JacobiSVD.h (32988B)
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: 55 void allocate(const JacobiSVD<MatrixType, QRPreconditioner>&) {} 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 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 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 Options = MatrixType::Options 116 }; 117 118 typedef typename internal::make_proper_matrix_type< 119 Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime 120 >::type TransposeTypeWithSameStorageOrder; 121 122 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 } 132 133 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: 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 } 169 170 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 Options = MatrixType::Options 206 }; 207 208 typedef typename internal::make_proper_matrix_type< 209 Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime 210 >::type TransposeTypeWithSameStorageOrder; 211 212 void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd) 213 { 214 if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols()) 215 { 216 m_qr.~QRType(); 217 ::new (&m_qr) QRType(svd.cols(), svd.rows()); 218 } 219 if (svd.m_computeFullV) m_workspace.resize(svd.cols()); 220 else if (svd.m_computeThinV) m_workspace.resize(svd.rows()); 221 m_adjoint.resize(svd.cols(), svd.rows()); 222 } 223 224 bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix) 225 { 226 if(matrix.cols() > matrix.rows()) 227 { 228 m_adjoint = matrix.adjoint(); 229 m_qr.compute(m_adjoint); 230 231 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint(); 232 if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace); 233 else if(svd.m_computeThinV) 234 { 235 svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows()); 236 m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace); 237 } 238 if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation(); 239 return true; 240 } 241 else return false; 242 } 243 244 private: 245 typedef ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType; 246 QRType m_qr; 247 TransposeTypeWithSameStorageOrder m_adjoint; 248 typename internal::plain_row_type<MatrixType>::type m_workspace; 249 }; 250 251 /*** preconditioner using HouseholderQR ***/ 252 253 template<typename MatrixType> 254 class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true> 255 { 256 public: 257 void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd) 258 { 259 if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols()) 260 { 261 m_qr.~QRType(); 262 ::new (&m_qr) QRType(svd.rows(), svd.cols()); 263 } 264 if (svd.m_computeFullU) m_workspace.resize(svd.rows()); 265 else if (svd.m_computeThinU) m_workspace.resize(svd.cols()); 266 } 267 268 bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix) 269 { 270 if(matrix.rows() > matrix.cols()) 271 { 272 m_qr.compute(matrix); 273 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>(); 274 if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace); 275 else if(svd.m_computeThinU) 276 { 277 svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols()); 278 m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace); 279 } 280 if(svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols()); 281 return true; 282 } 283 return false; 284 } 285 private: 286 typedef HouseholderQR<MatrixType> QRType; 287 QRType m_qr; 288 typename internal::plain_col_type<MatrixType>::type m_workspace; 289 }; 290 291 template<typename MatrixType> 292 class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true> 293 { 294 public: 295 typedef typename MatrixType::Scalar Scalar; 296 enum 297 { 298 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 299 ColsAtCompileTime = MatrixType::ColsAtCompileTime, 300 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 301 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, 302 Options = MatrixType::Options 303 }; 304 305 typedef typename internal::make_proper_matrix_type< 306 Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime 307 >::type TransposeTypeWithSameStorageOrder; 308 309 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 } 320 321 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 : traits<_MatrixType> 429 { 430 typedef _MatrixType MatrixType; 431 }; 432 433 } // end namespace internal 434 435 /** \ingroup SVD_Module 436 * 437 * 438 * \class JacobiSVD 439 * 440 * \brief Two-sided Jacobi SVD decomposition of a rectangular matrix 441 * 442 * \tparam _MatrixType the type of the matrix of which we are computing the SVD decomposition 443 * \tparam QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally 444 * for the R-SVD step for non-square matrices. See discussion of possible values below. 445 * 446 * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product 447 * \f[ A = U S V^* \f] 448 * 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; 449 * 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 450 * and right \em singular \em vectors of \a A respectively. 451 * 452 * Singular values are always sorted in decreasing order. 453 * 454 * 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. 455 * 456 * 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 457 * 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 458 * 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, 459 * 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. 460 * 461 * Here's an example demonstrating basic usage: 462 * \include JacobiSVD_basic.cpp 463 * Output: \verbinclude JacobiSVD_basic.out 464 * 465 * This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than 466 * 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 467 * \a p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms. 468 * In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension. 469 * 470 * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to 471 * terminate in finite (and reasonable) time. 472 * 473 * The possible values for QRPreconditioner are: 474 * \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR. 475 * \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR. 476 * Contrary to other QRs, it doesn't allow computing thin unitaries. 477 * \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR. 478 * This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization 479 * is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive 480 * process is more reliable than the optimized bidiagonal SVD iterations. 481 * \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing 482 * JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in 483 * faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking 484 * if QR preconditioning is needed before applying it anyway. 485 * 486 * \sa MatrixBase::jacobiSvd() 487 */ 488 template<typename _MatrixType, int QRPreconditioner> class JacobiSVD 489 : public SVDBase<JacobiSVD<_MatrixType,QRPreconditioner> > 490 { 491 typedef SVDBase<JacobiSVD> Base; 492 public: 493 494 typedef _MatrixType MatrixType; 495 typedef typename MatrixType::Scalar Scalar; 496 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; 497 enum { 498 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 499 ColsAtCompileTime = MatrixType::ColsAtCompileTime, 500 DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime), 501 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 502 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, 503 MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime), 504 MatrixOptions = MatrixType::Options 505 }; 506 507 typedef typename Base::MatrixUType MatrixUType; 508 typedef typename Base::MatrixVType MatrixVType; 509 typedef typename Base::SingularValuesType SingularValuesType; 510 511 typedef typename internal::plain_row_type<MatrixType>::type RowType; 512 typedef typename internal::plain_col_type<MatrixType>::type ColType; 513 typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, 514 MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> 515 WorkMatrixType; 516 517 /** \brief Default Constructor. 518 * 519 * The default constructor is useful in cases in which the user intends to 520 * perform decompositions via JacobiSVD::compute(const MatrixType&). 521 */ 522 JacobiSVD() 523 {} 524 525 526 /** \brief Default Constructor with memory preallocation 527 * 528 * Like the default constructor but with preallocation of the internal data 529 * according to the specified problem size. 530 * \sa JacobiSVD() 531 */ 532 JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0) 533 { 534 allocate(rows, cols, computationOptions); 535 } 536 537 /** \brief Constructor performing the decomposition of given matrix. 538 * 539 * \param matrix the matrix to decompose 540 * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. 541 * By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU, 542 * #ComputeFullV, #ComputeThinV. 543 * 544 * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not 545 * available with the (non-default) FullPivHouseholderQR preconditioner. 546 */ 547 explicit JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0) 548 { 549 compute(matrix, computationOptions); 550 } 551 552 /** \brief Method performing the decomposition of given matrix using custom options. 553 * 554 * \param matrix the matrix to decompose 555 * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. 556 * By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU, 557 * #ComputeFullV, #ComputeThinV. 558 * 559 * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not 560 * available with the (non-default) FullPivHouseholderQR preconditioner. 561 */ 562 JacobiSVD& compute(const MatrixType& matrix, unsigned int computationOptions); 563 564 /** \brief Method performing the decomposition of given matrix using current options. 565 * 566 * \param matrix the matrix to decompose 567 * 568 * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int). 569 */ 570 JacobiSVD& compute(const MatrixType& matrix) 571 { 572 return compute(matrix, m_computationOptions); 573 } 574 575 using Base::computeU; 576 using Base::computeV; 577 using Base::rows; 578 using Base::cols; 579 using Base::rank; 580 581 private: 582 void allocate(Index rows, Index cols, unsigned int computationOptions); 583 584 protected: 585 using Base::m_matrixU; 586 using Base::m_matrixV; 587 using Base::m_singularValues; 588 using Base::m_info; 589 using Base::m_isInitialized; 590 using Base::m_isAllocated; 591 using Base::m_usePrescribedThreshold; 592 using Base::m_computeFullU; 593 using Base::m_computeThinU; 594 using Base::m_computeFullV; 595 using Base::m_computeThinV; 596 using Base::m_computationOptions; 597 using Base::m_nonzeroSingularValues; 598 using Base::m_rows; 599 using Base::m_cols; 600 using Base::m_diagSize; 601 using Base::m_prescribedThreshold; 602 WorkMatrixType m_workMatrix; 603 604 template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex> 605 friend struct internal::svd_precondition_2x2_block_to_be_real; 606 template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything> 607 friend struct internal::qr_preconditioner_impl; 608 609 internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols; 610 internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows; 611 MatrixType m_scaledMatrix; 612 }; 613 614 template<typename MatrixType, int QRPreconditioner> 615 void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Eigen::Index rows, Eigen::Index cols, unsigned int computationOptions) 616 { 617 eigen_assert(rows >= 0 && cols >= 0); 618 619 if (m_isAllocated && 620 rows == m_rows && 621 cols == m_cols && 622 computationOptions == m_computationOptions) 623 { 624 return; 625 } 626 627 m_rows = rows; 628 m_cols = cols; 629 m_info = Success; 630 m_isInitialized = false; 631 m_isAllocated = true; 632 m_computationOptions = computationOptions; 633 m_computeFullU = (computationOptions & ComputeFullU) != 0; 634 m_computeThinU = (computationOptions & ComputeThinU) != 0; 635 m_computeFullV = (computationOptions & ComputeFullV) != 0; 636 m_computeThinV = (computationOptions & ComputeThinV) != 0; 637 eigen_assert(!(m_computeFullU && m_computeThinU) && "JacobiSVD: you can't ask for both full and thin U"); 638 eigen_assert(!(m_computeFullV && m_computeThinV) && "JacobiSVD: you can't ask for both full and thin V"); 639 eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime==Dynamic) && 640 "JacobiSVD: thin U and V are only available when your matrix has a dynamic number of columns."); 641 if (QRPreconditioner == FullPivHouseholderQRPreconditioner) 642 { 643 eigen_assert(!(m_computeThinU || m_computeThinV) && 644 "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. " 645 "Use the ColPivHouseholderQR preconditioner instead."); 646 } 647 m_diagSize = (std::min)(m_rows, m_cols); 648 m_singularValues.resize(m_diagSize); 649 if(RowsAtCompileTime==Dynamic) 650 m_matrixU.resize(m_rows, m_computeFullU ? m_rows 651 : m_computeThinU ? m_diagSize 652 : 0); 653 if(ColsAtCompileTime==Dynamic) 654 m_matrixV.resize(m_cols, m_computeFullV ? m_cols 655 : m_computeThinV ? m_diagSize 656 : 0); 657 m_workMatrix.resize(m_diagSize, m_diagSize); 658 659 if(m_cols>m_rows) m_qr_precond_morecols.allocate(*this); 660 if(m_rows>m_cols) m_qr_precond_morerows.allocate(*this); 661 if(m_rows!=m_cols) m_scaledMatrix.resize(rows,cols); 662 } 663 664 template<typename MatrixType, int QRPreconditioner> 665 JacobiSVD<MatrixType, QRPreconditioner>& 666 JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions) 667 { 668 using std::abs; 669 allocate(matrix.rows(), matrix.cols(), computationOptions); 670 671 // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations, 672 // only worsening the precision of U and V as we accumulate more rotations 673 const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon(); 674 675 // limit for denormal numbers to be considered zero in order to avoid infinite loops (see bug 286) 676 const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)(); 677 678 // Scaling factor to reduce over/under-flows 679 RealScalar scale = matrix.cwiseAbs().template maxCoeff<PropagateNaN>(); 680 if (!(numext::isfinite)(scale)) { 681 m_isInitialized = true; 682 m_info = InvalidInput; 683 return *this; 684 } 685 if(scale==RealScalar(0)) scale = RealScalar(1); 686 687 /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */ 688 689 if(m_rows!=m_cols) 690 { 691 m_scaledMatrix = matrix / scale; 692 m_qr_precond_morecols.run(*this, m_scaledMatrix); 693 m_qr_precond_morerows.run(*this, m_scaledMatrix); 694 } 695 else 696 { 697 m_workMatrix = matrix.block(0,0,m_diagSize,m_diagSize) / scale; 698 if(m_computeFullU) m_matrixU.setIdentity(m_rows,m_rows); 699 if(m_computeThinU) m_matrixU.setIdentity(m_rows,m_diagSize); 700 if(m_computeFullV) m_matrixV.setIdentity(m_cols,m_cols); 701 if(m_computeThinV) m_matrixV.setIdentity(m_cols, m_diagSize); 702 } 703 704 /*** step 2. The main Jacobi SVD iteration. ***/ 705 RealScalar maxDiagEntry = m_workMatrix.cwiseAbs().diagonal().maxCoeff(); 706 707 bool finished = false; 708 while(!finished) 709 { 710 finished = true; 711 712 // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix 713 714 for(Index p = 1; p < m_diagSize; ++p) 715 { 716 for(Index q = 0; q < p; ++q) 717 { 718 // if this 2x2 sub-matrix is not diagonal already... 719 // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't 720 // keep us iterating forever. Similarly, small denormal numbers are considered zero. 721 RealScalar threshold = numext::maxi<RealScalar>(considerAsZero, precision * maxDiagEntry); 722 if(abs(m_workMatrix.coeff(p,q))>threshold || abs(m_workMatrix.coeff(q,p)) > threshold) 723 { 724 finished = false; 725 // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal 726 // the complex to real operation returns true if the updated 2x2 block is not already diagonal 727 if(internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q, maxDiagEntry)) 728 { 729 JacobiRotation<RealScalar> j_left, j_right; 730 internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right); 731 732 // accumulate resulting Jacobi rotations 733 m_workMatrix.applyOnTheLeft(p,q,j_left); 734 if(computeU()) m_matrixU.applyOnTheRight(p,q,j_left.transpose()); 735 736 m_workMatrix.applyOnTheRight(p,q,j_right); 737 if(computeV()) m_matrixV.applyOnTheRight(p,q,j_right); 738 739 // keep track of the largest diagonal coefficient 740 maxDiagEntry = numext::maxi<RealScalar>(maxDiagEntry,numext::maxi<RealScalar>(abs(m_workMatrix.coeff(p,p)), abs(m_workMatrix.coeff(q,q)))); 741 } 742 } 743 } 744 } 745 } 746 747 /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/ 748 749 for(Index i = 0; i < m_diagSize; ++i) 750 { 751 // For a complex matrix, some diagonal coefficients might note have been 752 // treated by svd_precondition_2x2_block_to_be_real, and the imaginary part 753 // of some diagonal entry might not be null. 754 if(NumTraits<Scalar>::IsComplex && abs(numext::imag(m_workMatrix.coeff(i,i)))>considerAsZero) 755 { 756 RealScalar a = abs(m_workMatrix.coeff(i,i)); 757 m_singularValues.coeffRef(i) = abs(a); 758 if(computeU()) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a; 759 } 760 else 761 { 762 // m_workMatrix.coeff(i,i) is already real, no difficulty: 763 RealScalar a = numext::real(m_workMatrix.coeff(i,i)); 764 m_singularValues.coeffRef(i) = abs(a); 765 if(computeU() && (a<RealScalar(0))) m_matrixU.col(i) = -m_matrixU.col(i); 766 } 767 } 768 769 m_singularValues *= scale; 770 771 /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/ 772 773 m_nonzeroSingularValues = m_diagSize; 774 for(Index i = 0; i < m_diagSize; i++) 775 { 776 Index pos; 777 RealScalar maxRemainingSingularValue = m_singularValues.tail(m_diagSize-i).maxCoeff(&pos); 778 if(maxRemainingSingularValue == RealScalar(0)) 779 { 780 m_nonzeroSingularValues = i; 781 break; 782 } 783 if(pos) 784 { 785 pos += i; 786 std::swap(m_singularValues.coeffRef(i), m_singularValues.coeffRef(pos)); 787 if(computeU()) m_matrixU.col(pos).swap(m_matrixU.col(i)); 788 if(computeV()) m_matrixV.col(pos).swap(m_matrixV.col(i)); 789 } 790 } 791 792 m_isInitialized = true; 793 return *this; 794 } 795 796 /** \svd_module 797 * 798 * \return the singular value decomposition of \c *this computed by two-sided 799 * Jacobi transformations. 800 * 801 * \sa class JacobiSVD 802 */ 803 template<typename Derived> 804 JacobiSVD<typename MatrixBase<Derived>::PlainObject> 805 MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const 806 { 807 return JacobiSVD<PlainObject>(*this, computationOptions); 808 } 809 810 } // end namespace Eigen 811 812 #endif // EIGEN_JACOBISVD_H