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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 
13 namespace Eigen {
14 
15 namespace internal {
16 // forward declaration (needed by ICC)
17 // the empty body is required by MSVC
18 template<typename MatrixType, int QRPreconditioner,
19          bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
20 struct 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 
29 enum { PreconditionIfMoreColsThanRows, PreconditionIfMoreRowsThanCols };
30 
31 template<typename MatrixType, int QRPreconditioner, int Case>
32 struct 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 
46 template<typename MatrixType, int QRPreconditioner, int Case,
47          bool DoAnything = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret
48 > struct qr_preconditioner_impl {};
49 
50 template<typename MatrixType, int QRPreconditioner, int Case>
51 class qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false>
52 {
53 public:
54   typedef typename MatrixType::Index Index;
allocate(const JacobiSVD<MatrixType,QRPreconditioner> &)55   void allocate(const JacobiSVD<MatrixType, QRPreconditioner>&) {}
run(JacobiSVD<MatrixType,QRPreconditioner> &,const MatrixType &)56   bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&)
57   {
58     return false;
59   }
60 };
61 
62 /*** preconditioner using FullPivHouseholderQR ***/
63 
64 template<typename MatrixType>
65 class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
66 {
67 public:
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 
allocate(const JacobiSVD<MatrixType,FullPivHouseholderQRPreconditioner> & svd)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 
run(JacobiSVD<MatrixType,FullPivHouseholderQRPreconditioner> & svd,const MatrixType & matrix)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   }
99 private:
100   typedef FullPivHouseholderQR<MatrixType> QRType;
101   QRType m_qr;
102   WorkspaceType m_workspace;
103 };
104 
105 template<typename MatrixType>
106 class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
107 {
108 public:
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 
allocate(const JacobiSVD<MatrixType,FullPivHouseholderQRPreconditioner> & svd)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 
run(JacobiSVD<MatrixType,FullPivHouseholderQRPreconditioner> & svd,const MatrixType & matrix)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   }
146 private:
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 
155 template<typename MatrixType>
156 class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
157 {
158 public:
159   typedef typename MatrixType::Index Index;
160 
allocate(const JacobiSVD<MatrixType,ColPivHouseholderQRPreconditioner> & svd)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 
run(JacobiSVD<MatrixType,ColPivHouseholderQRPreconditioner> & svd,const MatrixType & matrix)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 
190 private:
191   typedef ColPivHouseholderQR<MatrixType> QRType;
192   QRType m_qr;
193   typename internal::plain_col_type<MatrixType>::type m_workspace;
194 };
195 
196 template<typename MatrixType>
197 class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
198 {
199 public:
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 
allocate(const JacobiSVD<MatrixType,ColPivHouseholderQRPreconditioner> & svd)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 
run(JacobiSVD<MatrixType,ColPivHouseholderQRPreconditioner> & svd,const MatrixType & matrix)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 
246 private:
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 
255 template<typename MatrixType>
256 class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
257 {
258 public:
259   typedef typename MatrixType::Index Index;
260 
allocate(const JacobiSVD<MatrixType,HouseholderQRPreconditioner> & svd)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 
run(JacobiSVD<MatrixType,HouseholderQRPreconditioner> & svd,const MatrixType & matrix)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   }
289 private:
290   typedef HouseholderQR<MatrixType> QRType;
291   QRType m_qr;
292   typename internal::plain_col_type<MatrixType>::type m_workspace;
293 };
294 
295 template<typename MatrixType>
296 class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
297 {
298 public:
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 
allocate(const JacobiSVD<MatrixType,HouseholderQRPreconditioner> & svd)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 
run(JacobiSVD<MatrixType,HouseholderQRPreconditioner> & svd,const MatrixType & matrix)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 
345 private:
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 
357 template<typename MatrixType, int QRPreconditioner>
358 struct 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 
365 template<typename MatrixType, int QRPreconditioner>
366 struct 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 
409 template<typename MatrixType, typename RealScalar, typename Index>
410 void 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   */
492 template<typename _MatrixType, int QRPreconditioner>
493 class 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 
620 template<typename MatrixType, int QRPreconditioner>
621 void 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 
638 template<typename MatrixType, int QRPreconditioner>
639 SVDBase<MatrixType>&
640 JacobiSVD<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 
736 namespace internal {
737 template<typename _MatrixType, int QRPreconditioner, typename Rhs>
738 struct 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   */
773 template<typename Derived>
774 JacobiSVD<typename MatrixBase<Derived>::PlainObject>
775 MatrixBase<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
783