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) 2014-2017 Gael Guennebaud <gael.guennebaud@inria.fr>
15//
16// Source Code Form is subject to the terms of the Mozilla
17// Public License v. 2.0. If a copy of the MPL was not distributed
18// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
19
20#ifndef EIGEN_BDCSVD_H
21#define EIGEN_BDCSVD_H
22// #define EIGEN_BDCSVD_DEBUG_VERBOSE
23// #define EIGEN_BDCSVD_SANITY_CHECKS
24
25namespace Eigen {
26
27#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
28IOFormat bdcsvdfmt(8, 0, ", ", "\n", " [", "]");
29#endif
30
31template<typename _MatrixType> class BDCSVD;
32
33namespace internal {
34
35template<typename _MatrixType>
36struct traits<BDCSVD<_MatrixType> >
37{
38 typedef _MatrixType MatrixType;
39};
40
41} // end namespace internal
42
43
44/** \ingroup SVD_Module
45 *
46 *
47 * \class BDCSVD
48 *
49 * \brief class Bidiagonal Divide and Conquer SVD
50 *
51 * \tparam _MatrixType the type of the matrix of which we are computing the SVD decomposition
52 *
53 * This class first reduces the input matrix to bi-diagonal form using class UpperBidiagonalization,
54 * and then performs a divide-and-conquer diagonalization. Small blocks are diagonalized using class JacobiSVD.
55 * You can control the switching size with the setSwitchSize() method, default is 16.
56 * For small matrice (<16), it is thus preferable to directly use JacobiSVD. For larger ones, BDCSVD is highly
57 * recommended and can several order of magnitude faster.
58 *
59 * \warning this algorithm is unlikely to provide accurate result when compiled with unsafe math optimizations.
60 * For instance, this concerns Intel's compiler (ICC), which perfroms such optimization by default unless
61 * you compile with the \c -fp-model \c precise option. Likewise, the \c -ffast-math option of GCC or clang will
62 * significantly degrade the accuracy.
63 *
64 * \sa class JacobiSVD
65 */
66template<typename _MatrixType>
67class BDCSVD : public SVDBase<BDCSVD<_MatrixType> >
68{
69 typedef SVDBase<BDCSVD> Base;
70
71public:
72 using Base::rows;
73 using Base::cols;
74 using Base::computeU;
75 using Base::computeV;
76
77 typedef _MatrixType MatrixType;
78 typedef typename MatrixType::Scalar Scalar;
79 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
80 typedef typename NumTraits<RealScalar>::Literal Literal;
81 enum {
82 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
83 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
84 DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime, ColsAtCompileTime),
85 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
86 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
87 MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime, MaxColsAtCompileTime),
88 MatrixOptions = MatrixType::Options
89 };
90
91 typedef typename Base::MatrixUType MatrixUType;
92 typedef typename Base::MatrixVType MatrixVType;
93 typedef typename Base::SingularValuesType SingularValuesType;
94
95 typedef Matrix<Scalar, Dynamic, Dynamic, ColMajor> MatrixX;
96 typedef Matrix<RealScalar, Dynamic, Dynamic, ColMajor> MatrixXr;
97 typedef Matrix<RealScalar, Dynamic, 1> VectorType;
98 typedef Array<RealScalar, Dynamic, 1> ArrayXr;
99 typedef Array<Index,1,Dynamic> ArrayXi;
100 typedef Ref<ArrayXr> ArrayRef;
101 typedef Ref<ArrayXi> IndicesRef;
102
103 /** \brief Default Constructor.
104 *
105 * The default constructor is useful in cases in which the user intends to
106 * perform decompositions via BDCSVD::compute(const MatrixType&).
107 */
108 BDCSVD() : m_algoswap(16), m_numIters(0)
109 {}
110
111
112 /** \brief Default Constructor with memory preallocation
113 *
114 * Like the default constructor but with preallocation of the internal data
115 * according to the specified problem size.
116 * \sa BDCSVD()
117 */
118 BDCSVD(Index rows, Index cols, unsigned int computationOptions = 0)
119 : m_algoswap(16), m_numIters(0)
120 {
121 allocate(rows, cols, computationOptions);
122 }
123
124 /** \brief Constructor performing the decomposition of given matrix.
125 *
126 * \param matrix the matrix to decompose
127 * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
128 * By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU,
129 * #ComputeFullV, #ComputeThinV.
130 *
131 * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
132 * available with the (non - default) FullPivHouseholderQR preconditioner.
133 */
134 BDCSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
135 : m_algoswap(16), m_numIters(0)
136 {
137 compute(matrix, computationOptions);
138 }
139
140 ~BDCSVD()
141 {
142 }
143
144 /** \brief Method performing the decomposition of given matrix using custom options.
145 *
146 * \param matrix the matrix to decompose
147 * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
148 * By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU,
149 * #ComputeFullV, #ComputeThinV.
150 *
151 * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
152 * available with the (non - default) FullPivHouseholderQR preconditioner.
153 */
154 BDCSVD& compute(const MatrixType& matrix, unsigned int computationOptions);
155
156 /** \brief Method performing the decomposition of given matrix using current options.
157 *
158 * \param matrix the matrix to decompose
159 *
160 * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
161 */
162 BDCSVD& compute(const MatrixType& matrix)
163 {
164 return compute(matrix, this->m_computationOptions);
165 }
166
167 void setSwitchSize(int s)
168 {
169 eigen_assert(s>3 && "BDCSVD the size of the algo switch has to be greater than 3");
170 m_algoswap = s;
171 }
172
173private:
174 void allocate(Index rows, Index cols, unsigned int computationOptions);
175 void divide(Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift);
176 void computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V);
177 void computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, VectorType& singVals, ArrayRef shifts, ArrayRef mus);
178 void perturbCol0(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat);
179 void computeSingVecs(const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V);
180 void deflation43(Index firstCol, Index shift, Index i, Index size);
181 void deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size);
182 void deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift);
183 template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
184 void copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naivev);
185 void structured_update(Block<MatrixXr,Dynamic,Dynamic> A, const MatrixXr &B, Index n1);
186 static RealScalar secularEq(RealScalar x, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const ArrayRef& diagShifted, RealScalar shift);
187
188protected:
189 MatrixXr m_naiveU, m_naiveV;
190 MatrixXr m_computed;
191 Index m_nRec;
192 ArrayXr m_workspace;
193 ArrayXi m_workspaceI;
194 int m_algoswap;
195 bool m_isTranspose, m_compU, m_compV;
196
197 using Base::m_singularValues;
198 using Base::m_diagSize;
199 using Base::m_computeFullU;
200 using Base::m_computeFullV;
201 using Base::m_computeThinU;
202 using Base::m_computeThinV;
203 using Base::m_matrixU;
204 using Base::m_matrixV;
205 using Base::m_isInitialized;
206 using Base::m_nonzeroSingularValues;
207
208public:
209 int m_numIters;
210}; //end class BDCSVD
211
212
213// Method to allocate and initialize matrix and attributes
214template<typename MatrixType>
215void BDCSVD<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions)
216{
217 m_isTranspose = (cols > rows);
218
219 if (Base::allocate(rows, cols, computationOptions))
220 return;
221
222 m_computed = MatrixXr::Zero(m_diagSize + 1, m_diagSize );
223 m_compU = computeV();
224 m_compV = computeU();
225 if (m_isTranspose)
226 std::swap(m_compU, m_compV);
227
228 if (m_compU) m_naiveU = MatrixXr::Zero(m_diagSize + 1, m_diagSize + 1 );
229 else m_naiveU = MatrixXr::Zero(2, m_diagSize + 1 );
230
231 if (m_compV) m_naiveV = MatrixXr::Zero(m_diagSize, m_diagSize);
232
233 m_workspace.resize((m_diagSize+1)*(m_diagSize+1)*3);
234 m_workspaceI.resize(3*m_diagSize);
235}// end allocate
236
237template<typename MatrixType>
238BDCSVD<MatrixType>& BDCSVD<MatrixType>::compute(const MatrixType& matrix, unsigned int computationOptions)
239{
240#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
241 std::cout << "\n\n\n======================================================================================================================\n\n\n";
242#endif
243 allocate(matrix.rows(), matrix.cols(), computationOptions);
244 using std::abs;
245
246 const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)();
247
248 //**** step -1 - If the problem is too small, directly falls back to JacobiSVD and return
249 if(matrix.cols() < m_algoswap)
250 {
251 // FIXME this line involves temporaries
252 JacobiSVD<MatrixType> jsvd(matrix,computationOptions);
253 if(computeU()) m_matrixU = jsvd.matrixU();
254 if(computeV()) m_matrixV = jsvd.matrixV();
255 m_singularValues = jsvd.singularValues();
256 m_nonzeroSingularValues = jsvd.nonzeroSingularValues();
257 m_isInitialized = true;
258 return *this;
259 }
260
261 //**** step 0 - Copy the input matrix and apply scaling to reduce over/under-flows
262 RealScalar scale = matrix.cwiseAbs().maxCoeff();
263 if(scale==Literal(0)) scale = Literal(1);
264 MatrixX copy;
265 if (m_isTranspose) copy = matrix.adjoint()/scale;
266 else copy = matrix/scale;
267
268 //**** step 1 - Bidiagonalization
269 // FIXME this line involves temporaries
270 internal::UpperBidiagonalization<MatrixX> bid(copy);
271
272 //**** step 2 - Divide & Conquer
273 m_naiveU.setZero();
274 m_naiveV.setZero();
275 // FIXME this line involves a temporary matrix
276 m_computed.topRows(m_diagSize) = bid.bidiagonal().toDenseMatrix().transpose();
277 m_computed.template bottomRows<1>().setZero();
278 divide(0, m_diagSize - 1, 0, 0, 0);
279
280 //**** step 3 - Copy singular values and vectors
281 for (int i=0; i<m_diagSize; i++)
282 {
283 RealScalar a = abs(m_computed.coeff(i, i));
284 m_singularValues.coeffRef(i) = a * scale;
285 if (a<considerZero)
286 {
287 m_nonzeroSingularValues = i;
288 m_singularValues.tail(m_diagSize - i - 1).setZero();
289 break;
290 }
291 else if (i == m_diagSize - 1)
292 {
293 m_nonzeroSingularValues = i + 1;
294 break;
295 }
296 }
297
298#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
299// std::cout << "m_naiveU\n" << m_naiveU << "\n\n";
300// std::cout << "m_naiveV\n" << m_naiveV << "\n\n";
301#endif
302 if(m_isTranspose) copyUV(bid.householderV(), bid.householderU(), m_naiveV, m_naiveU);
303 else copyUV(bid.householderU(), bid.householderV(), m_naiveU, m_naiveV);
304
305 m_isInitialized = true;
306 return *this;
307}// end compute
308
309
310template<typename MatrixType>
311template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
312void BDCSVD<MatrixType>::copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naiveV)
313{
314 // Note exchange of U and V: m_matrixU is set from m_naiveV and vice versa
315 if (computeU())
316 {
317 Index Ucols = m_computeThinU ? m_diagSize : householderU.cols();
318 m_matrixU = MatrixX::Identity(householderU.cols(), Ucols);
319 m_matrixU.topLeftCorner(m_diagSize, m_diagSize) = naiveV.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize);
320 householderU.applyThisOnTheLeft(m_matrixU); // FIXME this line involves a temporary buffer
321 }
322 if (computeV())
323 {
324 Index Vcols = m_computeThinV ? m_diagSize : householderV.cols();
325 m_matrixV = MatrixX::Identity(householderV.cols(), Vcols);
326 m_matrixV.topLeftCorner(m_diagSize, m_diagSize) = naiveU.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize);
327 householderV.applyThisOnTheLeft(m_matrixV); // FIXME this line involves a temporary buffer
328 }
329}
330
331/** \internal
332 * Performs A = A * B exploiting the special structure of the matrix A. Splitting A as:
333 * A = [A1]
334 * [A2]
335 * such that A1.rows()==n1, then we assume that at least half of the columns of A1 and A2 are zeros.
336 * We can thus pack them prior to the the matrix product. However, this is only worth the effort if the matrix is large
337 * enough.
338 */
339template<typename MatrixType>
340void BDCSVD<MatrixType>::structured_update(Block<MatrixXr,Dynamic,Dynamic> A, const MatrixXr &B, Index n1)
341{
342 Index n = A.rows();
343 if(n>100)
344 {
345 // If the matrices are large enough, let's exploit the sparse structure of A by
346 // splitting it in half (wrt n1), and packing the non-zero columns.
347 Index n2 = n - n1;
348 Map<MatrixXr> A1(m_workspace.data() , n1, n);
349 Map<MatrixXr> A2(m_workspace.data()+ n1*n, n2, n);
350 Map<MatrixXr> B1(m_workspace.data()+ n*n, n, n);
351 Map<MatrixXr> B2(m_workspace.data()+2*n*n, n, n);
352 Index k1=0, k2=0;
353 for(Index j=0; j<n; ++j)
354 {
355 if( (A.col(j).head(n1).array()!=Literal(0)).any() )
356 {
357 A1.col(k1) = A.col(j).head(n1);
358 B1.row(k1) = B.row(j);
359 ++k1;
360 }
361 if( (A.col(j).tail(n2).array()!=Literal(0)).any() )
362 {
363 A2.col(k2) = A.col(j).tail(n2);
364 B2.row(k2) = B.row(j);
365 ++k2;
366 }
367 }
368
369 A.topRows(n1).noalias() = A1.leftCols(k1) * B1.topRows(k1);
370 A.bottomRows(n2).noalias() = A2.leftCols(k2) * B2.topRows(k2);
371 }
372 else
373 {
374 Map<MatrixXr,Aligned> tmp(m_workspace.data(),n,n);
375 tmp.noalias() = A*B;
376 A = tmp;
377 }
378}
379
380// The divide algorithm is done "in place", we are always working on subsets of the same matrix. The divide methods takes as argument the
381// place of the submatrix we are currently working on.
382
383//@param firstCol : The Index of the first column of the submatrix of m_computed and for m_naiveU;
384//@param lastCol : The Index of the last column of the submatrix of m_computed and for m_naiveU;
385// lastCol + 1 - firstCol is the size of the submatrix.
386//@param firstRowW : The Index of the first row of the matrix W that we are to change. (see the reference paper section 1 for more information on W)
387//@param firstRowW : Same as firstRowW with the column.
388//@param shift : Each time one takes the left submatrix, one must add 1 to the shift. Why? Because! We actually want the last column of the U submatrix
389// to become the first column (*coeff) and to shift all the other columns to the right. There are more details on the reference paper.
390template<typename MatrixType>
391void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift)
392{
393 // requires rows = cols + 1;
394 using std::pow;
395 using std::sqrt;
396 using std::abs;
397 const Index n = lastCol - firstCol + 1;
398 const Index k = n/2;
399 const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)();
400 RealScalar alphaK;
401 RealScalar betaK;
402 RealScalar r0;
403 RealScalar lambda, phi, c0, s0;
404 VectorType l, f;
405 // We use the other algorithm which is more efficient for small
406 // matrices.
407 if (n < m_algoswap)
408 {
409 // FIXME this line involves temporaries
410 JacobiSVD<MatrixXr> b(m_computed.block(firstCol, firstCol, n + 1, n), ComputeFullU | (m_compV ? ComputeFullV : 0));
411 if (m_compU)
412 m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() = b.matrixU();
413 else
414 {
415 m_naiveU.row(0).segment(firstCol, n + 1).real() = b.matrixU().row(0);
416 m_naiveU.row(1).segment(firstCol, n + 1).real() = b.matrixU().row(n);
417 }
418 if (m_compV) m_naiveV.block(firstRowW, firstColW, n, n).real() = b.matrixV();
419 m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero();
420 m_computed.diagonal().segment(firstCol + shift, n) = b.singularValues().head(n);
421 return;
422 }
423 // We use the divide and conquer algorithm
424 alphaK = m_computed(firstCol + k, firstCol + k);
425 betaK = m_computed(firstCol + k + 1, firstCol + k);
426 // The divide must be done in that order in order to have good results. Divide change the data inside the submatrices
427 // and the divide of the right submatrice reads one column of the left submatrice. That's why we need to treat the
428 // right submatrix before the left one.
429 divide(k + 1 + firstCol, lastCol, k + 1 + firstRowW, k + 1 + firstColW, shift);
430 divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1);
431
432 if (m_compU)
433 {
434 lambda = m_naiveU(firstCol + k, firstCol + k);
435 phi = m_naiveU(firstCol + k + 1, lastCol + 1);
436 }
437 else
438 {
439 lambda = m_naiveU(1, firstCol + k);
440 phi = m_naiveU(0, lastCol + 1);
441 }
442 r0 = sqrt((abs(alphaK * lambda) * abs(alphaK * lambda)) + abs(betaK * phi) * abs(betaK * phi));
443 if (m_compU)
444 {
445 l = m_naiveU.row(firstCol + k).segment(firstCol, k);
446 f = m_naiveU.row(firstCol + k + 1).segment(firstCol + k + 1, n - k - 1);
447 }
448 else
449 {
450 l = m_naiveU.row(1).segment(firstCol, k);
451 f = m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1);
452 }
453 if (m_compV) m_naiveV(firstRowW+k, firstColW) = Literal(1);
454 if (r0<considerZero)
455 {
456 c0 = Literal(1);
457 s0 = Literal(0);
458 }
459 else
460 {
461 c0 = alphaK * lambda / r0;
462 s0 = betaK * phi / r0;
463 }
464
465#ifdef EIGEN_BDCSVD_SANITY_CHECKS
466 assert(m_naiveU.allFinite());
467 assert(m_naiveV.allFinite());
468 assert(m_computed.allFinite());
469#endif
470
471 if (m_compU)
472 {
473 MatrixXr q1 (m_naiveU.col(firstCol + k).segment(firstCol, k + 1));
474 // we shiftW Q1 to the right
475 for (Index i = firstCol + k - 1; i >= firstCol; i--)
476 m_naiveU.col(i + 1).segment(firstCol, k + 1) = m_naiveU.col(i).segment(firstCol, k + 1);
477 // we shift q1 at the left with a factor c0
478 m_naiveU.col(firstCol).segment( firstCol, k + 1) = (q1 * c0);
479 // last column = q1 * - s0
480 m_naiveU.col(lastCol + 1).segment(firstCol, k + 1) = (q1 * ( - s0));
481 // first column = q2 * s0
482 m_naiveU.col(firstCol).segment(firstCol + k + 1, n - k) = m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) * s0;
483 // q2 *= c0
484 m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *= c0;
485 }
486 else
487 {
488 RealScalar q1 = m_naiveU(0, firstCol + k);
489 // we shift Q1 to the right
490 for (Index i = firstCol + k - 1; i >= firstCol; i--)
491 m_naiveU(0, i + 1) = m_naiveU(0, i);
492 // we shift q1 at the left with a factor c0
493 m_naiveU(0, firstCol) = (q1 * c0);
494 // last column = q1 * - s0
495 m_naiveU(0, lastCol + 1) = (q1 * ( - s0));
496 // first column = q2 * s0
497 m_naiveU(1, firstCol) = m_naiveU(1, lastCol + 1) *s0;
498 // q2 *= c0
499 m_naiveU(1, lastCol + 1) *= c0;
500 m_naiveU.row(1).segment(firstCol + 1, k).setZero();
501 m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1).setZero();
502 }
503
504#ifdef EIGEN_BDCSVD_SANITY_CHECKS
505 assert(m_naiveU.allFinite());
506 assert(m_naiveV.allFinite());
507 assert(m_computed.allFinite());
508#endif
509
510 m_computed(firstCol + shift, firstCol + shift) = r0;
511 m_computed.col(firstCol + shift).segment(firstCol + shift + 1, k) = alphaK * l.transpose().real();
512 m_computed.col(firstCol + shift).segment(firstCol + shift + k + 1, n - k - 1) = betaK * f.transpose().real();
513
514#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
515 ArrayXr tmp1 = (m_computed.block(firstCol+shift, firstCol+shift, n, n)).jacobiSvd().singularValues();
516#endif
517 // Second part: try to deflate singular values in combined matrix
518 deflation(firstCol, lastCol, k, firstRowW, firstColW, shift);
519#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
520 ArrayXr tmp2 = (m_computed.block(firstCol+shift, firstCol+shift, n, n)).jacobiSvd().singularValues();
521 std::cout << "\n\nj1 = " << tmp1.transpose().format(bdcsvdfmt) << "\n";
522 std::cout << "j2 = " << tmp2.transpose().format(bdcsvdfmt) << "\n\n";
523 std::cout << "err: " << ((tmp1-tmp2).abs()>1e-12*tmp2.abs()).transpose() << "\n";
524 static int count = 0;
525 std::cout << "# " << ++count << "\n\n";
526 assert((tmp1-tmp2).matrix().norm() < 1e-14*tmp2.matrix().norm());
527// assert(count<681);
528// assert(((tmp1-tmp2).abs()<1e-13*tmp2.abs()).all());
529#endif
530
531 // Third part: compute SVD of combined matrix
532 MatrixXr UofSVD, VofSVD;
533 VectorType singVals;
534 computeSVDofM(firstCol + shift, n, UofSVD, singVals, VofSVD);
535
536#ifdef EIGEN_BDCSVD_SANITY_CHECKS
537 assert(UofSVD.allFinite());
538 assert(VofSVD.allFinite());
539#endif
540
541 if (m_compU)
542 structured_update(m_naiveU.block(firstCol, firstCol, n + 1, n + 1), UofSVD, (n+2)/2);
543 else
544 {
545 Map<Matrix<RealScalar,2,Dynamic>,Aligned> tmp(m_workspace.data(),2,n+1);
546 tmp.noalias() = m_naiveU.middleCols(firstCol, n+1) * UofSVD;
547 m_naiveU.middleCols(firstCol, n + 1) = tmp;
548 }
549
550 if (m_compV) structured_update(m_naiveV.block(firstRowW, firstColW, n, n), VofSVD, (n+1)/2);
551
552#ifdef EIGEN_BDCSVD_SANITY_CHECKS
553 assert(m_naiveU.allFinite());
554 assert(m_naiveV.allFinite());
555 assert(m_computed.allFinite());
556#endif
557
558 m_computed.block(firstCol + shift, firstCol + shift, n, n).setZero();
559 m_computed.block(firstCol + shift, firstCol + shift, n, n).diagonal() = singVals;
560}// end divide
561
562// Compute SVD of m_computed.block(firstCol, firstCol, n + 1, n); this block only has non-zeros in
563// the first column and on the diagonal and has undergone deflation, so diagonal is in increasing
564// order except for possibly the (0,0) entry. The computed SVD is stored U, singVals and V, except
565// that if m_compV is false, then V is not computed. Singular values are sorted in decreasing order.
566//
567// TODO Opportunities for optimization: better root finding algo, better stopping criterion, better
568// handling of round-off errors, be consistent in ordering
569// For instance, to solve the secular equation using FMM, see http://www.stat.uchicago.edu/~lekheng/courses/302/classics/greengard-rokhlin.pdf
570template <typename MatrixType>
571void BDCSVD<MatrixType>::computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V)
572{
573 const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)();
574 using std::abs;
575 ArrayRef col0 = m_computed.col(firstCol).segment(firstCol, n);
576 m_workspace.head(n) = m_computed.block(firstCol, firstCol, n, n).diagonal();
577 ArrayRef diag = m_workspace.head(n);
578 diag(0) = Literal(0);
579
580 // Allocate space for singular values and vectors
581 singVals.resize(n);
582 U.resize(n+1, n+1);
583 if (m_compV) V.resize(n, n);
584
585#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
586 if (col0.hasNaN() || diag.hasNaN())
587 std::cout << "\n\nHAS NAN\n\n";
588#endif
589
590 // Many singular values might have been deflated, the zero ones have been moved to the end,
591 // but others are interleaved and we must ignore them at this stage.
592 // To this end, let's compute a permutation skipping them:
593 Index actual_n = n;
594 while(actual_n>1 && diag(actual_n-1)==Literal(0)) --actual_n;
595 Index m = 0; // size of the deflated problem
596 for(Index k=0;k<actual_n;++k)
597 if(abs(col0(k))>considerZero)
598 m_workspaceI(m++) = k;
599 Map<ArrayXi> perm(m_workspaceI.data(),m);
600
601 Map<ArrayXr> shifts(m_workspace.data()+1*n, n);
602 Map<ArrayXr> mus(m_workspace.data()+2*n, n);
603 Map<ArrayXr> zhat(m_workspace.data()+3*n, n);
604
605#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
606 std::cout << "computeSVDofM using:\n";
607 std::cout << " z: " << col0.transpose() << "\n";
608 std::cout << " d: " << diag.transpose() << "\n";
609#endif
610
611 // Compute singVals, shifts, and mus
612 computeSingVals(col0, diag, perm, singVals, shifts, mus);
613
614#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
615 std::cout << " j: " << (m_computed.block(firstCol, firstCol, n, n)).jacobiSvd().singularValues().transpose().reverse() << "\n\n";
616 std::cout << " sing-val: " << singVals.transpose() << "\n";
617 std::cout << " mu: " << mus.transpose() << "\n";
618 std::cout << " shift: " << shifts.transpose() << "\n";
619
620 {
621 Index actual_n = n;
622 while(actual_n>1 && abs(col0(actual_n-1))<considerZero) --actual_n;
623 std::cout << "\n\n mus: " << mus.head(actual_n).transpose() << "\n\n";
624 std::cout << " check1 (expect0) : " << ((singVals.array()-(shifts+mus)) / singVals.array()).head(actual_n).transpose() << "\n\n";
625 std::cout << " check2 (>0) : " << ((singVals.array()-diag) / singVals.array()).head(actual_n).transpose() << "\n\n";
626 std::cout << " check3 (>0) : " << ((diag.segment(1,actual_n-1)-singVals.head(actual_n-1).array()) / singVals.head(actual_n-1).array()).transpose() << "\n\n\n";
627 std::cout << " check4 (>0) : " << ((singVals.segment(1,actual_n-1)-singVals.head(actual_n-1))).transpose() << "\n\n\n";
628 }
629#endif
630
631#ifdef EIGEN_BDCSVD_SANITY_CHECKS
632 assert(singVals.allFinite());
633 assert(mus.allFinite());
634 assert(shifts.allFinite());
635#endif
636
637 // Compute zhat
638 perturbCol0(col0, diag, perm, singVals, shifts, mus, zhat);
639#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
640 std::cout << " zhat: " << zhat.transpose() << "\n";
641#endif
642
643#ifdef EIGEN_BDCSVD_SANITY_CHECKS
644 assert(zhat.allFinite());
645#endif
646
647 computeSingVecs(zhat, diag, perm, singVals, shifts, mus, U, V);
648
649#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
650 std::cout << "U^T U: " << (U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() << "\n";
651 std::cout << "V^T V: " << (V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() << "\n";
652#endif
653
654#ifdef EIGEN_BDCSVD_SANITY_CHECKS
655 assert(U.allFinite());
656 assert(V.allFinite());
657 assert((U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() < 1e-14 * n);
658 assert((V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() < 1e-14 * n);
659 assert(m_naiveU.allFinite());
660 assert(m_naiveV.allFinite());
661 assert(m_computed.allFinite());
662#endif
663
664 // Because of deflation, the singular values might not be completely sorted.
665 // Fortunately, reordering them is a O(n) problem
666 for(Index i=0; i<actual_n-1; ++i)
667 {
668 if(singVals(i)>singVals(i+1))
669 {
670 using std::swap;
671 swap(singVals(i),singVals(i+1));
672 U.col(i).swap(U.col(i+1));
673 if(m_compV) V.col(i).swap(V.col(i+1));
674 }
675 }
676
677 // Reverse order so that singular values in increased order
678 // Because of deflation, the zeros singular-values are already at the end
679 singVals.head(actual_n).reverseInPlace();
680 U.leftCols(actual_n).rowwise().reverseInPlace();
681 if (m_compV) V.leftCols(actual_n).rowwise().reverseInPlace();
682
683#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
684 JacobiSVD<MatrixXr> jsvd(m_computed.block(firstCol, firstCol, n, n) );
685 std::cout << " * j: " << jsvd.singularValues().transpose() << "\n\n";
686 std::cout << " * sing-val: " << singVals.transpose() << "\n";
687// std::cout << " * err: " << ((jsvd.singularValues()-singVals)>1e-13*singVals.norm()).transpose() << "\n";
688#endif
689}
690
691template <typename MatrixType>
692typename BDCSVD<MatrixType>::RealScalar BDCSVD<MatrixType>::secularEq(RealScalar mu, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const ArrayRef& diagShifted, RealScalar shift)
693{
694 Index m = perm.size();
695 RealScalar res = Literal(1);
696 for(Index i=0; i<m; ++i)
697 {
698 Index j = perm(i);
699 // The following expression could be rewritten to involve only a single division,
700 // but this would make the expression more sensitive to overflow.
701 res += (col0(j) / (diagShifted(j) - mu)) * (col0(j) / (diag(j) + shift + mu));
702 }
703 return res;
704
705}
706
707template <typename MatrixType>
708void BDCSVD<MatrixType>::computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm,
709 VectorType& singVals, ArrayRef shifts, ArrayRef mus)
710{
711 using std::abs;
712 using std::swap;
713 using std::sqrt;
714
715 Index n = col0.size();
716 Index actual_n = n;
717 // Note that here actual_n is computed based on col0(i)==0 instead of diag(i)==0 as above
718 // because 1) we have diag(i)==0 => col0(i)==0 and 2) if col0(i)==0, then diag(i) is already a singular value.
719 while(actual_n>1 && col0(actual_n-1)==Literal(0)) --actual_n;
720
721 for (Index k = 0; k < n; ++k)
722 {
723 if (col0(k) == Literal(0) || actual_n==1)
724 {
725 // if col0(k) == 0, then entry is deflated, so singular value is on diagonal
726 // if actual_n==1, then the deflated problem is already diagonalized
727 singVals(k) = k==0 ? col0(0) : diag(k);
728 mus(k) = Literal(0);
729 shifts(k) = k==0 ? col0(0) : diag(k);
730 continue;
731 }
732
733 // otherwise, use secular equation to find singular value
734 RealScalar left = diag(k);
735 RealScalar right; // was: = (k != actual_n-1) ? diag(k+1) : (diag(actual_n-1) + col0.matrix().norm());
736 if(k==actual_n-1)
737 right = (diag(actual_n-1) + col0.matrix().norm());
738 else
739 {
740 // Skip deflated singular values,
741 // recall that at this stage we assume that z[j]!=0 and all entries for which z[j]==0 have been put aside.
742 // This should be equivalent to using perm[]
743 Index l = k+1;
744 while(col0(l)==Literal(0)) { ++l; eigen_internal_assert(l<actual_n); }
745 right = diag(l);
746 }
747
748 // first decide whether it's closer to the left end or the right end
749 RealScalar mid = left + (right-left) / Literal(2);
750 RealScalar fMid = secularEq(mid, col0, diag, perm, diag, Literal(0));
751#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
752 std::cout << right-left << "\n";
753 std::cout << "fMid = " << fMid << " " << secularEq(mid-left, col0, diag, perm, diag-left, left) << " " << secularEq(mid-right, col0, diag, perm, diag-right, right) << "\n";
754 std::cout << " = " << secularEq(0.1*(left+right), col0, diag, perm, diag, 0)
755 << " " << secularEq(0.2*(left+right), col0, diag, perm, diag, 0)
756 << " " << secularEq(0.3*(left+right), col0, diag, perm, diag, 0)
757 << " " << secularEq(0.4*(left+right), col0, diag, perm, diag, 0)
758 << " " << secularEq(0.49*(left+right), col0, diag, perm, diag, 0)
759 << " " << secularEq(0.5*(left+right), col0, diag, perm, diag, 0)
760 << " " << secularEq(0.51*(left+right), col0, diag, perm, diag, 0)
761 << " " << secularEq(0.6*(left+right), col0, diag, perm, diag, 0)
762 << " " << secularEq(0.7*(left+right), col0, diag, perm, diag, 0)
763 << " " << secularEq(0.8*(left+right), col0, diag, perm, diag, 0)
764 << " " << secularEq(0.9*(left+right), col0, diag, perm, diag, 0) << "\n";
765#endif
766 RealScalar shift = (k == actual_n-1 || fMid > Literal(0)) ? left : right;
767
768 // measure everything relative to shift
769 Map<ArrayXr> diagShifted(m_workspace.data()+4*n, n);
770 diagShifted = diag - shift;
771
772 // initial guess
773 RealScalar muPrev, muCur;
774 if (shift == left)
775 {
776 muPrev = (right - left) * RealScalar(0.1);
777 if (k == actual_n-1) muCur = right - left;
778 else muCur = (right - left) * RealScalar(0.5);
779 }
780 else
781 {
782 muPrev = -(right - left) * RealScalar(0.1);
783 muCur = -(right - left) * RealScalar(0.5);
784 }
785
786 RealScalar fPrev = secularEq(muPrev, col0, diag, perm, diagShifted, shift);
787 RealScalar fCur = secularEq(muCur, col0, diag, perm, diagShifted, shift);
788 if (abs(fPrev) < abs(fCur))
789 {
790 swap(fPrev, fCur);
791 swap(muPrev, muCur);
792 }
793
794 // rational interpolation: fit a function of the form a / mu + b through the two previous
795 // iterates and use its zero to compute the next iterate
796 bool useBisection = fPrev*fCur>Literal(0);
797 while (fCur!=Literal(0) && abs(muCur - muPrev) > Literal(8) * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(muCur), abs(muPrev)) && abs(fCur - fPrev)>NumTraits<RealScalar>::epsilon() && !useBisection)
798 {
799 ++m_numIters;
800
801 // Find a and b such that the function f(mu) = a / mu + b matches the current and previous samples.
802 RealScalar a = (fCur - fPrev) / (Literal(1)/muCur - Literal(1)/muPrev);
803 RealScalar b = fCur - a / muCur;
804 // And find mu such that f(mu)==0:
805 RealScalar muZero = -a/b;
806 RealScalar fZero = secularEq(muZero, col0, diag, perm, diagShifted, shift);
807
808 muPrev = muCur;
809 fPrev = fCur;
810 muCur = muZero;
811 fCur = fZero;
812
813
814 if (shift == left && (muCur < Literal(0) || muCur > right - left)) useBisection = true;
815 if (shift == right && (muCur < -(right - left) || muCur > Literal(0))) useBisection = true;
816 if (abs(fCur)>abs(fPrev)) useBisection = true;
817 }
818
819 // fall back on bisection method if rational interpolation did not work
820 if (useBisection)
821 {
822#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
823 std::cout << "useBisection for k = " << k << ", actual_n = " << actual_n << "\n";
824#endif
825 RealScalar leftShifted, rightShifted;
826 if (shift == left)
827 {
828 // to avoid overflow, we must have mu > max(real_min, |z(k)|/sqrt(real_max)),
829 // the factor 2 is to be more conservative
830 leftShifted = numext::maxi<RealScalar>( (std::numeric_limits<RealScalar>::min)(), Literal(2) * abs(col0(k)) / sqrt((std::numeric_limits<RealScalar>::max)()) );
831
832 // check that we did it right:
833 eigen_internal_assert( (numext::isfinite)( (col0(k)/leftShifted)*(col0(k)/(diag(k)+shift+leftShifted)) ) );
834 // I don't understand why the case k==0 would be special there:
835 // if (k == 0) rightShifted = right - left; else
836 rightShifted = (k==actual_n-1) ? right : ((right - left) * RealScalar(0.51)); // theoretically we can take 0.5, but let's be safe
837 }
838 else
839 {
840 leftShifted = -(right - left) * RealScalar(0.51);
841 if(k+1<n)
842 rightShifted = -numext::maxi<RealScalar>( (std::numeric_limits<RealScalar>::min)(), abs(col0(k+1)) / sqrt((std::numeric_limits<RealScalar>::max)()) );
843 else
844 rightShifted = -(std::numeric_limits<RealScalar>::min)();
845 }
846
847 RealScalar fLeft = secularEq(leftShifted, col0, diag, perm, diagShifted, shift);
848
849#if defined EIGEN_INTERNAL_DEBUGGING || defined EIGEN_BDCSVD_DEBUG_VERBOSE
850 RealScalar fRight = secularEq(rightShifted, col0, diag, perm, diagShifted, shift);
851#endif
852
853#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
854 if(!(fLeft * fRight<0))
855 {
856 std::cout << "fLeft: " << leftShifted << " - " << diagShifted.head(10).transpose() << "\n ; " << bool(left==shift) << " " << (left-shift) << "\n";
857 std::cout << k << " : " << fLeft << " * " << fRight << " == " << fLeft * fRight << " ; " << left << " - " << right << " -> " << leftShifted << " " << rightShifted << " shift=" << shift << "\n";
858 }
859#endif
860 eigen_internal_assert(fLeft * fRight < Literal(0));
861
862 while (rightShifted - leftShifted > Literal(2) * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(leftShifted), abs(rightShifted)))
863 {
864 RealScalar midShifted = (leftShifted + rightShifted) / Literal(2);
865 fMid = secularEq(midShifted, col0, diag, perm, diagShifted, shift);
866 if (fLeft * fMid < Literal(0))
867 {
868 rightShifted = midShifted;
869 }
870 else
871 {
872 leftShifted = midShifted;
873 fLeft = fMid;
874 }
875 }
876
877 muCur = (leftShifted + rightShifted) / Literal(2);
878 }
879
880 singVals[k] = shift + muCur;
881 shifts[k] = shift;
882 mus[k] = muCur;
883
884 // perturb singular value slightly if it equals diagonal entry to avoid division by zero later
885 // (deflation is supposed to avoid this from happening)
886 // - this does no seem to be necessary anymore -
887// if (singVals[k] == left) singVals[k] *= 1 + NumTraits<RealScalar>::epsilon();
888// if (singVals[k] == right) singVals[k] *= 1 - NumTraits<RealScalar>::epsilon();
889 }
890}
891
892
893// zhat is perturbation of col0 for which singular vectors can be computed stably (see Section 3.1)
894template <typename MatrixType>
895void BDCSVD<MatrixType>::perturbCol0
896 (const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const VectorType& singVals,
897 const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat)
898{
899 using std::sqrt;
900 Index n = col0.size();
901 Index m = perm.size();
902 if(m==0)
903 {
904 zhat.setZero();
905 return;
906 }
907 Index last = perm(m-1);
908 // The offset permits to skip deflated entries while computing zhat
909 for (Index k = 0; k < n; ++k)
910 {
911 if (col0(k) == Literal(0)) // deflated
912 zhat(k) = Literal(0);
913 else
914 {
915 // see equation (3.6)
916 RealScalar dk = diag(k);
917 RealScalar prod = (singVals(last) + dk) * (mus(last) + (shifts(last) - dk));
918
919 for(Index l = 0; l<m; ++l)
920 {
921 Index i = perm(l);
922 if(i!=k)
923 {
924 Index j = i<k ? i : perm(l-1);
925 prod *= ((singVals(j)+dk) / ((diag(i)+dk))) * ((mus(j)+(shifts(j)-dk)) / ((diag(i)-dk)));
926#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
927 if(i!=k && std::abs(((singVals(j)+dk)*(mus(j)+(shifts(j)-dk)))/((diag(i)+dk)*(diag(i)-dk)) - 1) > 0.9 )
928 std::cout << " " << ((singVals(j)+dk)*(mus(j)+(shifts(j)-dk)))/((diag(i)+dk)*(diag(i)-dk)) << " == (" << (singVals(j)+dk) << " * " << (mus(j)+(shifts(j)-dk))
929 << ") / (" << (diag(i)+dk) << " * " << (diag(i)-dk) << ")\n";
930#endif
931 }
932 }
933#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
934 std::cout << "zhat(" << k << ") = sqrt( " << prod << ") ; " << (singVals(last) + dk) << " * " << mus(last) + shifts(last) << " - " << dk << "\n";
935#endif
936 RealScalar tmp = sqrt(prod);
937 zhat(k) = col0(k) > Literal(0) ? tmp : -tmp;
938 }
939 }
940}
941
942// compute singular vectors
943template <typename MatrixType>
944void BDCSVD<MatrixType>::computeSingVecs
945 (const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef &perm, const VectorType& singVals,
946 const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V)
947{
948 Index n = zhat.size();
949 Index m = perm.size();
950
951 for (Index k = 0; k < n; ++k)
952 {
953 if (zhat(k) == Literal(0))
954 {
955 U.col(k) = VectorType::Unit(n+1, k);
956 if (m_compV) V.col(k) = VectorType::Unit(n, k);
957 }
958 else
959 {
960 U.col(k).setZero();
961 for(Index l=0;l<m;++l)
962 {
963 Index i = perm(l);
964 U(i,k) = zhat(i)/(((diag(i) - shifts(k)) - mus(k)) )/( (diag(i) + singVals[k]));
965 }
966 U(n,k) = Literal(0);
967 U.col(k).normalize();
968
969 if (m_compV)
970 {
971 V.col(k).setZero();
972 for(Index l=1;l<m;++l)
973 {
974 Index i = perm(l);
975 V(i,k) = diag(i) * zhat(i) / (((diag(i) - shifts(k)) - mus(k)) )/( (diag(i) + singVals[k]));
976 }
977 V(0,k) = Literal(-1);
978 V.col(k).normalize();
979 }
980 }
981 }
982 U.col(n) = VectorType::Unit(n+1, n);
983}
984
985
986// page 12_13
987// i >= 1, di almost null and zi non null.
988// We use a rotation to zero out zi applied to the left of M
989template <typename MatrixType>
990void BDCSVD<MatrixType>::deflation43(Index firstCol, Index shift, Index i, Index size)
991{
992 using std::abs;
993 using std::sqrt;
994 using std::pow;
995 Index start = firstCol + shift;
996 RealScalar c = m_computed(start, start);
997 RealScalar s = m_computed(start+i, start);
998 RealScalar r = numext::hypot(c,s);
999 if (r == Literal(0))
1000 {
1001 m_computed(start+i, start+i) = Literal(0);
1002 return;
1003 }
1004 m_computed(start,start) = r;
1005 m_computed(start+i, start) = Literal(0);
1006 m_computed(start+i, start+i) = Literal(0);
1007
1008 JacobiRotation<RealScalar> J(c/r,-s/r);
1009 if (m_compU) m_naiveU.middleRows(firstCol, size+1).applyOnTheRight(firstCol, firstCol+i, J);
1010 else m_naiveU.applyOnTheRight(firstCol, firstCol+i, J);
1011}// end deflation 43
1012
1013
1014// page 13
1015// i,j >= 1, i!=j and |di - dj| < epsilon * norm2(M)
1016// We apply two rotations to have zj = 0;
1017// TODO deflation44 is still broken and not properly tested
1018template <typename MatrixType>
1019void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size)
1020{
1021 using std::abs;
1022 using std::sqrt;
1023 using std::conj;
1024 using std::pow;
1025 RealScalar c = m_computed(firstColm+i, firstColm);
1026 RealScalar s = m_computed(firstColm+j, firstColm);
1027 RealScalar r = sqrt(numext::abs2(c) + numext::abs2(s));
1028#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1029 std::cout << "deflation 4.4: " << i << "," << j << " -> " << c << " " << s << " " << r << " ; "
1030 << m_computed(firstColm + i-1, firstColm) << " "
1031 << m_computed(firstColm + i, firstColm) << " "
1032 << m_computed(firstColm + i+1, firstColm) << " "
1033 << m_computed(firstColm + i+2, firstColm) << "\n";
1034 std::cout << m_computed(firstColm + i-1, firstColm + i-1) << " "
1035 << m_computed(firstColm + i, firstColm+i) << " "
1036 << m_computed(firstColm + i+1, firstColm+i+1) << " "
1037 << m_computed(firstColm + i+2, firstColm+i+2) << "\n";
1038#endif
1039 if (r==Literal(0))
1040 {
1041 m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j);
1042 return;
1043 }
1044 c/=r;
1045 s/=r;
1046 m_computed(firstColm + i, firstColm) = r;
1047 m_computed(firstColm + j, firstColm + j) = m_computed(firstColm + i, firstColm + i);
1048 m_computed(firstColm + j, firstColm) = Literal(0);
1049
1050 JacobiRotation<RealScalar> J(c,-s);
1051 if (m_compU) m_naiveU.middleRows(firstColu, size+1).applyOnTheRight(firstColu + i, firstColu + j, J);
1052 else m_naiveU.applyOnTheRight(firstColu+i, firstColu+j, J);
1053 if (m_compV) m_naiveV.middleRows(firstRowW, size).applyOnTheRight(firstColW + i, firstColW + j, J);
1054}// end deflation 44
1055
1056
1057// acts on block from (firstCol+shift, firstCol+shift) to (lastCol+shift, lastCol+shift) [inclusive]
1058template <typename MatrixType>
1059void BDCSVD<MatrixType>::deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift)
1060{
1061 using std::sqrt;
1062 using std::abs;
1063 const Index length = lastCol + 1 - firstCol;
1064
1065 Block<MatrixXr,Dynamic,1> col0(m_computed, firstCol+shift, firstCol+shift, length, 1);
1066 Diagonal<MatrixXr> fulldiag(m_computed);
1067 VectorBlock<Diagonal<MatrixXr>,Dynamic> diag(fulldiag, firstCol+shift, length);
1068
1069 const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)();
1070 RealScalar maxDiag = diag.tail((std::max)(Index(1),length-1)).cwiseAbs().maxCoeff();
1071 RealScalar epsilon_strict = numext::maxi<RealScalar>(considerZero,NumTraits<RealScalar>::epsilon() * maxDiag);
1072 RealScalar epsilon_coarse = Literal(8) * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(col0.cwiseAbs().maxCoeff(), maxDiag);
1073
1074#ifdef EIGEN_BDCSVD_SANITY_CHECKS
1075 assert(m_naiveU.allFinite());
1076 assert(m_naiveV.allFinite());
1077 assert(m_computed.allFinite());
1078#endif
1079
1080#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1081 std::cout << "\ndeflate:" << diag.head(k+1).transpose() << " | " << diag.segment(k+1,length-k-1).transpose() << "\n";
1082#endif
1083
1084 //condition 4.1
1085 if (diag(0) < epsilon_coarse)
1086 {
1087#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1088 std::cout << "deflation 4.1, because " << diag(0) << " < " << epsilon_coarse << "\n";
1089#endif
1090 diag(0) = epsilon_coarse;
1091 }
1092
1093 //condition 4.2
1094 for (Index i=1;i<length;++i)
1095 if (abs(col0(i)) < epsilon_strict)
1096 {
1097#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1098 std::cout << "deflation 4.2, set z(" << i << ") to zero because " << abs(col0(i)) << " < " << epsilon_strict << " (diag(" << i << ")=" << diag(i) << ")\n";
1099#endif
1100 col0(i) = Literal(0);
1101 }
1102
1103 //condition 4.3
1104 for (Index i=1;i<length; i++)
1105 if (diag(i) < epsilon_coarse)
1106 {
1107#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1108 std::cout << "deflation 4.3, cancel z(" << i << ")=" << col0(i) << " because diag(" << i << ")=" << diag(i) << " < " << epsilon_coarse << "\n";
1109#endif
1110 deflation43(firstCol, shift, i, length);
1111 }
1112
1113#ifdef EIGEN_BDCSVD_SANITY_CHECKS
1114 assert(m_naiveU.allFinite());
1115 assert(m_naiveV.allFinite());
1116 assert(m_computed.allFinite());
1117#endif
1118#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1119 std::cout << "to be sorted: " << diag.transpose() << "\n\n";
1120#endif
1121 {
1122 // Check for total deflation
1123 // If we have a total deflation, then we have to consider col0(0)==diag(0) as a singular value during sorting
1124 bool total_deflation = (col0.tail(length-1).array()<considerZero).all();
1125
1126 // Sort the diagonal entries, since diag(1:k-1) and diag(k:length) are already sorted, let's do a sorted merge.
1127 // First, compute the respective permutation.
1128 Index *permutation = m_workspaceI.data();
1129 {
1130 permutation[0] = 0;
1131 Index p = 1;
1132
1133 // Move deflated diagonal entries at the end.
1134 for(Index i=1; i<length; ++i)
1135 if(abs(diag(i))<considerZero)
1136 permutation[p++] = i;
1137
1138 Index i=1, j=k+1;
1139 for( ; p < length; ++p)
1140 {
1141 if (i > k) permutation[p] = j++;
1142 else if (j >= length) permutation[p] = i++;
1143 else if (diag(i) < diag(j)) permutation[p] = j++;
1144 else permutation[p] = i++;
1145 }
1146 }
1147
1148 // If we have a total deflation, then we have to insert diag(0) at the right place
1149 if(total_deflation)
1150 {
1151 for(Index i=1; i<length; ++i)
1152 {
1153 Index pi = permutation[i];
1154 if(abs(diag(pi))<considerZero || diag(0)<diag(pi))
1155 permutation[i-1] = permutation[i];
1156 else
1157 {
1158 permutation[i-1] = 0;
1159 break;
1160 }
1161 }
1162 }
1163
1164 // Current index of each col, and current column of each index
1165 Index *realInd = m_workspaceI.data()+length;
1166 Index *realCol = m_workspaceI.data()+2*length;
1167
1168 for(int pos = 0; pos< length; pos++)
1169 {
1170 realCol[pos] = pos;
1171 realInd[pos] = pos;
1172 }
1173
1174 for(Index i = total_deflation?0:1; i < length; i++)
1175 {
1176 const Index pi = permutation[length - (total_deflation ? i+1 : i)];
1177 const Index J = realCol[pi];
1178
1179 using std::swap;
1180 // swap diagonal and first column entries:
1181 swap(diag(i), diag(J));
1182 if(i!=0 && J!=0) swap(col0(i), col0(J));
1183
1184 // change columns
1185 if (m_compU) m_naiveU.col(firstCol+i).segment(firstCol, length + 1).swap(m_naiveU.col(firstCol+J).segment(firstCol, length + 1));
1186 else m_naiveU.col(firstCol+i).segment(0, 2) .swap(m_naiveU.col(firstCol+J).segment(0, 2));
1187 if (m_compV) m_naiveV.col(firstColW + i).segment(firstRowW, length).swap(m_naiveV.col(firstColW + J).segment(firstRowW, length));
1188
1189 //update real pos
1190 const Index realI = realInd[i];
1191 realCol[realI] = J;
1192 realCol[pi] = i;
1193 realInd[J] = realI;
1194 realInd[i] = pi;
1195 }
1196 }
1197#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1198 std::cout << "sorted: " << diag.transpose().format(bdcsvdfmt) << "\n";
1199 std::cout << " : " << col0.transpose() << "\n\n";
1200#endif
1201
1202 //condition 4.4
1203 {
1204 Index i = length-1;
1205 while(i>0 && (abs(diag(i))<considerZero || abs(col0(i))<considerZero)) --i;
1206 for(; i>1;--i)
1207 if( (diag(i) - diag(i-1)) < NumTraits<RealScalar>::epsilon()*maxDiag )
1208 {
1209#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
1210 std::cout << "deflation 4.4 with i = " << i << " because " << (diag(i) - diag(i-1)) << " < " << NumTraits<RealScalar>::epsilon()*diag(i) << "\n";
1211#endif
1212 eigen_internal_assert(abs(diag(i) - diag(i-1))<epsilon_coarse && " diagonal entries are not properly sorted");
1213 deflation44(firstCol, firstCol + shift, firstRowW, firstColW, i-1, i, length);
1214 }
1215 }
1216
1217#ifdef EIGEN_BDCSVD_SANITY_CHECKS
1218 for(Index j=2;j<length;++j)
1219 assert(diag(j-1)<=diag(j) || abs(diag(j))<considerZero);
1220#endif
1221
1222#ifdef EIGEN_BDCSVD_SANITY_CHECKS
1223 assert(m_naiveU.allFinite());
1224 assert(m_naiveV.allFinite());
1225 assert(m_computed.allFinite());
1226#endif
1227}//end deflation
1228
1229#ifndef __CUDACC__
1230/** \svd_module
1231 *
1232 * \return the singular value decomposition of \c *this computed by Divide & Conquer algorithm
1233 *
1234 * \sa class BDCSVD
1235 */
1236template<typename Derived>
1237BDCSVD<typename MatrixBase<Derived>::PlainObject>
1238MatrixBase<Derived>::bdcSvd(unsigned int computationOptions) const
1239{
1240 return BDCSVD<PlainObject>(*this, computationOptions);
1241}
1242#endif
1243
1244} // end namespace Eigen
1245
1246#endif
1247