1 namespace Eigen { 2 3 /** \eigenManualPage TutorialSparse Sparse matrix manipulations 4 5 \eigenAutoToc 6 7 Manipulating and solving sparse problems involves various modules which are summarized below: 8 9 <table class="manual"> 10 <tr><th>Module</th><th>Header file</th><th>Contents</th></tr> 11 <tr><td>\link SparseCore_Module SparseCore \endlink</td><td>\code#include <Eigen/SparseCore>\endcode</td><td>SparseMatrix and SparseVector classes, matrix assembly, basic sparse linear algebra (including sparse triangular solvers)</td></tr> 12 <tr><td>\link SparseCholesky_Module SparseCholesky \endlink</td><td>\code#include <Eigen/SparseCholesky>\endcode</td><td>Direct sparse LLT and LDLT Cholesky factorization to solve sparse self-adjoint positive definite problems</td></tr> 13 <tr><td>\link SparseLU_Module SparseLU \endlink</td><td>\code #include<Eigen/SparseLU> \endcode</td> 14 <td>%Sparse LU factorization to solve general square sparse systems</td></tr> 15 <tr><td>\link SparseQR_Module SparseQR \endlink</td><td>\code #include<Eigen/SparseQR>\endcode </td><td>%Sparse QR factorization for solving sparse linear least-squares problems</td></tr> 16 <tr><td>\link IterativeLinearSolvers_Module IterativeLinearSolvers \endlink</td><td>\code#include <Eigen/IterativeLinearSolvers>\endcode</td><td>Iterative solvers to solve large general linear square problems (including self-adjoint positive definite problems)</td></tr> 17 <tr><td>\link Sparse_Module Sparse \endlink</td><td>\code#include <Eigen/Sparse>\endcode</td><td>Includes all the above modules</td></tr> 18 </table> 19 20 \section TutorialSparseIntro Sparse matrix format 21 22 In many applications (e.g., finite element methods) it is common to deal with very large matrices where only a few coefficients are different from zero. In such cases, memory consumption can be reduced and performance increased by using a specialized representation storing only the nonzero coefficients. Such a matrix is called a sparse matrix. 23 24 \b The \b %SparseMatrix \b class 25 26 The class SparseMatrix is the main sparse matrix representation of Eigen's sparse module; it offers high performance and low memory usage. 27 It implements a more versatile variant of the widely-used Compressed Column (or Row) Storage scheme. 28 It consists of four compact arrays: 29 - \c Values: stores the coefficient values of the non-zeros. 30 - \c InnerIndices: stores the row (resp. column) indices of the non-zeros. 31 - \c OuterStarts: stores for each column (resp. row) the index of the first non-zero in the previous two arrays. 32 - \c InnerNNZs: stores the number of non-zeros of each column (resp. row). 33 The word \c inner refers to an \em inner \em vector that is a column for a column-major matrix, or a row for a row-major matrix. 34 The word \c outer refers to the other direction. 35 36 This storage scheme is better explained on an example. The following matrix 37 <table class="manual"> 38 <tr><td> 0</td><td>3</td><td> 0</td><td>0</td><td> 0</td></tr> 39 <tr><td>22</td><td>0</td><td> 0</td><td>0</td><td>17</td></tr> 40 <tr><td> 7</td><td>5</td><td> 0</td><td>1</td><td> 0</td></tr> 41 <tr><td> 0</td><td>0</td><td> 0</td><td>0</td><td> 0</td></tr> 42 <tr><td> 0</td><td>0</td><td>14</td><td>0</td><td> 8</td></tr> 43 </table> 44 45 and one of its possible sparse, \b column \b major representation: 46 <table class="manual"> 47 <tr><td>Values:</td> <td>22</td><td>7</td><td>_</td><td>3</td><td>5</td><td>14</td><td>_</td><td>_</td><td>1</td><td>_</td><td>17</td><td>8</td></tr> 48 <tr><td>InnerIndices:</td> <td> 1</td><td>2</td><td>_</td><td>0</td><td>2</td><td> 4</td><td>_</td><td>_</td><td>2</td><td>_</td><td> 1</td><td>4</td></tr> 49 </table> 50 <table class="manual"> 51 <tr><td>OuterStarts:</td><td>0</td><td>3</td><td>5</td><td>8</td><td>10</td><td>\em 12 </td></tr> 52 <tr><td>InnerNNZs:</td> <td>2</td><td>2</td><td>1</td><td>1</td><td> 2</td><td></td></tr> 53 </table> 54 55 Currently the elements of a given inner vector are guaranteed to be always sorted by increasing inner indices. 56 The \c "_" indicates available free space to quickly insert new elements. 57 Assuming no reallocation is needed, the insertion of a random element is therefore in O(nnz_j) where nnz_j is the number of nonzeros of the respective inner vector. 58 On the other hand, inserting elements with increasing inner indices in a given inner vector is much more efficient since this only requires to increase the respective \c InnerNNZs entry that is a O(1) operation. 59 60 The case where no empty space is available is a special case, and is refered as the \em compressed mode. 61 It corresponds to the widely used Compressed Column (or Row) Storage schemes (CCS or CRS). 62 Any SparseMatrix can be turned to this form by calling the SparseMatrix::makeCompressed() function. 63 In this case, one can remark that the \c InnerNNZs array is redundant with \c OuterStarts because we the equality: \c InnerNNZs[j] = \c OuterStarts[j+1]-\c OuterStarts[j]. 64 Therefore, in practice a call to SparseMatrix::makeCompressed() frees this buffer. 65 66 It is worth noting that most of our wrappers to external libraries requires compressed matrices as inputs. 67 68 The results of %Eigen's operations always produces \b compressed sparse matrices. 69 On the other hand, the insertion of a new element into a SparseMatrix converts this later to the \b uncompressed mode. 70 71 Here is the previous matrix represented in compressed mode: 72 <table class="manual"> 73 <tr><td>Values:</td> <td>22</td><td>7</td><td>3</td><td>5</td><td>14</td><td>1</td><td>17</td><td>8</td></tr> 74 <tr><td>InnerIndices:</td> <td> 1</td><td>2</td><td>0</td><td>2</td><td> 4</td><td>2</td><td> 1</td><td>4</td></tr> 75 </table> 76 <table class="manual"> 77 <tr><td>OuterStarts:</td><td>0</td><td>2</td><td>4</td><td>5</td><td>6</td><td>\em 8 </td></tr> 78 </table> 79 80 A SparseVector is a special case of a SparseMatrix where only the \c Values and \c InnerIndices arrays are stored. 81 There is no notion of compressed/uncompressed mode for a SparseVector. 82 83 84 \section TutorialSparseExample First example 85 86 Before describing each individual class, let's start with the following typical example: solving the Laplace equation \f$ \nabla u = 0 \f$ on a regular 2D grid using a finite difference scheme and Dirichlet boundary conditions. 87 Such problem can be mathematically expressed as a linear problem of the form \f$ Ax=b \f$ where \f$ x \f$ is the vector of \c m unknowns (in our case, the values of the pixels), \f$ b \f$ is the right hand side vector resulting from the boundary conditions, and \f$ A \f$ is an \f$ m \times m \f$ matrix containing only a few non-zero elements resulting from the discretization of the Laplacian operator. 88 89 <table class="manual"> 90 <tr><td> 91 \include Tutorial_sparse_example.cpp 92 </td> 93 <td> 94 \image html Tutorial_sparse_example.jpeg 95 </td></tr></table> 96 97 In this example, we start by defining a column-major sparse matrix type of double \c SparseMatrix<double>, and a triplet list of the same scalar type \c Triplet<double>. A triplet is a simple object representing a non-zero entry as the triplet: \c row index, \c column index, \c value. 98 99 In the main function, we declare a list \c coefficients of triplets (as a std vector) and the right hand side vector \f$ b \f$ which are filled by the \a buildProblem function. 100 The raw and flat list of non-zero entries is then converted to a true SparseMatrix object \c A. 101 Note that the elements of the list do not have to be sorted, and possible duplicate entries will be summed up. 102 103 The last step consists of effectively solving the assembled problem. 104 Since the resulting matrix \c A is symmetric by construction, we can perform a direct Cholesky factorization via the SimplicialLDLT class which behaves like its LDLT counterpart for dense objects. 105 106 The resulting vector \c x contains the pixel values as a 1D array which is saved to a jpeg file shown on the right of the code above. 107 108 Describing the \a buildProblem and \a save functions is out of the scope of this tutorial. They are given \ref TutorialSparse_example_details "here" for the curious and reproducibility purpose. 109 110 111 112 113 \section TutorialSparseSparseMatrix The SparseMatrix class 114 115 \b %Matrix \b and \b vector \b properties \n 116 117 The SparseMatrix and SparseVector classes take three template arguments: 118 * the scalar type (e.g., double) 119 * the storage order (ColMajor or RowMajor, the default is ColMajor) 120 * the inner index type (default is \c int). 121 122 As for dense Matrix objects, constructors takes the size of the object. 123 Here are some examples: 124 125 \code 126 SparseMatrix<std::complex<float> > mat(1000,2000); // declares a 1000x2000 column-major compressed sparse matrix of complex<float> 127 SparseMatrix<double,RowMajor> mat(1000,2000); // declares a 1000x2000 row-major compressed sparse matrix of double 128 SparseVector<std::complex<float> > vec(1000); // declares a column sparse vector of complex<float> of size 1000 129 SparseVector<double,RowMajor> vec(1000); // declares a row sparse vector of double of size 1000 130 \endcode 131 132 In the rest of the tutorial, \c mat and \c vec represent any sparse-matrix and sparse-vector objects, respectively. 133 134 The dimensions of a matrix can be queried using the following functions: 135 <table class="manual"> 136 <tr><td>Standard \n dimensions</td><td>\code 137 mat.rows() 138 mat.cols()\endcode</td> 139 <td>\code 140 vec.size() \endcode</td> 141 </tr> 142 <tr><td>Sizes along the \n inner/outer dimensions</td><td>\code 143 mat.innerSize() 144 mat.outerSize()\endcode</td> 145 <td></td> 146 </tr> 147 <tr><td>Number of non \n zero coefficients</td><td>\code 148 mat.nonZeros() \endcode</td> 149 <td>\code 150 vec.nonZeros() \endcode</td></tr> 151 </table> 152 153 154 \b Iterating \b over \b the \b nonzero \b coefficients \n 155 156 Random access to the elements of a sparse object can be done through the \c coeffRef(i,j) function. 157 However, this function involves a quite expensive binary search. 158 In most cases, one only wants to iterate over the non-zeros elements. This is achieved by a standard loop over the outer dimension, and then by iterating over the non-zeros of the current inner vector via an InnerIterator. Thus, the non-zero entries have to be visited in the same order than the storage order. 159 Here is an example: 160 <table class="manual"> 161 <tr><td> 162 \code 163 SparseMatrix<double> mat(rows,cols); 164 for (int k=0; k<mat.outerSize(); ++k) 165 for (SparseMatrix<double>::InnerIterator it(mat,k); it; ++it) 166 { 167 it.value(); 168 it.row(); // row index 169 it.col(); // col index (here it is equal to k) 170 it.index(); // inner index, here it is equal to it.row() 171 } 172 \endcode 173 </td><td> 174 \code 175 SparseVector<double> vec(size); 176 for (SparseVector<double>::InnerIterator it(vec); it; ++it) 177 { 178 it.value(); // == vec[ it.index() ] 179 it.index(); 180 } 181 \endcode 182 </td></tr> 183 </table> 184 For a writable expression, the referenced value can be modified using the valueRef() function. 185 If the type of the sparse matrix or vector depends on a template parameter, then the \c typename keyword is 186 required to indicate that \c InnerIterator denotes a type; see \ref TopicTemplateKeyword for details. 187 188 189 \section TutorialSparseFilling Filling a sparse matrix 190 191 Because of the special storage scheme of a SparseMatrix, special care has to be taken when adding new nonzero entries. 192 For instance, the cost of a single purely random insertion into a SparseMatrix is \c O(nnz), where \c nnz is the current number of non-zero coefficients. 193 194 The simplest way to create a sparse matrix while guaranteeing good performance is thus to first build a list of so-called \em triplets, and then convert it to a SparseMatrix. 195 196 Here is a typical usage example: 197 \code 198 typedef Eigen::Triplet<double> T; 199 std::vector<T> tripletList; 200 tripletList.reserve(estimation_of_entries); 201 for(...) 202 { 203 // ... 204 tripletList.push_back(T(i,j,v_ij)); 205 } 206 SparseMatrixType mat(rows,cols); 207 mat.setFromTriplets(tripletList.begin(), tripletList.end()); 208 // mat is ready to go! 209 \endcode 210 The \c std::vector of triplets might contain the elements in arbitrary order, and might even contain duplicated elements that will be summed up by setFromTriplets(). 211 See the SparseMatrix::setFromTriplets() function and class Triplet for more details. 212 213 214 In some cases, however, slightly higher performance, and lower memory consumption can be reached by directly inserting the non-zeros into the destination matrix. 215 A typical scenario of this approach is illustrated bellow: 216 \code 217 1: SparseMatrix<double> mat(rows,cols); // default is column major 218 2: mat.reserve(VectorXi::Constant(cols,6)); 219 3: for each i,j such that v_ij != 0 220 4: mat.insert(i,j) = v_ij; // alternative: mat.coeffRef(i,j) += v_ij; 221 5: mat.makeCompressed(); // optional 222 \endcode 223 224 - The key ingredient here is the line 2 where we reserve room for 6 non-zeros per column. In many cases, the number of non-zeros per column or row can easily be known in advance. If it varies significantly for each inner vector, then it is possible to specify a reserve size for each inner vector by providing a vector object with an operator[](int j) returning the reserve size of the \c j-th inner vector (e.g., via a VectorXi or std::vector<int>). If only a rought estimate of the number of nonzeros per inner-vector can be obtained, it is highly recommended to overestimate it rather than the opposite. If this line is omitted, then the first insertion of a new element will reserve room for 2 elements per inner vector. 225 - The line 4 performs a sorted insertion. In this example, the ideal case is when the \c j-th column is not full and contains non-zeros whose inner-indices are smaller than \c i. In this case, this operation boils down to trivial O(1) operation. 226 - When calling insert(i,j) the element \c i \c ,j must not already exists, otherwise use the coeffRef(i,j) method that will allow to, e.g., accumulate values. This method first performs a binary search and finally calls insert(i,j) if the element does not already exist. It is more flexible than insert() but also more costly. 227 - The line 5 suppresses the remaining empty space and transforms the matrix into a compressed column storage. 228 229 230 231 \section TutorialSparseFeatureSet Supported operators and functions 232 233 Because of their special storage format, sparse matrices cannot offer the same level of flexibility than dense matrices. 234 In Eigen's sparse module we chose to expose only the subset of the dense matrix API which can be efficiently implemented. 235 In the following \em sm denotes a sparse matrix, \em sv a sparse vector, \em dm a dense matrix, and \em dv a dense vector. 236 237 \subsection TutorialSparse_BasicOps Basic operations 238 239 %Sparse expressions support most of the unary and binary coefficient wise operations: 240 \code 241 sm1.real() sm1.imag() -sm1 0.5*sm1 242 sm1+sm2 sm1-sm2 sm1.cwiseProduct(sm2) 243 \endcode 244 However, a strong restriction is that the storage orders must match. For instance, in the following example: 245 \code 246 sm4 = sm1 + sm2 + sm3; 247 \endcode 248 sm1, sm2, and sm3 must all be row-major or all column major. 249 On the other hand, there is no restriction on the target matrix sm4. 250 For instance, this means that for computing \f$ A^T + A \f$, the matrix \f$ A^T \f$ must be evaluated into a temporary matrix of compatible storage order: 251 \code 252 SparseMatrix<double> A, B; 253 B = SparseMatrix<double>(A.transpose()) + A; 254 \endcode 255 256 Some binary coefficient-wise operators can also mix sparse and dense expressions: 257 \code 258 sm2 = sm1.cwiseProduct(dm1); 259 dm1 += sm1; 260 \endcode 261 262 However, it is not yet possible to add a sparse and a dense matrix as in <tt>dm2 = sm1 + dm1</tt>. 263 Please write this as the equivalent <tt>dm2 = dm1; dm2 += sm1</tt> (we plan to lift this restriction 264 in the next release of %Eigen). 265 266 %Sparse expressions also support transposition: 267 \code 268 sm1 = sm2.transpose(); 269 sm1 = sm2.adjoint(); 270 \endcode 271 However, there is no transposeInPlace() method. 272 273 274 \subsection TutorialSparse_Products Matrix products 275 276 %Eigen supports various kind of sparse matrix products which are summarize below: 277 - \b sparse-dense: 278 \code 279 dv2 = sm1 * dv1; 280 dm2 = dm1 * sm1.adjoint(); 281 dm2 = 2. * sm1 * dm1; 282 \endcode 283 - \b symmetric \b sparse-dense. The product of a sparse symmetric matrix with a dense matrix (or vector) can also be optimized by specifying the symmetry with selfadjointView(): 284 \code 285 dm2 = sm1.selfadjointView<>() * dm1; // if all coefficients of A are stored 286 dm2 = A.selfadjointView<Upper>() * dm1; // if only the upper part of A is stored 287 dm2 = A.selfadjointView<Lower>() * dm1; // if only the lower part of A is stored 288 \endcode 289 - \b sparse-sparse. For sparse-sparse products, two different algorithms are available. The default one is conservative and preserve the explicit zeros that might appear: 290 \code 291 sm3 = sm1 * sm2; 292 sm3 = 4 * sm1.adjoint() * sm2; 293 \endcode 294 The second algorithm prunes on the fly the explicit zeros, or the values smaller than a given threshold. It is enabled and controlled through the prune() functions: 295 \code 296 sm3 = (sm1 * sm2).pruned(); // removes numerical zeros 297 sm3 = (sm1 * sm2).pruned(ref); // removes elements much smaller than ref 298 sm3 = (sm1 * sm2).pruned(ref,epsilon); // removes elements smaller than ref*epsilon 299 \endcode 300 301 - \b permutations. Finally, permutations can be applied to sparse matrices too: 302 \code 303 PermutationMatrix<Dynamic,Dynamic> P = ...; 304 sm2 = P * sm1; 305 sm2 = sm1 * P.inverse(); 306 sm2 = sm1.transpose() * P; 307 \endcode 308 309 310 \subsection TutorialSparse_TriangularSelfadjoint Triangular and selfadjoint views 311 312 Just as with dense matrices, the triangularView() function can be used to address a triangular part of the matrix, and perform triangular solves with a dense right hand side: 313 \code 314 dm2 = sm1.triangularView<Lower>(dm1); 315 dv2 = sm1.transpose().triangularView<Upper>(dv1); 316 \endcode 317 318 The selfadjointView() function permits various operations: 319 - optimized sparse-dense matrix products: 320 \code 321 dm2 = sm1.selfadjointView<>() * dm1; // if all coefficients of A are stored 322 dm2 = A.selfadjointView<Upper>() * dm1; // if only the upper part of A is stored 323 dm2 = A.selfadjointView<Lower>() * dm1; // if only the lower part of A is stored 324 \endcode 325 - copy of triangular parts: 326 \code 327 sm2 = sm1.selfadjointView<Upper>(); // makes a full selfadjoint matrix from the upper triangular part 328 sm2.selfadjointView<Lower>() = sm1.selfadjointView<Upper>(); // copies the upper triangular part to the lower triangular part 329 \endcode 330 - application of symmetric permutations: 331 \code 332 PermutationMatrix<Dynamic,Dynamic> P = ...; 333 sm2 = A.selfadjointView<Upper>().twistedBy(P); // compute P S P' from the upper triangular part of A, and make it a full matrix 334 sm2.selfadjointView<Lower>() = A.selfadjointView<Lower>().twistedBy(P); // compute P S P' from the lower triangular part of A, and then only compute the lower part 335 \endcode 336 337 Please, refer to the \link SparseQuickRefPage Quick Reference \endlink guide for the list of supported operations. The list of linear solvers available is \link TopicSparseSystems here. \endlink 338 339 */ 340 341 } 342
