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 25namespace Eigen { 26 27#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 28IOFormat bdcsvdfmt(8, 0, ", ", "\n", " [", "]"); 29#endif 30 31template<typename _MatrixType> class BDCSVD; 32 33namespace internal { 34 35template<typename _MatrixType> 36struct 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 */ 66template<typename _MatrixType> 67class BDCSVD : public SVDBase<BDCSVD<_MatrixType> > 68{ 69 typedef SVDBase<BDCSVD> Base; 70 71public: 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 typedef typename NumTraits<RealScalar>::Literal Literal; 81 enum { 82 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 83 ColsAtCompileTime = MatrixType::ColsAtCompileTime, 84 DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime, ColsAtCompileTime), 85 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 86 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, 87 MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime, MaxColsAtCompileTime), 88 MatrixOptions = MatrixType::Options 89 }; 90 91 typedef typename Base::MatrixUType MatrixUType; 92 typedef typename Base::MatrixVType MatrixVType; 93 typedef typename Base::SingularValuesType SingularValuesType; 94 95 typedef Matrix<Scalar, Dynamic, Dynamic, ColMajor> MatrixX; 96 typedef Matrix<RealScalar, Dynamic, Dynamic, ColMajor> MatrixXr; 97 typedef Matrix<RealScalar, Dynamic, 1> VectorType; 98 typedef Array<RealScalar, Dynamic, 1> ArrayXr; 99 typedef Array<Index,1,Dynamic> ArrayXi; 100 typedef Ref<ArrayXr> ArrayRef; 101 typedef Ref<ArrayXi> IndicesRef; 102 103 /** \brief Default Constructor. 104 * 105 * The default constructor is useful in cases in which the user intends to 106 * perform decompositions via BDCSVD::compute(const MatrixType&). 107 */ 108 BDCSVD() : m_algoswap(16), m_numIters(0) 109 {} 110 111 112 /** \brief Default Constructor with memory preallocation 113 * 114 * Like the default constructor but with preallocation of the internal data 115 * according to the specified problem size. 116 * \sa BDCSVD() 117 */ 118 BDCSVD(Index rows, Index cols, unsigned int computationOptions = 0) 119 : m_algoswap(16), m_numIters(0) 120 { 121 allocate(rows, cols, computationOptions); 122 } 123 124 /** \brief Constructor performing the decomposition of given matrix. 125 * 126 * \param matrix the matrix to decompose 127 * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. 128 * By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU, 129 * #ComputeFullV, #ComputeThinV. 130 * 131 * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not 132 * available with the (non - default) FullPivHouseholderQR preconditioner. 133 */ 134 BDCSVD(const MatrixType& matrix, unsigned int computationOptions = 0) 135 : m_algoswap(16), m_numIters(0) 136 { 137 compute(matrix, computationOptions); 138 } 139 140 ~BDCSVD() 141 { 142 } 143 144 /** \brief Method performing the decomposition of given matrix using custom options. 145 * 146 * \param matrix the matrix to decompose 147 * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. 148 * By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU, 149 * #ComputeFullV, #ComputeThinV. 150 * 151 * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not 152 * available with the (non - default) FullPivHouseholderQR preconditioner. 153 */ 154 BDCSVD& compute(const MatrixType& matrix, unsigned int computationOptions); 155 156 /** \brief Method performing the decomposition of given matrix using current options. 157 * 158 * \param matrix the matrix to decompose 159 * 160 * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int). 161 */ 162 BDCSVD& compute(const MatrixType& matrix) 163 { 164 return compute(matrix, this->m_computationOptions); 165 } 166 167 void setSwitchSize(int s) 168 { 169 eigen_assert(s>3 && "BDCSVD the size of the algo switch has to be greater than 3"); 170 m_algoswap = s; 171 } 172 173private: 174 void allocate(Index rows, Index cols, unsigned int computationOptions); 175 void divide(Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift); 176 void computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V); 177 void computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, VectorType& singVals, ArrayRef shifts, ArrayRef mus); 178 void perturbCol0(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat); 179 void computeSingVecs(const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V); 180 void deflation43(Index firstCol, Index shift, Index i, Index size); 181 void deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size); 182 void deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift); 183 template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV> 184 void copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naivev); 185 void structured_update(Block<MatrixXr,Dynamic,Dynamic> A, const MatrixXr &B, Index n1); 186 static RealScalar secularEq(RealScalar x, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const ArrayRef& diagShifted, RealScalar shift); 187 188protected: 189 MatrixXr m_naiveU, m_naiveV; 190 MatrixXr m_computed; 191 Index m_nRec; 192 ArrayXr m_workspace; 193 ArrayXi m_workspaceI; 194 int m_algoswap; 195 bool m_isTranspose, m_compU, m_compV; 196 197 using Base::m_singularValues; 198 using Base::m_diagSize; 199 using Base::m_computeFullU; 200 using Base::m_computeFullV; 201 using Base::m_computeThinU; 202 using Base::m_computeThinV; 203 using Base::m_matrixU; 204 using Base::m_matrixV; 205 using Base::m_isInitialized; 206 using Base::m_nonzeroSingularValues; 207 208public: 209 int m_numIters; 210}; //end class BDCSVD 211 212 213// Method to allocate and initialize matrix and attributes 214template<typename MatrixType> 215void BDCSVD<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions) 216{ 217 m_isTranspose = (cols > rows); 218 219 if (Base::allocate(rows, cols, computationOptions)) 220 return; 221 222 m_computed = MatrixXr::Zero(m_diagSize + 1, m_diagSize ); 223 m_compU = computeV(); 224 m_compV = computeU(); 225 if (m_isTranspose) 226 std::swap(m_compU, m_compV); 227 228 if (m_compU) m_naiveU = MatrixXr::Zero(m_diagSize + 1, m_diagSize + 1 ); 229 else m_naiveU = MatrixXr::Zero(2, m_diagSize + 1 ); 230 231 if (m_compV) m_naiveV = MatrixXr::Zero(m_diagSize, m_diagSize); 232 233 m_workspace.resize((m_diagSize+1)*(m_diagSize+1)*3); 234 m_workspaceI.resize(3*m_diagSize); 235}// end allocate 236 237template<typename MatrixType> 238BDCSVD<MatrixType>& BDCSVD<MatrixType>::compute(const MatrixType& matrix, unsigned int computationOptions) 239{ 240#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 241 std::cout << "\n\n\n======================================================================================================================\n\n\n"; 242#endif 243 allocate(matrix.rows(), matrix.cols(), computationOptions); 244 using std::abs; 245 246 const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)(); 247 248 //**** step -1 - If the problem is too small, directly falls back to JacobiSVD and return 249 if(matrix.cols() < m_algoswap) 250 { 251 // FIXME this line involves temporaries 252 JacobiSVD<MatrixType> jsvd(matrix,computationOptions); 253 if(computeU()) m_matrixU = jsvd.matrixU(); 254 if(computeV()) m_matrixV = jsvd.matrixV(); 255 m_singularValues = jsvd.singularValues(); 256 m_nonzeroSingularValues = jsvd.nonzeroSingularValues(); 257 m_isInitialized = true; 258 return *this; 259 } 260 261 //**** step 0 - Copy the input matrix and apply scaling to reduce over/under-flows 262 RealScalar scale = matrix.cwiseAbs().maxCoeff(); 263 if(scale==Literal(0)) scale = Literal(1); 264 MatrixX copy; 265 if (m_isTranspose) copy = matrix.adjoint()/scale; 266 else copy = matrix/scale; 267 268 //**** step 1 - Bidiagonalization 269 // FIXME this line involves temporaries 270 internal::UpperBidiagonalization<MatrixX> bid(copy); 271 272 //**** step 2 - Divide & Conquer 273 m_naiveU.setZero(); 274 m_naiveV.setZero(); 275 // FIXME this line involves a temporary matrix 276 m_computed.topRows(m_diagSize) = bid.bidiagonal().toDenseMatrix().transpose(); 277 m_computed.template bottomRows<1>().setZero(); 278 divide(0, m_diagSize - 1, 0, 0, 0); 279 280 //**** step 3 - Copy singular values and vectors 281 for (int i=0; i<m_diagSize; i++) 282 { 283 RealScalar a = abs(m_computed.coeff(i, i)); 284 m_singularValues.coeffRef(i) = a * scale; 285 if (a<considerZero) 286 { 287 m_nonzeroSingularValues = i; 288 m_singularValues.tail(m_diagSize - i - 1).setZero(); 289 break; 290 } 291 else if (i == m_diagSize - 1) 292 { 293 m_nonzeroSingularValues = i + 1; 294 break; 295 } 296 } 297 298#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 299// std::cout << "m_naiveU\n" << m_naiveU << "\n\n"; 300// std::cout << "m_naiveV\n" << m_naiveV << "\n\n"; 301#endif 302 if(m_isTranspose) copyUV(bid.householderV(), bid.householderU(), m_naiveV, m_naiveU); 303 else copyUV(bid.householderU(), bid.householderV(), m_naiveU, m_naiveV); 304 305 m_isInitialized = true; 306 return *this; 307}// end compute 308 309 310template<typename MatrixType> 311template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV> 312void BDCSVD<MatrixType>::copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naiveV) 313{ 314 // Note exchange of U and V: m_matrixU is set from m_naiveV and vice versa 315 if (computeU()) 316 { 317 Index Ucols = m_computeThinU ? m_diagSize : householderU.cols(); 318 m_matrixU = MatrixX::Identity(householderU.cols(), Ucols); 319 m_matrixU.topLeftCorner(m_diagSize, m_diagSize) = naiveV.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize); 320 householderU.applyThisOnTheLeft(m_matrixU); // FIXME this line involves a temporary buffer 321 } 322 if (computeV()) 323 { 324 Index Vcols = m_computeThinV ? m_diagSize : householderV.cols(); 325 m_matrixV = MatrixX::Identity(householderV.cols(), Vcols); 326 m_matrixV.topLeftCorner(m_diagSize, m_diagSize) = naiveU.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize); 327 householderV.applyThisOnTheLeft(m_matrixV); // FIXME this line involves a temporary buffer 328 } 329} 330 331/** \internal 332 * Performs A = A * B exploiting the special structure of the matrix A. Splitting A as: 333 * A = [A1] 334 * [A2] 335 * such that A1.rows()==n1, then we assume that at least half of the columns of A1 and A2 are zeros. 336 * We can thus pack them prior to the the matrix product. However, this is only worth the effort if the matrix is large 337 * enough. 338 */ 339template<typename MatrixType> 340void BDCSVD<MatrixType>::structured_update(Block<MatrixXr,Dynamic,Dynamic> A, const MatrixXr &B, Index n1) 341{ 342 Index n = A.rows(); 343 if(n>100) 344 { 345 // If the matrices are large enough, let's exploit the sparse structure of A by 346 // splitting it in half (wrt n1), and packing the non-zero columns. 347 Index n2 = n - n1; 348 Map<MatrixXr> A1(m_workspace.data() , n1, n); 349 Map<MatrixXr> A2(m_workspace.data()+ n1*n, n2, n); 350 Map<MatrixXr> B1(m_workspace.data()+ n*n, n, n); 351 Map<MatrixXr> B2(m_workspace.data()+2*n*n, n, n); 352 Index k1=0, k2=0; 353 for(Index j=0; j<n; ++j) 354 { 355 if( (A.col(j).head(n1).array()!=Literal(0)).any() ) 356 { 357 A1.col(k1) = A.col(j).head(n1); 358 B1.row(k1) = B.row(j); 359 ++k1; 360 } 361 if( (A.col(j).tail(n2).array()!=Literal(0)).any() ) 362 { 363 A2.col(k2) = A.col(j).tail(n2); 364 B2.row(k2) = B.row(j); 365 ++k2; 366 } 367 } 368 369 A.topRows(n1).noalias() = A1.leftCols(k1) * B1.topRows(k1); 370 A.bottomRows(n2).noalias() = A2.leftCols(k2) * B2.topRows(k2); 371 } 372 else 373 { 374 Map<MatrixXr,Aligned> tmp(m_workspace.data(),n,n); 375 tmp.noalias() = A*B; 376 A = tmp; 377 } 378} 379 380// The divide algorithm is done "in place", we are always working on subsets of the same matrix. The divide methods takes as argument the 381// place of the submatrix we are currently working on. 382 383//@param firstCol : The Index of the first column of the submatrix of m_computed and for m_naiveU; 384//@param lastCol : The Index of the last column of the submatrix of m_computed and for m_naiveU; 385// lastCol + 1 - firstCol is the size of the submatrix. 386//@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) 387//@param firstRowW : Same as firstRowW with the column. 388//@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 389// to become the first column (*coeff) and to shift all the other columns to the right. There are more details on the reference paper. 390template<typename MatrixType> 391void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift) 392{ 393 // requires rows = cols + 1; 394 using std::pow; 395 using std::sqrt; 396 using std::abs; 397 const Index n = lastCol - firstCol + 1; 398 const Index k = n/2; 399 const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)(); 400 RealScalar alphaK; 401 RealScalar betaK; 402 RealScalar r0; 403 RealScalar lambda, phi, c0, s0; 404 VectorType l, f; 405 // We use the other algorithm which is more efficient for small 406 // matrices. 407 if (n < m_algoswap) 408 { 409 // FIXME this line involves temporaries 410 JacobiSVD<MatrixXr> b(m_computed.block(firstCol, firstCol, n + 1, n), ComputeFullU | (m_compV ? ComputeFullV : 0)); 411 if (m_compU) 412 m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() = b.matrixU(); 413 else 414 { 415 m_naiveU.row(0).segment(firstCol, n + 1).real() = b.matrixU().row(0); 416 m_naiveU.row(1).segment(firstCol, n + 1).real() = b.matrixU().row(n); 417 } 418 if (m_compV) m_naiveV.block(firstRowW, firstColW, n, n).real() = b.matrixV(); 419 m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero(); 420 m_computed.diagonal().segment(firstCol + shift, n) = b.singularValues().head(n); 421 return; 422 } 423 // We use the divide and conquer algorithm 424 alphaK = m_computed(firstCol + k, firstCol + k); 425 betaK = m_computed(firstCol + k + 1, firstCol + k); 426 // The divide must be done in that order in order to have good results. Divide change the data inside the submatrices 427 // and the divide of the right submatrice reads one column of the left submatrice. That's why we need to treat the 428 // right submatrix before the left one. 429 divide(k + 1 + firstCol, lastCol, k + 1 + firstRowW, k + 1 + firstColW, shift); 430 divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1); 431 432 if (m_compU) 433 { 434 lambda = m_naiveU(firstCol + k, firstCol + k); 435 phi = m_naiveU(firstCol + k + 1, lastCol + 1); 436 } 437 else 438 { 439 lambda = m_naiveU(1, firstCol + k); 440 phi = m_naiveU(0, lastCol + 1); 441 } 442 r0 = sqrt((abs(alphaK * lambda) * abs(alphaK * lambda)) + abs(betaK * phi) * abs(betaK * phi)); 443 if (m_compU) 444 { 445 l = m_naiveU.row(firstCol + k).segment(firstCol, k); 446 f = m_naiveU.row(firstCol + k + 1).segment(firstCol + k + 1, n - k - 1); 447 } 448 else 449 { 450 l = m_naiveU.row(1).segment(firstCol, k); 451 f = m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1); 452 } 453 if (m_compV) m_naiveV(firstRowW+k, firstColW) = Literal(1); 454 if (r0<considerZero) 455 { 456 c0 = Literal(1); 457 s0 = Literal(0); 458 } 459 else 460 { 461 c0 = alphaK * lambda / r0; 462 s0 = betaK * phi / r0; 463 } 464 465#ifdef EIGEN_BDCSVD_SANITY_CHECKS 466 assert(m_naiveU.allFinite()); 467 assert(m_naiveV.allFinite()); 468 assert(m_computed.allFinite()); 469#endif 470 471 if (m_compU) 472 { 473 MatrixXr q1 (m_naiveU.col(firstCol + k).segment(firstCol, k + 1)); 474 // we shiftW Q1 to the right 475 for (Index i = firstCol + k - 1; i >= firstCol; i--) 476 m_naiveU.col(i + 1).segment(firstCol, k + 1) = m_naiveU.col(i).segment(firstCol, k + 1); 477 // we shift q1 at the left with a factor c0 478 m_naiveU.col(firstCol).segment( firstCol, k + 1) = (q1 * c0); 479 // last column = q1 * - s0 480 m_naiveU.col(lastCol + 1).segment(firstCol, k + 1) = (q1 * ( - s0)); 481 // first column = q2 * s0 482 m_naiveU.col(firstCol).segment(firstCol + k + 1, n - k) = m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) * s0; 483 // q2 *= c0 484 m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *= c0; 485 } 486 else 487 { 488 RealScalar q1 = m_naiveU(0, firstCol + k); 489 // we shift Q1 to the right 490 for (Index i = firstCol + k - 1; i >= firstCol; i--) 491 m_naiveU(0, i + 1) = m_naiveU(0, i); 492 // we shift q1 at the left with a factor c0 493 m_naiveU(0, firstCol) = (q1 * c0); 494 // last column = q1 * - s0 495 m_naiveU(0, lastCol + 1) = (q1 * ( - s0)); 496 // first column = q2 * s0 497 m_naiveU(1, firstCol) = m_naiveU(1, lastCol + 1) *s0; 498 // q2 *= c0 499 m_naiveU(1, lastCol + 1) *= c0; 500 m_naiveU.row(1).segment(firstCol + 1, k).setZero(); 501 m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1).setZero(); 502 } 503 504#ifdef EIGEN_BDCSVD_SANITY_CHECKS 505 assert(m_naiveU.allFinite()); 506 assert(m_naiveV.allFinite()); 507 assert(m_computed.allFinite()); 508#endif 509 510 m_computed(firstCol + shift, firstCol + shift) = r0; 511 m_computed.col(firstCol + shift).segment(firstCol + shift + 1, k) = alphaK * l.transpose().real(); 512 m_computed.col(firstCol + shift).segment(firstCol + shift + k + 1, n - k - 1) = betaK * f.transpose().real(); 513 514#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 515 ArrayXr tmp1 = (m_computed.block(firstCol+shift, firstCol+shift, n, n)).jacobiSvd().singularValues(); 516#endif 517 // Second part: try to deflate singular values in combined matrix 518 deflation(firstCol, lastCol, k, firstRowW, firstColW, shift); 519#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 520 ArrayXr tmp2 = (m_computed.block(firstCol+shift, firstCol+shift, n, n)).jacobiSvd().singularValues(); 521 std::cout << "\n\nj1 = " << tmp1.transpose().format(bdcsvdfmt) << "\n"; 522 std::cout << "j2 = " << tmp2.transpose().format(bdcsvdfmt) << "\n\n"; 523 std::cout << "err: " << ((tmp1-tmp2).abs()>1e-12*tmp2.abs()).transpose() << "\n"; 524 static int count = 0; 525 std::cout << "# " << ++count << "\n\n"; 526 assert((tmp1-tmp2).matrix().norm() < 1e-14*tmp2.matrix().norm()); 527// assert(count<681); 528// assert(((tmp1-tmp2).abs()<1e-13*tmp2.abs()).all()); 529#endif 530 531 // Third part: compute SVD of combined matrix 532 MatrixXr UofSVD, VofSVD; 533 VectorType singVals; 534 computeSVDofM(firstCol + shift, n, UofSVD, singVals, VofSVD); 535 536#ifdef EIGEN_BDCSVD_SANITY_CHECKS 537 assert(UofSVD.allFinite()); 538 assert(VofSVD.allFinite()); 539#endif 540 541 if (m_compU) 542 structured_update(m_naiveU.block(firstCol, firstCol, n + 1, n + 1), UofSVD, (n+2)/2); 543 else 544 { 545 Map<Matrix<RealScalar,2,Dynamic>,Aligned> tmp(m_workspace.data(),2,n+1); 546 tmp.noalias() = m_naiveU.middleCols(firstCol, n+1) * UofSVD; 547 m_naiveU.middleCols(firstCol, n + 1) = tmp; 548 } 549 550 if (m_compV) structured_update(m_naiveV.block(firstRowW, firstColW, n, n), VofSVD, (n+1)/2); 551 552#ifdef EIGEN_BDCSVD_SANITY_CHECKS 553 assert(m_naiveU.allFinite()); 554 assert(m_naiveV.allFinite()); 555 assert(m_computed.allFinite()); 556#endif 557 558 m_computed.block(firstCol + shift, firstCol + shift, n, n).setZero(); 559 m_computed.block(firstCol + shift, firstCol + shift, n, n).diagonal() = singVals; 560}// end divide 561 562// Compute SVD of m_computed.block(firstCol, firstCol, n + 1, n); this block only has non-zeros in 563// the first column and on the diagonal and has undergone deflation, so diagonal is in increasing 564// order except for possibly the (0,0) entry. The computed SVD is stored U, singVals and V, except 565// that if m_compV is false, then V is not computed. Singular values are sorted in decreasing order. 566// 567// TODO Opportunities for optimization: better root finding algo, better stopping criterion, better 568// handling of round-off errors, be consistent in ordering 569// For instance, to solve the secular equation using FMM, see http://www.stat.uchicago.edu/~lekheng/courses/302/classics/greengard-rokhlin.pdf 570template <typename MatrixType> 571void BDCSVD<MatrixType>::computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V) 572{ 573 const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)(); 574 using std::abs; 575 ArrayRef col0 = m_computed.col(firstCol).segment(firstCol, n); 576 m_workspace.head(n) = m_computed.block(firstCol, firstCol, n, n).diagonal(); 577 ArrayRef diag = m_workspace.head(n); 578 diag(0) = Literal(0); 579 580 // Allocate space for singular values and vectors 581 singVals.resize(n); 582 U.resize(n+1, n+1); 583 if (m_compV) V.resize(n, n); 584 585#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 586 if (col0.hasNaN() || diag.hasNaN()) 587 std::cout << "\n\nHAS NAN\n\n"; 588#endif 589 590 // Many singular values might have been deflated, the zero ones have been moved to the end, 591 // but others are interleaved and we must ignore them at this stage. 592 // To this end, let's compute a permutation skipping them: 593 Index actual_n = n; 594 while(actual_n>1 && diag(actual_n-1)==Literal(0)) --actual_n; 595 Index m = 0; // size of the deflated problem 596 for(Index k=0;k<actual_n;++k) 597 if(abs(col0(k))>considerZero) 598 m_workspaceI(m++) = k; 599 Map<ArrayXi> perm(m_workspaceI.data(),m); 600 601 Map<ArrayXr> shifts(m_workspace.data()+1*n, n); 602 Map<ArrayXr> mus(m_workspace.data()+2*n, n); 603 Map<ArrayXr> zhat(m_workspace.data()+3*n, n); 604 605#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 606 std::cout << "computeSVDofM using:\n"; 607 std::cout << " z: " << col0.transpose() << "\n"; 608 std::cout << " d: " << diag.transpose() << "\n"; 609#endif 610 611 // Compute singVals, shifts, and mus 612 computeSingVals(col0, diag, perm, singVals, shifts, mus); 613 614#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 615 std::cout << " j: " << (m_computed.block(firstCol, firstCol, n, n)).jacobiSvd().singularValues().transpose().reverse() << "\n\n"; 616 std::cout << " sing-val: " << singVals.transpose() << "\n"; 617 std::cout << " mu: " << mus.transpose() << "\n"; 618 std::cout << " shift: " << shifts.transpose() << "\n"; 619 620 { 621 Index actual_n = n; 622 while(actual_n>1 && abs(col0(actual_n-1))<considerZero) --actual_n; 623 std::cout << "\n\n mus: " << mus.head(actual_n).transpose() << "\n\n"; 624 std::cout << " check1 (expect0) : " << ((singVals.array()-(shifts+mus)) / singVals.array()).head(actual_n).transpose() << "\n\n"; 625 std::cout << " check2 (>0) : " << ((singVals.array()-diag) / singVals.array()).head(actual_n).transpose() << "\n\n"; 626 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"; 627 std::cout << " check4 (>0) : " << ((singVals.segment(1,actual_n-1)-singVals.head(actual_n-1))).transpose() << "\n\n\n"; 628 } 629#endif 630 631#ifdef EIGEN_BDCSVD_SANITY_CHECKS 632 assert(singVals.allFinite()); 633 assert(mus.allFinite()); 634 assert(shifts.allFinite()); 635#endif 636 637 // Compute zhat 638 perturbCol0(col0, diag, perm, singVals, shifts, mus, zhat); 639#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 640 std::cout << " zhat: " << zhat.transpose() << "\n"; 641#endif 642 643#ifdef EIGEN_BDCSVD_SANITY_CHECKS 644 assert(zhat.allFinite()); 645#endif 646 647 computeSingVecs(zhat, diag, perm, singVals, shifts, mus, U, V); 648 649#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 650 std::cout << "U^T U: " << (U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() << "\n"; 651 std::cout << "V^T V: " << (V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() << "\n"; 652#endif 653 654#ifdef EIGEN_BDCSVD_SANITY_CHECKS 655 assert(U.allFinite()); 656 assert(V.allFinite()); 657 assert((U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() < 1e-14 * n); 658 assert((V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() < 1e-14 * n); 659 assert(m_naiveU.allFinite()); 660 assert(m_naiveV.allFinite()); 661 assert(m_computed.allFinite()); 662#endif 663 664 // Because of deflation, the singular values might not be completely sorted. 665 // Fortunately, reordering them is a O(n) problem 666 for(Index i=0; i<actual_n-1; ++i) 667 { 668 if(singVals(i)>singVals(i+1)) 669 { 670 using std::swap; 671 swap(singVals(i),singVals(i+1)); 672 U.col(i).swap(U.col(i+1)); 673 if(m_compV) V.col(i).swap(V.col(i+1)); 674 } 675 } 676 677 // Reverse order so that singular values in increased order 678 // Because of deflation, the zeros singular-values are already at the end 679 singVals.head(actual_n).reverseInPlace(); 680 U.leftCols(actual_n).rowwise().reverseInPlace(); 681 if (m_compV) V.leftCols(actual_n).rowwise().reverseInPlace(); 682 683#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 684 JacobiSVD<MatrixXr> jsvd(m_computed.block(firstCol, firstCol, n, n) ); 685 std::cout << " * j: " << jsvd.singularValues().transpose() << "\n\n"; 686 std::cout << " * sing-val: " << singVals.transpose() << "\n"; 687// std::cout << " * err: " << ((jsvd.singularValues()-singVals)>1e-13*singVals.norm()).transpose() << "\n"; 688#endif 689} 690 691template <typename MatrixType> 692typename BDCSVD<MatrixType>::RealScalar BDCSVD<MatrixType>::secularEq(RealScalar mu, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const ArrayRef& diagShifted, RealScalar shift) 693{ 694 Index m = perm.size(); 695 RealScalar res = Literal(1); 696 for(Index i=0; i<m; ++i) 697 { 698 Index j = perm(i); 699 res += numext::abs2(col0(j)) / ((diagShifted(j) - mu) * (diag(j) + shift + mu)); 700 } 701 return res; 702 703} 704 705template <typename MatrixType> 706void BDCSVD<MatrixType>::computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, 707 VectorType& singVals, ArrayRef shifts, ArrayRef mus) 708{ 709 using std::abs; 710 using std::swap; 711 712 Index n = col0.size(); 713 Index actual_n = n; 714 while(actual_n>1 && col0(actual_n-1)==Literal(0)) --actual_n; 715 716 for (Index k = 0; k < n; ++k) 717 { 718 if (col0(k) == Literal(0) || actual_n==1) 719 { 720 // if col0(k) == 0, then entry is deflated, so singular value is on diagonal 721 // if actual_n==1, then the deflated problem is already diagonalized 722 singVals(k) = k==0 ? col0(0) : diag(k); 723 mus(k) = Literal(0); 724 shifts(k) = k==0 ? col0(0) : diag(k); 725 continue; 726 } 727 728 // otherwise, use secular equation to find singular value 729 RealScalar left = diag(k); 730 RealScalar right; // was: = (k != actual_n-1) ? diag(k+1) : (diag(actual_n-1) + col0.matrix().norm()); 731 if(k==actual_n-1) 732 right = (diag(actual_n-1) + col0.matrix().norm()); 733 else 734 { 735 // Skip deflated singular values 736 Index l = k+1; 737 while(col0(l)==Literal(0)) { ++l; eigen_internal_assert(l<actual_n); } 738 right = diag(l); 739 } 740 741 // first decide whether it's closer to the left end or the right end 742 RealScalar mid = left + (right-left) / Literal(2); 743 RealScalar fMid = secularEq(mid, col0, diag, perm, diag, Literal(0)); 744#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 745 std::cout << right-left << "\n"; 746 std::cout << "fMid = " << fMid << " " << secularEq(mid-left, col0, diag, perm, diag-left, left) << " " << secularEq(mid-right, col0, diag, perm, diag-right, right) << "\n"; 747 std::cout << " = " << secularEq(0.1*(left+right), col0, diag, perm, diag, 0) 748 << " " << secularEq(0.2*(left+right), col0, diag, perm, diag, 0) 749 << " " << secularEq(0.3*(left+right), col0, diag, perm, diag, 0) 750 << " " << secularEq(0.4*(left+right), col0, diag, perm, diag, 0) 751 << " " << secularEq(0.49*(left+right), col0, diag, perm, diag, 0) 752 << " " << secularEq(0.5*(left+right), col0, diag, perm, diag, 0) 753 << " " << secularEq(0.51*(left+right), col0, diag, perm, diag, 0) 754 << " " << secularEq(0.6*(left+right), col0, diag, perm, diag, 0) 755 << " " << secularEq(0.7*(left+right), col0, diag, perm, diag, 0) 756 << " " << secularEq(0.8*(left+right), col0, diag, perm, diag, 0) 757 << " " << secularEq(0.9*(left+right), col0, diag, perm, diag, 0) << "\n"; 758#endif 759 RealScalar shift = (k == actual_n-1 || fMid > Literal(0)) ? left : right; 760 761 // measure everything relative to shift 762 Map<ArrayXr> diagShifted(m_workspace.data()+4*n, n); 763 diagShifted = diag - shift; 764 765 // initial guess 766 RealScalar muPrev, muCur; 767 if (shift == left) 768 { 769 muPrev = (right - left) * RealScalar(0.1); 770 if (k == actual_n-1) muCur = right - left; 771 else muCur = (right - left) * RealScalar(0.5); 772 } 773 else 774 { 775 muPrev = -(right - left) * RealScalar(0.1); 776 muCur = -(right - left) * RealScalar(0.5); 777 } 778 779 RealScalar fPrev = secularEq(muPrev, col0, diag, perm, diagShifted, shift); 780 RealScalar fCur = secularEq(muCur, col0, diag, perm, diagShifted, shift); 781 if (abs(fPrev) < abs(fCur)) 782 { 783 swap(fPrev, fCur); 784 swap(muPrev, muCur); 785 } 786 787 // rational interpolation: fit a function of the form a / mu + b through the two previous 788 // iterates and use its zero to compute the next iterate 789 bool useBisection = fPrev*fCur>Literal(0); 790 while (fCur!=Literal(0) && abs(muCur - muPrev) > Literal(8) * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(muCur), abs(muPrev)) && abs(fCur - fPrev)>NumTraits<RealScalar>::epsilon() && !useBisection) 791 { 792 ++m_numIters; 793 794 // Find a and b such that the function f(mu) = a / mu + b matches the current and previous samples. 795 RealScalar a = (fCur - fPrev) / (Literal(1)/muCur - Literal(1)/muPrev); 796 RealScalar b = fCur - a / muCur; 797 // And find mu such that f(mu)==0: 798 RealScalar muZero = -a/b; 799 RealScalar fZero = secularEq(muZero, col0, diag, perm, diagShifted, shift); 800 801 muPrev = muCur; 802 fPrev = fCur; 803 muCur = muZero; 804 fCur = fZero; 805 806 807 if (shift == left && (muCur < Literal(0) || muCur > right - left)) useBisection = true; 808 if (shift == right && (muCur < -(right - left) || muCur > Literal(0))) useBisection = true; 809 if (abs(fCur)>abs(fPrev)) useBisection = true; 810 } 811 812 // fall back on bisection method if rational interpolation did not work 813 if (useBisection) 814 { 815#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 816 std::cout << "useBisection for k = " << k << ", actual_n = " << actual_n << "\n"; 817#endif 818 RealScalar leftShifted, rightShifted; 819 if (shift == left) 820 { 821 leftShifted = (std::numeric_limits<RealScalar>::min)(); 822 // I don't understand why the case k==0 would be special there: 823 // if (k == 0) rightShifted = right - left; else 824 rightShifted = (k==actual_n-1) ? right : ((right - left) * RealScalar(0.6)); // theoretically we can take 0.5, but let's be safe 825 } 826 else 827 { 828 leftShifted = -(right - left) * RealScalar(0.6); 829 rightShifted = -(std::numeric_limits<RealScalar>::min)(); 830 } 831 832 RealScalar fLeft = secularEq(leftShifted, col0, diag, perm, diagShifted, shift); 833 834#if defined EIGEN_INTERNAL_DEBUGGING || defined EIGEN_BDCSVD_DEBUG_VERBOSE 835 RealScalar fRight = secularEq(rightShifted, col0, diag, perm, diagShifted, shift); 836#endif 837 838#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 839 if(!(fLeft * fRight<0)) 840 { 841 std::cout << "fLeft: " << leftShifted << " - " << diagShifted.head(10).transpose() << "\n ; " << bool(left==shift) << " " << (left-shift) << "\n"; 842 std::cout << k << " : " << fLeft << " * " << fRight << " == " << fLeft * fRight << " ; " << left << " - " << right << " -> " << leftShifted << " " << rightShifted << " shift=" << shift << "\n"; 843 } 844#endif 845 eigen_internal_assert(fLeft * fRight < Literal(0)); 846 847 while (rightShifted - leftShifted > Literal(2) * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(leftShifted), abs(rightShifted))) 848 { 849 RealScalar midShifted = (leftShifted + rightShifted) / Literal(2); 850 fMid = secularEq(midShifted, col0, diag, perm, diagShifted, shift); 851 if (fLeft * fMid < Literal(0)) 852 { 853 rightShifted = midShifted; 854 } 855 else 856 { 857 leftShifted = midShifted; 858 fLeft = fMid; 859 } 860 } 861 862 muCur = (leftShifted + rightShifted) / Literal(2); 863 } 864 865 singVals[k] = shift + muCur; 866 shifts[k] = shift; 867 mus[k] = muCur; 868 869 // perturb singular value slightly if it equals diagonal entry to avoid division by zero later 870 // (deflation is supposed to avoid this from happening) 871 // - this does no seem to be necessary anymore - 872// if (singVals[k] == left) singVals[k] *= 1 + NumTraits<RealScalar>::epsilon(); 873// if (singVals[k] == right) singVals[k] *= 1 - NumTraits<RealScalar>::epsilon(); 874 } 875} 876 877 878// zhat is perturbation of col0 for which singular vectors can be computed stably (see Section 3.1) 879template <typename MatrixType> 880void BDCSVD<MatrixType>::perturbCol0 881 (const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const VectorType& singVals, 882 const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat) 883{ 884 using std::sqrt; 885 Index n = col0.size(); 886 Index m = perm.size(); 887 if(m==0) 888 { 889 zhat.setZero(); 890 return; 891 } 892 Index last = perm(m-1); 893 // The offset permits to skip deflated entries while computing zhat 894 for (Index k = 0; k < n; ++k) 895 { 896 if (col0(k) == Literal(0)) // deflated 897 zhat(k) = Literal(0); 898 else 899 { 900 // see equation (3.6) 901 RealScalar dk = diag(k); 902 RealScalar prod = (singVals(last) + dk) * (mus(last) + (shifts(last) - dk)); 903 904 for(Index l = 0; l<m; ++l) 905 { 906 Index i = perm(l); 907 if(i!=k) 908 { 909 Index j = i<k ? i : perm(l-1); 910 prod *= ((singVals(j)+dk) / ((diag(i)+dk))) * ((mus(j)+(shifts(j)-dk)) / ((diag(i)-dk))); 911#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 912 if(i!=k && std::abs(((singVals(j)+dk)*(mus(j)+(shifts(j)-dk)))/((diag(i)+dk)*(diag(i)-dk)) - 1) > 0.9 ) 913 std::cout << " " << ((singVals(j)+dk)*(mus(j)+(shifts(j)-dk)))/((diag(i)+dk)*(diag(i)-dk)) << " == (" << (singVals(j)+dk) << " * " << (mus(j)+(shifts(j)-dk)) 914 << ") / (" << (diag(i)+dk) << " * " << (diag(i)-dk) << ")\n"; 915#endif 916 } 917 } 918#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 919 std::cout << "zhat(" << k << ") = sqrt( " << prod << ") ; " << (singVals(last) + dk) << " * " << mus(last) + shifts(last) << " - " << dk << "\n"; 920#endif 921 RealScalar tmp = sqrt(prod); 922 zhat(k) = col0(k) > Literal(0) ? tmp : -tmp; 923 } 924 } 925} 926 927// compute singular vectors 928template <typename MatrixType> 929void BDCSVD<MatrixType>::computeSingVecs 930 (const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef &perm, const VectorType& singVals, 931 const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V) 932{ 933 Index n = zhat.size(); 934 Index m = perm.size(); 935 936 for (Index k = 0; k < n; ++k) 937 { 938 if (zhat(k) == Literal(0)) 939 { 940 U.col(k) = VectorType::Unit(n+1, k); 941 if (m_compV) V.col(k) = VectorType::Unit(n, k); 942 } 943 else 944 { 945 U.col(k).setZero(); 946 for(Index l=0;l<m;++l) 947 { 948 Index i = perm(l); 949 U(i,k) = zhat(i)/(((diag(i) - shifts(k)) - mus(k)) )/( (diag(i) + singVals[k])); 950 } 951 U(n,k) = Literal(0); 952 U.col(k).normalize(); 953 954 if (m_compV) 955 { 956 V.col(k).setZero(); 957 for(Index l=1;l<m;++l) 958 { 959 Index i = perm(l); 960 V(i,k) = diag(i) * zhat(i) / (((diag(i) - shifts(k)) - mus(k)) )/( (diag(i) + singVals[k])); 961 } 962 V(0,k) = Literal(-1); 963 V.col(k).normalize(); 964 } 965 } 966 } 967 U.col(n) = VectorType::Unit(n+1, n); 968} 969 970 971// page 12_13 972// i >= 1, di almost null and zi non null. 973// We use a rotation to zero out zi applied to the left of M 974template <typename MatrixType> 975void BDCSVD<MatrixType>::deflation43(Index firstCol, Index shift, Index i, Index size) 976{ 977 using std::abs; 978 using std::sqrt; 979 using std::pow; 980 Index start = firstCol + shift; 981 RealScalar c = m_computed(start, start); 982 RealScalar s = m_computed(start+i, start); 983 RealScalar r = sqrt(numext::abs2(c) + numext::abs2(s)); 984 if (r == Literal(0)) 985 { 986 m_computed(start+i, start+i) = Literal(0); 987 return; 988 } 989 m_computed(start,start) = r; 990 m_computed(start+i, start) = Literal(0); 991 m_computed(start+i, start+i) = Literal(0); 992 993 JacobiRotation<RealScalar> J(c/r,-s/r); 994 if (m_compU) m_naiveU.middleRows(firstCol, size+1).applyOnTheRight(firstCol, firstCol+i, J); 995 else m_naiveU.applyOnTheRight(firstCol, firstCol+i, J); 996}// end deflation 43 997 998 999// page 13 1000// i,j >= 1, i!=j and |di - dj| < epsilon * norm2(M) 1001// We apply two rotations to have zj = 0; 1002// TODO deflation44 is still broken and not properly tested 1003template <typename MatrixType> 1004void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size) 1005{ 1006 using std::abs; 1007 using std::sqrt; 1008 using std::conj; 1009 using std::pow; 1010 RealScalar c = m_computed(firstColm+i, firstColm); 1011 RealScalar s = m_computed(firstColm+j, firstColm); 1012 RealScalar r = sqrt(numext::abs2(c) + numext::abs2(s)); 1013#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1014 std::cout << "deflation 4.4: " << i << "," << j << " -> " << c << " " << s << " " << r << " ; " 1015 << m_computed(firstColm + i-1, firstColm) << " " 1016 << m_computed(firstColm + i, firstColm) << " " 1017 << m_computed(firstColm + i+1, firstColm) << " " 1018 << m_computed(firstColm + i+2, firstColm) << "\n"; 1019 std::cout << m_computed(firstColm + i-1, firstColm + i-1) << " " 1020 << m_computed(firstColm + i, firstColm+i) << " " 1021 << m_computed(firstColm + i+1, firstColm+i+1) << " " 1022 << m_computed(firstColm + i+2, firstColm+i+2) << "\n"; 1023#endif 1024 if (r==Literal(0)) 1025 { 1026 m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j); 1027 return; 1028 } 1029 c/=r; 1030 s/=r; 1031 m_computed(firstColm + i, firstColm) = r; 1032 m_computed(firstColm + j, firstColm + j) = m_computed(firstColm + i, firstColm + i); 1033 m_computed(firstColm + j, firstColm) = Literal(0); 1034 1035 JacobiRotation<RealScalar> J(c,-s); 1036 if (m_compU) m_naiveU.middleRows(firstColu, size+1).applyOnTheRight(firstColu + i, firstColu + j, J); 1037 else m_naiveU.applyOnTheRight(firstColu+i, firstColu+j, J); 1038 if (m_compV) m_naiveV.middleRows(firstRowW, size).applyOnTheRight(firstColW + i, firstColW + j, J); 1039}// end deflation 44 1040 1041 1042// acts on block from (firstCol+shift, firstCol+shift) to (lastCol+shift, lastCol+shift) [inclusive] 1043template <typename MatrixType> 1044void BDCSVD<MatrixType>::deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift) 1045{ 1046 using std::sqrt; 1047 using std::abs; 1048 const Index length = lastCol + 1 - firstCol; 1049 1050 Block<MatrixXr,Dynamic,1> col0(m_computed, firstCol+shift, firstCol+shift, length, 1); 1051 Diagonal<MatrixXr> fulldiag(m_computed); 1052 VectorBlock<Diagonal<MatrixXr>,Dynamic> diag(fulldiag, firstCol+shift, length); 1053 1054 const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)(); 1055 RealScalar maxDiag = diag.tail((std::max)(Index(1),length-1)).cwiseAbs().maxCoeff(); 1056 RealScalar epsilon_strict = numext::maxi<RealScalar>(considerZero,NumTraits<RealScalar>::epsilon() * maxDiag); 1057 RealScalar epsilon_coarse = Literal(8) * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(col0.cwiseAbs().maxCoeff(), maxDiag); 1058 1059#ifdef EIGEN_BDCSVD_SANITY_CHECKS 1060 assert(m_naiveU.allFinite()); 1061 assert(m_naiveV.allFinite()); 1062 assert(m_computed.allFinite()); 1063#endif 1064 1065#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1066 std::cout << "\ndeflate:" << diag.head(k+1).transpose() << " | " << diag.segment(k+1,length-k-1).transpose() << "\n"; 1067#endif 1068 1069 //condition 4.1 1070 if (diag(0) < epsilon_coarse) 1071 { 1072#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1073 std::cout << "deflation 4.1, because " << diag(0) << " < " << epsilon_coarse << "\n"; 1074#endif 1075 diag(0) = epsilon_coarse; 1076 } 1077 1078 //condition 4.2 1079 for (Index i=1;i<length;++i) 1080 if (abs(col0(i)) < epsilon_strict) 1081 { 1082#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1083 std::cout << "deflation 4.2, set z(" << i << ") to zero because " << abs(col0(i)) << " < " << epsilon_strict << " (diag(" << i << ")=" << diag(i) << ")\n"; 1084#endif 1085 col0(i) = Literal(0); 1086 } 1087 1088 //condition 4.3 1089 for (Index i=1;i<length; i++) 1090 if (diag(i) < epsilon_coarse) 1091 { 1092#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1093 std::cout << "deflation 4.3, cancel z(" << i << ")=" << col0(i) << " because diag(" << i << ")=" << diag(i) << " < " << epsilon_coarse << "\n"; 1094#endif 1095 deflation43(firstCol, shift, i, length); 1096 } 1097 1098#ifdef EIGEN_BDCSVD_SANITY_CHECKS 1099 assert(m_naiveU.allFinite()); 1100 assert(m_naiveV.allFinite()); 1101 assert(m_computed.allFinite()); 1102#endif 1103#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1104 std::cout << "to be sorted: " << diag.transpose() << "\n\n"; 1105#endif 1106 { 1107 // Check for total deflation 1108 // If we have a total deflation, then we have to consider col0(0)==diag(0) as a singular value during sorting 1109 bool total_deflation = (col0.tail(length-1).array()<considerZero).all(); 1110 1111 // Sort the diagonal entries, since diag(1:k-1) and diag(k:length) are already sorted, let's do a sorted merge. 1112 // First, compute the respective permutation. 1113 Index *permutation = m_workspaceI.data(); 1114 { 1115 permutation[0] = 0; 1116 Index p = 1; 1117 1118 // Move deflated diagonal entries at the end. 1119 for(Index i=1; i<length; ++i) 1120 if(abs(diag(i))<considerZero) 1121 permutation[p++] = i; 1122 1123 Index i=1, j=k+1; 1124 for( ; p < length; ++p) 1125 { 1126 if (i > k) permutation[p] = j++; 1127 else if (j >= length) permutation[p] = i++; 1128 else if (diag(i) < diag(j)) permutation[p] = j++; 1129 else permutation[p] = i++; 1130 } 1131 } 1132 1133 // If we have a total deflation, then we have to insert diag(0) at the right place 1134 if(total_deflation) 1135 { 1136 for(Index i=1; i<length; ++i) 1137 { 1138 Index pi = permutation[i]; 1139 if(abs(diag(pi))<considerZero || diag(0)<diag(pi)) 1140 permutation[i-1] = permutation[i]; 1141 else 1142 { 1143 permutation[i-1] = 0; 1144 break; 1145 } 1146 } 1147 } 1148 1149 // Current index of each col, and current column of each index 1150 Index *realInd = m_workspaceI.data()+length; 1151 Index *realCol = m_workspaceI.data()+2*length; 1152 1153 for(int pos = 0; pos< length; pos++) 1154 { 1155 realCol[pos] = pos; 1156 realInd[pos] = pos; 1157 } 1158 1159 for(Index i = total_deflation?0:1; i < length; i++) 1160 { 1161 const Index pi = permutation[length - (total_deflation ? i+1 : i)]; 1162 const Index J = realCol[pi]; 1163 1164 using std::swap; 1165 // swap diagonal and first column entries: 1166 swap(diag(i), diag(J)); 1167 if(i!=0 && J!=0) swap(col0(i), col0(J)); 1168 1169 // change columns 1170 if (m_compU) m_naiveU.col(firstCol+i).segment(firstCol, length + 1).swap(m_naiveU.col(firstCol+J).segment(firstCol, length + 1)); 1171 else m_naiveU.col(firstCol+i).segment(0, 2) .swap(m_naiveU.col(firstCol+J).segment(0, 2)); 1172 if (m_compV) m_naiveV.col(firstColW + i).segment(firstRowW, length).swap(m_naiveV.col(firstColW + J).segment(firstRowW, length)); 1173 1174 //update real pos 1175 const Index realI = realInd[i]; 1176 realCol[realI] = J; 1177 realCol[pi] = i; 1178 realInd[J] = realI; 1179 realInd[i] = pi; 1180 } 1181 } 1182#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1183 std::cout << "sorted: " << diag.transpose().format(bdcsvdfmt) << "\n"; 1184 std::cout << " : " << col0.transpose() << "\n\n"; 1185#endif 1186 1187 //condition 4.4 1188 { 1189 Index i = length-1; 1190 while(i>0 && (abs(diag(i))<considerZero || abs(col0(i))<considerZero)) --i; 1191 for(; i>1;--i) 1192 if( (diag(i) - diag(i-1)) < NumTraits<RealScalar>::epsilon()*maxDiag ) 1193 { 1194#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE 1195 std::cout << "deflation 4.4 with i = " << i << " because " << (diag(i) - diag(i-1)) << " < " << NumTraits<RealScalar>::epsilon()*diag(i) << "\n"; 1196#endif 1197 eigen_internal_assert(abs(diag(i) - diag(i-1))<epsilon_coarse && " diagonal entries are not properly sorted"); 1198 deflation44(firstCol, firstCol + shift, firstRowW, firstColW, i-1, i, length); 1199 } 1200 } 1201 1202#ifdef EIGEN_BDCSVD_SANITY_CHECKS 1203 for(Index j=2;j<length;++j) 1204 assert(diag(j-1)<=diag(j) || abs(diag(j))<considerZero); 1205#endif 1206 1207#ifdef EIGEN_BDCSVD_SANITY_CHECKS 1208 assert(m_naiveU.allFinite()); 1209 assert(m_naiveV.allFinite()); 1210 assert(m_computed.allFinite()); 1211#endif 1212}//end deflation 1213 1214#ifndef __CUDACC__ 1215/** \svd_module 1216 * 1217 * \return the singular value decomposition of \c *this computed by Divide & Conquer algorithm 1218 * 1219 * \sa class BDCSVD 1220 */ 1221template<typename Derived> 1222BDCSVD<typename MatrixBase<Derived>::PlainObject> 1223MatrixBase<Derived>::bdcSvd(unsigned int computationOptions) const 1224{ 1225 return BDCSVD<PlainObject>(*this, computationOptions); 1226} 1227#endif 1228 1229} // end namespace Eigen 1230 1231#endif 1232