source: pacpussensors/trunk/Vislab/lib3dv/eigen/unsupported/Eigen/src/SVD/JacobiSVD.h@ 136

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1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
5//
6// This Source Code Form is subject to the terms of the Mozilla
7// Public License v. 2.0. If a copy of the MPL was not distributed
8// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9
10#ifndef EIGEN_JACOBISVD_H
11#define EIGEN_JACOBISVD_H
12
13namespace Eigen {
14
15namespace internal {
16// forward declaration (needed by ICC)
17// the empty body is required by MSVC
18template<typename MatrixType, int QRPreconditioner,
19 bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
20struct svd_precondition_2x2_block_to_be_real {};
21
22/*** QR preconditioners (R-SVD)
23 ***
24 *** Their role is to reduce the problem of computing the SVD to the case of a square matrix.
25 *** This approach, known as R-SVD, is an optimization for rectangular-enough matrices, and is a requirement for
26 *** JacobiSVD which by itself is only able to work on square matrices.
27 ***/
28
29enum { PreconditionIfMoreColsThanRows, PreconditionIfMoreRowsThanCols };
30
31template<typename MatrixType, int QRPreconditioner, int Case>
32struct qr_preconditioner_should_do_anything
33{
34 enum { a = MatrixType::RowsAtCompileTime != Dynamic &&
35 MatrixType::ColsAtCompileTime != Dynamic &&
36 MatrixType::ColsAtCompileTime <= MatrixType::RowsAtCompileTime,
37 b = MatrixType::RowsAtCompileTime != Dynamic &&
38 MatrixType::ColsAtCompileTime != Dynamic &&
39 MatrixType::RowsAtCompileTime <= MatrixType::ColsAtCompileTime,
40 ret = !( (QRPreconditioner == NoQRPreconditioner) ||
41 (Case == PreconditionIfMoreColsThanRows && bool(a)) ||
42 (Case == PreconditionIfMoreRowsThanCols && bool(b)) )
43 };
44};
45
46template<typename MatrixType, int QRPreconditioner, int Case,
47 bool DoAnything = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret
48> struct qr_preconditioner_impl {};
49
50template<typename MatrixType, int QRPreconditioner, int Case>
51class qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false>
52{
53public:
54 typedef typename MatrixType::Index Index;
55 void allocate(const JacobiSVD<MatrixType, QRPreconditioner>&) {}
56 bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&)
57 {
58 return false;
59 }
60};
61
62/*** preconditioner using FullPivHouseholderQR ***/
63
64template<typename MatrixType>
65class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
66{
67public:
68 typedef typename MatrixType::Index Index;
69 typedef typename MatrixType::Scalar Scalar;
70 enum
71 {
72 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
73 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
74 };
75 typedef Matrix<Scalar, 1, RowsAtCompileTime, RowMajor, 1, MaxRowsAtCompileTime> WorkspaceType;
76
77 void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
78 {
79 if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
80 {
81 m_qr.~QRType();
82 ::new (&m_qr) QRType(svd.rows(), svd.cols());
83 }
84 if (svd.m_computeFullU) m_workspace.resize(svd.rows());
85 }
86
87 bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
88 {
89 if(matrix.rows() > matrix.cols())
90 {
91 m_qr.compute(matrix);
92 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
93 if(svd.m_computeFullU) m_qr.matrixQ().evalTo(svd.m_matrixU, m_workspace);
94 if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
95 return true;
96 }
97 return false;
98 }
99private:
100 typedef FullPivHouseholderQR<MatrixType> QRType;
101 QRType m_qr;
102 WorkspaceType m_workspace;
103};
104
105template<typename MatrixType>
106class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
107{
108public:
109 typedef typename MatrixType::Index Index;
110 typedef typename MatrixType::Scalar Scalar;
111 enum
112 {
113 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
114 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
115 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
116 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
117 Options = MatrixType::Options
118 };
119 typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
120 TransposeTypeWithSameStorageOrder;
121
122 void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
123 {
124 if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
125 {
126 m_qr.~QRType();
127 ::new (&m_qr) QRType(svd.cols(), svd.rows());
128 }
129 m_adjoint.resize(svd.cols(), svd.rows());
130 if (svd.m_computeFullV) m_workspace.resize(svd.cols());
131 }
132
133 bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
134 {
135 if(matrix.cols() > matrix.rows())
136 {
137 m_adjoint = matrix.adjoint();
138 m_qr.compute(m_adjoint);
139 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
140 if(svd.m_computeFullV) m_qr.matrixQ().evalTo(svd.m_matrixV, m_workspace);
141 if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
142 return true;
143 }
144 else return false;
145 }
146private:
147 typedef FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
148 QRType m_qr;
149 TransposeTypeWithSameStorageOrder m_adjoint;
150 typename internal::plain_row_type<MatrixType>::type m_workspace;
151};
152
153/*** preconditioner using ColPivHouseholderQR ***/
154
155template<typename MatrixType>
156class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
157{
158public:
159 typedef typename MatrixType::Index Index;
160
161 void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
162 {
163 if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
164 {
165 m_qr.~QRType();
166 ::new (&m_qr) QRType(svd.rows(), svd.cols());
167 }
168 if (svd.m_computeFullU) m_workspace.resize(svd.rows());
169 else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
170 }
171
172 bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
173 {
174 if(matrix.rows() > matrix.cols())
175 {
176 m_qr.compute(matrix);
177 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
178 if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
179 else if(svd.m_computeThinU)
180 {
181 svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
182 m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
183 }
184 if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
185 return true;
186 }
187 return false;
188 }
189
190private:
191 typedef ColPivHouseholderQR<MatrixType> QRType;
192 QRType m_qr;
193 typename internal::plain_col_type<MatrixType>::type m_workspace;
194};
195
196template<typename MatrixType>
197class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
198{
199public:
200 typedef typename MatrixType::Index Index;
201 typedef typename MatrixType::Scalar Scalar;
202 enum
203 {
204 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
205 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
206 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
207 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
208 Options = MatrixType::Options
209 };
210
211 typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
212 TransposeTypeWithSameStorageOrder;
213
214 void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
215 {
216 if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
217 {
218 m_qr.~QRType();
219 ::new (&m_qr) QRType(svd.cols(), svd.rows());
220 }
221 if (svd.m_computeFullV) m_workspace.resize(svd.cols());
222 else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
223 m_adjoint.resize(svd.cols(), svd.rows());
224 }
225
226 bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
227 {
228 if(matrix.cols() > matrix.rows())
229 {
230 m_adjoint = matrix.adjoint();
231 m_qr.compute(m_adjoint);
232
233 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
234 if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
235 else if(svd.m_computeThinV)
236 {
237 svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
238 m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
239 }
240 if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
241 return true;
242 }
243 else return false;
244 }
245
246private:
247 typedef ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
248 QRType m_qr;
249 TransposeTypeWithSameStorageOrder m_adjoint;
250 typename internal::plain_row_type<MatrixType>::type m_workspace;
251};
252
253/*** preconditioner using HouseholderQR ***/
254
255template<typename MatrixType>
256class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
257{
258public:
259 typedef typename MatrixType::Index Index;
260
261 void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
262 {
263 if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
264 {
265 m_qr.~QRType();
266 ::new (&m_qr) QRType(svd.rows(), svd.cols());
267 }
268 if (svd.m_computeFullU) m_workspace.resize(svd.rows());
269 else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
270 }
271
272 bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
273 {
274 if(matrix.rows() > matrix.cols())
275 {
276 m_qr.compute(matrix);
277 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
278 if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
279 else if(svd.m_computeThinU)
280 {
281 svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
282 m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
283 }
284 if(svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols());
285 return true;
286 }
287 return false;
288 }
289private:
290 typedef HouseholderQR<MatrixType> QRType;
291 QRType m_qr;
292 typename internal::plain_col_type<MatrixType>::type m_workspace;
293};
294
295template<typename MatrixType>
296class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
297{
298public:
299 typedef typename MatrixType::Index Index;
300 typedef typename MatrixType::Scalar Scalar;
301 enum
302 {
303 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
304 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
305 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
306 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
307 Options = MatrixType::Options
308 };
309
310 typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
311 TransposeTypeWithSameStorageOrder;
312
313 void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
314 {
315 if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
316 {
317 m_qr.~QRType();
318 ::new (&m_qr) QRType(svd.cols(), svd.rows());
319 }
320 if (svd.m_computeFullV) m_workspace.resize(svd.cols());
321 else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
322 m_adjoint.resize(svd.cols(), svd.rows());
323 }
324
325 bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
326 {
327 if(matrix.cols() > matrix.rows())
328 {
329 m_adjoint = matrix.adjoint();
330 m_qr.compute(m_adjoint);
331
332 svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
333 if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
334 else if(svd.m_computeThinV)
335 {
336 svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
337 m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
338 }
339 if(svd.computeU()) svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows());
340 return true;
341 }
342 else return false;
343 }
344
345private:
346 typedef HouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
347 QRType m_qr;
348 TransposeTypeWithSameStorageOrder m_adjoint;
349 typename internal::plain_row_type<MatrixType>::type m_workspace;
350};
351
352/*** 2x2 SVD implementation
353 ***
354 *** JacobiSVD consists in performing a series of 2x2 SVD subproblems
355 ***/
356
357template<typename MatrixType, int QRPreconditioner>
358struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, false>
359{
360 typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
361 typedef typename SVD::Index Index;
362 static void run(typename SVD::WorkMatrixType&, SVD&, Index, Index) {}
363};
364
365template<typename MatrixType, int QRPreconditioner>
366struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true>
367{
368 typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
369 typedef typename MatrixType::Scalar Scalar;
370 typedef typename MatrixType::RealScalar RealScalar;
371 typedef typename SVD::Index Index;
372 static void run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q)
373 {
374 using std::sqrt;
375 Scalar z;
376 JacobiRotation<Scalar> rot;
377 RealScalar n = sqrt(numext::abs2(work_matrix.coeff(p,p)) + numext::abs2(work_matrix.coeff(q,p)));
378 if(n==0)
379 {
380 z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
381 work_matrix.row(p) *= z;
382 if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z);
383 z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
384 work_matrix.row(q) *= z;
385 if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
386 }
387 else
388 {
389 rot.c() = conj(work_matrix.coeff(p,p)) / n;
390 rot.s() = work_matrix.coeff(q,p) / n;
391 work_matrix.applyOnTheLeft(p,q,rot);
392 if(svd.computeU()) svd.m_matrixU.applyOnTheRight(p,q,rot.adjoint());
393 if(work_matrix.coeff(p,q) != Scalar(0))
394 {
395 Scalar z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
396 work_matrix.col(q) *= z;
397 if(svd.computeV()) svd.m_matrixV.col(q) *= z;
398 }
399 if(work_matrix.coeff(q,q) != Scalar(0))
400 {
401 z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
402 work_matrix.row(q) *= z;
403 if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
404 }
405 }
406 }
407};
408
409template<typename MatrixType, typename RealScalar, typename Index>
410void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q,
411 JacobiRotation<RealScalar> *j_left,
412 JacobiRotation<RealScalar> *j_right)
413{
414 using std::sqrt;
415 Matrix<RealScalar,2,2> m;
416 m << numext::real(matrix.coeff(p,p)), numext::real(matrix.coeff(p,q)),
417 numext::real(matrix.coeff(q,p)), numext::real(matrix.coeff(q,q));
418 JacobiRotation<RealScalar> rot1;
419 RealScalar t = m.coeff(0,0) + m.coeff(1,1);
420 RealScalar d = m.coeff(1,0) - m.coeff(0,1);
421 if(t == RealScalar(0))
422 {
423 rot1.c() = RealScalar(0);
424 rot1.s() = d > RealScalar(0) ? RealScalar(1) : RealScalar(-1);
425 }
426 else
427 {
428 RealScalar u = d / t;
429 rot1.c() = RealScalar(1) / sqrt(RealScalar(1) + numext::abs2(u));
430 rot1.s() = rot1.c() * u;
431 }
432 m.applyOnTheLeft(0,1,rot1);
433 j_right->makeJacobi(m,0,1);
434 *j_left = rot1 * j_right->transpose();
435}
436
437} // end namespace internal
438
439/** \ingroup SVD_Module
440 *
441 *
442 * \class JacobiSVD
443 *
444 * \brief Two-sided Jacobi SVD decomposition of a rectangular matrix
445 *
446 * \param MatrixType the type of the matrix of which we are computing the SVD decomposition
447 * \param QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally
448 * for the R-SVD step for non-square matrices. See discussion of possible values below.
449 *
450 * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product
451 * \f[ A = U S V^* \f]
452 * where \a U is a n-by-n unitary, \a V is a p-by-p unitary, and \a S is a n-by-p real positive matrix which is zero outside of its main diagonal;
453 * the diagonal entries of S are known as the \em singular \em values of \a A and the columns of \a U and \a V are known as the left
454 * and right \em singular \em vectors of \a A respectively.
455 *
456 * Singular values are always sorted in decreasing order.
457 *
458 * This JacobiSVD decomposition computes only the singular values by default. If you want \a U or \a V, you need to ask for them explicitly.
459 *
460 * You can ask for only \em thin \a U or \a V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting \a m be the
461 * smaller value among \a n and \a p, there are only \a m singular vectors; the remaining columns of \a U and \a V do not correspond to actual
462 * singular vectors. Asking for \em thin \a U or \a V means asking for only their \a m first columns to be formed. So \a U is then a n-by-m matrix,
463 * and \a V is then a p-by-m matrix. Notice that thin \a U and \a V are all you need for (least squares) solving.
464 *
465 * Here's an example demonstrating basic usage:
466 * \include JacobiSVD_basic.cpp
467 * Output: \verbinclude JacobiSVD_basic.out
468 *
469 * This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than
470 * bidiagonalizing SVD algorithms for large square matrices; however its complexity is still \f$ O(n^2p) \f$ where \a n is the smaller dimension and
471 * \a p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms.
472 * In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.
473 *
474 * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to
475 * terminate in finite (and reasonable) time.
476 *
477 * The possible values for QRPreconditioner are:
478 * \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
479 * \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR.
480 * Contrary to other QRs, it doesn't allow computing thin unitaries.
481 * \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR.
482 * This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization
483 * is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive
484 * process is more reliable than the optimized bidiagonal SVD iterations.
485 * \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing
486 * JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in
487 * faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking
488 * if QR preconditioning is needed before applying it anyway.
489 *
490 * \sa MatrixBase::jacobiSvd()
491 */
492template<typename _MatrixType, int QRPreconditioner>
493class JacobiSVD : public SVDBase<_MatrixType>
494{
495 public:
496
497 typedef _MatrixType MatrixType;
498 typedef typename MatrixType::Scalar Scalar;
499 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
500 typedef typename MatrixType::Index Index;
501 enum {
502 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
503 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
504 DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime),
505 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
506 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
507 MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime),
508 MatrixOptions = MatrixType::Options
509 };
510
511 typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime,
512 MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime>
513 MatrixUType;
514 typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime,
515 MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime>
516 MatrixVType;
517 typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;
518 typedef typename internal::plain_row_type<MatrixType>::type RowType;
519 typedef typename internal::plain_col_type<MatrixType>::type ColType;
520 typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime,
521 MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime>
522 WorkMatrixType;
523
524 /** \brief Default Constructor.
525 *
526 * The default constructor is useful in cases in which the user intends to
527 * perform decompositions via JacobiSVD::compute(const MatrixType&).
528 */
529 JacobiSVD()
530 : SVDBase<_MatrixType>::SVDBase()
531 {}
532
533
534 /** \brief Default Constructor with memory preallocation
535 *
536 * Like the default constructor but with preallocation of the internal data
537 * according to the specified problem size.
538 * \sa JacobiSVD()
539 */
540 JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0)
541 : SVDBase<_MatrixType>::SVDBase()
542 {
543 allocate(rows, cols, computationOptions);
544 }
545
546 /** \brief Constructor performing the decomposition of given matrix.
547 *
548 * \param matrix the matrix to decompose
549 * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
550 * By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
551 * #ComputeFullV, #ComputeThinV.
552 *
553 * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
554 * available with the (non-default) FullPivHouseholderQR preconditioner.
555 */
556 JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
557 : SVDBase<_MatrixType>::SVDBase()
558 {
559 compute(matrix, computationOptions);
560 }
561
562 /** \brief Method performing the decomposition of given matrix using custom options.
563 *
564 * \param matrix the matrix to decompose
565 * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
566 * By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
567 * #ComputeFullV, #ComputeThinV.
568 *
569 * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
570 * available with the (non-default) FullPivHouseholderQR preconditioner.
571 */
572 SVDBase<MatrixType>& compute(const MatrixType& matrix, unsigned int computationOptions);
573
574 /** \brief Method performing the decomposition of given matrix using current options.
575 *
576 * \param matrix the matrix to decompose
577 *
578 * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
579 */
580 SVDBase<MatrixType>& compute(const MatrixType& matrix)
581 {
582 return compute(matrix, this->m_computationOptions);
583 }
584
585 /** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A.
586 *
587 * \param b the right-hand-side of the equation to solve.
588 *
589 * \note Solving requires both U and V to be computed. Thin U and V are enough, there is no need for full U or V.
590 *
591 * \note SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and least-squares solving.
592 * In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$.
593 */
594 template<typename Rhs>
595 inline const internal::solve_retval<JacobiSVD, Rhs>
596 solve(const MatrixBase<Rhs>& b) const
597 {
598 eigen_assert(this->m_isInitialized && "JacobiSVD is not initialized.");
599 eigen_assert(SVDBase<MatrixType>::computeU() && SVDBase<MatrixType>::computeV() && "JacobiSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
600 return internal::solve_retval<JacobiSVD, Rhs>(*this, b.derived());
601 }
602
603
604
605 private:
606 void allocate(Index rows, Index cols, unsigned int computationOptions);
607
608 protected:
609 WorkMatrixType m_workMatrix;
610
611 template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex>
612 friend struct internal::svd_precondition_2x2_block_to_be_real;
613 template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything>
614 friend struct internal::qr_preconditioner_impl;
615
616 internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols;
617 internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows;
618};
619
620template<typename MatrixType, int QRPreconditioner>
621void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, unsigned int computationOptions)
622{
623 if (SVDBase<MatrixType>::allocate(rows, cols, computationOptions)) return;
624
625 if (QRPreconditioner == FullPivHouseholderQRPreconditioner)
626 {
627 eigen_assert(!(this->m_computeThinU || this->m_computeThinV) &&
628 "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
629 "Use the ColPivHouseholderQR preconditioner instead.");
630 }
631
632 m_workMatrix.resize(this->m_diagSize, this->m_diagSize);
633
634 if(this->m_cols>this->m_rows) m_qr_precond_morecols.allocate(*this);
635 if(this->m_rows>this->m_cols) m_qr_precond_morerows.allocate(*this);
636}
637
638template<typename MatrixType, int QRPreconditioner>
639SVDBase<MatrixType>&
640JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions)
641{
642 using std::abs;
643 allocate(matrix.rows(), matrix.cols(), computationOptions);
644
645 // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations,
646 // only worsening the precision of U and V as we accumulate more rotations
647 const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon();
648
649 // limit for very small denormal numbers to be considered zero in order to avoid infinite loops (see bug 286)
650 const RealScalar considerAsZero = RealScalar(2) * std::numeric_limits<RealScalar>::denorm_min();
651
652 /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
653
654 if(!m_qr_precond_morecols.run(*this, matrix) && !m_qr_precond_morerows.run(*this, matrix))
655 {
656 m_workMatrix = matrix.block(0,0,this->m_diagSize,this->m_diagSize);
657 if(this->m_computeFullU) this->m_matrixU.setIdentity(this->m_rows,this->m_rows);
658 if(this->m_computeThinU) this->m_matrixU.setIdentity(this->m_rows,this->m_diagSize);
659 if(this->m_computeFullV) this->m_matrixV.setIdentity(this->m_cols,this->m_cols);
660 if(this->m_computeThinV) this->m_matrixV.setIdentity(this->m_cols, this->m_diagSize);
661 }
662
663 /*** step 2. The main Jacobi SVD iteration. ***/
664
665 bool finished = false;
666 while(!finished)
667 {
668 finished = true;
669
670 // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix
671
672 for(Index p = 1; p < this->m_diagSize; ++p)
673 {
674 for(Index q = 0; q < p; ++q)
675 {
676 // if this 2x2 sub-matrix is not diagonal already...
677 // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't
678 // keep us iterating forever. Similarly, small denormal numbers are considered zero.
679 using std::max;
680 RealScalar threshold = (max)(considerAsZero, precision * (max)(abs(m_workMatrix.coeff(p,p)),
681 abs(m_workMatrix.coeff(q,q))));
682 if((max)(abs(m_workMatrix.coeff(p,q)),abs(m_workMatrix.coeff(q,p))) > threshold)
683 {
684 finished = false;
685
686 // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
687 internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q);
688 JacobiRotation<RealScalar> j_left, j_right;
689 internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right);
690
691 // accumulate resulting Jacobi rotations
692 m_workMatrix.applyOnTheLeft(p,q,j_left);
693 if(SVDBase<MatrixType>::computeU()) this->m_matrixU.applyOnTheRight(p,q,j_left.transpose());
694
695 m_workMatrix.applyOnTheRight(p,q,j_right);
696 if(SVDBase<MatrixType>::computeV()) this->m_matrixV.applyOnTheRight(p,q,j_right);
697 }
698 }
699 }
700 }
701
702 /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/
703
704 for(Index i = 0; i < this->m_diagSize; ++i)
705 {
706 RealScalar a = abs(m_workMatrix.coeff(i,i));
707 this->m_singularValues.coeffRef(i) = a;
708 if(SVDBase<MatrixType>::computeU() && (a!=RealScalar(0))) this->m_matrixU.col(i) *= this->m_workMatrix.coeff(i,i)/a;
709 }
710
711 /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/
712
713 this->m_nonzeroSingularValues = this->m_diagSize;
714 for(Index i = 0; i < this->m_diagSize; i++)
715 {
716 Index pos;
717 RealScalar maxRemainingSingularValue = this->m_singularValues.tail(this->m_diagSize-i).maxCoeff(&pos);
718 if(maxRemainingSingularValue == RealScalar(0))
719 {
720 this->m_nonzeroSingularValues = i;
721 break;
722 }
723 if(pos)
724 {
725 pos += i;
726 std::swap(this->m_singularValues.coeffRef(i), this->m_singularValues.coeffRef(pos));
727 if(SVDBase<MatrixType>::computeU()) this->m_matrixU.col(pos).swap(this->m_matrixU.col(i));
728 if(SVDBase<MatrixType>::computeV()) this->m_matrixV.col(pos).swap(this->m_matrixV.col(i));
729 }
730 }
731
732 this->m_isInitialized = true;
733 return *this;
734}
735
736namespace internal {
737template<typename _MatrixType, int QRPreconditioner, typename Rhs>
738struct solve_retval<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
739 : solve_retval_base<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
740{
741 typedef JacobiSVD<_MatrixType, QRPreconditioner> JacobiSVDType;
742 EIGEN_MAKE_SOLVE_HELPERS(JacobiSVDType,Rhs)
743
744 template<typename Dest> void evalTo(Dest& dst) const
745 {
746 eigen_assert(rhs().rows() == dec().rows());
747
748 // A = U S V^*
749 // So A^{-1} = V S^{-1} U^*
750
751 Index diagSize = (std::min)(dec().rows(), dec().cols());
752 typename JacobiSVDType::SingularValuesType invertedSingVals(diagSize);
753
754 Index nonzeroSingVals = dec().nonzeroSingularValues();
755 invertedSingVals.head(nonzeroSingVals) = dec().singularValues().head(nonzeroSingVals).array().inverse();
756 invertedSingVals.tail(diagSize - nonzeroSingVals).setZero();
757
758 dst = dec().matrixV().leftCols(diagSize)
759 * invertedSingVals.asDiagonal()
760 * dec().matrixU().leftCols(diagSize).adjoint()
761 * rhs();
762 }
763};
764} // end namespace internal
765
766/** \svd_module
767 *
768 * \return the singular value decomposition of \c *this computed by two-sided
769 * Jacobi transformations.
770 *
771 * \sa class JacobiSVD
772 */
773template<typename Derived>
774JacobiSVD<typename MatrixBase<Derived>::PlainObject>
775MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const
776{
777 return JacobiSVD<PlainObject>(*this, computationOptions);
778}
779
780} // end namespace Eigen
781
782#endif // EIGEN_JACOBISVD_H
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