Eigen  3.2.91
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 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 namespace Eigen {
25 
26 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
27 IOFormat bdcsvdfmt(8, 0, ", ", "\n", " [", "]");
28 #endif
29 
30 template<typename _MatrixType> class BDCSVD;
31 
32 namespace internal {
33 
34 template<typename _MatrixType>
35 struct traits<BDCSVD<_MatrixType> >
36 {
37  typedef _MatrixType MatrixType;
38 };
39 
40 } // end namespace internal
41 
42 
54 template<typename _MatrixType>
55 class BDCSVD : public SVDBase<BDCSVD<_MatrixType> >
56 {
57  typedef SVDBase<BDCSVD> Base;
58 
59 public:
60  using Base::rows;
61  using Base::cols;
62  using Base::computeU;
63  using Base::computeV;
64 
65  typedef _MatrixType MatrixType;
66  typedef typename MatrixType::Scalar Scalar;
67  typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
68  enum {
69  RowsAtCompileTime = MatrixType::RowsAtCompileTime,
70  ColsAtCompileTime = MatrixType::ColsAtCompileTime,
71  DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime, ColsAtCompileTime),
72  MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
73  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
74  MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime, MaxColsAtCompileTime),
75  MatrixOptions = MatrixType::Options
76  };
77 
78  typedef typename Base::MatrixUType MatrixUType;
79  typedef typename Base::MatrixVType MatrixVType;
80  typedef typename Base::SingularValuesType SingularValuesType;
81 
82  typedef Matrix<Scalar, Dynamic, Dynamic, ColMajor> MatrixX;
83  typedef Matrix<RealScalar, Dynamic, Dynamic, ColMajor> MatrixXr;
84  typedef Matrix<RealScalar, Dynamic, 1> VectorType;
85  typedef Array<RealScalar, Dynamic, 1> ArrayXr;
86  typedef Array<Index,1,Dynamic> ArrayXi;
87  typedef Ref<ArrayXr> ArrayRef;
88  typedef Ref<ArrayXi> IndicesRef;
89 
95  BDCSVD() : m_algoswap(16), m_numIters(0)
96  {}
97 
98 
105  BDCSVD(Index rows, Index cols, unsigned int computationOptions = 0)
106  : m_algoswap(16), m_numIters(0)
107  {
108  allocate(rows, cols, computationOptions);
109  }
110 
121  BDCSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
122  : m_algoswap(16), m_numIters(0)
123  {
124  compute(matrix, computationOptions);
125  }
126 
127  ~BDCSVD()
128  {
129  }
130 
141  BDCSVD& compute(const MatrixType& matrix, unsigned int computationOptions);
142 
149  BDCSVD& compute(const MatrixType& matrix)
150  {
151  return compute(matrix, this->m_computationOptions);
152  }
153 
154  void setSwitchSize(int s)
155  {
156  eigen_assert(s>3 && "BDCSVD the size of the algo switch has to be greater than 3");
157  m_algoswap = s;
158  }
159 
160 private:
161  void allocate(Index rows, Index cols, unsigned int computationOptions);
162  void divide(Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift);
163  void computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V);
164  void computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, VectorType& singVals, ArrayRef shifts, ArrayRef mus);
165  void perturbCol0(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat);
166  void computeSingVecs(const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V);
167  void deflation43(Index firstCol, Index shift, Index i, Index size);
168  void deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size);
169  void deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift);
170  template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
171  void copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naivev);
172  void structured_update(Block<MatrixXr,Dynamic,Dynamic> A, const MatrixXr &B, Index n1);
173  static RealScalar secularEq(RealScalar x, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const ArrayRef& diagShifted, RealScalar shift);
174 
175 protected:
176  MatrixXr m_naiveU, m_naiveV;
177  MatrixXr m_computed;
178  Index m_nRec;
179  ArrayXr m_workspace;
180  ArrayXi m_workspaceI;
181  int m_algoswap;
182  bool m_isTranspose, m_compU, m_compV;
183 
184  using Base::m_singularValues;
185  using Base::m_diagSize;
186  using Base::m_computeFullU;
187  using Base::m_computeFullV;
188  using Base::m_computeThinU;
189  using Base::m_computeThinV;
190  using Base::m_matrixU;
191  using Base::m_matrixV;
192  using Base::m_isInitialized;
193  using Base::m_nonzeroSingularValues;
194 
195 public:
196  int m_numIters;
197 }; //end class BDCSVD
198 
199 
200 // Method to allocate and initialize matrix and attributes
201 template<typename MatrixType>
202 void BDCSVD<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions)
203 {
204  m_isTranspose = (cols > rows);
205 
206  if (Base::allocate(rows, cols, computationOptions))
207  return;
208 
209  m_computed = MatrixXr::Zero(m_diagSize + 1, m_diagSize );
210  m_compU = computeV();
211  m_compV = computeU();
212  if (m_isTranspose)
213  std::swap(m_compU, m_compV);
214 
215  if (m_compU) m_naiveU = MatrixXr::Zero(m_diagSize + 1, m_diagSize + 1 );
216  else m_naiveU = MatrixXr::Zero(2, m_diagSize + 1 );
217 
218  if (m_compV) m_naiveV = MatrixXr::Zero(m_diagSize, m_diagSize);
219 
220  m_workspace.resize((m_diagSize+1)*(m_diagSize+1)*3);
221  m_workspaceI.resize(3*m_diagSize);
222 }// end allocate
223 
224 template<typename MatrixType>
225 BDCSVD<MatrixType>& BDCSVD<MatrixType>::compute(const MatrixType& matrix, unsigned int computationOptions)
226 {
227 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
228  std::cout << "\n\n\n======================================================================================================================\n\n\n";
229 #endif
230  allocate(matrix.rows(), matrix.cols(), computationOptions);
231  using std::abs;
232 
233  //**** step -1 - If the problem is too small, directly falls back to JacobiSVD and return
234  if(matrix.cols() < m_algoswap)
235  {
236  // FIXME this line involves temporaries
237  JacobiSVD<MatrixType> jsvd(matrix,computationOptions);
238  if(computeU()) m_matrixU = jsvd.matrixU();
239  if(computeV()) m_matrixV = jsvd.matrixV();
240  m_singularValues = jsvd.singularValues();
241  m_nonzeroSingularValues = jsvd.nonzeroSingularValues();
242  m_isInitialized = true;
243  return *this;
244  }
245 
246  //**** step 0 - Copy the input matrix and apply scaling to reduce over/under-flows
247  RealScalar scale = matrix.cwiseAbs().maxCoeff();
248  if(scale==RealScalar(0)) scale = RealScalar(1);
249  MatrixX copy;
250  if (m_isTranspose) copy = matrix.adjoint()/scale;
251  else copy = matrix/scale;
252 
253  //**** step 1 - Bidiagonalization
254  // FIXME this line involves temporaries
255  internal::UpperBidiagonalization<MatrixX> bid(copy);
256 
257  //**** step 2 - Divide & Conquer
258  m_naiveU.setZero();
259  m_naiveV.setZero();
260  // FIXME this line involves a temporary matrix
261  m_computed.topRows(m_diagSize) = bid.bidiagonal().toDenseMatrix().transpose();
262  m_computed.template bottomRows<1>().setZero();
263  divide(0, m_diagSize - 1, 0, 0, 0);
264 
265  //**** step 3 - Copy singular values and vectors
266  for (int i=0; i<m_diagSize; i++)
267  {
268  RealScalar a = abs(m_computed.coeff(i, i));
269  m_singularValues.coeffRef(i) = a * scale;
270  if (a == 0)
271  {
272  m_nonzeroSingularValues = i;
273  m_singularValues.tail(m_diagSize - i - 1).setZero();
274  break;
275  }
276  else if (i == m_diagSize - 1)
277  {
278  m_nonzeroSingularValues = i + 1;
279  break;
280  }
281  }
282 
283 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
284 // std::cout << "m_naiveU\n" << m_naiveU << "\n\n";
285 // std::cout << "m_naiveV\n" << m_naiveV << "\n\n";
286 #endif
287  if(m_isTranspose) copyUV(bid.householderV(), bid.householderU(), m_naiveV, m_naiveU);
288  else copyUV(bid.householderU(), bid.householderV(), m_naiveU, m_naiveV);
289 
290  m_isInitialized = true;
291  return *this;
292 }// end compute
293 
294 
295 template<typename MatrixType>
296 template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
297 void BDCSVD<MatrixType>::copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naiveV)
298 {
299  // Note exchange of U and V: m_matrixU is set from m_naiveV and vice versa
300  if (computeU())
301  {
302  Index Ucols = m_computeThinU ? m_diagSize : householderU.cols();
303  m_matrixU = MatrixX::Identity(householderU.cols(), Ucols);
304  m_matrixU.topLeftCorner(m_diagSize, m_diagSize) = naiveV.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize);
305  householderU.applyThisOnTheLeft(m_matrixU); // FIXME this line involves a temporary buffer
306  }
307  if (computeV())
308  {
309  Index Vcols = m_computeThinV ? m_diagSize : householderV.cols();
310  m_matrixV = MatrixX::Identity(householderV.cols(), Vcols);
311  m_matrixV.topLeftCorner(m_diagSize, m_diagSize) = naiveU.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize);
312  householderV.applyThisOnTheLeft(m_matrixV); // FIXME this line involves a temporary buffer
313  }
314 }
315 
324 template<typename MatrixType>
325 void BDCSVD<MatrixType>::structured_update(Block<MatrixXr,Dynamic,Dynamic> A, const MatrixXr &B, Index n1)
326 {
327  Index n = A.rows();
328  if(n>100)
329  {
330  // If the matrices are large enough, let's exploit the sparse structure of A by
331  // splitting it in half (wrt n1), and packing the non-zero columns.
332  Index n2 = n - n1;
333  Map<MatrixXr> A1(m_workspace.data() , n1, n);
334  Map<MatrixXr> A2(m_workspace.data()+ n1*n, n2, n);
335  Map<MatrixXr> B1(m_workspace.data()+ n*n, n, n);
336  Map<MatrixXr> B2(m_workspace.data()+2*n*n, n, n);
337  Index k1=0, k2=0;
338  for(Index j=0; j<n; ++j)
339  {
340  if( (A.col(j).head(n1).array()!=0).any() )
341  {
342  A1.col(k1) = A.col(j).head(n1);
343  B1.row(k1) = B.row(j);
344  ++k1;
345  }
346  if( (A.col(j).tail(n2).array()!=0).any() )
347  {
348  A2.col(k2) = A.col(j).tail(n2);
349  B2.row(k2) = B.row(j);
350  ++k2;
351  }
352  }
353 
354  A.topRows(n1).noalias() = A1.leftCols(k1) * B1.topRows(k1);
355  A.bottomRows(n2).noalias() = A2.leftCols(k2) * B2.topRows(k2);
356  }
357  else
358  {
359  Map<MatrixXr,Aligned> tmp(m_workspace.data(),n,n);
360  tmp.noalias() = A*B;
361  A = tmp;
362  }
363 }
364 
365 // The divide algorithm is done "in place", we are always working on subsets of the same matrix. The divide methods takes as argument the
366 // place of the submatrix we are currently working on.
367 
368 //@param firstCol : The Index of the first column of the submatrix of m_computed and for m_naiveU;
369 //@param lastCol : The Index of the last column of the submatrix of m_computed and for m_naiveU;
370 // lastCol + 1 - firstCol is the size of the submatrix.
371 //@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)
372 //@param firstRowW : Same as firstRowW with the column.
373 //@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
374 // to become the first column (*coeff) and to shift all the other columns to the right. There are more details on the reference paper.
375 template<typename MatrixType>
376 void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift)
377 {
378  // requires rows = cols + 1;
379  using std::pow;
380  using std::sqrt;
381  using std::abs;
382  const Index n = lastCol - firstCol + 1;
383  const Index k = n/2;
384  RealScalar alphaK;
385  RealScalar betaK;
386  RealScalar r0;
387  RealScalar lambda, phi, c0, s0;
388  VectorType l, f;
389  // We use the other algorithm which is more efficient for small
390  // matrices.
391  if (n < m_algoswap)
392  {
393  // FIXME this line involves temporaries
394  JacobiSVD<MatrixXr> b(m_computed.block(firstCol, firstCol, n + 1, n), ComputeFullU | (m_compV ? ComputeFullV : 0));
395  if (m_compU)
396  m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() = b.matrixU();
397  else
398  {
399  m_naiveU.row(0).segment(firstCol, n + 1).real() = b.matrixU().row(0);
400  m_naiveU.row(1).segment(firstCol, n + 1).real() = b.matrixU().row(n);
401  }
402  if (m_compV) m_naiveV.block(firstRowW, firstColW, n, n).real() = b.matrixV();
403  m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero();
404  m_computed.diagonal().segment(firstCol + shift, n) = b.singularValues().head(n);
405  return;
406  }
407  // We use the divide and conquer algorithm
408  alphaK = m_computed(firstCol + k, firstCol + k);
409  betaK = m_computed(firstCol + k + 1, firstCol + k);
410  // The divide must be done in that order in order to have good results. Divide change the data inside the submatrices
411  // and the divide of the right submatrice reads one column of the left submatrice. That's why we need to treat the
412  // right submatrix before the left one.
413  divide(k + 1 + firstCol, lastCol, k + 1 + firstRowW, k + 1 + firstColW, shift);
414  divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1);
415 
416  if (m_compU)
417  {
418  lambda = m_naiveU(firstCol + k, firstCol + k);
419  phi = m_naiveU(firstCol + k + 1, lastCol + 1);
420  }
421  else
422  {
423  lambda = m_naiveU(1, firstCol + k);
424  phi = m_naiveU(0, lastCol + 1);
425  }
426  r0 = sqrt((abs(alphaK * lambda) * abs(alphaK * lambda)) + abs(betaK * phi) * abs(betaK * phi));
427  if (m_compU)
428  {
429  l = m_naiveU.row(firstCol + k).segment(firstCol, k);
430  f = m_naiveU.row(firstCol + k + 1).segment(firstCol + k + 1, n - k - 1);
431  }
432  else
433  {
434  l = m_naiveU.row(1).segment(firstCol, k);
435  f = m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1);
436  }
437  if (m_compV) m_naiveV(firstRowW+k, firstColW) = 1;
438  if (r0 == 0)
439  {
440  c0 = 1;
441  s0 = 0;
442  }
443  else
444  {
445  c0 = alphaK * lambda / r0;
446  s0 = betaK * phi / r0;
447  }
448 
449 #ifdef EIGEN_BDCSVD_SANITY_CHECKS
450  assert(m_naiveU.allFinite());
451  assert(m_naiveV.allFinite());
452  assert(m_computed.allFinite());
453 #endif
454 
455  if (m_compU)
456  {
457  MatrixXr q1 (m_naiveU.col(firstCol + k).segment(firstCol, k + 1));
458  // we shiftW Q1 to the right
459  for (Index i = firstCol + k - 1; i >= firstCol; i--)
460  m_naiveU.col(i + 1).segment(firstCol, k + 1) = m_naiveU.col(i).segment(firstCol, k + 1);
461  // we shift q1 at the left with a factor c0
462  m_naiveU.col(firstCol).segment( firstCol, k + 1) = (q1 * c0);
463  // last column = q1 * - s0
464  m_naiveU.col(lastCol + 1).segment(firstCol, k + 1) = (q1 * ( - s0));
465  // first column = q2 * s0
466  m_naiveU.col(firstCol).segment(firstCol + k + 1, n - k) = m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) * s0;
467  // q2 *= c0
468  m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *= c0;
469  }
470  else
471  {
472  RealScalar q1 = m_naiveU(0, firstCol + k);
473  // we shift Q1 to the right
474  for (Index i = firstCol + k - 1; i >= firstCol; i--)
475  m_naiveU(0, i + 1) = m_naiveU(0, i);
476  // we shift q1 at the left with a factor c0
477  m_naiveU(0, firstCol) = (q1 * c0);
478  // last column = q1 * - s0
479  m_naiveU(0, lastCol + 1) = (q1 * ( - s0));
480  // first column = q2 * s0
481  m_naiveU(1, firstCol) = m_naiveU(1, lastCol + 1) *s0;
482  // q2 *= c0
483  m_naiveU(1, lastCol + 1) *= c0;
484  m_naiveU.row(1).segment(firstCol + 1, k).setZero();
485  m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1).setZero();
486  }
487 
488 #ifdef EIGEN_BDCSVD_SANITY_CHECKS
489  assert(m_naiveU.allFinite());
490  assert(m_naiveV.allFinite());
491  assert(m_computed.allFinite());
492 #endif
493 
494  m_computed(firstCol + shift, firstCol + shift) = r0;
495  m_computed.col(firstCol + shift).segment(firstCol + shift + 1, k) = alphaK * l.transpose().real();
496  m_computed.col(firstCol + shift).segment(firstCol + shift + k + 1, n - k - 1) = betaK * f.transpose().real();
497 
498 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
499  ArrayXr tmp1 = (m_computed.block(firstCol+shift, firstCol+shift, n, n)).jacobiSvd().singularValues();
500 #endif
501  // Second part: try to deflate singular values in combined matrix
502  deflation(firstCol, lastCol, k, firstRowW, firstColW, shift);
503 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
504  ArrayXr tmp2 = (m_computed.block(firstCol+shift, firstCol+shift, n, n)).jacobiSvd().singularValues();
505  std::cout << "\n\nj1 = " << tmp1.transpose().format(bdcsvdfmt) << "\n";
506  std::cout << "j2 = " << tmp2.transpose().format(bdcsvdfmt) << "\n\n";
507  std::cout << "err: " << ((tmp1-tmp2).abs()>1e-12*tmp2.abs()).transpose() << "\n";
508  static int count = 0;
509  std::cout << "# " << ++count << "\n\n";
510  assert((tmp1-tmp2).matrix().norm() < 1e-14*tmp2.matrix().norm());
511 // assert(count<681);
512 // assert(((tmp1-tmp2).abs()<1e-13*tmp2.abs()).all());
513 #endif
514 
515  // Third part: compute SVD of combined matrix
516  MatrixXr UofSVD, VofSVD;
517  VectorType singVals;
518  computeSVDofM(firstCol + shift, n, UofSVD, singVals, VofSVD);
519 
520 #ifdef EIGEN_BDCSVD_SANITY_CHECKS
521  assert(UofSVD.allFinite());
522  assert(VofSVD.allFinite());
523 #endif
524 
525  if (m_compU)
526  structured_update(m_naiveU.block(firstCol, firstCol, n + 1, n + 1), UofSVD, (n+2)/2);
527  else
528  {
529  Map<Matrix<RealScalar,2,Dynamic>,Aligned> tmp(m_workspace.data(),2,n+1);
530  tmp.noalias() = m_naiveU.middleCols(firstCol, n+1) * UofSVD;
531  m_naiveU.middleCols(firstCol, n + 1) = tmp;
532  }
533 
534  if (m_compV) structured_update(m_naiveV.block(firstRowW, firstColW, n, n), VofSVD, (n+1)/2);
535 
536 #ifdef EIGEN_BDCSVD_SANITY_CHECKS
537  assert(m_naiveU.allFinite());
538  assert(m_naiveV.allFinite());
539  assert(m_computed.allFinite());
540 #endif
541 
542  m_computed.block(firstCol + shift, firstCol + shift, n, n).setZero();
543  m_computed.block(firstCol + shift, firstCol + shift, n, n).diagonal() = singVals;
544 }// end divide
545 
546 // Compute SVD of m_computed.block(firstCol, firstCol, n + 1, n); this block only has non-zeros in
547 // the first column and on the diagonal and has undergone deflation, so diagonal is in increasing
548 // order except for possibly the (0,0) entry. The computed SVD is stored U, singVals and V, except
549 // that if m_compV is false, then V is not computed. Singular values are sorted in decreasing order.
550 //
551 // TODO Opportunities for optimization: better root finding algo, better stopping criterion, better
552 // handling of round-off errors, be consistent in ordering
553 // For instance, to solve the secular equation using FMM, see http://www.stat.uchicago.edu/~lekheng/courses/302/classics/greengard-rokhlin.pdf
554 template <typename MatrixType>
555 void BDCSVD<MatrixType>::computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V)
556 {
557  ArrayRef col0 = m_computed.col(firstCol).segment(firstCol, n);
558  m_workspace.head(n) = m_computed.block(firstCol, firstCol, n, n).diagonal();
559  ArrayRef diag = m_workspace.head(n);
560  diag(0) = 0;
561 
562  // Allocate space for singular values and vectors
563  singVals.resize(n);
564  U.resize(n+1, n+1);
565  if (m_compV) V.resize(n, n);
566 
567 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
568  if (col0.hasNaN() || diag.hasNaN())
569  std::cout << "\n\nHAS NAN\n\n";
570 #endif
571 
572  // Many singular values might have been deflated, the zero ones have been moved to the end,
573  // but others are interleaved and we must ignore them at this stage.
574  // To this end, let's compute a permutation skipping them:
575  Index actual_n = n;
576  while(actual_n>1 && diag(actual_n-1)==0) --actual_n;
577  Index m = 0; // size of the deflated problem
578  for(Index k=0;k<actual_n;++k)
579  if(col0(k)!=0)
580  m_workspaceI(m++) = k;
581  Map<ArrayXi> perm(m_workspaceI.data(),m);
582 
583  Map<ArrayXr> shifts(m_workspace.data()+1*n, n);
584  Map<ArrayXr> mus(m_workspace.data()+2*n, n);
585  Map<ArrayXr> zhat(m_workspace.data()+3*n, n);
586 
587 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
588  std::cout << "computeSVDofM using:\n";
589  std::cout << " z: " << col0.transpose() << "\n";
590  std::cout << " d: " << diag.transpose() << "\n";
591 #endif
592 
593  // Compute singVals, shifts, and mus
594  computeSingVals(col0, diag, perm, singVals, shifts, mus);
595 
596 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
597  std::cout << " j: " << (m_computed.block(firstCol, firstCol, n, n)).jacobiSvd().singularValues().transpose().reverse() << "\n\n";
598  std::cout << " sing-val: " << singVals.transpose() << "\n";
599  std::cout << " mu: " << mus.transpose() << "\n";
600  std::cout << " shift: " << shifts.transpose() << "\n";
601 
602  {
603  Index actual_n = n;
604  while(actual_n>1 && col0(actual_n-1)==0) --actual_n;
605  std::cout << "\n\n mus: " << mus.head(actual_n).transpose() << "\n\n";
606  std::cout << " check1 (expect0) : " << ((singVals.array()-(shifts+mus)) / singVals.array()).head(actual_n).transpose() << "\n\n";
607  std::cout << " check2 (>0) : " << ((singVals.array()-diag) / singVals.array()).head(actual_n).transpose() << "\n\n";
608  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";
609  std::cout << " check4 (>0) : " << ((singVals.segment(1,actual_n-1)-singVals.head(actual_n-1))).transpose() << "\n\n\n";
610  }
611 #endif
612 
613 #ifdef EIGEN_BDCSVD_SANITY_CHECKS
614  assert(singVals.allFinite());
615  assert(mus.allFinite());
616  assert(shifts.allFinite());
617 #endif
618 
619  // Compute zhat
620  perturbCol0(col0, diag, perm, singVals, shifts, mus, zhat);
621 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
622  std::cout << " zhat: " << zhat.transpose() << "\n";
623 #endif
624 
625 #ifdef EIGEN_BDCSVD_SANITY_CHECKS
626  assert(zhat.allFinite());
627 #endif
628 
629  computeSingVecs(zhat, diag, perm, singVals, shifts, mus, U, V);
630 
631 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
632  std::cout << "U^T U: " << (U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() << "\n";
633  std::cout << "V^T V: " << (V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() << "\n";
634 #endif
635 
636 #ifdef EIGEN_BDCSVD_SANITY_CHECKS
637  assert(U.allFinite());
638  assert(V.allFinite());
639  assert((U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() < 1e-14 * n);
640  assert((V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() < 1e-14 * n);
641  assert(m_naiveU.allFinite());
642  assert(m_naiveV.allFinite());
643  assert(m_computed.allFinite());
644 #endif
645 
646  // Because of deflation, the singular values might not be completely sorted.
647  // Fortunately, reordering them is a O(n) problem
648  for(Index i=0; i<actual_n-1; ++i)
649  {
650  if(singVals(i)>singVals(i+1))
651  {
652  using std::swap;
653  swap(singVals(i),singVals(i+1));
654  U.col(i).swap(U.col(i+1));
655  if(m_compV) V.col(i).swap(V.col(i+1));
656  }
657  }
658 
659  // Reverse order so that singular values in increased order
660  // Because of deflation, the zeros singular-values are already at the end
661  singVals.head(actual_n).reverseInPlace();
662  U.leftCols(actual_n).rowwise().reverseInPlace();
663  if (m_compV) V.leftCols(actual_n).rowwise().reverseInPlace();
664 
665 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
666  JacobiSVD<MatrixXr> jsvd(m_computed.block(firstCol, firstCol, n, n) );
667  std::cout << " * j: " << jsvd.singularValues().transpose() << "\n\n";
668  std::cout << " * sing-val: " << singVals.transpose() << "\n";
669 // std::cout << " * err: " << ((jsvd.singularValues()-singVals)>1e-13*singVals.norm()).transpose() << "\n";
670 #endif
671 }
672 
673 template <typename MatrixType>
674 typename BDCSVD<MatrixType>::RealScalar BDCSVD<MatrixType>::secularEq(RealScalar mu, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const ArrayRef& diagShifted, RealScalar shift)
675 {
676  Index m = perm.size();
677  RealScalar res = 1;
678  for(Index i=0; i<m; ++i)
679  {
680  Index j = perm(i);
681  res += numext::abs2(col0(j)) / ((diagShifted(j) - mu) * (diag(j) + shift + mu));
682  }
683  return res;
684 }
685 
686 template <typename MatrixType>
687 void BDCSVD<MatrixType>::computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm,
688  VectorType& singVals, ArrayRef shifts, ArrayRef mus)
689 {
690  using std::abs;
691  using std::swap;
692 
693  Index n = col0.size();
694  Index actual_n = n;
695  while(actual_n>1 && col0(actual_n-1)==0) --actual_n;
696 
697  for (Index k = 0; k < n; ++k)
698  {
699  if (col0(k) == 0 || actual_n==1)
700  {
701  // if col0(k) == 0, then entry is deflated, so singular value is on diagonal
702  // if actual_n==1, then the deflated problem is already diagonalized
703  singVals(k) = k==0 ? col0(0) : diag(k);
704  mus(k) = 0;
705  shifts(k) = k==0 ? col0(0) : diag(k);
706  continue;
707  }
708 
709  // otherwise, use secular equation to find singular value
710  RealScalar left = diag(k);
711  RealScalar right; // was: = (k != actual_n-1) ? diag(k+1) : (diag(actual_n-1) + col0.matrix().norm());
712  if(k==actual_n-1)
713  right = (diag(actual_n-1) + col0.matrix().norm());
714  else
715  {
716  // Skip deflated singular values
717  Index l = k+1;
718  while(col0(l)==0) { ++l; eigen_internal_assert(l<actual_n); }
719  right = diag(l);
720  }
721 
722  // first decide whether it's closer to the left end or the right end
723  RealScalar mid = left + (right-left) / 2;
724  RealScalar fMid = secularEq(mid, col0, diag, perm, diag, 0);
725 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
726  std::cout << right-left << "\n";
727  std::cout << "fMid = " << fMid << " " << secularEq(mid-left, col0, diag, perm, diag-left, left) << " " << secularEq(mid-right, col0, diag, perm, diag-right, right) << "\n";
728  std::cout << " = " << secularEq(0.1*(left+right), col0, diag, perm, diag, 0)
729  << " " << secularEq(0.2*(left+right), col0, diag, perm, diag, 0)
730  << " " << secularEq(0.3*(left+right), col0, diag, perm, diag, 0)
731  << " " << secularEq(0.4*(left+right), col0, diag, perm, diag, 0)
732  << " " << secularEq(0.49*(left+right), col0, diag, perm, diag, 0)
733  << " " << secularEq(0.5*(left+right), col0, diag, perm, diag, 0)
734  << " " << secularEq(0.51*(left+right), col0, diag, perm, diag, 0)
735  << " " << secularEq(0.6*(left+right), col0, diag, perm, diag, 0)
736  << " " << secularEq(0.7*(left+right), col0, diag, perm, diag, 0)
737  << " " << secularEq(0.8*(left+right), col0, diag, perm, diag, 0)
738  << " " << secularEq(0.9*(left+right), col0, diag, perm, diag, 0) << "\n";
739 #endif
740  RealScalar shift = (k == actual_n-1 || fMid > 0) ? left : right;
741 
742  // measure everything relative to shift
743  Map<ArrayXr> diagShifted(m_workspace.data()+4*n, n);
744  diagShifted = diag - shift;
745 
746  // initial guess
747  RealScalar muPrev, muCur;
748  if (shift == left)
749  {
750  muPrev = (right - left) * 0.1;
751  if (k == actual_n-1) muCur = right - left;
752  else muCur = (right - left) * 0.5;
753  }
754  else
755  {
756  muPrev = -(right - left) * 0.1;
757  muCur = -(right - left) * 0.5;
758  }
759 
760  RealScalar fPrev = secularEq(muPrev, col0, diag, perm, diagShifted, shift);
761  RealScalar fCur = secularEq(muCur, col0, diag, perm, diagShifted, shift);
762  if (abs(fPrev) < abs(fCur))
763  {
764  swap(fPrev, fCur);
765  swap(muPrev, muCur);
766  }
767 
768  // rational interpolation: fit a function of the form a / mu + b through the two previous
769  // iterates and use its zero to compute the next iterate
770  bool useBisection = fPrev*fCur>0;
771  while (fCur!=0 && abs(muCur - muPrev) > 8 * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(muCur), abs(muPrev)) && abs(fCur - fPrev)>NumTraits<RealScalar>::epsilon() && !useBisection)
772  {
773  ++m_numIters;
774 
775  // Find a and b such that the function f(mu) = a / mu + b matches the current and previous samples.
776  RealScalar a = (fCur - fPrev) / (1/muCur - 1/muPrev);
777  RealScalar b = fCur - a / muCur;
778  // And find mu such that f(mu)==0:
779  RealScalar muZero = -a/b;
780  RealScalar fZero = secularEq(muZero, col0, diag, perm, diagShifted, shift);
781 
782  muPrev = muCur;
783  fPrev = fCur;
784  muCur = muZero;
785  fCur = fZero;
786 
787 
788  if (shift == left && (muCur < 0 || muCur > right - left)) useBisection = true;
789  if (shift == right && (muCur < -(right - left) || muCur > 0)) useBisection = true;
790  if (abs(fCur)>abs(fPrev)) useBisection = true;
791  }
792 
793  // fall back on bisection method if rational interpolation did not work
794  if (useBisection)
795  {
796 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
797  std::cout << "useBisection for k = " << k << ", actual_n = " << actual_n << "\n";
798 #endif
799  RealScalar leftShifted, rightShifted;
800  if (shift == left)
801  {
802  leftShifted = RealScalar(1)/NumTraits<RealScalar>::highest();
803  // I don't understand why the case k==0 would be special there:
804  // if (k == 0) rightShifted = right - left; else
805  rightShifted = (k==actual_n-1) ? right : ((right - left) * 0.6); // theoretically we can take 0.5, but let's be safe
806  }
807  else
808  {
809  leftShifted = -(right - left) * 0.6;
810  rightShifted = -RealScalar(1)/NumTraits<RealScalar>::highest();
811  }
812 
813  RealScalar fLeft = secularEq(leftShifted, col0, diag, perm, diagShifted, shift);
814 
815 #if defined EIGEN_INTERNAL_DEBUGGING || defined EIGEN_BDCSVD_DEBUG_VERBOSE
816  RealScalar fRight = secularEq(rightShifted, col0, diag, perm, diagShifted, shift);
817 #endif
818 
819 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
820  if(!(fLeft * fRight<0))
821  std::cout << k << " : " << fLeft << " * " << fRight << " == " << fLeft * fRight << " ; " << left << " - " << right << " -> " << leftShifted << " " << rightShifted << " shift=" << shift << "\n";
822 #endif
823  eigen_internal_assert(fLeft * fRight < 0);
824 
825  while (rightShifted - leftShifted > 2 * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(leftShifted), abs(rightShifted)))
826  {
827  RealScalar midShifted = (leftShifted + rightShifted) / 2;
828  fMid = secularEq(midShifted, col0, diag, perm, diagShifted, shift);
829  if (fLeft * fMid < 0)
830  {
831  rightShifted = midShifted;
832  }
833  else
834  {
835  leftShifted = midShifted;
836  fLeft = fMid;
837  }
838  }
839 
840  muCur = (leftShifted + rightShifted) / 2;
841  }
842 
843  singVals[k] = shift + muCur;
844  shifts[k] = shift;
845  mus[k] = muCur;
846 
847  // perturb singular value slightly if it equals diagonal entry to avoid division by zero later
848  // (deflation is supposed to avoid this from happening)
849  // - this does no seem to be necessary anymore -
850 // if (singVals[k] == left) singVals[k] *= 1 + NumTraits<RealScalar>::epsilon();
851 // if (singVals[k] == right) singVals[k] *= 1 - NumTraits<RealScalar>::epsilon();
852  }
853 }
854 
855 
856 // zhat is perturbation of col0 for which singular vectors can be computed stably (see Section 3.1)
857 template <typename MatrixType>
858 void BDCSVD<MatrixType>::perturbCol0
859  (const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const VectorType& singVals,
860  const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat)
861 {
862  using std::sqrt;
863  Index n = col0.size();
864  Index m = perm.size();
865  if(m==0)
866  {
867  zhat.setZero();
868  return;
869  }
870  Index last = perm(m-1);
871  // The offset permits to skip deflated entries while computing zhat
872  for (Index k = 0; k < n; ++k)
873  {
874  if (col0(k) == 0) // deflated
875  zhat(k) = 0;
876  else
877  {
878  // see equation (3.6)
879  RealScalar dk = diag(k);
880  RealScalar prod = (singVals(last) + dk) * (mus(last) + (shifts(last) - dk));
881 
882  for(Index l = 0; l<m; ++l)
883  {
884  Index i = perm(l);
885  if(i!=k)
886  {
887  Index j = i<k ? i : perm(l-1);
888  prod *= ((singVals(j)+dk) / ((diag(i)+dk))) * ((mus(j)+(shifts(j)-dk)) / ((diag(i)-dk)));
889 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
890  if(i!=k && std::abs(((singVals(j)+dk)*(mus(j)+(shifts(j)-dk)))/((diag(i)+dk)*(diag(i)-dk)) - 1) > 0.9 )
891  std::cout << " " << ((singVals(j)+dk)*(mus(j)+(shifts(j)-dk)))/((diag(i)+dk)*(diag(i)-dk)) << " == (" << (singVals(j)+dk) << " * " << (mus(j)+(shifts(j)-dk))
892  << ") / (" << (diag(i)+dk) << " * " << (diag(i)-dk) << ")\n";
893 #endif
894  }
895  }
896 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
897  std::cout << "zhat(" << k << ") = sqrt( " << prod << ") ; " << (singVals(last) + dk) << " * " << mus(last) + shifts(last) << " - " << dk << "\n";
898 #endif
899  RealScalar tmp = sqrt(prod);
900  zhat(k) = col0(k) > 0 ? tmp : -tmp;
901  }
902  }
903 }
904 
905 // compute singular vectors
906 template <typename MatrixType>
907 void BDCSVD<MatrixType>::computeSingVecs
908  (const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef &perm, const VectorType& singVals,
909  const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V)
910 {
911  Index n = zhat.size();
912  Index m = perm.size();
913 
914  for (Index k = 0; k < n; ++k)
915  {
916  if (zhat(k) == 0)
917  {
918  U.col(k) = VectorType::Unit(n+1, k);
919  if (m_compV) V.col(k) = VectorType::Unit(n, k);
920  }
921  else
922  {
923  U.col(k).setZero();
924  for(Index l=0;l<m;++l)
925  {
926  Index i = perm(l);
927  U(i,k) = zhat(i)/(((diag(i) - shifts(k)) - mus(k)) )/( (diag(i) + singVals[k]));
928  }
929  U(n,k) = 0;
930  U.col(k).normalize();
931 
932  if (m_compV)
933  {
934  V.col(k).setZero();
935  for(Index l=1;l<m;++l)
936  {
937  Index i = perm(l);
938  V(i,k) = diag(i) * zhat(i) / (((diag(i) - shifts(k)) - mus(k)) )/( (diag(i) + singVals[k]));
939  }
940  V(0,k) = -1;
941  V.col(k).normalize();
942  }
943  }
944  }
945  U.col(n) = VectorType::Unit(n+1, n);
946 }
947 
948 
949 // page 12_13
950 // i >= 1, di almost null and zi non null.
951 // We use a rotation to zero out zi applied to the left of M
952 template <typename MatrixType>
953 void BDCSVD<MatrixType>::deflation43(Index firstCol, Index shift, Index i, Index size)
954 {
955  using std::abs;
956  using std::sqrt;
957  using std::pow;
958  Index start = firstCol + shift;
959  RealScalar c = m_computed(start, start);
960  RealScalar s = m_computed(start+i, start);
961  RealScalar r = sqrt(numext::abs2(c) + numext::abs2(s));
962  if (r == 0)
963  {
964  m_computed(start+i, start+i) = 0;
965  return;
966  }
967  m_computed(start,start) = r;
968  m_computed(start+i, start) = 0;
969  m_computed(start+i, start+i) = 0;
970 
971  JacobiRotation<RealScalar> J(c/r,-s/r);
972  if (m_compU) m_naiveU.middleRows(firstCol, size+1).applyOnTheRight(firstCol, firstCol+i, J);
973  else m_naiveU.applyOnTheRight(firstCol, firstCol+i, J);
974 }// end deflation 43
975 
976 
977 // page 13
978 // i,j >= 1, i!=j and |di - dj| < epsilon * norm2(M)
979 // We apply two rotations to have zj = 0;
980 // TODO deflation44 is still broken and not properly tested
981 template <typename MatrixType>
982 void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size)
983 {
984  using std::abs;
985  using std::sqrt;
986  using std::conj;
987  using std::pow;
988  RealScalar c = m_computed(firstColm+i, firstColm);
989  RealScalar s = m_computed(firstColm+j, firstColm);
990  RealScalar r = sqrt(numext::abs2(c) + numext::abs2(s));
991 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
992  std::cout << "deflation 4.4: " << i << "," << j << " -> " << c << " " << s << " " << r << " ; "
993  << m_computed(firstColm + i-1, firstColm) << " "
994  << m_computed(firstColm + i, firstColm) << " "
995  << m_computed(firstColm + i+1, firstColm) << " "
996  << m_computed(firstColm + i+2, firstColm) << "\n";
997  std::cout << m_computed(firstColm + i-1, firstColm + i-1) << " "
998  << m_computed(firstColm + i, firstColm+i) << " "
999  << m_computed(firstColm + i+1, firstColm+i+1) << " "
1000  << m_computed(firstColm + i+2, firstColm+i+2) << "\n";
1001 #endif
1002  if (r==0)
1003  {
1004  m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j);
1005  return;
1006  }
1007  c/=r;
1008  s/=r;
1009  m_computed(firstColm + i, firstColm) = r;
1010  m_computed(firstColm + j, firstColm + j) = m_computed(firstColm + i, firstColm + i);
1011  m_computed(firstColm + j, firstColm) = 0;
1012 
1013  JacobiRotation<RealScalar> J(c,-s);
1014  if (m_compU) m_naiveU.middleRows(firstColu, size+1).applyOnTheRight(firstColu + i, firstColu + j, J);
1015  else m_naiveU.applyOnTheRight(firstColu+i, firstColu+j, J);
1016  if (m_compV) m_naiveV.middleRows(firstRowW, size).applyOnTheRight(firstColW + i, firstColW + j, J);
1017 }// end deflation 44
1018 
1019 
1020 // acts on block from (firstCol+shift, firstCol+shift) to (lastCol+shift, lastCol+shift) [inclusive]
1021 template <typename MatrixType>
1022 void BDCSVD<MatrixType>::deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift)
1023 {
1024  using std::sqrt;
1025  using std::abs;
1026  const Index length = lastCol + 1 - firstCol;
1027 
1028  Block<MatrixXr,Dynamic,1> col0(m_computed, firstCol+shift, firstCol+shift, length, 1);
1029  Diagonal<MatrixXr> fulldiag(m_computed);
1030  VectorBlock<Diagonal<MatrixXr>,Dynamic> diag(fulldiag, firstCol+shift, length);
1031 
1032  RealScalar maxDiag = diag.tail((std::max)(Index(1),length-1)).cwiseAbs().maxCoeff();
1033  RealScalar epsilon_strict = NumTraits<RealScalar>::epsilon() * maxDiag;
1034  RealScalar epsilon_coarse = 8 * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(col0.cwiseAbs().maxCoeff(), maxDiag);
1035 
1036 #ifdef EIGEN_BDCSVD_SANITY_CHECKS
1037  assert(m_naiveU.allFinite());
1038  assert(m_naiveV.allFinite());
1039  assert(m_computed.allFinite());
1040 #endif
1041 
1042 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1043  std::cout << "\ndeflate:" << diag.head(k+1).transpose() << " | " << diag.segment(k+1,length-k-1).transpose() << "\n";
1044 #endif
1045 
1046  //condition 4.1
1047  if (diag(0) < epsilon_coarse)
1048  {
1049 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1050  std::cout << "deflation 4.1, because " << diag(0) << " < " << epsilon_coarse << "\n";
1051 #endif
1052  diag(0) = epsilon_coarse;
1053  }
1054 
1055  //condition 4.2
1056  for (Index i=1;i<length;++i)
1057  if (abs(col0(i)) < epsilon_strict)
1058  {
1059 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1060  std::cout << "deflation 4.2, set z(" << i << ") to zero because " << abs(col0(i)) << " < " << epsilon_strict << " (diag(" << i << ")=" << diag(i) << ")\n";
1061 #endif
1062  col0(i) = 0;
1063  }
1064 
1065  //condition 4.3
1066  for (Index i=1;i<length; i++)
1067  if (diag(i) < epsilon_coarse)
1068  {
1069 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1070  std::cout << "deflation 4.3, cancel z(" << i << ")=" << col0(i) << " because diag(" << i << ")=" << diag(i) << " < " << epsilon_coarse << "\n";
1071 #endif
1072  deflation43(firstCol, shift, i, length);
1073  }
1074 
1075 #ifdef EIGEN_BDCSVD_SANITY_CHECKS
1076  assert(m_naiveU.allFinite());
1077  assert(m_naiveV.allFinite());
1078  assert(m_computed.allFinite());
1079 #endif
1080 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1081  std::cout << "to be sorted: " << diag.transpose() << "\n\n";
1082 #endif
1083  {
1084  // Check for total deflation
1085  // If we have a total deflation, then we have to consider col0(0)==diag(0) as a singular value during sorting
1086  bool total_deflation = (col0.tail(length-1).array()==RealScalar(0)).all();
1087 
1088  // Sort the diagonal entries, since diag(1:k-1) and diag(k:length) are already sorted, let's do a sorted merge.
1089  // First, compute the respective permutation.
1090  Index *permutation = m_workspaceI.data();
1091  {
1092  permutation[0] = 0;
1093  Index p = 1;
1094 
1095  // Move deflated diagonal entries at the end.
1096  for(Index i=1; i<length; ++i)
1097  if(diag(i)==0)
1098  permutation[p++] = i;
1099 
1100  Index i=1, j=k+1;
1101  for( ; p < length; ++p)
1102  {
1103  if (i > k) permutation[p] = j++;
1104  else if (j >= length) permutation[p] = i++;
1105  else if (diag(i) < diag(j)) permutation[p] = j++;
1106  else permutation[p] = i++;
1107  }
1108  }
1109 
1110  // If we have a total deflation, then we have to insert diag(0) at the right place
1111  if(total_deflation)
1112  {
1113  for(Index i=1; i<length; ++i)
1114  {
1115  Index pi = permutation[i];
1116  if(diag(pi)==0 || diag(0)<diag(pi))
1117  permutation[i-1] = permutation[i];
1118  else
1119  {
1120  permutation[i-1] = 0;
1121  break;
1122  }
1123  }
1124  }
1125 
1126  // Current index of each col, and current column of each index
1127  Index *realInd = m_workspaceI.data()+length;
1128  Index *realCol = m_workspaceI.data()+2*length;
1129 
1130  for(int pos = 0; pos< length; pos++)
1131  {
1132  realCol[pos] = pos;
1133  realInd[pos] = pos;
1134  }
1135 
1136  for(Index i = total_deflation?0:1; i < length; i++)
1137  {
1138  const Index pi = permutation[length - (total_deflation ? i+1 : i)];
1139  const Index J = realCol[pi];
1140 
1141  using std::swap;
1142  // swap diagonal and first column entries:
1143  swap(diag(i), diag(J));
1144  if(i!=0 && J!=0) swap(col0(i), col0(J));
1145 
1146  // change columns
1147  if (m_compU) m_naiveU.col(firstCol+i).segment(firstCol, length + 1).swap(m_naiveU.col(firstCol+J).segment(firstCol, length + 1));
1148  else m_naiveU.col(firstCol+i).segment(0, 2) .swap(m_naiveU.col(firstCol+J).segment(0, 2));
1149  if (m_compV) m_naiveV.col(firstColW + i).segment(firstRowW, length).swap(m_naiveV.col(firstColW + J).segment(firstRowW, length));
1150 
1151  //update real pos
1152  const Index realI = realInd[i];
1153  realCol[realI] = J;
1154  realCol[pi] = i;
1155  realInd[J] = realI;
1156  realInd[i] = pi;
1157  }
1158  }
1159 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1160  std::cout << "sorted: " << diag.transpose().format(bdcsvdfmt) << "\n";
1161  std::cout << " : " << col0.transpose() << "\n\n";
1162 #endif
1163 
1164  //condition 4.4
1165  {
1166  Index i = length-1;
1167  while(i>0 && (diag(i)==0 || col0(i)==0)) --i;
1168  for(; i>1;--i)
1169  if( (diag(i) - diag(i-1)) < NumTraits<RealScalar>::epsilon()*maxDiag )
1170  {
1171 #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1172  std::cout << "deflation 4.4 with i = " << i << " because " << (diag(i) - diag(i-1)) << " < " << NumTraits<RealScalar>::epsilon()*diag(i) << "\n";
1173 #endif
1174  eigen_internal_assert(abs(diag(i) - diag(i-1))<epsilon_coarse && " diagonal entries are not properly sorted");
1175  deflation44(firstCol, firstCol + shift, firstRowW, firstColW, i-1, i, length);
1176  }
1177  }
1178 
1179 #ifdef EIGEN_BDCSVD_SANITY_CHECKS
1180  for(Index j=2;j<length;++j)
1181  assert(diag(j-1)<=diag(j) || diag(j)==0);
1182 #endif
1183 
1184 #ifdef EIGEN_BDCSVD_SANITY_CHECKS
1185  assert(m_naiveU.allFinite());
1186  assert(m_naiveV.allFinite());
1187  assert(m_computed.allFinite());
1188 #endif
1189 }//end deflation
1190 
1191 #ifndef __CUDACC__
1192 
1198 template<typename Derived>
1199 BDCSVD<typename MatrixBase<Derived>::PlainObject>
1200 MatrixBase<Derived>::bdcSvd(unsigned int computationOptions) const
1201 {
1202  return BDCSVD<PlainObject>(*this, computationOptions);
1203 }
1204 #endif
1205 
1206 } // end namespace Eigen
1207 
1208 #endif
BDCSVD & compute(const MatrixType &matrix)
Method performing the decomposition of given matrix using current options.
Definition: BDCSVD.h:149
Definition: Constants.h:375
const SingularValuesType & singularValues() const
Definition: SVDBase.h:111
Eigen::Index Index
Definition: SVDBase.h:56
Definition: LDLT.h:16
bool computeV() const
Definition: SVDBase.h:191
BDCSVD< PlainObject > bdcSvd(unsigned int computationOptions=0) const
Definition: BDCSVD.h:1200
BDCSVD(Index rows, Index cols, unsigned int computationOptions=0)
Default Constructor with memory preallocation.
Definition: BDCSVD.h:105
BDCSVD & compute(const MatrixType &matrix, unsigned int computationOptions)
Method performing the decomposition of given matrix using custom options.
Definition: BDCSVD.h:225
const MatrixVType & matrixV() const
Definition: SVDBase.h:99
BDCSVD(const MatrixType &matrix, unsigned int computationOptions=0)
Constructor performing the decomposition of given matrix.
Definition: BDCSVD.h:121
const MatrixUType & matrixU() const
Definition: SVDBase.h:83
bool computeU() const
Definition: SVDBase.h:189
class Bidiagonal Divide and Conquer SVD
Definition: ForwardDeclarations.h:255
Index nonzeroSingularValues() const
Definition: SVDBase.h:118
Definition: Eigen_Colamd.h:54
BDCSVD()
Default Constructor.
Definition: BDCSVD.h:95
Two-sided Jacobi SVD decomposition of a rectangular matrix.
Definition: ForwardDeclarations.h:254
Definition: Constants.h:379
The matrix class, also used for vectors and row-vectors.
Definition: Matrix.h:178
Definition: Constants.h:227
const AdjointReturnType adjoint() const
Definition: Transpose.h:204