1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // We used the "A Divide-And-Conquer Algorithm for the Bidiagonal SVD" 5 // research report written by Ming Gu and Stanley C.Eisenstat 6 // The code variable names correspond to the names they used in their 7 // report 8 // 9 // Copyright (C) 2013 Gauthier Brun <brun.gauthier@gmail.com> 10 // Copyright (C) 2013 Nicolas Carre <nicolas.carre@ensimag.fr> 11 // Copyright (C) 2013 Jean Ceccato <jean.ceccato@ensimag.fr> 12 // Copyright (C) 2013 Pierre Zoppitelli <pierre.zoppitelli@ensimag.fr> 13 // Copyright (C) 2013 Jitse Niesen <jitse@maths.leeds.ac.uk> 14 // Copyright (C) 2014-2016 Gael Guennebaud <gael.guennebaud@inria.fr> 15 // 16 // Source Code Form is subject to the terms of the Mozilla 17 // Public License v. 2.0. If a copy of the MPL was not distributed 18 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 19 20 #ifndef EIGEN_BDCSVD_H 21 #define EIGEN_BDCSVD_H 22 // #define EIGEN_BDCSVD_DEBUG_VERBOSE 23 // #define EIGEN_BDCSVD_SANITY_CHECKS 24 25 namespace Eigen { 26 27 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 28 IOFormat bdcsvdfmt(8, 0, ", ", "\n", " [", "]"); 29 #endif 30 31 template<typename _MatrixType> class BDCSVD; 32 33 namespace internal { 34 35 template<typename _MatrixType> 36 struct traits<BDCSVD<_MatrixType> > 37 { 38 typedef _MatrixType MatrixType; 39 }; 40 41 } // end namespace internal 42 43 44 /** \ingroup SVD_Module 45 * 46 * 47 * \class BDCSVD 48 * 49 * \brief class Bidiagonal Divide and Conquer SVD 50 * 51 * \tparam _MatrixType the type of the matrix of which we are computing the SVD decomposition 52 * 53 * This class first reduces the input matrix to bi-diagonal form using class UpperBidiagonalization, 54 * and then performs a divide-and-conquer diagonalization. Small blocks are diagonalized using class JacobiSVD. 55 * You can control the switching size with the setSwitchSize() method, default is 16. 56 * For small matrice (<16), it is thus preferable to directly use JacobiSVD. For larger ones, BDCSVD is highly 57 * recommended and can several order of magnitude faster. 58 * 59 * \warning this algorithm is unlikely to provide accurate result when compiled with unsafe math optimizations. 60 * For instance, this concerns Intel's compiler (ICC), which perfroms such optimization by default unless 61 * you compile with the \c -fp-model \c precise option. Likewise, the \c -ffast-math option of GCC or clang will 62 * significantly degrade the accuracy. 63 * 64 * \sa class JacobiSVD 65 */ 66 template<typename _MatrixType> 67 class BDCSVD : public SVDBase<BDCSVD<_MatrixType> > 68 { 69 typedef SVDBase<BDCSVD> Base; 70 71 public: 72 using Base::rows; 73 using Base::cols; 74 using Base::computeU; 75 using Base::computeV; 76 77 typedef _MatrixType MatrixType; 78 typedef typename MatrixType::Scalar Scalar; 79 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; 80 enum { 81 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 82 ColsAtCompileTime = MatrixType::ColsAtCompileTime, 83 DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime, ColsAtCompileTime), 84 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 85 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, 86 MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime, MaxColsAtCompileTime), 87 MatrixOptions = MatrixType::Options 88 }; 89 90 typedef typename Base::MatrixUType MatrixUType; 91 typedef typename Base::MatrixVType MatrixVType; 92 typedef typename Base::SingularValuesType SingularValuesType; 93 94 typedef Matrix<Scalar, Dynamic, Dynamic, ColMajor> MatrixX; 95 typedef Matrix<RealScalar, Dynamic, Dynamic, ColMajor> MatrixXr; 96 typedef Matrix<RealScalar, Dynamic, 1> VectorType; 97 typedef Array<RealScalar, Dynamic, 1> ArrayXr; 98 typedef Array<Index,1,Dynamic> ArrayXi; 99 typedef Ref<ArrayXr> ArrayRef; 100 typedef Ref<ArrayXi> IndicesRef; 101 102 /** \brief Default Constructor. 103 * 104 * The default constructor is useful in cases in which the user intends to 105 * perform decompositions via BDCSVD::compute(const MatrixType&). 106 */ 107 BDCSVD() : m_algoswap(16), m_numIters(0) 108 {} 109 110 111 /** \brief Default Constructor with memory preallocation 112 * 113 * Like the default constructor but with preallocation of the internal data 114 * according to the specified problem size. 115 * \sa BDCSVD() 116 */ 117 BDCSVD(Index rows, Index cols, unsigned int computationOptions = 0) 118 : m_algoswap(16), m_numIters(0) 119 { 120 allocate(rows, cols, computationOptions); 121 } 122 123 /** \brief Constructor performing the decomposition of given matrix. 124 * 125 * \param matrix the matrix to decompose 126 * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. 127 * By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU, 128 * #ComputeFullV, #ComputeThinV. 129 * 130 * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not 131 * available with the (non - default) FullPivHouseholderQR preconditioner. 132 */ 133 BDCSVD(const MatrixType& matrix, unsigned int computationOptions = 0) 134 : m_algoswap(16), m_numIters(0) 135 { 136 compute(matrix, computationOptions); 137 } 138 139 ~BDCSVD() 140 { 141 } 142 143 /** \brief Method performing the decomposition of given matrix using custom options. 144 * 145 * \param matrix the matrix to decompose 146 * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. 147 * By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU, 148 * #ComputeFullV, #ComputeThinV. 149 * 150 * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not 151 * available with the (non - default) FullPivHouseholderQR preconditioner. 152 */ 153 BDCSVD& compute(const MatrixType& matrix, unsigned int computationOptions); 154 155 /** \brief Method performing the decomposition of given matrix using current options. 156 * 157 * \param matrix the matrix to decompose 158 * 159 * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int). 160 */ 161 BDCSVD& compute(const MatrixType& matrix) 162 { 163 return compute(matrix, this->m_computationOptions); 164 } 165 166 void setSwitchSize(int s) 167 { 168 eigen_assert(s>3 && "BDCSVD the size of the algo switch has to be greater than 3"); 169 m_algoswap = s; 170 } 171 172 private: 173 void allocate(Index rows, Index cols, unsigned int computationOptions); 174 void divide(Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift); 175 void computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V); 176 void computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, VectorType& singVals, ArrayRef shifts, ArrayRef mus); 177 void perturbCol0(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat); 178 void computeSingVecs(const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V); 179 void deflation43(Index firstCol, Index shift, Index i, Index size); 180 void deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size); 181 void deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift); 182 template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV> 183 void copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naivev); 184 void structured_update(Block<MatrixXr,Dynamic,Dynamic> A, const MatrixXr &B, Index n1); 185 static RealScalar secularEq(RealScalar x, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const ArrayRef& diagShifted, RealScalar shift); 186 187 protected: 188 MatrixXr m_naiveU, m_naiveV; 189 MatrixXr m_computed; 190 Index m_nRec; 191 ArrayXr m_workspace; 192 ArrayXi m_workspaceI; 193 int m_algoswap; 194 bool m_isTranspose, m_compU, m_compV; 195 196 using Base::m_singularValues; 197 using Base::m_diagSize; 198 using Base::m_computeFullU; 199 using Base::m_computeFullV; 200 using Base::m_computeThinU; 201 using Base::m_computeThinV; 202 using Base::m_matrixU; 203 using Base::m_matrixV; 204 using Base::m_isInitialized; 205 using Base::m_nonzeroSingularValues; 206 207 public: 208 int m_numIters; 209 }; //end class BDCSVD 210 211 212 // Method to allocate and initialize matrix and attributes 213 template<typename MatrixType> 214 void BDCSVD<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions) 215 { 216 m_isTranspose = (cols > rows); 217 218 if (Base::allocate(rows, cols, computationOptions)) 219 return; 220 221 m_computed = MatrixXr::Zero(m_diagSize + 1, m_diagSize ); 222 m_compU = computeV(); 223 m_compV = computeU(); 224 if (m_isTranspose) 225 std::swap(m_compU, m_compV); 226 227 if (m_compU) m_naiveU = MatrixXr::Zero(m_diagSize + 1, m_diagSize + 1 ); 228 else m_naiveU = MatrixXr::Zero(2, m_diagSize + 1 ); 229 230 if (m_compV) m_naiveV = MatrixXr::Zero(m_diagSize, m_diagSize); 231 232 m_workspace.resize((m_diagSize+1)*(m_diagSize+1)*3); 233 m_workspaceI.resize(3*m_diagSize); 234 }// end allocate 235 236 template<typename MatrixType> 237 BDCSVD<MatrixType>& BDCSVD<MatrixType>::compute(const MatrixType& matrix, unsigned int computationOptions) 238 { 239 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 240 std::cout << "\n\n\n======================================================================================================================\n\n\n"; 241 #endif 242 allocate(matrix.rows(), matrix.cols(), computationOptions); 243 using std::abs; 244 245 const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)(); 246 247 //**** step -1 - If the problem is too small, directly falls back to JacobiSVD and return 248 if(matrix.cols() < m_algoswap) 249 { 250 // FIXME this line involves temporaries 251 JacobiSVD<MatrixType> jsvd(matrix,computationOptions); 252 if(computeU()) m_matrixU = jsvd.matrixU(); 253 if(computeV()) m_matrixV = jsvd.matrixV(); 254 m_singularValues = jsvd.singularValues(); 255 m_nonzeroSingularValues = jsvd.nonzeroSingularValues(); 256 m_isInitialized = true; 257 return *this; 258 } 259 260 //**** step 0 - Copy the input matrix and apply scaling to reduce over/under-flows 261 RealScalar scale = matrix.cwiseAbs().maxCoeff(); 262 if(scale==RealScalar(0)) scale = RealScalar(1); 263 MatrixX copy; 264 if (m_isTranspose) copy = matrix.adjoint()/scale; 265 else copy = matrix/scale; 266 267 //**** step 1 - Bidiagonalization 268 // FIXME this line involves temporaries 269 internal::UpperBidiagonalization<MatrixX> bid(copy); 270 271 //**** step 2 - Divide & Conquer 272 m_naiveU.setZero(); 273 m_naiveV.setZero(); 274 // FIXME this line involves a temporary matrix 275 m_computed.topRows(m_diagSize) = bid.bidiagonal().toDenseMatrix().transpose(); 276 m_computed.template bottomRows<1>().setZero(); 277 divide(0, m_diagSize - 1, 0, 0, 0); 278 279 //**** step 3 - Copy singular values and vectors 280 for (int i=0; i<m_diagSize; i++) 281 { 282 RealScalar a = abs(m_computed.coeff(i, i)); 283 m_singularValues.coeffRef(i) = a * scale; 284 if (a<considerZero) 285 { 286 m_nonzeroSingularValues = i; 287 m_singularValues.tail(m_diagSize - i - 1).setZero(); 288 break; 289 } 290 else if (i == m_diagSize - 1) 291 { 292 m_nonzeroSingularValues = i + 1; 293 break; 294 } 295 } 296 297 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 298 // std::cout << "m_naiveU\n" << m_naiveU << "\n\n"; 299 // std::cout << "m_naiveV\n" << m_naiveV << "\n\n"; 300 #endif 301 if(m_isTranspose) copyUV(bid.householderV(), bid.householderU(), m_naiveV, m_naiveU); 302 else copyUV(bid.householderU(), bid.householderV(), m_naiveU, m_naiveV); 303 304 m_isInitialized = true; 305 return *this; 306 }// end compute 307 308 309 template<typename MatrixType> 310 template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV> 311 void BDCSVD<MatrixType>::copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naiveV) 312 { 313 // Note exchange of U and V: m_matrixU is set from m_naiveV and vice versa 314 if (computeU()) 315 { 316 Index Ucols = m_computeThinU ? m_diagSize : householderU.cols(); 317 m_matrixU = MatrixX::Identity(householderU.cols(), Ucols); 318 m_matrixU.topLeftCorner(m_diagSize, m_diagSize) = naiveV.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize); 319 householderU.applyThisOnTheLeft(m_matrixU); // FIXME this line involves a temporary buffer 320 } 321 if (computeV()) 322 { 323 Index Vcols = m_computeThinV ? m_diagSize : householderV.cols(); 324 m_matrixV = MatrixX::Identity(householderV.cols(), Vcols); 325 m_matrixV.topLeftCorner(m_diagSize, m_diagSize) = naiveU.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize); 326 householderV.applyThisOnTheLeft(m_matrixV); // FIXME this line involves a temporary buffer 327 } 328 } 329 330 /** \internal 331 * Performs A = A * B exploiting the special structure of the matrix A. Splitting A as: 332 * A = [A1] 333 * [A2] 334 * such that A1.rows()==n1, then we assume that at least half of the columns of A1 and A2 are zeros. 335 * We can thus pack them prior to the the matrix product. However, this is only worth the effort if the matrix is large 336 * enough. 337 */ 338 template<typename MatrixType> 339 void BDCSVD<MatrixType>::structured_update(Block<MatrixXr,Dynamic,Dynamic> A, const MatrixXr &B, Index n1) 340 { 341 Index n = A.rows(); 342 if(n>100) 343 { 344 // If the matrices are large enough, let's exploit the sparse structure of A by 345 // splitting it in half (wrt n1), and packing the non-zero columns. 346 Index n2 = n - n1; 347 Map<MatrixXr> A1(m_workspace.data() , n1, n); 348 Map<MatrixXr> A2(m_workspace.data()+ n1*n, n2, n); 349 Map<MatrixXr> B1(m_workspace.data()+ n*n, n, n); 350 Map<MatrixXr> B2(m_workspace.data()+2*n*n, n, n); 351 Index k1=0, k2=0; 352 for(Index j=0; j<n; ++j) 353 { 354 if( (A.col(j).head(n1).array()!=0).any() ) 355 { 356 A1.col(k1) = A.col(j).head(n1); 357 B1.row(k1) = B.row(j); 358 ++k1; 359 } 360 if( (A.col(j).tail(n2).array()!=0).any() ) 361 { 362 A2.col(k2) = A.col(j).tail(n2); 363 B2.row(k2) = B.row(j); 364 ++k2; 365 } 366 } 367 368 A.topRows(n1).noalias() = A1.leftCols(k1) * B1.topRows(k1); 369 A.bottomRows(n2).noalias() = A2.leftCols(k2) * B2.topRows(k2); 370 } 371 else 372 { 373 Map<MatrixXr,Aligned> tmp(m_workspace.data(),n,n); 374 tmp.noalias() = A*B; 375 A = tmp; 376 } 377 } 378 379 // The divide algorithm is done "in place", we are always working on subsets of the same matrix. The divide methods takes as argument the 380 // place of the submatrix we are currently working on. 381 382 //@param firstCol : The Index of the first column of the submatrix of m_computed and for m_naiveU; 383 //@param lastCol : The Index of the last column of the submatrix of m_computed and for m_naiveU; 384 // lastCol + 1 - firstCol is the size of the submatrix. 385 //@param firstRowW : The Index of the first row of the matrix W that we are to change. (see the reference paper section 1 for more information on W) 386 //@param firstRowW : Same as firstRowW with the column. 387 //@param shift : Each time one takes the left submatrix, one must add 1 to the shift. Why? Because! We actually want the last column of the U submatrix 388 // to become the first column (*coeff) and to shift all the other columns to the right. There are more details on the reference paper. 389 template<typename MatrixType> 390 void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift) 391 { 392 // requires rows = cols + 1; 393 using std::pow; 394 using std::sqrt; 395 using std::abs; 396 const Index n = lastCol - firstCol + 1; 397 const Index k = n/2; 398 const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)(); 399 RealScalar alphaK; 400 RealScalar betaK; 401 RealScalar r0; 402 RealScalar lambda, phi, c0, s0; 403 VectorType l, f; 404 // We use the other algorithm which is more efficient for small 405 // matrices. 406 if (n < m_algoswap) 407 { 408 // FIXME this line involves temporaries 409 JacobiSVD<MatrixXr> b(m_computed.block(firstCol, firstCol, n + 1, n), ComputeFullU | (m_compV ? ComputeFullV : 0)); 410 if (m_compU) 411 m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() = b.matrixU(); 412 else 413 { 414 m_naiveU.row(0).segment(firstCol, n + 1).real() = b.matrixU().row(0); 415 m_naiveU.row(1).segment(firstCol, n + 1).real() = b.matrixU().row(n); 416 } 417 if (m_compV) m_naiveV.block(firstRowW, firstColW, n, n).real() = b.matrixV(); 418 m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero(); 419 m_computed.diagonal().segment(firstCol + shift, n) = b.singularValues().head(n); 420 return; 421 } 422 // We use the divide and conquer algorithm 423 alphaK = m_computed(firstCol + k, firstCol + k); 424 betaK = m_computed(firstCol + k + 1, firstCol + k); 425 // The divide must be done in that order in order to have good results. Divide change the data inside the submatrices 426 // and the divide of the right submatrice reads one column of the left submatrice. That's why we need to treat the 427 // right submatrix before the left one. 428 divide(k + 1 + firstCol, lastCol, k + 1 + firstRowW, k + 1 + firstColW, shift); 429 divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1); 430 431 if (m_compU) 432 { 433 lambda = m_naiveU(firstCol + k, firstCol + k); 434 phi = m_naiveU(firstCol + k + 1, lastCol + 1); 435 } 436 else 437 { 438 lambda = m_naiveU(1, firstCol + k); 439 phi = m_naiveU(0, lastCol + 1); 440 } 441 r0 = sqrt((abs(alphaK * lambda) * abs(alphaK * lambda)) + abs(betaK * phi) * abs(betaK * phi)); 442 if (m_compU) 443 { 444 l = m_naiveU.row(firstCol + k).segment(firstCol, k); 445 f = m_naiveU.row(firstCol + k + 1).segment(firstCol + k + 1, n - k - 1); 446 } 447 else 448 { 449 l = m_naiveU.row(1).segment(firstCol, k); 450 f = m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1); 451 } 452 if (m_compV) m_naiveV(firstRowW+k, firstColW) = 1; 453 if (r0<considerZero) 454 { 455 c0 = 1; 456 s0 = 0; 457 } 458 else 459 { 460 c0 = alphaK * lambda / r0; 461 s0 = betaK * phi / r0; 462 } 463 464 #ifdef EIGEN_BDCSVD_SANITY_CHECKS 465 assert(m_naiveU.allFinite()); 466 assert(m_naiveV.allFinite()); 467 assert(m_computed.allFinite()); 468 #endif 469 470 if (m_compU) 471 { 472 MatrixXr q1 (m_naiveU.col(firstCol + k).segment(firstCol, k + 1)); 473 // we shiftW Q1 to the right 474 for (Index i = firstCol + k - 1; i >= firstCol; i--) 475 m_naiveU.col(i + 1).segment(firstCol, k + 1) = m_naiveU.col(i).segment(firstCol, k + 1); 476 // we shift q1 at the left with a factor c0 477 m_naiveU.col(firstCol).segment( firstCol, k + 1) = (q1 * c0); 478 // last column = q1 * - s0 479 m_naiveU.col(lastCol + 1).segment(firstCol, k + 1) = (q1 * ( - s0)); 480 // first column = q2 * s0 481 m_naiveU.col(firstCol).segment(firstCol + k + 1, n - k) = m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) * s0; 482 // q2 *= c0 483 m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *= c0; 484 } 485 else 486 { 487 RealScalar q1 = m_naiveU(0, firstCol + k); 488 // we shift Q1 to the right 489 for (Index i = firstCol + k - 1; i >= firstCol; i--) 490 m_naiveU(0, i + 1) = m_naiveU(0, i); 491 // we shift q1 at the left with a factor c0 492 m_naiveU(0, firstCol) = (q1 * c0); 493 // last column = q1 * - s0 494 m_naiveU(0, lastCol + 1) = (q1 * ( - s0)); 495 // first column = q2 * s0 496 m_naiveU(1, firstCol) = m_naiveU(1, lastCol + 1) *s0; 497 // q2 *= c0 498 m_naiveU(1, lastCol + 1) *= c0; 499 m_naiveU.row(1).segment(firstCol + 1, k).setZero(); 500 m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1).setZero(); 501 } 502 503 #ifdef EIGEN_BDCSVD_SANITY_CHECKS 504 assert(m_naiveU.allFinite()); 505 assert(m_naiveV.allFinite()); 506 assert(m_computed.allFinite()); 507 #endif 508 509 m_computed(firstCol + shift, firstCol + shift) = r0; 510 m_computed.col(firstCol + shift).segment(firstCol + shift + 1, k) = alphaK * l.transpose().real(); 511 m_computed.col(firstCol + shift).segment(firstCol + shift + k + 1, n - k - 1) = betaK * f.transpose().real(); 512 513 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 514 ArrayXr tmp1 = (m_computed.block(firstCol+shift, firstCol+shift, n, n)).jacobiSvd().singularValues(); 515 #endif 516 // Second part: try to deflate singular values in combined matrix 517 deflation(firstCol, lastCol, k, firstRowW, firstColW, shift); 518 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 519 ArrayXr tmp2 = (m_computed.block(firstCol+shift, firstCol+shift, n, n)).jacobiSvd().singularValues(); 520 std::cout << "\n\nj1 = " << tmp1.transpose().format(bdcsvdfmt) << "\n"; 521 std::cout << "j2 = " << tmp2.transpose().format(bdcsvdfmt) << "\n\n"; 522 std::cout << "err: " << ((tmp1-tmp2).abs()>1e-12*tmp2.abs()).transpose() << "\n"; 523 static int count = 0; 524 std::cout << "# " << ++count << "\n\n"; 525 assert((tmp1-tmp2).matrix().norm() < 1e-14*tmp2.matrix().norm()); 526 // assert(count<681); 527 // assert(((tmp1-tmp2).abs()<1e-13*tmp2.abs()).all()); 528 #endif 529 530 // Third part: compute SVD of combined matrix 531 MatrixXr UofSVD, VofSVD; 532 VectorType singVals; 533 computeSVDofM(firstCol + shift, n, UofSVD, singVals, VofSVD); 534 535 #ifdef EIGEN_BDCSVD_SANITY_CHECKS 536 assert(UofSVD.allFinite()); 537 assert(VofSVD.allFinite()); 538 #endif 539 540 if (m_compU) 541 structured_update(m_naiveU.block(firstCol, firstCol, n + 1, n + 1), UofSVD, (n+2)/2); 542 else 543 { 544 Map<Matrix<RealScalar,2,Dynamic>,Aligned> tmp(m_workspace.data(),2,n+1); 545 tmp.noalias() = m_naiveU.middleCols(firstCol, n+1) * UofSVD; 546 m_naiveU.middleCols(firstCol, n + 1) = tmp; 547 } 548 549 if (m_compV) structured_update(m_naiveV.block(firstRowW, firstColW, n, n), VofSVD, (n+1)/2); 550 551 #ifdef EIGEN_BDCSVD_SANITY_CHECKS 552 assert(m_naiveU.allFinite()); 553 assert(m_naiveV.allFinite()); 554 assert(m_computed.allFinite()); 555 #endif 556 557 m_computed.block(firstCol + shift, firstCol + shift, n, n).setZero(); 558 m_computed.block(firstCol + shift, firstCol + shift, n, n).diagonal() = singVals; 559 }// end divide 560 561 // Compute SVD of m_computed.block(firstCol, firstCol, n + 1, n); this block only has non-zeros in 562 // the first column and on the diagonal and has undergone deflation, so diagonal is in increasing 563 // order except for possibly the (0,0) entry. The computed SVD is stored U, singVals and V, except 564 // that if m_compV is false, then V is not computed. Singular values are sorted in decreasing order. 565 // 566 // TODO Opportunities for optimization: better root finding algo, better stopping criterion, better 567 // handling of round-off errors, be consistent in ordering 568 // For instance, to solve the secular equation using FMM, see http://www.stat.uchicago.edu/~lekheng/courses/302/classics/greengard-rokhlin.pdf 569 template <typename MatrixType> 570 void BDCSVD<MatrixType>::computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V) 571 { 572 const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)(); 573 using std::abs; 574 ArrayRef col0 = m_computed.col(firstCol).segment(firstCol, n); 575 m_workspace.head(n) = m_computed.block(firstCol, firstCol, n, n).diagonal(); 576 ArrayRef diag = m_workspace.head(n); 577 diag(0) = 0; 578 579 // Allocate space for singular values and vectors 580 singVals.resize(n); 581 U.resize(n+1, n+1); 582 if (m_compV) V.resize(n, n); 583 584 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 585 if (col0.hasNaN() || diag.hasNaN()) 586 std::cout << "\n\nHAS NAN\n\n"; 587 #endif 588 589 // Many singular values might have been deflated, the zero ones have been moved to the end, 590 // but others are interleaved and we must ignore them at this stage. 591 // To this end, let's compute a permutation skipping them: 592 Index actual_n = n; 593 while(actual_n>1 && diag(actual_n-1)==0) --actual_n; 594 Index m = 0; // size of the deflated problem 595 for(Index k=0;k<actual_n;++k) 596 if(abs(col0(k))>considerZero) 597 m_workspaceI(m++) = k; 598 Map<ArrayXi> perm(m_workspaceI.data(),m); 599 600 Map<ArrayXr> shifts(m_workspace.data()+1*n, n); 601 Map<ArrayXr> mus(m_workspace.data()+2*n, n); 602 Map<ArrayXr> zhat(m_workspace.data()+3*n, n); 603 604 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 605 std::cout << "computeSVDofM using:\n"; 606 std::cout << " z: " << col0.transpose() << "\n"; 607 std::cout << " d: " << diag.transpose() << "\n"; 608 #endif 609 610 // Compute singVals, shifts, and mus 611 computeSingVals(col0, diag, perm, singVals, shifts, mus); 612 613 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 614 std::cout << " j: " << (m_computed.block(firstCol, firstCol, n, n)).jacobiSvd().singularValues().transpose().reverse() << "\n\n"; 615 std::cout << " sing-val: " << singVals.transpose() << "\n"; 616 std::cout << " mu: " << mus.transpose() << "\n"; 617 std::cout << " shift: " << shifts.transpose() << "\n"; 618 619 { 620 Index actual_n = n; 621 while(actual_n>1 && abs(col0(actual_n-1))<considerZero) --actual_n; 622 std::cout << "\n\n mus: " << mus.head(actual_n).transpose() << "\n\n"; 623 std::cout << " check1 (expect0) : " << ((singVals.array()-(shifts+mus)) / singVals.array()).head(actual_n).transpose() << "\n\n"; 624 std::cout << " check2 (>0) : " << ((singVals.array()-diag) / singVals.array()).head(actual_n).transpose() << "\n\n"; 625 std::cout << " check3 (>0) : " << ((diag.segment(1,actual_n-1)-singVals.head(actual_n-1).array()) / singVals.head(actual_n-1).array()).transpose() << "\n\n\n"; 626 std::cout << " check4 (>0) : " << ((singVals.segment(1,actual_n-1)-singVals.head(actual_n-1))).transpose() << "\n\n\n"; 627 } 628 #endif 629 630 #ifdef EIGEN_BDCSVD_SANITY_CHECKS 631 assert(singVals.allFinite()); 632 assert(mus.allFinite()); 633 assert(shifts.allFinite()); 634 #endif 635 636 // Compute zhat 637 perturbCol0(col0, diag, perm, singVals, shifts, mus, zhat); 638 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 639 std::cout << " zhat: " << zhat.transpose() << "\n"; 640 #endif 641 642 #ifdef EIGEN_BDCSVD_SANITY_CHECKS 643 assert(zhat.allFinite()); 644 #endif 645 646 computeSingVecs(zhat, diag, perm, singVals, shifts, mus, U, V); 647 648 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 649 std::cout << "U^T U: " << (U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() << "\n"; 650 std::cout << "V^T V: " << (V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() << "\n"; 651 #endif 652 653 #ifdef EIGEN_BDCSVD_SANITY_CHECKS 654 assert(U.allFinite()); 655 assert(V.allFinite()); 656 assert((U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() < 1e-14 * n); 657 assert((V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() < 1e-14 * n); 658 assert(m_naiveU.allFinite()); 659 assert(m_naiveV.allFinite()); 660 assert(m_computed.allFinite()); 661 #endif 662 663 // Because of deflation, the singular values might not be completely sorted. 664 // Fortunately, reordering them is a O(n) problem 665 for(Index i=0; i<actual_n-1; ++i) 666 { 667 if(singVals(i)>singVals(i+1)) 668 { 669 using std::swap; 670 swap(singVals(i),singVals(i+1)); 671 U.col(i).swap(U.col(i+1)); 672 if(m_compV) V.col(i).swap(V.col(i+1)); 673 } 674 } 675 676 // Reverse order so that singular values in increased order 677 // Because of deflation, the zeros singular-values are already at the end 678 singVals.head(actual_n).reverseInPlace(); 679 U.leftCols(actual_n).rowwise().reverseInPlace(); 680 if (m_compV) V.leftCols(actual_n).rowwise().reverseInPlace(); 681 682 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 683 JacobiSVD<MatrixXr> jsvd(m_computed.block(firstCol, firstCol, n, n) ); 684 std::cout << " * j: " << jsvd.singularValues().transpose() << "\n\n"; 685 std::cout << " * sing-val: " << singVals.transpose() << "\n"; 686 // std::cout << " * err: " << ((jsvd.singularValues()-singVals)>1e-13*singVals.norm()).transpose() << "\n"; 687 #endif 688 } 689 690 template <typename MatrixType> 691 typename BDCSVD<MatrixType>::RealScalar BDCSVD<MatrixType>::secularEq(RealScalar mu, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const ArrayRef& diagShifted, RealScalar shift) 692 { 693 Index m = perm.size(); 694 RealScalar res = 1; 695 for(Index i=0; i<m; ++i) 696 { 697 Index j = perm(i); 698 res += numext::abs2(col0(j)) / ((diagShifted(j) - mu) * (diag(j) + shift + mu)); 699 } 700 return res; 701 702 } 703 704 template <typename MatrixType> 705 void BDCSVD<MatrixType>::computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, 706 VectorType& singVals, ArrayRef shifts, ArrayRef mus) 707 { 708 using std::abs; 709 using std::swap; 710 711 Index n = col0.size(); 712 Index actual_n = n; 713 while(actual_n>1 && col0(actual_n-1)==0) --actual_n; 714 715 for (Index k = 0; k < n; ++k) 716 { 717 if (col0(k) == 0 || actual_n==1) 718 { 719 // if col0(k) == 0, then entry is deflated, so singular value is on diagonal 720 // if actual_n==1, then the deflated problem is already diagonalized 721 singVals(k) = k==0 ? col0(0) : diag(k); 722 mus(k) = 0; 723 shifts(k) = k==0 ? col0(0) : diag(k); 724 continue; 725 } 726 727 // otherwise, use secular equation to find singular value 728 RealScalar left = diag(k); 729 RealScalar right; // was: = (k != actual_n-1) ? diag(k+1) : (diag(actual_n-1) + col0.matrix().norm()); 730 if(k==actual_n-1) 731 right = (diag(actual_n-1) + col0.matrix().norm()); 732 else 733 { 734 // Skip deflated singular values 735 Index l = k+1; 736 while(col0(l)==0) { ++l; eigen_internal_assert(l<actual_n); } 737 right = diag(l); 738 } 739 740 // first decide whether it's closer to the left end or the right end 741 RealScalar mid = left + (right-left) / 2; 742 RealScalar fMid = secularEq(mid, col0, diag, perm, diag, 0); 743 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 744 std::cout << right-left << "\n"; 745 std::cout << "fMid = " << fMid << " " << secularEq(mid-left, col0, diag, perm, diag-left, left) << " " << secularEq(mid-right, col0, diag, perm, diag-right, right) << "\n"; 746 std::cout << " = " << secularEq(0.1*(left+right), col0, diag, perm, diag, 0) 747 << " " << secularEq(0.2*(left+right), col0, diag, perm, diag, 0) 748 << " " << secularEq(0.3*(left+right), col0, diag, perm, diag, 0) 749 << " " << secularEq(0.4*(left+right), col0, diag, perm, diag, 0) 750 << " " << secularEq(0.49*(left+right), col0, diag, perm, diag, 0) 751 << " " << secularEq(0.5*(left+right), col0, diag, perm, diag, 0) 752 << " " << secularEq(0.51*(left+right), col0, diag, perm, diag, 0) 753 << " " << secularEq(0.6*(left+right), col0, diag, perm, diag, 0) 754 << " " << secularEq(0.7*(left+right), col0, diag, perm, diag, 0) 755 << " " << secularEq(0.8*(left+right), col0, diag, perm, diag, 0) 756 << " " << secularEq(0.9*(left+right), col0, diag, perm, diag, 0) << "\n"; 757 #endif 758 RealScalar shift = (k == actual_n-1 || fMid > 0) ? left : right; 759 760 // measure everything relative to shift 761 Map<ArrayXr> diagShifted(m_workspace.data()+4*n, n); 762 diagShifted = diag - shift; 763 764 // initial guess 765 RealScalar muPrev, muCur; 766 if (shift == left) 767 { 768 muPrev = (right - left) * RealScalar(0.1); 769 if (k == actual_n-1) muCur = right - left; 770 else muCur = (right - left) * RealScalar(0.5); 771 } 772 else 773 { 774 muPrev = -(right - left) * RealScalar(0.1); 775 muCur = -(right - left) * RealScalar(0.5); 776 } 777 778 RealScalar fPrev = secularEq(muPrev, col0, diag, perm, diagShifted, shift); 779 RealScalar fCur = secularEq(muCur, col0, diag, perm, diagShifted, shift); 780 if (abs(fPrev) < abs(fCur)) 781 { 782 swap(fPrev, fCur); 783 swap(muPrev, muCur); 784 } 785 786 // rational interpolation: fit a function of the form a / mu + b through the two previous 787 // iterates and use its zero to compute the next iterate 788 bool useBisection = fPrev*fCur>0; 789 while (fCur!=0 && abs(muCur - muPrev) > 8 * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(muCur), abs(muPrev)) && abs(fCur - fPrev)>NumTraits<RealScalar>::epsilon() && !useBisection) 790 { 791 ++m_numIters; 792 793 // Find a and b such that the function f(mu) = a / mu + b matches the current and previous samples. 794 RealScalar a = (fCur - fPrev) / (1/muCur - 1/muPrev); 795 RealScalar b = fCur - a / muCur; 796 // And find mu such that f(mu)==0: 797 RealScalar muZero = -a/b; 798 RealScalar fZero = secularEq(muZero, col0, diag, perm, diagShifted, shift); 799 800 muPrev = muCur; 801 fPrev = fCur; 802 muCur = muZero; 803 fCur = fZero; 804 805 806 if (shift == left && (muCur < 0 || muCur > right - left)) useBisection = true; 807 if (shift == right && (muCur < -(right - left) || muCur > 0)) useBisection = true; 808 if (abs(fCur)>abs(fPrev)) useBisection = true; 809 } 810 811 // fall back on bisection method if rational interpolation did not work 812 if (useBisection) 813 { 814 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 815 std::cout << "useBisection for k = " << k << ", actual_n = " << actual_n << "\n"; 816 #endif 817 RealScalar leftShifted, rightShifted; 818 if (shift == left) 819 { 820 leftShifted = (std::numeric_limits<RealScalar>::min)(); 821 // I don't understand why the case k==0 would be special there: 822 // if (k == 0) rightShifted = right - left; else 823 rightShifted = (k==actual_n-1) ? right : ((right - left) * RealScalar(0.6)); // theoretically we can take 0.5, but let's be safe 824 } 825 else 826 { 827 leftShifted = -(right - left) * RealScalar(0.6); 828 rightShifted = -(std::numeric_limits<RealScalar>::min)(); 829 } 830 831 RealScalar fLeft = secularEq(leftShifted, col0, diag, perm, diagShifted, shift); 832 833 #if defined EIGEN_INTERNAL_DEBUGGING || defined EIGEN_BDCSVD_DEBUG_VERBOSE 834 RealScalar fRight = secularEq(rightShifted, col0, diag, perm, diagShifted, shift); 835 #endif 836 837 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 838 if(!(fLeft * fRight<0)) 839 { 840 std::cout << "fLeft: " << leftShifted << " - " << diagShifted.head(10).transpose() << "\n ; " << bool(left==shift) << " " << (left-shift) << "\n"; 841 std::cout << k << " : " << fLeft << " * " << fRight << " == " << fLeft * fRight << " ; " << left << " - " << right << " -> " << leftShifted << " " << rightShifted << " shift=" << shift << "\n"; 842 } 843 #endif 844 eigen_internal_assert(fLeft * fRight < 0); 845 846 while (rightShifted - leftShifted > 2 * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(leftShifted), abs(rightShifted))) 847 { 848 RealScalar midShifted = (leftShifted + rightShifted) / 2; 849 fMid = secularEq(midShifted, col0, diag, perm, diagShifted, shift); 850 if (fLeft * fMid < 0) 851 { 852 rightShifted = midShifted; 853 } 854 else 855 { 856 leftShifted = midShifted; 857 fLeft = fMid; 858 } 859 } 860 861 muCur = (leftShifted + rightShifted) / 2; 862 } 863 864 singVals[k] = shift + muCur; 865 shifts[k] = shift; 866 mus[k] = muCur; 867 868 // perturb singular value slightly if it equals diagonal entry to avoid division by zero later 869 // (deflation is supposed to avoid this from happening) 870 // - this does no seem to be necessary anymore - 871 // if (singVals[k] == left) singVals[k] *= 1 + NumTraits<RealScalar>::epsilon(); 872 // if (singVals[k] == right) singVals[k] *= 1 - NumTraits<RealScalar>::epsilon(); 873 } 874 } 875 876 877 // zhat is perturbation of col0 for which singular vectors can be computed stably (see Section 3.1) 878 template <typename MatrixType> 879 void BDCSVD<MatrixType>::perturbCol0 880 (const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const VectorType& singVals, 881 const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat) 882 { 883 using std::sqrt; 884 Index n = col0.size(); 885 Index m = perm.size(); 886 if(m==0) 887 { 888 zhat.setZero(); 889 return; 890 } 891 Index last = perm(m-1); 892 // The offset permits to skip deflated entries while computing zhat 893 for (Index k = 0; k < n; ++k) 894 { 895 if (col0(k) == 0) // deflated 896 zhat(k) = 0; 897 else 898 { 899 // see equation (3.6) 900 RealScalar dk = diag(k); 901 RealScalar prod = (singVals(last) + dk) * (mus(last) + (shifts(last) - dk)); 902 903 for(Index l = 0; l<m; ++l) 904 { 905 Index i = perm(l); 906 if(i!=k) 907 { 908 Index j = i<k ? i : perm(l-1); 909 prod *= ((singVals(j)+dk) / ((diag(i)+dk))) * ((mus(j)+(shifts(j)-dk)) / ((diag(i)-dk))); 910 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 911 if(i!=k && std::abs(((singVals(j)+dk)*(mus(j)+(shifts(j)-dk)))/((diag(i)+dk)*(diag(i)-dk)) - 1) > 0.9 ) 912 std::cout << " " << ((singVals(j)+dk)*(mus(j)+(shifts(j)-dk)))/((diag(i)+dk)*(diag(i)-dk)) << " == (" << (singVals(j)+dk) << " * " << (mus(j)+(shifts(j)-dk)) 913 << ") / (" << (diag(i)+dk) << " * " << (diag(i)-dk) << ")\n"; 914 #endif 915 } 916 } 917 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 918 std::cout << "zhat(" << k << ") = sqrt( " << prod << ") ; " << (singVals(last) + dk) << " * " << mus(last) + shifts(last) << " - " << dk << "\n"; 919 #endif 920 RealScalar tmp = sqrt(prod); 921 zhat(k) = col0(k) > 0 ? tmp : -tmp; 922 } 923 } 924 } 925 926 // compute singular vectors 927 template <typename MatrixType> 928 void BDCSVD<MatrixType>::computeSingVecs 929 (const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef &perm, const VectorType& singVals, 930 const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V) 931 { 932 Index n = zhat.size(); 933 Index m = perm.size(); 934 935 for (Index k = 0; k < n; ++k) 936 { 937 if (zhat(k) == 0) 938 { 939 U.col(k) = VectorType::Unit(n+1, k); 940 if (m_compV) V.col(k) = VectorType::Unit(n, k); 941 } 942 else 943 { 944 U.col(k).setZero(); 945 for(Index l=0;l<m;++l) 946 { 947 Index i = perm(l); 948 U(i,k) = zhat(i)/(((diag(i) - shifts(k)) - mus(k)) )/( (diag(i) + singVals[k])); 949 } 950 U(n,k) = 0; 951 U.col(k).normalize(); 952 953 if (m_compV) 954 { 955 V.col(k).setZero(); 956 for(Index l=1;l<m;++l) 957 { 958 Index i = perm(l); 959 V(i,k) = diag(i) * zhat(i) / (((diag(i) - shifts(k)) - mus(k)) )/( (diag(i) + singVals[k])); 960 } 961 V(0,k) = -1; 962 V.col(k).normalize(); 963 } 964 } 965 } 966 U.col(n) = VectorType::Unit(n+1, n); 967 } 968 969 970 // page 12_13 971 // i >= 1, di almost null and zi non null. 972 // We use a rotation to zero out zi applied to the left of M 973 template <typename MatrixType> 974 void BDCSVD<MatrixType>::deflation43(Index firstCol, Index shift, Index i, Index size) 975 { 976 using std::abs; 977 using std::sqrt; 978 using std::pow; 979 Index start = firstCol + shift; 980 RealScalar c = m_computed(start, start); 981 RealScalar s = m_computed(start+i, start); 982 RealScalar r = sqrt(numext::abs2(c) + numext::abs2(s)); 983 if (r == 0) 984 { 985 m_computed(start+i, start+i) = 0; 986 return; 987 } 988 m_computed(start,start) = r; 989 m_computed(start+i, start) = 0; 990 m_computed(start+i, start+i) = 0; 991 992 JacobiRotation<RealScalar> J(c/r,-s/r); 993 if (m_compU) m_naiveU.middleRows(firstCol, size+1).applyOnTheRight(firstCol, firstCol+i, J); 994 else m_naiveU.applyOnTheRight(firstCol, firstCol+i, J); 995 }// end deflation 43 996 997 998 // page 13 999 // i,j >= 1, i!=j and |di - dj| < epsilon * norm2(M) 1000 // We apply two rotations to have zj = 0; 1001 // TODO deflation44 is still broken and not properly tested 1002 template <typename MatrixType> 1003 void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size) 1004 { 1005 using std::abs; 1006 using std::sqrt; 1007 using std::conj; 1008 using std::pow; 1009 RealScalar c = m_computed(firstColm+i, firstColm); 1010 RealScalar s = m_computed(firstColm+j, firstColm); 1011 RealScalar r = sqrt(numext::abs2(c) + numext::abs2(s)); 1012 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1013 std::cout << "deflation 4.4: " << i << "," << j << " -> " << c << " " << s << " " << r << " ; " 1014 << m_computed(firstColm + i-1, firstColm) << " " 1015 << m_computed(firstColm + i, firstColm) << " " 1016 << m_computed(firstColm + i+1, firstColm) << " " 1017 << m_computed(firstColm + i+2, firstColm) << "\n"; 1018 std::cout << m_computed(firstColm + i-1, firstColm + i-1) << " " 1019 << m_computed(firstColm + i, firstColm+i) << " " 1020 << m_computed(firstColm + i+1, firstColm+i+1) << " " 1021 << m_computed(firstColm + i+2, firstColm+i+2) << "\n"; 1022 #endif 1023 if (r==0) 1024 { 1025 m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j); 1026 return; 1027 } 1028 c/=r; 1029 s/=r; 1030 m_computed(firstColm + i, firstColm) = r; 1031 m_computed(firstColm + j, firstColm + j) = m_computed(firstColm + i, firstColm + i); 1032 m_computed(firstColm + j, firstColm) = 0; 1033 1034 JacobiRotation<RealScalar> J(c,-s); 1035 if (m_compU) m_naiveU.middleRows(firstColu, size+1).applyOnTheRight(firstColu + i, firstColu + j, J); 1036 else m_naiveU.applyOnTheRight(firstColu+i, firstColu+j, J); 1037 if (m_compV) m_naiveV.middleRows(firstRowW, size).applyOnTheRight(firstColW + i, firstColW + j, J); 1038 }// end deflation 44 1039 1040 1041 // acts on block from (firstCol+shift, firstCol+shift) to (lastCol+shift, lastCol+shift) [inclusive] 1042 template <typename MatrixType> 1043 void BDCSVD<MatrixType>::deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift) 1044 { 1045 using std::sqrt; 1046 using std::abs; 1047 const Index length = lastCol + 1 - firstCol; 1048 1049 Block<MatrixXr,Dynamic,1> col0(m_computed, firstCol+shift, firstCol+shift, length, 1); 1050 Diagonal<MatrixXr> fulldiag(m_computed); 1051 VectorBlock<Diagonal<MatrixXr>,Dynamic> diag(fulldiag, firstCol+shift, length); 1052 1053 const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)(); 1054 RealScalar maxDiag = diag.tail((std::max)(Index(1),length-1)).cwiseAbs().maxCoeff(); 1055 RealScalar epsilon_strict = numext::maxi<RealScalar>(considerZero,NumTraits<RealScalar>::epsilon() * maxDiag); 1056 RealScalar epsilon_coarse = 8 * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(col0.cwiseAbs().maxCoeff(), maxDiag); 1057 1058 #ifdef EIGEN_BDCSVD_SANITY_CHECKS 1059 assert(m_naiveU.allFinite()); 1060 assert(m_naiveV.allFinite()); 1061 assert(m_computed.allFinite()); 1062 #endif 1063 1064 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1065 std::cout << "\ndeflate:" << diag.head(k+1).transpose() << " | " << diag.segment(k+1,length-k-1).transpose() << "\n"; 1066 #endif 1067 1068 //condition 4.1 1069 if (diag(0) < epsilon_coarse) 1070 { 1071 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1072 std::cout << "deflation 4.1, because " << diag(0) << " < " << epsilon_coarse << "\n"; 1073 #endif 1074 diag(0) = epsilon_coarse; 1075 } 1076 1077 //condition 4.2 1078 for (Index i=1;i<length;++i) 1079 if (abs(col0(i)) < epsilon_strict) 1080 { 1081 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1082 std::cout << "deflation 4.2, set z(" << i << ") to zero because " << abs(col0(i)) << " < " << epsilon_strict << " (diag(" << i << ")=" << diag(i) << ")\n"; 1083 #endif 1084 col0(i) = 0; 1085 } 1086 1087 //condition 4.3 1088 for (Index i=1;i<length; i++) 1089 if (diag(i) < epsilon_coarse) 1090 { 1091 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1092 std::cout << "deflation 4.3, cancel z(" << i << ")=" << col0(i) << " because diag(" << i << ")=" << diag(i) << " < " << epsilon_coarse << "\n"; 1093 #endif 1094 deflation43(firstCol, shift, i, length); 1095 } 1096 1097 #ifdef EIGEN_BDCSVD_SANITY_CHECKS 1098 assert(m_naiveU.allFinite()); 1099 assert(m_naiveV.allFinite()); 1100 assert(m_computed.allFinite()); 1101 #endif 1102 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1103 std::cout << "to be sorted: " << diag.transpose() << "\n\n"; 1104 #endif 1105 { 1106 // Check for total deflation 1107 // If we have a total deflation, then we have to consider col0(0)==diag(0) as a singular value during sorting 1108 bool total_deflation = (col0.tail(length-1).array()<considerZero).all(); 1109 1110 // Sort the diagonal entries, since diag(1:k-1) and diag(k:length) are already sorted, let's do a sorted merge. 1111 // First, compute the respective permutation. 1112 Index *permutation = m_workspaceI.data(); 1113 { 1114 permutation[0] = 0; 1115 Index p = 1; 1116 1117 // Move deflated diagonal entries at the end. 1118 for(Index i=1; i<length; ++i) 1119 if(abs(diag(i))<considerZero) 1120 permutation[p++] = i; 1121 1122 Index i=1, j=k+1; 1123 for( ; p < length; ++p) 1124 { 1125 if (i > k) permutation[p] = j++; 1126 else if (j >= length) permutation[p] = i++; 1127 else if (diag(i) < diag(j)) permutation[p] = j++; 1128 else permutation[p] = i++; 1129 } 1130 } 1131 1132 // If we have a total deflation, then we have to insert diag(0) at the right place 1133 if(total_deflation) 1134 { 1135 for(Index i=1; i<length; ++i) 1136 { 1137 Index pi = permutation[i]; 1138 if(abs(diag(pi))<considerZero || diag(0)<diag(pi)) 1139 permutation[i-1] = permutation[i]; 1140 else 1141 { 1142 permutation[i-1] = 0; 1143 break; 1144 } 1145 } 1146 } 1147 1148 // Current index of each col, and current column of each index 1149 Index *realInd = m_workspaceI.data()+length; 1150 Index *realCol = m_workspaceI.data()+2*length; 1151 1152 for(int pos = 0; pos< length; pos++) 1153 { 1154 realCol[pos] = pos; 1155 realInd[pos] = pos; 1156 } 1157 1158 for(Index i = total_deflation?0:1; i < length; i++) 1159 { 1160 const Index pi = permutation[length - (total_deflation ? i+1 : i)]; 1161 const Index J = realCol[pi]; 1162 1163 using std::swap; 1164 // swap diagonal and first column entries: 1165 swap(diag(i), diag(J)); 1166 if(i!=0 && J!=0) swap(col0(i), col0(J)); 1167 1168 // change columns 1169 if (m_compU) m_naiveU.col(firstCol+i).segment(firstCol, length + 1).swap(m_naiveU.col(firstCol+J).segment(firstCol, length + 1)); 1170 else m_naiveU.col(firstCol+i).segment(0, 2) .swap(m_naiveU.col(firstCol+J).segment(0, 2)); 1171 if (m_compV) m_naiveV.col(firstColW + i).segment(firstRowW, length).swap(m_naiveV.col(firstColW + J).segment(firstRowW, length)); 1172 1173 //update real pos 1174 const Index realI = realInd[i]; 1175 realCol[realI] = J; 1176 realCol[pi] = i; 1177 realInd[J] = realI; 1178 realInd[i] = pi; 1179 } 1180 } 1181 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1182 std::cout << "sorted: " << diag.transpose().format(bdcsvdfmt) << "\n"; 1183 std::cout << " : " << col0.transpose() << "\n\n"; 1184 #endif 1185 1186 //condition 4.4 1187 { 1188 Index i = length-1; 1189 while(i>0 && (abs(diag(i))<considerZero || abs(col0(i))<considerZero)) --i; 1190 for(; i>1;--i) 1191 if( (diag(i) - diag(i-1)) < NumTraits<RealScalar>::epsilon()*maxDiag ) 1192 { 1193 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1194 std::cout << "deflation 4.4 with i = " << i << " because " << (diag(i) - diag(i-1)) << " < " << NumTraits<RealScalar>::epsilon()*diag(i) << "\n"; 1195 #endif 1196 eigen_internal_assert(abs(diag(i) - diag(i-1))<epsilon_coarse && " diagonal entries are not properly sorted"); 1197 deflation44(firstCol, firstCol + shift, firstRowW, firstColW, i-1, i, length); 1198 } 1199 } 1200 1201 #ifdef EIGEN_BDCSVD_SANITY_CHECKS 1202 for(Index j=2;j<length;++j) 1203 assert(diag(j-1)<=diag(j) || abs(diag(j))<considerZero); 1204 #endif 1205 1206 #ifdef EIGEN_BDCSVD_SANITY_CHECKS 1207 assert(m_naiveU.allFinite()); 1208 assert(m_naiveV.allFinite()); 1209 assert(m_computed.allFinite()); 1210 #endif 1211 }//end deflation 1212 1213 #ifndef __CUDACC__ 1214 /** \svd_module 1215 * 1216 * \return the singular value decomposition of \c *this computed by Divide & Conquer algorithm 1217 * 1218 * \sa class BDCSVD 1219 */ 1220 template<typename Derived> 1221 BDCSVD<typename MatrixBase<Derived>::PlainObject> 1222 MatrixBase<Derived>::bdcSvd(unsigned int computationOptions) const 1223 { 1224 return BDCSVD<PlainObject>(*this, computationOptions); 1225 } 1226 #endif 1227 1228 } // end namespace Eigen 1229 1230 #endif 1231