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