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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 // Copyright (C) 2013-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
6 //
7 // This Source Code Form is subject to the terms of the Mozilla
8 // Public License v. 2.0. If a copy of the MPL was not distributed
9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
10 
11 #ifndef EIGEN_JACOBISVD_H
12 #define EIGEN_JACOBISVD_H
13 
14 namespace Eigen {
15 
16 namespace internal {
17 // forward declaration (needed by ICC)
18 // the empty body is required by MSVC
19 template<typename MatrixType, int QRPreconditioner,
20          bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
21 struct svd_precondition_2x2_block_to_be_real {};
22 
23 /*** QR preconditioners (R-SVD)
24  ***
25  *** Their role is to reduce the problem of computing the SVD to the case of a square matrix.
26  *** This approach, known as R-SVD, is an optimization for rectangular-enough matrices, and is a requirement for
27  *** JacobiSVD which by itself is only able to work on square matrices.
28  ***/
29 
30 enum { PreconditionIfMoreColsThanRows, PreconditionIfMoreRowsThanCols };
31 
32 template<typename MatrixType, int QRPreconditioner, int Case>
33 struct qr_preconditioner_should_do_anything
34 {
35   enum { a = MatrixType::RowsAtCompileTime != Dynamic &&
36              MatrixType::ColsAtCompileTime != Dynamic &&
37              MatrixType::ColsAtCompileTime <= MatrixType::RowsAtCompileTime,
38          b = MatrixType::RowsAtCompileTime != Dynamic &&
39              MatrixType::ColsAtCompileTime != Dynamic &&
40              MatrixType::RowsAtCompileTime <= MatrixType::ColsAtCompileTime,
41          ret = !( (QRPreconditioner == NoQRPreconditioner) ||
42                   (Case == PreconditionIfMoreColsThanRows && bool(a)) ||
43                   (Case == PreconditionIfMoreRowsThanCols && bool(b)) )
44   };
45 };
46 
47 template<typename MatrixType, int QRPreconditioner, int Case,
48          bool DoAnything = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret
49 > struct qr_preconditioner_impl {};
50 
51 template<typename MatrixType, int QRPreconditioner, int Case>
52 class qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false>
53 {
54 public:
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::Scalar Scalar;
69   enum
70   {
71     RowsAtCompileTime = MatrixType::RowsAtCompileTime,
72     MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
73   };
74   typedef Matrix<Scalar, 1, RowsAtCompileTime, RowMajor, 1, MaxRowsAtCompileTime> WorkspaceType;
75 
allocate(const JacobiSVD<MatrixType,FullPivHouseholderQRPreconditioner> & svd)76   void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
77   {
78     if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
79     {
80       m_qr.~QRType();
81       ::new (&m_qr) QRType(svd.rows(), svd.cols());
82     }
83     if (svd.m_computeFullU) m_workspace.resize(svd.rows());
84   }
85 
run(JacobiSVD<MatrixType,FullPivHouseholderQRPreconditioner> & svd,const MatrixType & matrix)86   bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
87   {
88     if(matrix.rows() > matrix.cols())
89     {
90       m_qr.compute(matrix);
91       svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
92       if(svd.m_computeFullU) m_qr.matrixQ().evalTo(svd.m_matrixU, m_workspace);
93       if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
94       return true;
95     }
96     return false;
97   }
98 private:
99   typedef FullPivHouseholderQR<MatrixType> QRType;
100   QRType m_qr;
101   WorkspaceType m_workspace;
102 };
103 
104 template<typename MatrixType>
105 class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
106 {
107 public:
108   typedef typename MatrixType::Scalar Scalar;
109   enum
110   {
111     RowsAtCompileTime = MatrixType::RowsAtCompileTime,
112     ColsAtCompileTime = MatrixType::ColsAtCompileTime,
113     MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
114     MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
115     TrOptions = RowsAtCompileTime==1 ? (MatrixType::Options & ~(RowMajor))
116               : ColsAtCompileTime==1 ? (MatrixType::Options |   RowMajor)
117               : MatrixType::Options
118   };
119   typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, TrOptions, 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:
allocate(const JacobiSVD<MatrixType,ColPivHouseholderQRPreconditioner> & svd)159   void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
160   {
161     if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
162     {
163       m_qr.~QRType();
164       ::new (&m_qr) QRType(svd.rows(), svd.cols());
165     }
166     if (svd.m_computeFullU) m_workspace.resize(svd.rows());
167     else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
168   }
169 
run(JacobiSVD<MatrixType,ColPivHouseholderQRPreconditioner> & svd,const MatrixType & matrix)170   bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
171   {
172     if(matrix.rows() > matrix.cols())
173     {
174       m_qr.compute(matrix);
175       svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
176       if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
177       else if(svd.m_computeThinU)
178       {
179         svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
180         m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
181       }
182       if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
183       return true;
184     }
185     return false;
186   }
187 
188 private:
189   typedef ColPivHouseholderQR<MatrixType> QRType;
190   QRType m_qr;
191   typename internal::plain_col_type<MatrixType>::type m_workspace;
192 };
193 
194 template<typename MatrixType>
195 class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
196 {
197 public:
198   typedef typename MatrixType::Scalar Scalar;
199   enum
200   {
201     RowsAtCompileTime = MatrixType::RowsAtCompileTime,
202     ColsAtCompileTime = MatrixType::ColsAtCompileTime,
203     MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
204     MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
205     TrOptions = RowsAtCompileTime==1 ? (MatrixType::Options & ~(RowMajor))
206               : ColsAtCompileTime==1 ? (MatrixType::Options |   RowMajor)
207               : MatrixType::Options
208   };
209 
210   typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, TrOptions, MaxColsAtCompileTime, MaxRowsAtCompileTime>
211           TransposeTypeWithSameStorageOrder;
212 
allocate(const JacobiSVD<MatrixType,ColPivHouseholderQRPreconditioner> & svd)213   void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
214   {
215     if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
216     {
217       m_qr.~QRType();
218       ::new (&m_qr) QRType(svd.cols(), svd.rows());
219     }
220     if (svd.m_computeFullV) m_workspace.resize(svd.cols());
221     else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
222     m_adjoint.resize(svd.cols(), svd.rows());
223   }
224 
run(JacobiSVD<MatrixType,ColPivHouseholderQRPreconditioner> & svd,const MatrixType & matrix)225   bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
226   {
227     if(matrix.cols() > matrix.rows())
228     {
229       m_adjoint = matrix.adjoint();
230       m_qr.compute(m_adjoint);
231 
232       svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
233       if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
234       else if(svd.m_computeThinV)
235       {
236         svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
237         m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
238       }
239       if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
240       return true;
241     }
242     else return false;
243   }
244 
245 private:
246   typedef ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
247   QRType m_qr;
248   TransposeTypeWithSameStorageOrder m_adjoint;
249   typename internal::plain_row_type<MatrixType>::type m_workspace;
250 };
251 
252 /*** preconditioner using HouseholderQR ***/
253 
254 template<typename MatrixType>
255 class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
256 {
257 public:
allocate(const JacobiSVD<MatrixType,HouseholderQRPreconditioner> & svd)258   void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
259   {
260     if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
261     {
262       m_qr.~QRType();
263       ::new (&m_qr) QRType(svd.rows(), svd.cols());
264     }
265     if (svd.m_computeFullU) m_workspace.resize(svd.rows());
266     else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
267   }
268 
run(JacobiSVD<MatrixType,HouseholderQRPreconditioner> & svd,const MatrixType & matrix)269   bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
270   {
271     if(matrix.rows() > matrix.cols())
272     {
273       m_qr.compute(matrix);
274       svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
275       if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
276       else if(svd.m_computeThinU)
277       {
278         svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
279         m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
280       }
281       if(svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols());
282       return true;
283     }
284     return false;
285   }
286 private:
287   typedef HouseholderQR<MatrixType> QRType;
288   QRType m_qr;
289   typename internal::plain_col_type<MatrixType>::type m_workspace;
290 };
291 
292 template<typename MatrixType>
293 class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
294 {
295 public:
296   typedef typename MatrixType::Scalar Scalar;
297   enum
298   {
299     RowsAtCompileTime = MatrixType::RowsAtCompileTime,
300     ColsAtCompileTime = MatrixType::ColsAtCompileTime,
301     MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
302     MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
303     Options = MatrixType::Options
304   };
305 
306   typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
307           TransposeTypeWithSameStorageOrder;
308 
allocate(const JacobiSVD<MatrixType,HouseholderQRPreconditioner> & svd)309   void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
310   {
311     if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
312     {
313       m_qr.~QRType();
314       ::new (&m_qr) QRType(svd.cols(), svd.rows());
315     }
316     if (svd.m_computeFullV) m_workspace.resize(svd.cols());
317     else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
318     m_adjoint.resize(svd.cols(), svd.rows());
319   }
320 
run(JacobiSVD<MatrixType,HouseholderQRPreconditioner> & svd,const MatrixType & matrix)321   bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
322   {
323     if(matrix.cols() > matrix.rows())
324     {
325       m_adjoint = matrix.adjoint();
326       m_qr.compute(m_adjoint);
327 
328       svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
329       if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
330       else if(svd.m_computeThinV)
331       {
332         svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
333         m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
334       }
335       if(svd.computeU()) svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows());
336       return true;
337     }
338     else return false;
339   }
340 
341 private:
342   typedef HouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
343   QRType m_qr;
344   TransposeTypeWithSameStorageOrder m_adjoint;
345   typename internal::plain_row_type<MatrixType>::type m_workspace;
346 };
347 
348 /*** 2x2 SVD implementation
349  ***
350  *** JacobiSVD consists in performing a series of 2x2 SVD subproblems
351  ***/
352 
353 template<typename MatrixType, int QRPreconditioner>
354 struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, false>
355 {
356   typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
357   typedef typename MatrixType::RealScalar RealScalar;
358   static bool run(typename SVD::WorkMatrixType&, SVD&, Index, Index, RealScalar&) { return true; }
359 };
360 
361 template<typename MatrixType, int QRPreconditioner>
362 struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true>
363 {
364   typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
365   typedef typename MatrixType::Scalar Scalar;
366   typedef typename MatrixType::RealScalar RealScalar;
367   static bool run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q, RealScalar& maxDiagEntry)
368   {
369     using std::sqrt;
370     using std::abs;
371     Scalar z;
372     JacobiRotation<Scalar> rot;
373     RealScalar n = sqrt(numext::abs2(work_matrix.coeff(p,p)) + numext::abs2(work_matrix.coeff(q,p)));
374 
375     const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
376     const RealScalar precision = NumTraits<Scalar>::epsilon();
377 
378     if(n==0)
379     {
380       // make sure first column is zero
381       work_matrix.coeffRef(p,p) = work_matrix.coeffRef(q,p) = Scalar(0);
382 
383       if(abs(numext::imag(work_matrix.coeff(p,q)))>considerAsZero)
384       {
385         // work_matrix.coeff(p,q) can be zero if work_matrix.coeff(q,p) is not zero but small enough to underflow when computing n
386         z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
387         work_matrix.row(p) *= z;
388         if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z);
389       }
390       if(abs(numext::imag(work_matrix.coeff(q,q)))>considerAsZero)
391       {
392         z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
393         work_matrix.row(q) *= z;
394         if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
395       }
396       // otherwise the second row is already zero, so we have nothing to do.
397     }
398     else
399     {
400       rot.c() = conj(work_matrix.coeff(p,p)) / n;
401       rot.s() = work_matrix.coeff(q,p) / n;
402       work_matrix.applyOnTheLeft(p,q,rot);
403       if(svd.computeU()) svd.m_matrixU.applyOnTheRight(p,q,rot.adjoint());
404       if(abs(numext::imag(work_matrix.coeff(p,q)))>considerAsZero)
405       {
406         z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
407         work_matrix.col(q) *= z;
408         if(svd.computeV()) svd.m_matrixV.col(q) *= z;
409       }
410       if(abs(numext::imag(work_matrix.coeff(q,q)))>considerAsZero)
411       {
412         z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
413         work_matrix.row(q) *= z;
414         if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
415       }
416     }
417 
418     // update largest diagonal entry
419     maxDiagEntry = numext::maxi<RealScalar>(maxDiagEntry,numext::maxi<RealScalar>(abs(work_matrix.coeff(p,p)), abs(work_matrix.coeff(q,q))));
420     // and check whether the 2x2 block is already diagonal
421     RealScalar threshold = numext::maxi<RealScalar>(considerAsZero, precision * maxDiagEntry);
422     return abs(work_matrix.coeff(p,q))>threshold || abs(work_matrix.coeff(q,p)) > threshold;
423   }
424 };
425 
426 template<typename _MatrixType, int QRPreconditioner>
427 struct traits<JacobiSVD<_MatrixType,QRPreconditioner> >
428 {
429   typedef _MatrixType MatrixType;
430 };
431 
432 } // end namespace internal
433 
434 /** \ingroup SVD_Module
435   *
436   *
437   * \class JacobiSVD
438   *
439   * \brief Two-sided Jacobi SVD decomposition of a rectangular matrix
440   *
441   * \tparam _MatrixType the type of the matrix of which we are computing the SVD decomposition
442   * \tparam QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally
443   *                        for the R-SVD step for non-square matrices. See discussion of possible values below.
444   *
445   * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product
446   *   \f[ A = U S V^* \f]
447   * 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;
448   * 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
449   * and right \em singular \em vectors of \a A respectively.
450   *
451   * Singular values are always sorted in decreasing order.
452   *
453   * 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.
454   *
455   * 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
456   * 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
457   * 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,
458   * 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.
459   *
460   * Here's an example demonstrating basic usage:
461   * \include JacobiSVD_basic.cpp
462   * Output: \verbinclude JacobiSVD_basic.out
463   *
464   * This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than
465   * 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
466   * \a p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms.
467   * In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.
468   *
469   * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to
470   * terminate in finite (and reasonable) time.
471   *
472   * The possible values for QRPreconditioner are:
473   * \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
474   * \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR.
475   *     Contrary to other QRs, it doesn't allow computing thin unitaries.
476   * \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR.
477   *     This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization
478   *     is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive
479   *     process is more reliable than the optimized bidiagonal SVD iterations.
480   * \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing
481   *     JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in
482   *     faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking
483   *     if QR preconditioning is needed before applying it anyway.
484   *
485   * \sa MatrixBase::jacobiSvd()
486   */
487 template<typename _MatrixType, int QRPreconditioner> class JacobiSVD
488  : public SVDBase<JacobiSVD<_MatrixType,QRPreconditioner> >
489 {
490     typedef SVDBase<JacobiSVD> Base;
491   public:
492 
493     typedef _MatrixType MatrixType;
494     typedef typename MatrixType::Scalar Scalar;
495     typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
496     enum {
497       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
498       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
499       DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime),
500       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
501       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
502       MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime),
503       MatrixOptions = MatrixType::Options
504     };
505 
506     typedef typename Base::MatrixUType MatrixUType;
507     typedef typename Base::MatrixVType MatrixVType;
508     typedef typename Base::SingularValuesType SingularValuesType;
509 
510     typedef typename internal::plain_row_type<MatrixType>::type RowType;
511     typedef typename internal::plain_col_type<MatrixType>::type ColType;
512     typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime,
513                    MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime>
514             WorkMatrixType;
515 
516     /** \brief Default Constructor.
517       *
518       * The default constructor is useful in cases in which the user intends to
519       * perform decompositions via JacobiSVD::compute(const MatrixType&).
520       */
521     JacobiSVD()
522     {}
523 
524 
525     /** \brief Default Constructor with memory preallocation
526       *
527       * Like the default constructor but with preallocation of the internal data
528       * according to the specified problem size.
529       * \sa JacobiSVD()
530       */
531     JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0)
532     {
533       allocate(rows, cols, computationOptions);
534     }
535 
536     /** \brief Constructor performing the decomposition of given matrix.
537      *
538      * \param matrix the matrix to decompose
539      * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
540      *                           By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
541      *                           #ComputeFullV, #ComputeThinV.
542      *
543      * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
544      * available with the (non-default) FullPivHouseholderQR preconditioner.
545      */
546     explicit JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
547     {
548       compute(matrix, computationOptions);
549     }
550 
551     /** \brief Method performing the decomposition of given matrix using custom options.
552      *
553      * \param matrix the matrix to decompose
554      * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
555      *                           By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
556      *                           #ComputeFullV, #ComputeThinV.
557      *
558      * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
559      * available with the (non-default) FullPivHouseholderQR preconditioner.
560      */
561     JacobiSVD& compute(const MatrixType& matrix, unsigned int computationOptions);
562 
563     /** \brief Method performing the decomposition of given matrix using current options.
564      *
565      * \param matrix the matrix to decompose
566      *
567      * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
568      */
569     JacobiSVD& compute(const MatrixType& matrix)
570     {
571       return compute(matrix, m_computationOptions);
572     }
573 
574     using Base::computeU;
575     using Base::computeV;
576     using Base::rows;
577     using Base::cols;
578     using Base::rank;
579 
580   private:
581     void allocate(Index rows, Index cols, unsigned int computationOptions);
582 
583   protected:
584     using Base::m_matrixU;
585     using Base::m_matrixV;
586     using Base::m_singularValues;
587     using Base::m_isInitialized;
588     using Base::m_isAllocated;
589     using Base::m_usePrescribedThreshold;
590     using Base::m_computeFullU;
591     using Base::m_computeThinU;
592     using Base::m_computeFullV;
593     using Base::m_computeThinV;
594     using Base::m_computationOptions;
595     using Base::m_nonzeroSingularValues;
596     using Base::m_rows;
597     using Base::m_cols;
598     using Base::m_diagSize;
599     using Base::m_prescribedThreshold;
600     WorkMatrixType m_workMatrix;
601 
602     template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex>
603     friend struct internal::svd_precondition_2x2_block_to_be_real;
604     template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything>
605     friend struct internal::qr_preconditioner_impl;
606 
607     internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols;
608     internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows;
609     MatrixType m_scaledMatrix;
610 };
611 
612 template<typename MatrixType, int QRPreconditioner>
613 void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, unsigned int computationOptions)
614 {
615   eigen_assert(rows >= 0 && cols >= 0);
616 
617   if (m_isAllocated &&
618       rows == m_rows &&
619       cols == m_cols &&
620       computationOptions == m_computationOptions)
621   {
622     return;
623   }
624 
625   m_rows = rows;
626   m_cols = cols;
627   m_isInitialized = false;
628   m_isAllocated = true;
629   m_computationOptions = computationOptions;
630   m_computeFullU = (computationOptions & ComputeFullU) != 0;
631   m_computeThinU = (computationOptions & ComputeThinU) != 0;
632   m_computeFullV = (computationOptions & ComputeFullV) != 0;
633   m_computeThinV = (computationOptions & ComputeThinV) != 0;
634   eigen_assert(!(m_computeFullU && m_computeThinU) && "JacobiSVD: you can't ask for both full and thin U");
635   eigen_assert(!(m_computeFullV && m_computeThinV) && "JacobiSVD: you can't ask for both full and thin V");
636   eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime==Dynamic) &&
637               "JacobiSVD: thin U and V are only available when your matrix has a dynamic number of columns.");
638   if (QRPreconditioner == FullPivHouseholderQRPreconditioner)
639   {
640       eigen_assert(!(m_computeThinU || m_computeThinV) &&
641               "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
642               "Use the ColPivHouseholderQR preconditioner instead.");
643   }
644   m_diagSize = (std::min)(m_rows, m_cols);
645   m_singularValues.resize(m_diagSize);
646   if(RowsAtCompileTime==Dynamic)
647     m_matrixU.resize(m_rows, m_computeFullU ? m_rows
648                             : m_computeThinU ? m_diagSize
649                             : 0);
650   if(ColsAtCompileTime==Dynamic)
651     m_matrixV.resize(m_cols, m_computeFullV ? m_cols
652                             : m_computeThinV ? m_diagSize
653                             : 0);
654   m_workMatrix.resize(m_diagSize, m_diagSize);
655 
656   if(m_cols>m_rows)   m_qr_precond_morecols.allocate(*this);
657   if(m_rows>m_cols)   m_qr_precond_morerows.allocate(*this);
658   if(m_rows!=m_cols)  m_scaledMatrix.resize(rows,cols);
659 }
660 
661 template<typename MatrixType, int QRPreconditioner>
662 JacobiSVD<MatrixType, QRPreconditioner>&
663 JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions)
664 {
665   using std::abs;
666   allocate(matrix.rows(), matrix.cols(), computationOptions);
667 
668   // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations,
669   // only worsening the precision of U and V as we accumulate more rotations
670   const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon();
671 
672   // limit for denormal numbers to be considered zero in order to avoid infinite loops (see bug 286)
673   const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
674 
675   // Scaling factor to reduce over/under-flows
676   RealScalar scale = matrix.cwiseAbs().maxCoeff();
677   if(scale==RealScalar(0)) scale = RealScalar(1);
678 
679   /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
680 
681   if(m_rows!=m_cols)
682   {
683     m_scaledMatrix = matrix / scale;
684     m_qr_precond_morecols.run(*this, m_scaledMatrix);
685     m_qr_precond_morerows.run(*this, m_scaledMatrix);
686   }
687   else
688   {
689     m_workMatrix = matrix.block(0,0,m_diagSize,m_diagSize) / scale;
690     if(m_computeFullU) m_matrixU.setIdentity(m_rows,m_rows);
691     if(m_computeThinU) m_matrixU.setIdentity(m_rows,m_diagSize);
692     if(m_computeFullV) m_matrixV.setIdentity(m_cols,m_cols);
693     if(m_computeThinV) m_matrixV.setIdentity(m_cols, m_diagSize);
694   }
695 
696   /*** step 2. The main Jacobi SVD iteration. ***/
697   RealScalar maxDiagEntry = m_workMatrix.cwiseAbs().diagonal().maxCoeff();
698 
699   bool finished = false;
700   while(!finished)
701   {
702     finished = true;
703 
704     // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix
705 
706     for(Index p = 1; p < m_diagSize; ++p)
707     {
708       for(Index q = 0; q < p; ++q)
709       {
710         // if this 2x2 sub-matrix is not diagonal already...
711         // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't
712         // keep us iterating forever. Similarly, small denormal numbers are considered zero.
713         RealScalar threshold = numext::maxi<RealScalar>(considerAsZero, precision * maxDiagEntry);
714         if(abs(m_workMatrix.coeff(p,q))>threshold || abs(m_workMatrix.coeff(q,p)) > threshold)
715         {
716           finished = false;
717           // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
718           // the complex to real operation returns true if the updated 2x2 block is not already diagonal
719           if(internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q, maxDiagEntry))
720           {
721             JacobiRotation<RealScalar> j_left, j_right;
722             internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right);
723 
724             // accumulate resulting Jacobi rotations
725             m_workMatrix.applyOnTheLeft(p,q,j_left);
726             if(computeU()) m_matrixU.applyOnTheRight(p,q,j_left.transpose());
727 
728             m_workMatrix.applyOnTheRight(p,q,j_right);
729             if(computeV()) m_matrixV.applyOnTheRight(p,q,j_right);
730 
731             // keep track of the largest diagonal coefficient
732             maxDiagEntry = numext::maxi<RealScalar>(maxDiagEntry,numext::maxi<RealScalar>(abs(m_workMatrix.coeff(p,p)), abs(m_workMatrix.coeff(q,q))));
733           }
734         }
735       }
736     }
737   }
738 
739   /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/
740 
741   for(Index i = 0; i < m_diagSize; ++i)
742   {
743     // For a complex matrix, some diagonal coefficients might note have been
744     // treated by svd_precondition_2x2_block_to_be_real, and the imaginary part
745     // of some diagonal entry might not be null.
746     if(NumTraits<Scalar>::IsComplex && abs(numext::imag(m_workMatrix.coeff(i,i)))>considerAsZero)
747     {
748       RealScalar a = abs(m_workMatrix.coeff(i,i));
749       m_singularValues.coeffRef(i) = abs(a);
750       if(computeU()) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a;
751     }
752     else
753     {
754       // m_workMatrix.coeff(i,i) is already real, no difficulty:
755       RealScalar a = numext::real(m_workMatrix.coeff(i,i));
756       m_singularValues.coeffRef(i) = abs(a);
757       if(computeU() && (a<RealScalar(0))) m_matrixU.col(i) = -m_matrixU.col(i);
758     }
759   }
760 
761   m_singularValues *= scale;
762 
763   /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/
764 
765   m_nonzeroSingularValues = m_diagSize;
766   for(Index i = 0; i < m_diagSize; i++)
767   {
768     Index pos;
769     RealScalar maxRemainingSingularValue = m_singularValues.tail(m_diagSize-i).maxCoeff(&pos);
770     if(maxRemainingSingularValue == RealScalar(0))
771     {
772       m_nonzeroSingularValues = i;
773       break;
774     }
775     if(pos)
776     {
777       pos += i;
778       std::swap(m_singularValues.coeffRef(i), m_singularValues.coeffRef(pos));
779       if(computeU()) m_matrixU.col(pos).swap(m_matrixU.col(i));
780       if(computeV()) m_matrixV.col(pos).swap(m_matrixV.col(i));
781     }
782   }
783 
784   m_isInitialized = true;
785   return *this;
786 }
787 
788 /** \svd_module
789   *
790   * \return the singular value decomposition of \c *this computed by two-sided
791   * Jacobi transformations.
792   *
793   * \sa class JacobiSVD
794   */
795 template<typename Derived>
796 JacobiSVD<typename MatrixBase<Derived>::PlainObject>
797 MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const
798 {
799   return JacobiSVD<PlainObject>(*this, computationOptions);
800 }
801 
802 } // end namespace Eigen
803 
804 #endif // EIGEN_JACOBISVD_H
805