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BDCSVD.h
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 
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  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 
108  BDCSVD() : m_algoswap(16), m_numIters(0)
109  {}
110 
111 
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 
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 
154  BDCSVD& compute(const MatrixType& matrix, unsigned int computationOptions);
155 
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 
173 private:
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 
188 protected:
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 
208 public:
209  int m_numIters;
210 }; //end class BDCSVD
211 
212 
213 // Method to allocate and initialize matrix and attributes
214 template<typename MatrixType>
215 void 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 
237 template<typename MatrixType>
238 BDCSVD<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 
310 template<typename MatrixType>
311 template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
312 void 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 
339 template<typename MatrixType>
340 void 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.
390 template<typename MatrixType>
391 void 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
570 template <typename MatrixType>
571 void 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 
691 template <typename MatrixType>
692 typename 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 
705 template <typename MatrixType>
706 void 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)
879 template <typename MatrixType>
880 void 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
928 template <typename MatrixType>
929 void 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
974 template <typename MatrixType>
975 void 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
1003 template <typename MatrixType>
1004 void 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]
1043 template <typename MatrixType>
1044 void 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 EIGEN_CUDACC
1215 
1221 template<typename Derived>
1222 BDCSVD<typename MatrixBase<Derived>::PlainObject>
1223 MatrixBase<Derived>::bdcSvd(unsigned int computationOptions) const
1224 {
1225  return BDCSVD<PlainObject>(*this, computationOptions);
1226 }
1227 #endif
1228 
1229 } // end namespace Eigen
1230 
1231 #endif
BDCSVD(Index rows, Index cols, unsigned int computationOptions=0)
Default Constructor with memory preallocation.
Definition: BDCSVD.h:118
static const Eigen::internal::all_t all
Definition: IndexedViewHelper.h:181
Definition: Constants.h:387
const SingularValuesType & singularValues() const
Definition: SVDBase.h:111
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_sqrt_op< typename Derived::Scalar >, const Derived > sqrt(const Eigen::ArrayBase< Derived > &x)
Eigen::Index Index
Definition: SVDBase.h:56
bool computeV() const
Definition: SVDBase.h:190
BDCSVD< PlainObject > bdcSvd(unsigned int computationOptions=0) const
Definition: BDCSVD.h:1223
BDCSVD & compute(const MatrixType &matrix)
Method performing the decomposition of given matrix using current options.
Definition: BDCSVD.h:162
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
The Index type as used for the API.
Definition: Meta.h:33
Definition: Constants.h:239
BDCSVD & compute(const MatrixType &matrix, unsigned int computationOptions)
Method performing the decomposition of given matrix using custom options.
Definition: BDCSVD.h:238
const MatrixVType & matrixV() const
Definition: SVDBase.h:99
const MatrixUType & matrixU() const
Definition: SVDBase.h:83
bool computeU() const
Definition: SVDBase.h:188
class Bidiagonal Divide and Conquer SVD
Definition: ForwardDeclarations.h:260
Index nonzeroSingularValues() const
Definition: SVDBase.h:118
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_abs_op< typename Derived::Scalar >, const Derived > abs(const Eigen::ArrayBase< Derived > &x)
BDCSVD()
Default Constructor.
Definition: BDCSVD.h:108
Two-sided Jacobi SVD decomposition of a rectangular matrix.
Definition: ForwardDeclarations.h:259
const int Dynamic
Definition: Constants.h:21
Definition: Constants.h:391
The matrix class, also used for vectors and row-vectors.
Definition: Matrix.h:178
const AdjointReturnType adjoint() const
Definition: Transpose.h:210
static const Symbolic::SymbolExpr< internal::symbolic_last_tag > last
Definition: IndexedViewHelper.h:45
BDCSVD(const MatrixType &matrix, unsigned int computationOptions=0)
Constructor performing the decomposition of given matrix.
Definition: BDCSVD.h:134