1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com> 5 // Copyright (C) 2013-2014 Gael Guennebaud <gael.guennebaud@inria.fr> 6 // 7 // This Source Code Form is subject to the terms of the Mozilla 8 // Public License v. 2.0. If a copy of the MPL was not distributed 9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 10 11 #ifndef EIGEN_JACOBISVD_H 12 #define EIGEN_JACOBISVD_H 13 14 namespace Eigen { 15 16 namespace internal { 17 // forward declaration (needed by ICC) 18 // the empty body is required by MSVC 19 template<typename MatrixType, int QRPreconditioner, 20 bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex> 21 struct svd_precondition_2x2_block_to_be_real {}; 22 23 /*** QR preconditioners (R-SVD) 24 *** 25 *** Their role is to reduce the problem of computing the SVD to the case of a square matrix. 26 *** This approach, known as R-SVD, is an optimization for rectangular-enough matrices, and is a requirement for 27 *** JacobiSVD which by itself is only able to work on square matrices. 28 ***/ 29 30 enum { PreconditionIfMoreColsThanRows, PreconditionIfMoreRowsThanCols }; 31 32 template<typename MatrixType, int QRPreconditioner, int Case> 33 struct qr_preconditioner_should_do_anything 34 { 35 enum { a = MatrixType::RowsAtCompileTime != Dynamic && 36 MatrixType::ColsAtCompileTime != Dynamic && 37 MatrixType::ColsAtCompileTime <= MatrixType::RowsAtCompileTime, 38 b = MatrixType::RowsAtCompileTime != Dynamic && 39 MatrixType::ColsAtCompileTime != Dynamic && 40 MatrixType::RowsAtCompileTime <= MatrixType::ColsAtCompileTime, 41 ret = !( (QRPreconditioner == NoQRPreconditioner) || 42 (Case == PreconditionIfMoreColsThanRows && bool(a)) || 43 (Case == PreconditionIfMoreRowsThanCols && bool(b)) ) 44 }; 45 }; 46 47 template<typename MatrixType, int QRPreconditioner, int Case, 48 bool DoAnything = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret 49 > struct qr_preconditioner_impl {}; 50 51 template<typename MatrixType, int QRPreconditioner, int Case> 52 class qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false> 53 { 54 public: allocate(const JacobiSVD<MatrixType,QRPreconditioner> &)55 void allocate(const JacobiSVD<MatrixType, QRPreconditioner>&) {} run(JacobiSVD<MatrixType,QRPreconditioner> &,const MatrixType &)56 bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&) 57 { 58 return false; 59 } 60 }; 61 62 /*** preconditioner using FullPivHouseholderQR ***/ 63 64 template<typename MatrixType> 65 class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true> 66 { 67 public: 68 typedef typename MatrixType::Scalar Scalar; 69 enum 70 { 71 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 72 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime 73 }; 74 typedef Matrix<Scalar, 1, RowsAtCompileTime, RowMajor, 1, MaxRowsAtCompileTime> WorkspaceType; 75 allocate(const JacobiSVD<MatrixType,FullPivHouseholderQRPreconditioner> & svd)76 void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd) 77 { 78 if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols()) 79 { 80 m_qr.~QRType(); 81 ::new (&m_qr) QRType(svd.rows(), svd.cols()); 82 } 83 if (svd.m_computeFullU) m_workspace.resize(svd.rows()); 84 } 85 run(JacobiSVD<MatrixType,FullPivHouseholderQRPreconditioner> & svd,const MatrixType & matrix)86 bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix) 87 { 88 if(matrix.rows() > matrix.cols()) 89 { 90 m_qr.compute(matrix); 91 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>(); 92 if(svd.m_computeFullU) m_qr.matrixQ().evalTo(svd.m_matrixU, m_workspace); 93 if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation(); 94 return true; 95 } 96 return false; 97 } 98 private: 99 typedef FullPivHouseholderQR<MatrixType> QRType; 100 QRType m_qr; 101 WorkspaceType m_workspace; 102 }; 103 104 template<typename MatrixType> 105 class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true> 106 { 107 public: 108 typedef typename MatrixType::Scalar Scalar; 109 enum 110 { 111 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 112 ColsAtCompileTime = MatrixType::ColsAtCompileTime, 113 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 114 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, 115 TrOptions = RowsAtCompileTime==1 ? (MatrixType::Options & ~(RowMajor)) 116 : ColsAtCompileTime==1 ? (MatrixType::Options | RowMajor) 117 : MatrixType::Options 118 }; 119 typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, TrOptions, MaxColsAtCompileTime, MaxRowsAtCompileTime> 120 TransposeTypeWithSameStorageOrder; 121 allocate(const JacobiSVD<MatrixType,FullPivHouseholderQRPreconditioner> & svd)122 void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd) 123 { 124 if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols()) 125 { 126 m_qr.~QRType(); 127 ::new (&m_qr) QRType(svd.cols(), svd.rows()); 128 } 129 m_adjoint.resize(svd.cols(), svd.rows()); 130 if (svd.m_computeFullV) m_workspace.resize(svd.cols()); 131 } 132 run(JacobiSVD<MatrixType,FullPivHouseholderQRPreconditioner> & svd,const MatrixType & matrix)133 bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix) 134 { 135 if(matrix.cols() > matrix.rows()) 136 { 137 m_adjoint = matrix.adjoint(); 138 m_qr.compute(m_adjoint); 139 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint(); 140 if(svd.m_computeFullV) m_qr.matrixQ().evalTo(svd.m_matrixV, m_workspace); 141 if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation(); 142 return true; 143 } 144 else return false; 145 } 146 private: 147 typedef FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType; 148 QRType m_qr; 149 TransposeTypeWithSameStorageOrder m_adjoint; 150 typename internal::plain_row_type<MatrixType>::type m_workspace; 151 }; 152 153 /*** preconditioner using ColPivHouseholderQR ***/ 154 155 template<typename MatrixType> 156 class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true> 157 { 158 public: allocate(const JacobiSVD<MatrixType,ColPivHouseholderQRPreconditioner> & svd)159 void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd) 160 { 161 if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols()) 162 { 163 m_qr.~QRType(); 164 ::new (&m_qr) QRType(svd.rows(), svd.cols()); 165 } 166 if (svd.m_computeFullU) m_workspace.resize(svd.rows()); 167 else if (svd.m_computeThinU) m_workspace.resize(svd.cols()); 168 } 169 run(JacobiSVD<MatrixType,ColPivHouseholderQRPreconditioner> & svd,const MatrixType & matrix)170 bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix) 171 { 172 if(matrix.rows() > matrix.cols()) 173 { 174 m_qr.compute(matrix); 175 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>(); 176 if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace); 177 else if(svd.m_computeThinU) 178 { 179 svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols()); 180 m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace); 181 } 182 if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation(); 183 return true; 184 } 185 return false; 186 } 187 188 private: 189 typedef ColPivHouseholderQR<MatrixType> QRType; 190 QRType m_qr; 191 typename internal::plain_col_type<MatrixType>::type m_workspace; 192 }; 193 194 template<typename MatrixType> 195 class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true> 196 { 197 public: 198 typedef typename MatrixType::Scalar Scalar; 199 enum 200 { 201 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 202 ColsAtCompileTime = MatrixType::ColsAtCompileTime, 203 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 204 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, 205 TrOptions = RowsAtCompileTime==1 ? (MatrixType::Options & ~(RowMajor)) 206 : ColsAtCompileTime==1 ? (MatrixType::Options | RowMajor) 207 : MatrixType::Options 208 }; 209 210 typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, TrOptions, MaxColsAtCompileTime, MaxRowsAtCompileTime> 211 TransposeTypeWithSameStorageOrder; 212 allocate(const JacobiSVD<MatrixType,ColPivHouseholderQRPreconditioner> & svd)213 void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd) 214 { 215 if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols()) 216 { 217 m_qr.~QRType(); 218 ::new (&m_qr) QRType(svd.cols(), svd.rows()); 219 } 220 if (svd.m_computeFullV) m_workspace.resize(svd.cols()); 221 else if (svd.m_computeThinV) m_workspace.resize(svd.rows()); 222 m_adjoint.resize(svd.cols(), svd.rows()); 223 } 224 run(JacobiSVD<MatrixType,ColPivHouseholderQRPreconditioner> & svd,const MatrixType & matrix)225 bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix) 226 { 227 if(matrix.cols() > matrix.rows()) 228 { 229 m_adjoint = matrix.adjoint(); 230 m_qr.compute(m_adjoint); 231 232 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint(); 233 if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace); 234 else if(svd.m_computeThinV) 235 { 236 svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows()); 237 m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace); 238 } 239 if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation(); 240 return true; 241 } 242 else return false; 243 } 244 245 private: 246 typedef ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType; 247 QRType m_qr; 248 TransposeTypeWithSameStorageOrder m_adjoint; 249 typename internal::plain_row_type<MatrixType>::type m_workspace; 250 }; 251 252 /*** preconditioner using HouseholderQR ***/ 253 254 template<typename MatrixType> 255 class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true> 256 { 257 public: allocate(const JacobiSVD<MatrixType,HouseholderQRPreconditioner> & svd)258 void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd) 259 { 260 if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols()) 261 { 262 m_qr.~QRType(); 263 ::new (&m_qr) QRType(svd.rows(), svd.cols()); 264 } 265 if (svd.m_computeFullU) m_workspace.resize(svd.rows()); 266 else if (svd.m_computeThinU) m_workspace.resize(svd.cols()); 267 } 268 run(JacobiSVD<MatrixType,HouseholderQRPreconditioner> & svd,const MatrixType & matrix)269 bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix) 270 { 271 if(matrix.rows() > matrix.cols()) 272 { 273 m_qr.compute(matrix); 274 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>(); 275 if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace); 276 else if(svd.m_computeThinU) 277 { 278 svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols()); 279 m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace); 280 } 281 if(svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols()); 282 return true; 283 } 284 return false; 285 } 286 private: 287 typedef HouseholderQR<MatrixType> QRType; 288 QRType m_qr; 289 typename internal::plain_col_type<MatrixType>::type m_workspace; 290 }; 291 292 template<typename MatrixType> 293 class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true> 294 { 295 public: 296 typedef typename MatrixType::Scalar Scalar; 297 enum 298 { 299 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 300 ColsAtCompileTime = MatrixType::ColsAtCompileTime, 301 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 302 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, 303 Options = MatrixType::Options 304 }; 305 306 typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime> 307 TransposeTypeWithSameStorageOrder; 308 allocate(const JacobiSVD<MatrixType,HouseholderQRPreconditioner> & svd)309 void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd) 310 { 311 if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols()) 312 { 313 m_qr.~QRType(); 314 ::new (&m_qr) QRType(svd.cols(), svd.rows()); 315 } 316 if (svd.m_computeFullV) m_workspace.resize(svd.cols()); 317 else if (svd.m_computeThinV) m_workspace.resize(svd.rows()); 318 m_adjoint.resize(svd.cols(), svd.rows()); 319 } 320 run(JacobiSVD<MatrixType,HouseholderQRPreconditioner> & svd,const MatrixType & matrix)321 bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix) 322 { 323 if(matrix.cols() > matrix.rows()) 324 { 325 m_adjoint = matrix.adjoint(); 326 m_qr.compute(m_adjoint); 327 328 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint(); 329 if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace); 330 else if(svd.m_computeThinV) 331 { 332 svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows()); 333 m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace); 334 } 335 if(svd.computeU()) svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows()); 336 return true; 337 } 338 else return false; 339 } 340 341 private: 342 typedef HouseholderQR<TransposeTypeWithSameStorageOrder> QRType; 343 QRType m_qr; 344 TransposeTypeWithSameStorageOrder m_adjoint; 345 typename internal::plain_row_type<MatrixType>::type m_workspace; 346 }; 347 348 /*** 2x2 SVD implementation 349 *** 350 *** JacobiSVD consists in performing a series of 2x2 SVD subproblems 351 ***/ 352 353 template<typename MatrixType, int QRPreconditioner> 354 struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, false> 355 { 356 typedef JacobiSVD<MatrixType, QRPreconditioner> SVD; 357 typedef typename MatrixType::RealScalar RealScalar; 358 static bool run(typename SVD::WorkMatrixType&, SVD&, Index, Index, RealScalar&) { return true; } 359 }; 360 361 template<typename MatrixType, int QRPreconditioner> 362 struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true> 363 { 364 typedef JacobiSVD<MatrixType, QRPreconditioner> SVD; 365 typedef typename MatrixType::Scalar Scalar; 366 typedef typename MatrixType::RealScalar RealScalar; 367 static bool run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q, RealScalar& maxDiagEntry) 368 { 369 using std::sqrt; 370 using std::abs; 371 Scalar z; 372 JacobiRotation<Scalar> rot; 373 RealScalar n = sqrt(numext::abs2(work_matrix.coeff(p,p)) + numext::abs2(work_matrix.coeff(q,p))); 374 375 const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)(); 376 const RealScalar precision = NumTraits<Scalar>::epsilon(); 377 378 if(n==0) 379 { 380 // make sure first column is zero 381 work_matrix.coeffRef(p,p) = work_matrix.coeffRef(q,p) = Scalar(0); 382 383 if(abs(numext::imag(work_matrix.coeff(p,q)))>considerAsZero) 384 { 385 // work_matrix.coeff(p,q) can be zero if work_matrix.coeff(q,p) is not zero but small enough to underflow when computing n 386 z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q); 387 work_matrix.row(p) *= z; 388 if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z); 389 } 390 if(abs(numext::imag(work_matrix.coeff(q,q)))>considerAsZero) 391 { 392 z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q); 393 work_matrix.row(q) *= z; 394 if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z); 395 } 396 // otherwise the second row is already zero, so we have nothing to do. 397 } 398 else 399 { 400 rot.c() = conj(work_matrix.coeff(p,p)) / n; 401 rot.s() = work_matrix.coeff(q,p) / n; 402 work_matrix.applyOnTheLeft(p,q,rot); 403 if(svd.computeU()) svd.m_matrixU.applyOnTheRight(p,q,rot.adjoint()); 404 if(abs(numext::imag(work_matrix.coeff(p,q)))>considerAsZero) 405 { 406 z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q); 407 work_matrix.col(q) *= z; 408 if(svd.computeV()) svd.m_matrixV.col(q) *= z; 409 } 410 if(abs(numext::imag(work_matrix.coeff(q,q)))>considerAsZero) 411 { 412 z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q); 413 work_matrix.row(q) *= z; 414 if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z); 415 } 416 } 417 418 // update largest diagonal entry 419 maxDiagEntry = numext::maxi<RealScalar>(maxDiagEntry,numext::maxi<RealScalar>(abs(work_matrix.coeff(p,p)), abs(work_matrix.coeff(q,q)))); 420 // and check whether the 2x2 block is already diagonal 421 RealScalar threshold = numext::maxi<RealScalar>(considerAsZero, precision * maxDiagEntry); 422 return abs(work_matrix.coeff(p,q))>threshold || abs(work_matrix.coeff(q,p)) > threshold; 423 } 424 }; 425 426 template<typename _MatrixType, int QRPreconditioner> 427 struct traits<JacobiSVD<_MatrixType,QRPreconditioner> > 428 { 429 typedef _MatrixType MatrixType; 430 }; 431 432 } // end namespace internal 433 434 /** \ingroup SVD_Module 435 * 436 * 437 * \class JacobiSVD 438 * 439 * \brief Two-sided Jacobi SVD decomposition of a rectangular matrix 440 * 441 * \tparam _MatrixType the type of the matrix of which we are computing the SVD decomposition 442 * \tparam QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally 443 * for the R-SVD step for non-square matrices. See discussion of possible values below. 444 * 445 * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product 446 * \f[ A = U S V^* \f] 447 * where \a U is a n-by-n unitary, \a V is a p-by-p unitary, and \a S is a n-by-p real positive matrix which is zero outside of its main diagonal; 448 * the diagonal entries of S are known as the \em singular \em values of \a A and the columns of \a U and \a V are known as the left 449 * and right \em singular \em vectors of \a A respectively. 450 * 451 * Singular values are always sorted in decreasing order. 452 * 453 * This JacobiSVD decomposition computes only the singular values by default. If you want \a U or \a V, you need to ask for them explicitly. 454 * 455 * You can ask for only \em thin \a U or \a V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting \a m be the 456 * smaller value among \a n and \a p, there are only \a m singular vectors; the remaining columns of \a U and \a V do not correspond to actual 457 * singular vectors. Asking for \em thin \a U or \a V means asking for only their \a m first columns to be formed. So \a U is then a n-by-m matrix, 458 * and \a V is then a p-by-m matrix. Notice that thin \a U and \a V are all you need for (least squares) solving. 459 * 460 * Here's an example demonstrating basic usage: 461 * \include JacobiSVD_basic.cpp 462 * Output: \verbinclude JacobiSVD_basic.out 463 * 464 * This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than 465 * bidiagonalizing SVD algorithms for large square matrices; however its complexity is still \f$ O(n^2p) \f$ where \a n is the smaller dimension and 466 * \a p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms. 467 * In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension. 468 * 469 * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to 470 * terminate in finite (and reasonable) time. 471 * 472 * The possible values for QRPreconditioner are: 473 * \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR. 474 * \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR. 475 * Contrary to other QRs, it doesn't allow computing thin unitaries. 476 * \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR. 477 * This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization 478 * is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive 479 * process is more reliable than the optimized bidiagonal SVD iterations. 480 * \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing 481 * JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in 482 * faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking 483 * if QR preconditioning is needed before applying it anyway. 484 * 485 * \sa MatrixBase::jacobiSvd() 486 */ 487 template<typename _MatrixType, int QRPreconditioner> class JacobiSVD 488 : public SVDBase<JacobiSVD<_MatrixType,QRPreconditioner> > 489 { 490 typedef SVDBase<JacobiSVD> Base; 491 public: 492 493 typedef _MatrixType MatrixType; 494 typedef typename MatrixType::Scalar Scalar; 495 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; 496 enum { 497 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 498 ColsAtCompileTime = MatrixType::ColsAtCompileTime, 499 DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime), 500 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 501 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, 502 MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime), 503 MatrixOptions = MatrixType::Options 504 }; 505 506 typedef typename Base::MatrixUType MatrixUType; 507 typedef typename Base::MatrixVType MatrixVType; 508 typedef typename Base::SingularValuesType SingularValuesType; 509 510 typedef typename internal::plain_row_type<MatrixType>::type RowType; 511 typedef typename internal::plain_col_type<MatrixType>::type ColType; 512 typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, 513 MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> 514 WorkMatrixType; 515 516 /** \brief Default Constructor. 517 * 518 * The default constructor is useful in cases in which the user intends to 519 * perform decompositions via JacobiSVD::compute(const MatrixType&). 520 */ 521 JacobiSVD() 522 {} 523 524 525 /** \brief Default Constructor with memory preallocation 526 * 527 * Like the default constructor but with preallocation of the internal data 528 * according to the specified problem size. 529 * \sa JacobiSVD() 530 */ 531 JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0) 532 { 533 allocate(rows, cols, computationOptions); 534 } 535 536 /** \brief Constructor performing the decomposition of given matrix. 537 * 538 * \param matrix the matrix to decompose 539 * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. 540 * By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU, 541 * #ComputeFullV, #ComputeThinV. 542 * 543 * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not 544 * available with the (non-default) FullPivHouseholderQR preconditioner. 545 */ 546 explicit JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0) 547 { 548 compute(matrix, computationOptions); 549 } 550 551 /** \brief Method performing the decomposition of given matrix using custom options. 552 * 553 * \param matrix the matrix to decompose 554 * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. 555 * By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU, 556 * #ComputeFullV, #ComputeThinV. 557 * 558 * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not 559 * available with the (non-default) FullPivHouseholderQR preconditioner. 560 */ 561 JacobiSVD& compute(const MatrixType& matrix, unsigned int computationOptions); 562 563 /** \brief Method performing the decomposition of given matrix using current options. 564 * 565 * \param matrix the matrix to decompose 566 * 567 * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int). 568 */ 569 JacobiSVD& compute(const MatrixType& matrix) 570 { 571 return compute(matrix, m_computationOptions); 572 } 573 574 using Base::computeU; 575 using Base::computeV; 576 using Base::rows; 577 using Base::cols; 578 using Base::rank; 579 580 private: 581 void allocate(Index rows, Index cols, unsigned int computationOptions); 582 583 protected: 584 using Base::m_matrixU; 585 using Base::m_matrixV; 586 using Base::m_singularValues; 587 using Base::m_isInitialized; 588 using Base::m_isAllocated; 589 using Base::m_usePrescribedThreshold; 590 using Base::m_computeFullU; 591 using Base::m_computeThinU; 592 using Base::m_computeFullV; 593 using Base::m_computeThinV; 594 using Base::m_computationOptions; 595 using Base::m_nonzeroSingularValues; 596 using Base::m_rows; 597 using Base::m_cols; 598 using Base::m_diagSize; 599 using Base::m_prescribedThreshold; 600 WorkMatrixType m_workMatrix; 601 602 template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex> 603 friend struct internal::svd_precondition_2x2_block_to_be_real; 604 template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything> 605 friend struct internal::qr_preconditioner_impl; 606 607 internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols; 608 internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows; 609 MatrixType m_scaledMatrix; 610 }; 611 612 template<typename MatrixType, int QRPreconditioner> 613 void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, unsigned int computationOptions) 614 { 615 eigen_assert(rows >= 0 && cols >= 0); 616 617 if (m_isAllocated && 618 rows == m_rows && 619 cols == m_cols && 620 computationOptions == m_computationOptions) 621 { 622 return; 623 } 624 625 m_rows = rows; 626 m_cols = cols; 627 m_isInitialized = false; 628 m_isAllocated = true; 629 m_computationOptions = computationOptions; 630 m_computeFullU = (computationOptions & ComputeFullU) != 0; 631 m_computeThinU = (computationOptions & ComputeThinU) != 0; 632 m_computeFullV = (computationOptions & ComputeFullV) != 0; 633 m_computeThinV = (computationOptions & ComputeThinV) != 0; 634 eigen_assert(!(m_computeFullU && m_computeThinU) && "JacobiSVD: you can't ask for both full and thin U"); 635 eigen_assert(!(m_computeFullV && m_computeThinV) && "JacobiSVD: you can't ask for both full and thin V"); 636 eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime==Dynamic) && 637 "JacobiSVD: thin U and V are only available when your matrix has a dynamic number of columns."); 638 if (QRPreconditioner == FullPivHouseholderQRPreconditioner) 639 { 640 eigen_assert(!(m_computeThinU || m_computeThinV) && 641 "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. " 642 "Use the ColPivHouseholderQR preconditioner instead."); 643 } 644 m_diagSize = (std::min)(m_rows, m_cols); 645 m_singularValues.resize(m_diagSize); 646 if(RowsAtCompileTime==Dynamic) 647 m_matrixU.resize(m_rows, m_computeFullU ? m_rows 648 : m_computeThinU ? m_diagSize 649 : 0); 650 if(ColsAtCompileTime==Dynamic) 651 m_matrixV.resize(m_cols, m_computeFullV ? m_cols 652 : m_computeThinV ? m_diagSize 653 : 0); 654 m_workMatrix.resize(m_diagSize, m_diagSize); 655 656 if(m_cols>m_rows) m_qr_precond_morecols.allocate(*this); 657 if(m_rows>m_cols) m_qr_precond_morerows.allocate(*this); 658 if(m_rows!=m_cols) m_scaledMatrix.resize(rows,cols); 659 } 660 661 template<typename MatrixType, int QRPreconditioner> 662 JacobiSVD<MatrixType, QRPreconditioner>& 663 JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions) 664 { 665 using std::abs; 666 allocate(matrix.rows(), matrix.cols(), computationOptions); 667 668 // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations, 669 // only worsening the precision of U and V as we accumulate more rotations 670 const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon(); 671 672 // limit for denormal numbers to be considered zero in order to avoid infinite loops (see bug 286) 673 const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)(); 674 675 // Scaling factor to reduce over/under-flows 676 RealScalar scale = matrix.cwiseAbs().maxCoeff(); 677 if(scale==RealScalar(0)) scale = RealScalar(1); 678 679 /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */ 680 681 if(m_rows!=m_cols) 682 { 683 m_scaledMatrix = matrix / scale; 684 m_qr_precond_morecols.run(*this, m_scaledMatrix); 685 m_qr_precond_morerows.run(*this, m_scaledMatrix); 686 } 687 else 688 { 689 m_workMatrix = matrix.block(0,0,m_diagSize,m_diagSize) / scale; 690 if(m_computeFullU) m_matrixU.setIdentity(m_rows,m_rows); 691 if(m_computeThinU) m_matrixU.setIdentity(m_rows,m_diagSize); 692 if(m_computeFullV) m_matrixV.setIdentity(m_cols,m_cols); 693 if(m_computeThinV) m_matrixV.setIdentity(m_cols, m_diagSize); 694 } 695 696 /*** step 2. The main Jacobi SVD iteration. ***/ 697 RealScalar maxDiagEntry = m_workMatrix.cwiseAbs().diagonal().maxCoeff(); 698 699 bool finished = false; 700 while(!finished) 701 { 702 finished = true; 703 704 // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix 705 706 for(Index p = 1; p < m_diagSize; ++p) 707 { 708 for(Index q = 0; q < p; ++q) 709 { 710 // if this 2x2 sub-matrix is not diagonal already... 711 // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't 712 // keep us iterating forever. Similarly, small denormal numbers are considered zero. 713 RealScalar threshold = numext::maxi<RealScalar>(considerAsZero, precision * maxDiagEntry); 714 if(abs(m_workMatrix.coeff(p,q))>threshold || abs(m_workMatrix.coeff(q,p)) > threshold) 715 { 716 finished = false; 717 // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal 718 // the complex to real operation returns true if the updated 2x2 block is not already diagonal 719 if(internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q, maxDiagEntry)) 720 { 721 JacobiRotation<RealScalar> j_left, j_right; 722 internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right); 723 724 // accumulate resulting Jacobi rotations 725 m_workMatrix.applyOnTheLeft(p,q,j_left); 726 if(computeU()) m_matrixU.applyOnTheRight(p,q,j_left.transpose()); 727 728 m_workMatrix.applyOnTheRight(p,q,j_right); 729 if(computeV()) m_matrixV.applyOnTheRight(p,q,j_right); 730 731 // keep track of the largest diagonal coefficient 732 maxDiagEntry = numext::maxi<RealScalar>(maxDiagEntry,numext::maxi<RealScalar>(abs(m_workMatrix.coeff(p,p)), abs(m_workMatrix.coeff(q,q)))); 733 } 734 } 735 } 736 } 737 } 738 739 /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/ 740 741 for(Index i = 0; i < m_diagSize; ++i) 742 { 743 // For a complex matrix, some diagonal coefficients might note have been 744 // treated by svd_precondition_2x2_block_to_be_real, and the imaginary part 745 // of some diagonal entry might not be null. 746 if(NumTraits<Scalar>::IsComplex && abs(numext::imag(m_workMatrix.coeff(i,i)))>considerAsZero) 747 { 748 RealScalar a = abs(m_workMatrix.coeff(i,i)); 749 m_singularValues.coeffRef(i) = abs(a); 750 if(computeU()) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a; 751 } 752 else 753 { 754 // m_workMatrix.coeff(i,i) is already real, no difficulty: 755 RealScalar a = numext::real(m_workMatrix.coeff(i,i)); 756 m_singularValues.coeffRef(i) = abs(a); 757 if(computeU() && (a<RealScalar(0))) m_matrixU.col(i) = -m_matrixU.col(i); 758 } 759 } 760 761 m_singularValues *= scale; 762 763 /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/ 764 765 m_nonzeroSingularValues = m_diagSize; 766 for(Index i = 0; i < m_diagSize; i++) 767 { 768 Index pos; 769 RealScalar maxRemainingSingularValue = m_singularValues.tail(m_diagSize-i).maxCoeff(&pos); 770 if(maxRemainingSingularValue == RealScalar(0)) 771 { 772 m_nonzeroSingularValues = i; 773 break; 774 } 775 if(pos) 776 { 777 pos += i; 778 std::swap(m_singularValues.coeffRef(i), m_singularValues.coeffRef(pos)); 779 if(computeU()) m_matrixU.col(pos).swap(m_matrixU.col(i)); 780 if(computeV()) m_matrixV.col(pos).swap(m_matrixV.col(i)); 781 } 782 } 783 784 m_isInitialized = true; 785 return *this; 786 } 787 788 /** \svd_module 789 * 790 * \return the singular value decomposition of \c *this computed by two-sided 791 * Jacobi transformations. 792 * 793 * \sa class JacobiSVD 794 */ 795 template<typename Derived> 796 JacobiSVD<typename MatrixBase<Derived>::PlainObject> 797 MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const 798 { 799 return JacobiSVD<PlainObject>(*this, computationOptions); 800 } 801 802 } // end namespace Eigen 803 804 #endif // EIGEN_JACOBISVD_H 805