end group |
| 5164 | 589 @end example |
| 590 | |
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591 @noindent |
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592 will give. The first example of @var{s} raised to the power of 2 causes |
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593 no problems. However @var{s} raised element-wise to itself involves a |
| 6431 | 594 large number of terms @code{0 .^ 0} which is 1. There @code{@var{s} .^ |
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595 @var{s}} is a full matrix. |
| 5164 | 596 |
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597 Likewise @code{@var{s} .^ -2} involves terms like @code{0 .^ -2} which |
| 5164 | 598 is infinity, and so @code{@var{s} .^ -2} is equally a full matrix. |
| 599 | |
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600 For the "./" operator @code{@var{s} ./ 2} has no problems, but |
| 5164 | 601 @code{2 ./ @var{s}} involves a large number of infinity terms as well |
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602 and is equally a full matrix. The case of @code{@var{s} ./ @var{s}} |
| 5164 | 603 involves terms like @code{0 ./ 0} which is a @code{NaN} and so this |
| 604 is equally a full matrix with the zero elements of @var{s} filled with | |
| 605 @code{NaN} values. | |
| 606 | |
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607 The above behavior is consistent with full matrices, but is not |
| 5164 | 608 consistent with sparse implementations in other products. |
| 609 | |
| 610 A particular problem of sparse matrices comes about due to the fact that | |
| 611 as the zeros are not stored, the sign-bit of these zeros is equally not | |
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612 stored. In certain cases the sign-bit of zero is important. For example: |
| 5164 | 613 |
| 614 @example | |
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615 @group |
| 5164 | 616 a = 0 ./ [-1, 1; 1, -1]; |
| 617 b = 1 ./ a | |
| 618 @result{} -Inf Inf | |
| 619 Inf -Inf | |
| 620 c = 1 ./ sparse (a) | |
| 621 @result{} Inf Inf | |
| 622 Inf Inf | |
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623 @end group |
| 5164 | 624 @end example |
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625 |
| 5648 | 626 To correct this behavior would mean that zero elements with a negative |
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627 sign-bit would need to be stored in the matrix to ensure that their |
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628 sign-bit was respected. This is not done at this time, for reasons of |
| 6750 | 629 efficiency, and so the user is warned that calculations where the sign-bit |
| 5164 | 630 of zero is important must not be done using sparse matrices. |
| 631 | |
| 5648 | 632 In general any function or operator used on a sparse matrix will |
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633 result in a sparse matrix with the same or a larger number of nonzero |
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634 elements than the original matrix. This is particularly true for the |
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635 important case of sparse matrix factorizations. The usual way to |
| 5648 | 636 address this is to reorder the matrix, such that its factorization is |
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637 sparser than the factorization of the original matrix. That is the |
| 5648 | 638 factorization of @code{L * U = P * S * Q} has sparser terms @code{L} |
| 639 and @code{U} than the equivalent factorization @code{L * U = S}. | |
| 640 | |
| 641 Several functions are available to reorder depending on the type of the | |
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642 matrix to be factorized. If the matrix is symmetric positive-definite, |
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643 then @dfn{symamd} or @dfn{csymamd} should be used. Otherwise |
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644 @dfn{amd}, @dfn{colamd} or @dfn{ccolamd} should be used. For completeness |
| 5648 | 645 the reordering functions @dfn{colperm} and @dfn{randperm} are |
| 646 also available. | |
| 647 | |
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648 @xref{fig:simplematrix}, for an example of the structure of a simple |
| 5648 | 649 positive definite matrix. |
| 5506 | 650 |
| 5648 | 651 @float Figure,fig:simplematrix |
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652 @center @image{spmatrix,4in} |
| 5648 | 653 @caption{Structure of simple sparse matrix.} |
| 654 @end float | |
| 5506 | 655 |
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656 The standard Cholesky@tie{}factorization of this matrix can be |
| 5648 | 657 obtained by the same command that would be used for a full |
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658 matrix. This can be visualized with the command |
|
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|
659 @code{r = chol (A); spy (r);}. |
| 5652 | 660 @xref{fig:simplechol}. |
|
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|
661 The original matrix had |
| 5648 | 662 @ifinfo |
| 663 @ifnothtml | |
| 664 43 | |
| 665 @end ifnothtml | |
| 666 @end ifinfo | |
| 667 @ifset htmltex | |
| 668 598 | |
| 669 @end ifset | |
|
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Rik <rik@octave.org>
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18845
diff
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|
670 nonzero terms, while this Cholesky@tie{}factorization has |
| 5648 | 671 @ifinfo |
| 672 @ifnothtml | |
| 673 71, | |
| 674 @end ifnothtml | |
| 675 @end ifinfo | |
| 676 @ifset htmltex | |
| 677 10200, | |
| 678 @end ifset | |
|
9066
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Cleanup documentation for diagperm.texi, sparse.texi
Rik <rdrider0-list@yahoo.com>
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diff
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|
679 with only half of the symmetric matrix being stored. This |
| 5648 | 680 is a significant level of fill in, and although not an issue |
|
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|
681 for such a small test case, can represents a large overhead |
| 5648 | 682 in working with other sparse matrices. |
| 5164 | 683 |
| 5648 | 684 The appropriate sparsity preserving permutation of the original |
| 685 matrix is given by @dfn{symamd} and the factorization using this | |
|
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parents:
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diff
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|
686 reordering can be visualized using the command @code{q = symamd (A); |
|
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John W. Eaton <jwe@octave.org>
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diff
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|
687 r = chol (A(q,q)); spy (r)}. This gives |
| 5648 | 688 @ifinfo |
| 689 @ifnothtml | |
| 690 29 | |
| 691 @end ifnothtml | |
| 692 @end ifinfo | |
| 693 @ifset htmltex | |
| 694 399 | |
| 695 @end ifset | |
|
18851
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doc: Replace "non-zero" with "nonzero" to match existing usage.
Rik <rik@octave.org>
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18845
diff
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|
696 nonzero terms which is a significant improvement. |
| 5164 | 697 |
|
11593
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|
698 The Cholesky@tie{}factorization itself can be used to determine the |
| 5648 | 699 appropriate sparsity preserving reordering of the matrix during the |
| 700 factorization, In that case this might be obtained with three return | |
|
14856
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parents:
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diff
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|
701 arguments as @code{[r, p, q] = chol (A); spy (r)}. |
| 5164 | 702 |
| 5648 | 703 @float Figure,fig:simplechol |
|
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|
704 @center @image{spchol,4in} |
|
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16816
diff
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|
705 @caption{Structure of the unpermuted Cholesky@tie{}factorization of the above matrix.} |
| 5648 | 706 @end float |
| 5164 | 707 |
| 5648 | 708 @float Figure,fig:simplecholperm |
|
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Fix documentation image printing under new development code
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diff
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|
709 @center @image{spcholperm,4in} |
|
11593
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diff
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|
710 @caption{Structure of the permuted Cholesky@tie{}factorization of the above matrix.} |
| 5648 | 711 @end float |
| 5164 | 712 |
| 5648 | 713 In the case of an asymmetric matrix, the appropriate sparsity |
| 714 preserving permutation is @dfn{colamd} and the factorization using | |
|
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Grammarcheck .txi documentation files.
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10791
diff
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|
715 this reordering can be visualized using the command |
|
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parents:
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diff
changeset
|
716 @code{q = colamd (A); [l, u, p] = lu (A(:,q)); spy (l+u)}. |
| 5164 | 717 |
| 5648 | 718 Finally, Octave implicitly reorders the matrix when using the div (/) |
| 719 and ldiv (\) operators, and so no the user does not need to explicitly | |
| 720 reorder the matrix to maximize performance. | |
| 721 | |
| 7619 | 722 @DOCSTRING(amd) |
| 723 | |
| 6620 | 724 @DOCSTRING(ccolamd) |
| 725 | |
| 726 @DOCSTRING(colamd) | |
| 727 | |
| 728 @DOCSTRING(colperm) | |
| 729 | |
| 730 @DOCSTRING(csymamd) | |
| 731 | |
| 732 @DOCSTRING(dmperm) | |
| 733 | |
| 734 @DOCSTRING(symamd) | |
| 735 | |
| 736 @DOCSTRING(symrcm) | |
| 737 | |
|
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include docstrings for new functions in the manual
John W. Eaton <jwe@octave.org>
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diff
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|
738 @node Sparse Linear Algebra |
| 5164 | 739 @section Linear Algebra on Sparse Matrices |
| 740 | |
|
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diff
changeset
|
741 Octave includes a polymorphic solver for sparse matrices, where |
| 5164 | 742 the exact solver used to factorize the matrix, depends on the properties |
|
9066
be150a172010
Cleanup documentation for diagperm.texi, sparse.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
743 of the sparse matrix itself. Generally, the cost of determining the matrix type |
| 5322 | 744 is small relative to the cost of factorizing the matrix itself, but in any |
| 745 case the matrix type is cached once it is calculated, so that it is not | |
| 746 re-determined each time it is used in a linear equation. | |
| 5164 | 747 |
| 748 The selection tree for how the linear equation is solve is | |
| 749 | |
| 750 @enumerate 1 | |
| 5648 | 751 @item If the matrix is diagonal, solve directly and goto 8 |
| 5164 | 752 |
| 753 @item If the matrix is a permuted diagonal, solve directly taking into | |
|
9066
be150a172010
Cleanup documentation for diagperm.texi, sparse.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
754 account the permutations. Goto 8 |
| 5164 | 755 |
| 5648 | 756 @item If the matrix is square, banded and if the band density is less |
| 757 than that given by @code{spparms ("bandden")} continue, else goto 4. | |
| 5164 | 758 |
| 759 @enumerate a | |
|
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John W. Eaton <jwe@octave.org>
parents:
18830
diff
changeset
|
760 @item If the matrix is tridiagonal and the right-hand side is not sparse |
| 5648 | 761 continue, else goto 3b. |
| 5164 | 762 |
| 763 @enumerate | |
|
10828
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Grammarcheck .txi documentation files.
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parents:
10791
diff
changeset
|
764 @item If the matrix is Hermitian, with a positive real diagonal, attempt |
|
11593
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Use non-breaking spaces between certain adjectives and their nouns in docstrings.
Rik <octave@nomad.inbox5.com>
parents:
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diff
changeset
|
765 Cholesky@tie{}factorization using @sc{lapack} xPTSV. |
| 5164 | 766 |
|
10828
322f43e0e170
Grammarcheck .txi documentation files.
Rik <octave@nomad.inbox5.com>
parents:
10791
diff
changeset
|
767 @item If the above failed or the matrix is not Hermitian with a positive |
|
19627
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John W. Eaton <jwe@octave.org>
parents:
18830
diff
changeset
|
768 real diagonal use Gaussian elimination with pivoting using |
|
9088
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Fix documentation image printing under new development code
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diff
changeset
|
769 @sc{lapack} xGTSV, and goto 8. |
| 5164 | 770 @end enumerate |
| 771 | |
|
10828
322f43e0e170
Grammarcheck .txi documentation files.
Rik <octave@nomad.inbox5.com>
parents:
10791
diff
changeset
|
772 @item If the matrix is Hermitian with a positive real diagonal, attempt |
|
11593
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Use non-breaking spaces between certain adjectives and their nouns in docstrings.
Rik <octave@nomad.inbox5.com>
parents:
11573
diff
changeset
|
773 Cholesky@tie{}factorization using @sc{lapack} xPBTRF. |
| 5164 | 774 |
|
10828
322f43e0e170
Grammarcheck .txi documentation files.
Rik <octave@nomad.inbox5.com>
parents:
10791
diff
changeset
|
775 @item if the above failed or the matrix is not Hermitian with a positive |
|
19627
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strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents:
18830
diff
changeset
|
776 real diagonal use Gaussian elimination with pivoting using |
|
9088
77e71f3da3d6
Fix documentation image printing under new development code
Rik <rdrider0-list@yahoo.com>
parents:
9066
diff
changeset
|
777 @sc{lapack} xGBTRF, and goto 8. |
| 5164 | 778 @end enumerate |
| 779 | |
| 780 @item If the matrix is upper or lower triangular perform a sparse forward | |
| 5648 | 781 or backward substitution, and goto 8 |
| 5164 | 782 |
|
11573
6f8ffe2c6f76
Grammarcheck txi files for 3.4 release.
Rik <octave@nomad.inbox5.com>
parents:
11523
diff
changeset
|
783 @item If the matrix is an upper triangular matrix with column permutations |
|
19627
446c46af4b42
strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents:
18830
diff
changeset
|
784 or lower triangular matrix with row permutations, perform a sparse forward |
| 5648 | 785 or backward substitution, and goto 8 |
| 5164 | 786 |
|
10828
322f43e0e170
Grammarcheck .txi documentation files.
Rik <octave@nomad.inbox5.com>
parents:
10791
diff
changeset
|
787 @item If the matrix is square, Hermitian with a real positive diagonal, attempt |
|
11593
1577c6f80926
Use non-breaking spaces between certain adjectives and their nouns in docstrings.
Rik <octave@nomad.inbox5.com>
parents:
11573
diff
changeset
|
788 sparse Cholesky@tie{}factorization using @sc{cholmod}. |
| 5164 | 789 |
|
11593
1577c6f80926
Use non-breaking spaces between certain adjectives and their nouns in docstrings.
Rik <octave@nomad.inbox5.com>
parents:
11573
diff
changeset
|
790 @item If the sparse Cholesky@tie{}factorization failed or the matrix is not |
|
19627
446c46af4b42
strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents:
18830
diff
changeset
|
791 Hermitian with a real positive diagonal, and the matrix is square, factorize |
|
10711
fbd7843974fa
Periodic grammar check of documentation files to ensure common format.
Rik <octave@nomad.inbox5.com>
parents:
10668
diff
changeset
|
792 using @sc{umfpack}. |
| 5164 | 793 |
| 794 @item If the matrix is not square, or any of the previous solvers flags | |
| 5648 | 795 a singular or near singular matrix, find a minimum norm solution using |
|
18713
a142f35f3cb6
doc: Fix unbalanced parentheses in documentation.
Rik <rik@octave.org>
parents:
17744
diff
changeset
|
796 @sc{cxsparse}@footnote{The @sc{cholmod}, @sc{umfpack} and @sc{cxsparse} |
|
a142f35f3cb6
doc: Fix unbalanced parentheses in documentation.
Rik <rik@octave.org>
parents:
17744
diff
changeset
|
797 packages were written by Tim Davis and are available at |
|
16792
b157ba28f123
doc: Use Texinfo @url command consistently throughout documentation.
Rik <rik@octave.org>
parents:
16601
diff
changeset
|
798 @url{http://www.cise.ufl.edu/research/sparse/}}. |
| 5164 | 799 @end enumerate |
| 800 | |
|
18851
9ac2357f19bc
doc: Replace "non-zero" with "nonzero" to match existing usage.
Rik <rik@octave.org>
parents:
18845
diff
changeset
|
801 The band density is defined as the number of nonzero values in the band |
|
16816
12005245b645
doc: Periodic grammarcheck of documentation.
Rik <rik@octave.org>
parents:
16792
diff
changeset
|
802 divided by the total number of values in the full band. The banded |
|
16600
f2f5dd09e97d
doc: fix some minor sparse documentation oversights
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
14138
diff
changeset
|
803 matrix solvers can be entirely disabled by using @dfn{spparms} to set |
|
f2f5dd09e97d
doc: fix some minor sparse documentation oversights
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
14138
diff
changeset
|
804 @code{bandden} to 1 (i.e., @code{spparms ("bandden", 1)}). |
| 5164 | 805 |
|
19074
0850b5212619
doc: Add @nospell macro around proper names in documentation.
Rik <rik@octave.org>
parents:
18851
diff
changeset
|
806 The QR@tie{}solver factorizes the problem with a @nospell{Dulmage-Mendelsohn} |
|
16600
f2f5dd09e97d
doc: fix some minor sparse documentation oversights
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
14138
diff
changeset
|
807 decomposition, to separate the problem into blocks that can be treated |
|
f2f5dd09e97d
doc: fix some minor sparse documentation oversights
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
14138
diff
changeset
|
808 as over-determined, multiple well determined blocks, and a final |
|
16816
12005245b645
doc: Periodic grammarcheck of documentation.
Rik <rik@octave.org>
parents:
16792
diff
changeset
|
809 over-determined block. For matrices with blocks of strongly connected |
|
16600
f2f5dd09e97d
doc: fix some minor sparse documentation oversights
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
14138
diff
changeset
|
810 nodes this is a big win as LU@tie{}decomposition can be used for many |
|
16816
12005245b645
doc: Periodic grammarcheck of documentation.
Rik <rik@octave.org>
parents:
16792
diff
changeset
|
811 blocks. It also significantly improves the chance of finding a solution |
|
16600
f2f5dd09e97d
doc: fix some minor sparse documentation oversights
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
14138
diff
changeset
|
812 to over-determined problems rather than just returning a vector of |
|
f2f5dd09e97d
doc: fix some minor sparse documentation oversights
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
14138
diff
changeset
|
813 @dfn{NaN}'s. |
| 5681 | 814 |
| 815 All of the solvers above, can calculate an estimate of the condition | |
|
9066
be150a172010
Cleanup documentation for diagperm.texi, sparse.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
816 number. This can be used to detect numerical stability problems in the |
|
be150a172010
Cleanup documentation for diagperm.texi, sparse.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
817 solution and force a minimum norm solution to be used. However, for |
| 5681 | 818 narrow banded, triangular or diagonal matrices, the cost of |
| 819 calculating the condition number is significant, and can in fact | |
|
9066
be150a172010
Cleanup documentation for diagperm.texi, sparse.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
820 exceed the cost of factoring the matrix. Therefore the condition |
| 6939 | 821 number is not calculated in these cases, and Octave relies on simpler |
|
9088
77e71f3da3d6
Fix documentation image printing under new development code
Rik <rdrider0-list@yahoo.com>
parents:
9066
diff
changeset
|
822 techniques to detect singular matrices or the underlying @sc{lapack} code in |
| 5681 | 823 the case of banded matrices. |
| 5164 | 824 |
| 5322 | 825 The user can force the type of the matrix with the @code{matrix_type} |
|
9066
be150a172010
Cleanup documentation for diagperm.texi, sparse.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
826 function. This overcomes the cost of discovering the type of the matrix. |
|
8347
fa78cb8d8a5c
corrections for typos
Brian Gough<bjg@network-theory.co.uk>
parents:
8286
diff
changeset
|
827 However, it should be noted that identifying the type of the matrix incorrectly |
| 5322 | 828 will lead to unpredictable results, and so @code{matrix_type} should be |
| 5506 | 829 used with care. |
| 5322 | 830 |
| 6620 | 831 @DOCSTRING(normest) |
| 832 | |
|
8286
6f2d95255911
fix @seealso references to point to existing anchors
Thorsten Meyer <thorsten.meyier@gmx.de>
parents:
8106
diff
changeset
|
833 @DOCSTRING(onenormest) |
|
6f2d95255911
fix @seealso references to point to existing anchors
Thorsten Meyer <thorsten.meyier@gmx.de>
parents:
8106
diff
changeset
|
834 |
| 7189 | 835 @DOCSTRING(condest) |
| 836 | |
| 6620 | 837 @DOCSTRING(spparms) |
| 838 | |
| 839 @DOCSTRING(sprank) | |
| 840 | |
| 841 @DOCSTRING(symbfact) | |
| 842 | |
|
7681
b1c1133641ee
Add the spaugment function
David Bateman <dbateman@free.fr>
parents:
7619
diff
changeset
|
843 For non square matrices, the user can also utilize the @code{spaugment} |
|
b1c1133641ee
Add the spaugment function
David Bateman <dbateman@free.fr>
parents:
7619
diff
changeset
|
844 function to find a least squares solution to a linear equation. |
|
b1c1133641ee
Add the spaugment function
David Bateman <dbateman@free.fr>
parents:
7619
diff
changeset
|
845 |
|
b1c1133641ee
Add the spaugment function
David Bateman <dbateman@free.fr>
parents:
7619
diff
changeset
|
846 @DOCSTRING(spaugment) |
|
b1c1133641ee
Add the spaugment function
David Bateman <dbateman@free.fr>
parents:
7619
diff
changeset
|
847 |
|
8417
654bcfb937bf
Add the eigs and svds functions
David Bateman <dbateman@free.fr>
parents:
8347
diff
changeset
|
848 Finally, the function @code{eigs} can be used to calculate a limited |
|
654bcfb937bf
Add the eigs and svds functions
David Bateman <dbateman@free.fr>
parents:
8347
diff
changeset
|
849 number of eigenvalues and eigenvectors based on a selection criteria |
|
654bcfb937bf
Add the eigs and svds functions
David Bateman <dbateman@free.fr>
parents:
8347
diff
changeset
|
850 and likewise for @code{svds} which calculates a limited number of |
|
654bcfb937bf
Add the eigs and svds functions
David Bateman <dbateman@free.fr>
parents:
8347
diff
changeset
|
851 singular values and vectors. |
|
654bcfb937bf
Add the eigs and svds functions
David Bateman <dbateman@free.fr>
parents:
8347
diff
changeset
|
852 |
|
654bcfb937bf
Add the eigs and svds functions
David Bateman <dbateman@free.fr>
parents:
8347
diff
changeset
|
853 @DOCSTRING(eigs) |
|
654bcfb937bf
Add the eigs and svds functions
David Bateman <dbateman@free.fr>
parents:
8347
diff
changeset
|
854 |
|
654bcfb937bf
Add the eigs and svds functions
David Bateman <dbateman@free.fr>
parents:
8347
diff
changeset
|
855 @DOCSTRING(svds) |
|
654bcfb937bf
Add the eigs and svds functions
David Bateman <dbateman@free.fr>
parents:
8347
diff
changeset
|
856 |
|
8817
03b7f618ab3d
include docstrings for new functions in the manual
John W. Eaton <jwe@octave.org>
parents:
8488
diff
changeset
|
857 @node Iterative Techniques |
|
17152
f2a8592b8fbd
doc: Shorten some long subsection names in Manual.
Rik <rik@octave.org>
parents:
17097
diff
changeset
|
858 @section Iterative Techniques Applied to Sparse Matrices |
| 5164 | 859 |
| 6620 | 860 The left division @code{\} and right division @code{/} operators, |
| 861 discussed in the previous section, use direct solvers to resolve a | |
| 862 linear equation of the form @code{@var{x} = @var{A} \ @var{b}} or | |
|
19089
38937efbee21
Incorporate new functions ichol and ilu into Octave.
Rik <rik@octave.org>
parents:
19074
diff
changeset
|
863 @code{@var{x} = @var{b} / @var{A}}. Octave also includes a number of |
| 6620 | 864 functions to solve sparse linear equations using iterative techniques. |
| 865 | |
| 866 @DOCSTRING(pcg) | |
| 867 | |
| 868 @DOCSTRING(pcr) | |
| 5837 | 869 |
| 6620 | 870 The speed with which an iterative solver converges to a solution can be |
|
9066
be150a172010
Cleanup documentation for diagperm.texi, sparse.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
871 accelerated with the use of a pre-conditioning matrix @var{M}. In this |
| 6620 | 872 case the linear equation @code{@var{M}^-1 * @var{x} = @var{M}^-1 * |
|
9066
be150a172010
Cleanup documentation for diagperm.texi, sparse.texi
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
873 @var{A} \ @var{b}} is solved instead. Typical pre-conditioning matrices |
| 6620 | 874 are partial factorizations of the original matrix. |
| 5648 | 875 |
|
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|
876 @DOCSTRING(ichol) |
|
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|
877 |
|
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|
878 @DOCSTRING(ilu) |
| 6620 | 879 |
|
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880 @node Real Life Example |
|
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881 @section Real Life Example using Sparse Matrices |
| 5648 | 882 |
| 883 A common application for sparse matrices is in the solution of Finite | |
|
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|
884 Element Models. Finite element models allow numerical solution of |
| 5648 | 885 partial differential equations that do not have closed form solutions, |
| 886 typically because of the complex shape of the domain. | |
| 887 | |
| 888 In order to motivate this application, we consider the boundary value | |
|
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|
889 Laplace equation. This system can model scalar potential fields, such |
|
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|
890 as heat or electrical potential. Given a medium |
| 5648 | 891 @tex |
|
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|
892 $\Omega$ with boundary $\partial\Omega$. At all points on the $\partial\Omega$ |
|
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|
893 the boundary conditions are known, and we wish to calculate the potential in |
|
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|
894 $\Omega$. |
| 5648 | 895 @end tex |
|
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|
896 @ifnottex |
|
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|
897 Omega with boundary dOmega. At all points on the dOmega |
| 5648 | 898 the boundary conditions are known, and we wish to calculate the potential in |
|
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|
899 Omega. |
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900 @end ifnottex |
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|
901 Boundary conditions may specify the potential (Dirichlet |
| 5648 | 902 boundary condition), its normal derivative across the boundary |
| 903 (Neumann boundary condition), or a weighted sum of the potential and | |
| 904 its derivative (Cauchy boundary condition). | |
| 905 | |
| 906 In a thermal model, we want to calculate the temperature in | |
| 907 @tex | |
|
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908 $\Omega$ |
| 5648 | 909 @end tex |
|
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910 @ifnottex |
| 5648 | 911 Omega |
|
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|
912 @end ifnottex |
| 5648 | 913 and know the boundary temperature (Dirichlet condition) |
| 914 or heat flux (from which we can calculate the Neumann condition | |
|
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915 by dividing by the thermal conductivity at the boundary). Similarly, |
| 5648 | 916 in an electrical model, we want to calculate the voltage in |
| 917 @tex | |
|
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918 $\Omega$ |
| 5648 | 919 @end tex |
|
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|
920 @ifnottex |
| 5648 | 921 Omega |
|
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|
922 @end ifnottex |
| 5648 | 923 and know the boundary voltage (Dirichlet) or current |
| 924 (Neumann condition after diving by the electrical conductivity). | |
| 925 In an electrical model, it is common for much of the boundary | |
| 926 to be electrically isolated; this is a Neumann boundary condition | |
| 927 with the current equal to zero. | |
| 928 | |
|
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|
929 The simplest finite element models will divide |
| 5648 | 930 @tex |
|
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931 $\Omega$ |
| 5648 | 932 @end tex |
|
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933 @ifnottex |
| 5648 | 934 Omega |
|
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935 @end ifnottex |
| 5648 | 936 into simplexes (triangles in 2D, pyramids in 3D). |
| 937 @ifset htmltex | |
|
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|
938 We take as a 3-D example a cylindrical liquid filled tank with a small |
|
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|
939 non-conductive ball from the EIDORS project@footnote{EIDORS - Electrical |
|
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|
940 Impedance Tomography and Diffuse optical Tomography Reconstruction Software |
|
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|
941 @url{http://eidors3d.sourceforge.net}}. This is model is designed to reflect |
|
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|
942 an application of electrical impedance tomography, where current patterns |
|
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|
943 are applied to such a tank in order to image the internal conductivity |
|
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|
944 distribution. In order to describe the FEM geometry, we have a matrix of |
| 5648 | 945 vertices @code{nodes} and simplices @code{elems}. |
| 946 @end ifset | |
| 947 | |
|
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948 The following example creates a simple rectangular 2-D electrically |
|
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949 conductive medium with 10 V and 20 V imposed on opposite sides |
|
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|
950 (Dirichlet boundary conditions). All other edges are electrically |
| 5648 | 951 isolated. |
| 952 | |
| 953 @example | |
|
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954 @group |
|
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955 node_y = [1;1.2;1.5;1.8;2]*ones(1,11); |
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|
956 node_x = ones(5,1)*[1,1.05,1.1,1.2, ... |
| 5648 | 957 1.3,1.5,1.7,1.8,1.9,1.95,2]; |
|
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958 nodes = [node_x(:), node_y(:)]; |
| 5648 | 959 |
|
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960 [h,w] = size (node_x); |
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961 elems = []; |
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|
962 for idx = 1:w-1 |
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963 widx = (idx-1)*h; |
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|
964 elems = [elems; ... |
| 5648 | 965 widx+[(1:h-1);(2:h);h+(1:h-1)]'; ... |
|
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966 widx+[(2:h);h+(2:h);h+(1:h-1)]' ]; |
| 5648 | 967 endfor |
| 968 | |
|
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969 E = size (elems,1); # No. of simplices |
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970 N = size (nodes,1); # No. of vertices |
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971 D = size (elems,2); # dimensions+1 |
|
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|
972 @end group |
| 5648 | 973 @end example |
| 974 | |
| 975 This creates a N-by-2 matrix @code{nodes} and a E-by-3 matrix | |
| 976 @code{elems} with values, which define finite element triangles: | |
| 5164 | 977 |
| 5648 | 978 @example |
|
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|
979 @group |
| 5648 | 980 nodes(1:7,:)' |
|
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981 1.00 1.00 1.00 1.00 1.00 1.05 1.05 @dots{} |
|
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|
982 1.00 1.20 1.50 1.80 2.00 1.00 1.20 @dots{} |
| 5648 | 983 |
| 984 elems(1:7,:)' | |
|
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|
985 1 2 3 4 2 3 4 @dots{} |
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|
986 2 3 4 5 7 8 9 @dots{} |
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987 6 7 8 9 6 7 8 @dots{} |
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988 @end group |
| 5648 | 989 @end example |
| 990 | |
|
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991 Using a first order FEM, we approximate the electrical conductivity |
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992 distribution in |
| 5648 | 993 @tex |
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994 $\Omega$ |
| 5648 | 995 @end tex |
|
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996 @ifnottex |
| 5648 | 997 Omega |
|
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998 @end ifnottex |
| 5648 | 999 as constant on each simplex (represented by the vector @code{conductivity}). |
| 1000 Based on the finite element geometry, we first calculate a system (or | |
| 1001 stiffness) matrix for each simplex (represented as 3-by-3 elements on the | |
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1002 diagonal of the element-wise system matrix @code{SE}). Based on @code{SE} |
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1003 and a N-by-DE connectivity matrix @code{C}, representing the connections |
| 7001 | 1004 between simplices and vertices, the global connectivity matrix @code{S} is |
| 5648 | 1005 calculated. |
| 1006 | |
| 1007 @example | |
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1008 ## Element conductivity |
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1009 conductivity = [1*ones(1,16), ... |
| 5648 | 1010 2*ones(1,48), 1*ones(1,16)]; |
| 1011 | |
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|
1012 ## Connectivity matrix |
| 5648 | 1013 C = sparse ((1:D*E), reshape (elems', ... |
| 1014 D*E, 1), 1, D*E, N); | |
| 1015 | |
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|
1016 ## Calculate system matrix |
| 5648 | 1017 Siidx = floor ([0:D*E-1]'/D) * D * ... |
| 1018 ones(1,D) + ones(D*E,1)*(1:D) ; | |
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1019 Sjidx = [1:D*E]'*ones (1,D); |
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1020 Sdata = zeros (D*E,D); |
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1021 dfact = factorial (D-1); |
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|
1022 for j = 1:E |
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1023 a = inv ([ones(D,1), ... |
| 5648 | 1024 nodes(elems(j,:), :)]); |
| 1025 const = conductivity(j) * 2 / ... | |
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|
1026 dfact / abs (det (a)); |
| 5648 | 1027 Sdata(D*(j-1)+(1:D),:) = const * ... |
| 1028 a(2:D,:)' * a(2:D,:); | |
| 1029 endfor | |
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1030 ## Element-wise system matrix |
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1031 SE = sparse(Siidx,Sjidx,Sdata); |
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1032 ## Global system matrix |
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1033 S = C'* SE *C; |
| 5648 | 1034 @end example |
| 1035 | |
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1036 The system matrix acts like the conductivity |
| 5648 | 1037 @tex |
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1038 $S$ |
| 5648 | 1039 @end tex |
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1040 @ifnottex |
| 5648 | 1041 @code{S} |
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1042 @end ifnottex |
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1043 in Ohm's law |
| 5648 | 1044 @tex |
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1045 $SV = I$. |
| 5648 | 1046 @end tex |
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|
1047 @ifnottex |
| 5648 | 1048 @code{S * V = I}. |
|
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1049 @end ifnottex |
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1050 Based on the Dirichlet and Neumann boundary conditions, we are able to |
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1051 solve for the voltages at each vertex @code{V}. |
| 5648 | 1052 |
| 1053 @example | |
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1054 ## Dirichlet boundary conditions |
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1055 D_nodes = [1:5, 51:55]; |
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1056 D_value = [10*ones(1,5), 20*ones(1,5)]; |
| 5648 | 1057 |
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1058 V = zeros (N,1); |
| 5648 | 1059 V(D_nodes) = D_value; |
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1060 idx = 1:N; # vertices without Dirichlet |
| 5648 | 1061 # boundary condns |
| 1062 idx(D_nodes) = []; | |
| 1063 | |
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1064 ## Neumann boundary conditions. Note that |
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1065 ## N_value must be normalized by the |
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1066 ## boundary length and element conductivity |
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1067 N_nodes = []; |
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1068 N_value = []; |
| 5648 | 1069 |
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1070 Q = zeros (N,1); |
| 5648 | 1071 Q(N_nodes) = N_value; |
| 1072 | |
| 1073 V(idx) = S(idx,idx) \ ( Q(idx) - ... | |
| 1074 S(idx,D_nodes) * V(D_nodes)); | |
| 1075 @end example | |
| 1076 | |
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1077 Finally, in order to display the solution, we show each solved voltage |
| 5648 | 1078 value in the z-axis for each simplex vertex. |
| 1079 @ifset htmltex | |
| 1080 @xref{fig:femmodel}. | |
| 1081 @end ifset | |
| 1082 | |
| 1083 @example | |
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1084 @group |
| 5648 | 1085 elemx = elems(:,[1,2,3,1])'; |
| 1086 xelems = reshape (nodes(elemx, 1), 4, E); | |
| 1087 yelems = reshape (nodes(elemx, 2), 4, E); | |
| 1088 velems = reshape (V(elemx), 4, E); | |
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1089 plot3 (xelems,yelems,velems,"k"); |
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1090 print "grid.eps"; |
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1091 @end group |
| 5648 | 1092 @end example |
| 1093 | |
| 1094 | |
| 1095 @ifset htmltex | |
| 1096 @float Figure,fig:femmodel | |
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1097 @center @image{grid,4in} |
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1098 @caption{Example finite element model the showing triangular elements. |
| 5648 | 1099 The height of each vertex corresponds to the solution value.} |
| 1100 @end float | |
| 1101 @end ifset |