JacobiSVD.h

00001 // This file is part of Eigen, a lightweight C++ template library
00002 // for linear algebra.
00003 //
00004 // Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
00005 //
00006 // This Source Code Form is subject to the terms of the Mozilla
00007 // Public License v. 2.0. If a copy of the MPL was not distributed
00008 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
00009 
00010 #ifndef EIGEN_JACOBISVD_H
00011 #define EIGEN_JACOBISVD_H
00012 
00013 namespace Eigen { 
00014 
00015 namespace internal {
00016 // forward declaration (needed by ICC)
00017 // the empty body is required by MSVC
00018 template<typename MatrixType, int QRPreconditioner,
00019          bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
00020 struct svd_precondition_2x2_block_to_be_real {};
00021 
00022 /*** QR preconditioners (R-SVD)
00023  ***
00024  *** Their role is to reduce the problem of computing the SVD to the case of a square matrix.
00025  *** This approach, known as R-SVD, is an optimization for rectangular-enough matrices, and is a requirement for
00026  *** JacobiSVD which by itself is only able to work on square matrices.
00027  ***/
00028 
00029 enum { PreconditionIfMoreColsThanRows, PreconditionIfMoreRowsThanCols };
00030 
00031 template<typename MatrixType, int QRPreconditioner, int Case>
00032 struct qr_preconditioner_should_do_anything
00033 {
00034   enum { a = MatrixType::RowsAtCompileTime != Dynamic &&
00035              MatrixType::ColsAtCompileTime != Dynamic &&
00036              MatrixType::ColsAtCompileTime <= MatrixType::RowsAtCompileTime,
00037          b = MatrixType::RowsAtCompileTime != Dynamic &&
00038              MatrixType::ColsAtCompileTime != Dynamic &&
00039              MatrixType::RowsAtCompileTime <= MatrixType::ColsAtCompileTime,
00040          ret = !( (QRPreconditioner == NoQRPreconditioner) ||
00041                   (Case == PreconditionIfMoreColsThanRows && bool(a)) ||
00042                   (Case == PreconditionIfMoreRowsThanCols && bool(b)) )
00043   };
00044 };
00045 
00046 template<typename MatrixType, int QRPreconditioner, int Case,
00047          bool DoAnything = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret
00048 > struct qr_preconditioner_impl {};
00049 
00050 template<typename MatrixType, int QRPreconditioner, int Case>
00051 class qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false>
00052 {
00053 public:
00054   typedef typename MatrixType::Index Index;
00055   void allocate(const JacobiSVD<MatrixType, QRPreconditioner>&) {}
00056   bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&)
00057   {
00058     return false;
00059   }
00060 };
00061 
00062 /*** preconditioner using FullPivHouseholderQR ***/
00063 
00064 template<typename MatrixType>
00065 class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
00066 {
00067 public:
00068   typedef typename MatrixType::Index Index;
00069   typedef typename MatrixType::Scalar Scalar;
00070   enum
00071   {
00072     RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00073     MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
00074   };
00075   typedef Matrix<Scalar, 1, RowsAtCompileTime, RowMajor, 1, MaxRowsAtCompileTime> WorkspaceType;
00076 
00077   void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
00078   {
00079     if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
00080     {
00081       m_qr.~QRType();
00082       ::new (&m_qr) QRType(svd.rows(), svd.cols());
00083     }
00084     if (svd.m_computeFullU) m_workspace.resize(svd.rows());
00085   }
00086 
00087   bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
00088   {
00089     if(matrix.rows() > matrix.cols())
00090     {
00091       m_qr.compute(matrix);
00092       svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
00093       if(svd.m_computeFullU) m_qr.matrixQ().evalTo(svd.m_matrixU, m_workspace);
00094       if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
00095       return true;
00096     }
00097     return false;
00098   }
00099 private:
00100   typedef FullPivHouseholderQR<MatrixType> QRType;
00101   QRType m_qr;
00102   WorkspaceType m_workspace;
00103 };
00104 
00105 template<typename MatrixType>
00106 class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
00107 {
00108 public:
00109   typedef typename MatrixType::Index Index;
00110   typedef typename MatrixType::Scalar Scalar;
00111   enum
00112   {
00113     RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00114     ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00115     MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00116     MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
00117     Options = MatrixType::Options
00118   };
00119   typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
00120           TransposeTypeWithSameStorageOrder;
00121 
00122   void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
00123   {
00124     if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
00125     {
00126       m_qr.~QRType();
00127       ::new (&m_qr) QRType(svd.cols(), svd.rows());
00128     }
00129     m_adjoint.resize(svd.cols(), svd.rows());
00130     if (svd.m_computeFullV) m_workspace.resize(svd.cols());
00131   }
00132 
00133   bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
00134   {
00135     if(matrix.cols() > matrix.rows())
00136     {
00137       m_adjoint = matrix.adjoint();
00138       m_qr.compute(m_adjoint);
00139       svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
00140       if(svd.m_computeFullV) m_qr.matrixQ().evalTo(svd.m_matrixV, m_workspace);
00141       if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
00142       return true;
00143     }
00144     else return false;
00145   }
00146 private:
00147   typedef FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
00148   QRType m_qr;
00149   TransposeTypeWithSameStorageOrder m_adjoint;
00150   typename internal::plain_row_type<MatrixType>::type m_workspace;
00151 };
00152 
00153 /*** preconditioner using ColPivHouseholderQR ***/
00154 
00155 template<typename MatrixType>
00156 class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
00157 {
00158 public:
00159   typedef typename MatrixType::Index Index;
00160 
00161   void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
00162   {
00163     if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
00164     {
00165       m_qr.~QRType();
00166       ::new (&m_qr) QRType(svd.rows(), svd.cols());
00167     }
00168     if (svd.m_computeFullU) m_workspace.resize(svd.rows());
00169     else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
00170   }
00171 
00172   bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
00173   {
00174     if(matrix.rows() > matrix.cols())
00175     {
00176       m_qr.compute(matrix);
00177       svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
00178       if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
00179       else if(svd.m_computeThinU)
00180       {
00181         svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
00182         m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
00183       }
00184       if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
00185       return true;
00186     }
00187     return false;
00188   }
00189 
00190 private:
00191   typedef ColPivHouseholderQR<MatrixType> QRType;
00192   QRType m_qr;
00193   typename internal::plain_col_type<MatrixType>::type m_workspace;
00194 };
00195 
00196 template<typename MatrixType>
00197 class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
00198 {
00199 public:
00200   typedef typename MatrixType::Index Index;
00201   typedef typename MatrixType::Scalar Scalar;
00202   enum
00203   {
00204     RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00205     ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00206     MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00207     MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
00208     Options = MatrixType::Options
00209   };
00210 
00211   typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
00212           TransposeTypeWithSameStorageOrder;
00213 
00214   void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
00215   {
00216     if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
00217     {
00218       m_qr.~QRType();
00219       ::new (&m_qr) QRType(svd.cols(), svd.rows());
00220     }
00221     if (svd.m_computeFullV) m_workspace.resize(svd.cols());
00222     else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
00223     m_adjoint.resize(svd.cols(), svd.rows());
00224   }
00225 
00226   bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
00227   {
00228     if(matrix.cols() > matrix.rows())
00229     {
00230       m_adjoint = matrix.adjoint();
00231       m_qr.compute(m_adjoint);
00232 
00233       svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
00234       if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
00235       else if(svd.m_computeThinV)
00236       {
00237         svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
00238         m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
00239       }
00240       if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
00241       return true;
00242     }
00243     else return false;
00244   }
00245 
00246 private:
00247   typedef ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
00248   QRType m_qr;
00249   TransposeTypeWithSameStorageOrder m_adjoint;
00250   typename internal::plain_row_type<MatrixType>::type m_workspace;
00251 };
00252 
00253 /*** preconditioner using HouseholderQR ***/
00254 
00255 template<typename MatrixType>
00256 class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
00257 {
00258 public:
00259   typedef typename MatrixType::Index Index;
00260 
00261   void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
00262   {
00263     if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
00264     {
00265       m_qr.~QRType();
00266       ::new (&m_qr) QRType(svd.rows(), svd.cols());
00267     }
00268     if (svd.m_computeFullU) m_workspace.resize(svd.rows());
00269     else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
00270   }
00271 
00272   bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
00273   {
00274     if(matrix.rows() > matrix.cols())
00275     {
00276       m_qr.compute(matrix);
00277       svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
00278       if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
00279       else if(svd.m_computeThinU)
00280       {
00281         svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
00282         m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
00283       }
00284       if(svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols());
00285       return true;
00286     }
00287     return false;
00288   }
00289 private:
00290   typedef HouseholderQR<MatrixType> QRType;
00291   QRType m_qr;
00292   typename internal::plain_col_type<MatrixType>::type m_workspace;
00293 };
00294 
00295 template<typename MatrixType>
00296 class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
00297 {
00298 public:
00299   typedef typename MatrixType::Index Index;
00300   typedef typename MatrixType::Scalar Scalar;
00301   enum
00302   {
00303     RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00304     ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00305     MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00306     MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
00307     Options = MatrixType::Options
00308   };
00309 
00310   typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
00311           TransposeTypeWithSameStorageOrder;
00312 
00313   void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
00314   {
00315     if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
00316     {
00317       m_qr.~QRType();
00318       ::new (&m_qr) QRType(svd.cols(), svd.rows());
00319     }
00320     if (svd.m_computeFullV) m_workspace.resize(svd.cols());
00321     else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
00322     m_adjoint.resize(svd.cols(), svd.rows());
00323   }
00324 
00325   bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
00326   {
00327     if(matrix.cols() > matrix.rows())
00328     {
00329       m_adjoint = matrix.adjoint();
00330       m_qr.compute(m_adjoint);
00331 
00332       svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
00333       if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
00334       else if(svd.m_computeThinV)
00335       {
00336         svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
00337         m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
00338       }
00339       if(svd.computeU()) svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows());
00340       return true;
00341     }
00342     else return false;
00343   }
00344 
00345 private:
00346   typedef HouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
00347   QRType m_qr;
00348   TransposeTypeWithSameStorageOrder m_adjoint;
00349   typename internal::plain_row_type<MatrixType>::type m_workspace;
00350 };
00351 
00352 /*** 2x2 SVD implementation
00353  ***
00354  *** JacobiSVD consists in performing a series of 2x2 SVD subproblems
00355  ***/
00356 
00357 template<typename MatrixType, int QRPreconditioner>
00358 struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, false>
00359 {
00360   typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
00361   typedef typename SVD::Index Index;
00362   static void run(typename SVD::WorkMatrixType&, SVD&, Index, Index) {}
00363 };
00364 
00365 template<typename MatrixType, int QRPreconditioner>
00366 struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true>
00367 {
00368   typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
00369   typedef typename MatrixType::Scalar Scalar;
00370   typedef typename MatrixType::RealScalar RealScalar;
00371   typedef typename SVD::Index Index;
00372   static void run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q)
00373   {
00374     using std::sqrt;
00375     Scalar z;
00376     JacobiRotation<Scalar> rot;
00377     RealScalar n = sqrt(numext::abs2(work_matrix.coeff(p,p)) + numext::abs2(work_matrix.coeff(q,p)));
00378     
00379     if(n==0)
00380     {
00381       z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
00382       work_matrix.row(p) *= z;
00383       if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z);
00384       if(work_matrix.coeff(q,q)!=Scalar(0))
00385       {
00386         z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
00387         work_matrix.row(q) *= z;
00388         if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
00389       }
00390       // otherwise the second row is already zero, so we have nothing to do.
00391     }
00392     else
00393     {
00394       rot.c() = conj(work_matrix.coeff(p,p)) / n;
00395       rot.s() = work_matrix.coeff(q,p) / n;
00396       work_matrix.applyOnTheLeft(p,q,rot);
00397       if(svd.computeU()) svd.m_matrixU.applyOnTheRight(p,q,rot.adjoint());
00398       if(work_matrix.coeff(p,q) != Scalar(0))
00399       {
00400         Scalar z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
00401         work_matrix.col(q) *= z;
00402         if(svd.computeV()) svd.m_matrixV.col(q) *= z;
00403       }
00404       if(work_matrix.coeff(q,q) != Scalar(0))
00405       {
00406         z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
00407         work_matrix.row(q) *= z;
00408         if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
00409       }
00410     }
00411   }
00412 };
00413 
00414 template<typename MatrixType, typename RealScalar, typename Index>
00415 void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q,
00416                             JacobiRotation<RealScalar> *j_left,
00417                             JacobiRotation<RealScalar> *j_right)
00418 {
00419   using std::sqrt;
00420   using std::abs;
00421   Matrix<RealScalar,2,2> m;
00422   m << numext::real(matrix.coeff(p,p)), numext::real(matrix.coeff(p,q)),
00423        numext::real(matrix.coeff(q,p)), numext::real(matrix.coeff(q,q));
00424   JacobiRotation<RealScalar> rot1;
00425   RealScalar t = m.coeff(0,0) + m.coeff(1,1);
00426   RealScalar d = m.coeff(1,0) - m.coeff(0,1);
00427   if(t == RealScalar(0))
00428   {
00429     rot1.c() = RealScalar(0);
00430     rot1.s() = d > RealScalar(0) ? RealScalar(1) : RealScalar(-1);
00431   }
00432   else
00433   {
00434     RealScalar t2d2 = numext::hypot(t,d);
00435     rot1.c() = abs(t)/t2d2;
00436     rot1.s() = d/t2d2;
00437     if(t<RealScalar(0))
00438       rot1.s() = -rot1.s();
00439   }
00440   m.applyOnTheLeft(0,1,rot1);
00441   j_right->makeJacobi(m,0,1);
00442   *j_left  = rot1 * j_right->transpose();
00443 }
00444 
00445 } // end namespace internal
00446 
00447 /** \ingroup SVD_Module
00448   *
00449   *
00450   * \class JacobiSVD
00451   *
00452   * \brief Two-sided Jacobi SVD decomposition of a rectangular matrix
00453   *
00454   * \param MatrixType the type of the matrix of which we are computing the SVD decomposition
00455   * \param QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally
00456   *                        for the R-SVD step for non-square matrices. See discussion of possible values below.
00457   *
00458   * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product
00459   *   \f[ A = U S V^* \f]
00460   * 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;
00461   * 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
00462   * and right \em singular \em vectors of \a A respectively.
00463   *
00464   * Singular values are always sorted in decreasing order.
00465   *
00466   * 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.
00467   *
00468   * 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
00469   * 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
00470   * 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,
00471   * 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.
00472   *
00473   * Here's an example demonstrating basic usage:
00474   * \include JacobiSVD_basic.cpp
00475   * Output: \verbinclude JacobiSVD_basic.out
00476   *
00477   * This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than
00478   * 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
00479   * \a p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms.
00480   * In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.
00481   *
00482   * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to
00483   * terminate in finite (and reasonable) time.
00484   *
00485   * The possible values for QRPreconditioner are:
00486   * \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
00487   * \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR.
00488   *     Contrary to other QRs, it doesn't allow computing thin unitaries.
00489   * \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR.
00490   *     This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization
00491   *     is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive
00492   *     process is more reliable than the optimized bidiagonal SVD iterations.
00493   * \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing
00494   *     JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in
00495   *     faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking
00496   *     if QR preconditioning is needed before applying it anyway.
00497   *
00498   * \sa MatrixBase::jacobiSvd()
00499   */
00500 template<typename _MatrixType, int QRPreconditioner> class JacobiSVD
00501 {
00502   public:
00503 
00504     typedef _MatrixType MatrixType;
00505     typedef typename MatrixType::Scalar Scalar;
00506     typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
00507     typedef typename MatrixType::Index Index;
00508     enum {
00509       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00510       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00511       DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime),
00512       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00513       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
00514       MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime),
00515       MatrixOptions = MatrixType::Options
00516     };
00517 
00518     typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime,
00519                    MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime>
00520             MatrixUType ;
00521     typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime,
00522                    MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime>
00523             MatrixVType ;
00524     typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;
00525     typedef typename internal::plain_row_type<MatrixType>::type RowType;
00526     typedef typename internal::plain_col_type<MatrixType>::type ColType;
00527     typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime,
00528                    MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime>
00529             WorkMatrixType ;
00530 
00531     /** \brief Default Constructor.
00532       *
00533       * The default constructor is useful in cases in which the user intends to
00534       * perform decompositions via JacobiSVD::compute(const MatrixType&).
00535       */
00536     JacobiSVD()
00537       : m_isInitialized(false),
00538         m_isAllocated(false),
00539         m_usePrescribedThreshold(false),
00540         m_computationOptions(0),
00541         m_rows(-1), m_cols(-1), m_diagSize(0)
00542     {}
00543 
00544 
00545     /** \brief Default Constructor with memory preallocation
00546       *
00547       * Like the default constructor but with preallocation of the internal data
00548       * according to the specified problem size.
00549       * \sa JacobiSVD()
00550       */
00551     JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0)
00552       : m_isInitialized(false),
00553         m_isAllocated(false),
00554         m_usePrescribedThreshold(false),
00555         m_computationOptions(0),
00556         m_rows(-1), m_cols(-1)
00557     {
00558       allocate(rows, cols, computationOptions);
00559     }
00560 
00561     /** \brief Constructor performing the decomposition of given matrix.
00562      *
00563      * \param matrix the matrix to decompose
00564      * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
00565      *                           By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
00566      *                           #ComputeFullV, #ComputeThinV.
00567      *
00568      * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
00569      * available with the (non-default) FullPivHouseholderQR preconditioner.
00570      */
00571     JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
00572       : m_isInitialized(false),
00573         m_isAllocated(false),
00574         m_usePrescribedThreshold(false),
00575         m_computationOptions(0),
00576         m_rows(-1), m_cols(-1)
00577     {
00578       compute(matrix, computationOptions);
00579     }
00580 
00581     /** \brief Method performing the decomposition of given matrix using custom options.
00582      *
00583      * \param matrix the matrix to decompose
00584      * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
00585      *                           By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
00586      *                           #ComputeFullV, #ComputeThinV.
00587      *
00588      * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
00589      * available with the (non-default) FullPivHouseholderQR preconditioner.
00590      */
00591     JacobiSVD& compute(const MatrixType& matrix, unsigned int computationOptions);
00592 
00593     /** \brief Method performing the decomposition of given matrix using current options.
00594      *
00595      * \param matrix the matrix to decompose
00596      *
00597      * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
00598      */
00599     JacobiSVD& compute(const MatrixType& matrix)
00600     {
00601       return compute(matrix, m_computationOptions);
00602     }
00603 
00604     /** \returns the \a U matrix.
00605      *
00606      * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p,
00607      * the U matrix is n-by-n if you asked for #ComputeFullU, and is n-by-m if you asked for #ComputeThinU.
00608      *
00609      * The \a m first columns of \a U are the left singular vectors of the matrix being decomposed.
00610      *
00611      * This method asserts that you asked for \a U to be computed.
00612      */
00613     const MatrixUType & matrixU () const
00614     {
00615       eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
00616       eigen_assert(computeU () && "This JacobiSVD decomposition didn't compute U. Did you ask for it?");
00617       return m_matrixU;
00618     }
00619 
00620     /** \returns the \a V matrix.
00621      *
00622      * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p,
00623      * the V matrix is p-by-p if you asked for #ComputeFullV, and is p-by-m if you asked for ComputeThinV.
00624      *
00625      * The \a m first columns of \a V are the right singular vectors of the matrix being decomposed.
00626      *
00627      * This method asserts that you asked for \a V to be computed.
00628      */
00629     const MatrixVType & matrixV () const
00630     {
00631       eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
00632       eigen_assert(computeV () && "This JacobiSVD decomposition didn't compute V. Did you ask for it?");
00633       return m_matrixV;
00634     }
00635 
00636     /** \returns the vector of singular values.
00637      *
00638      * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, the
00639      * returned vector has size \a m.  Singular values are always sorted in decreasing order.
00640      */
00641     const SingularValuesType& singularValues () const
00642     {
00643       eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
00644       return m_singularValues;
00645     }
00646 
00647     /** \returns true if \a U (full or thin) is asked for in this SVD decomposition */
00648     inline bool computeU () const { return m_computeFullU || m_computeThinU; }
00649     /** \returns true if \a V (full or thin) is asked for in this SVD decomposition */
00650     inline bool computeV () const { return m_computeFullV || m_computeThinV; }
00651 
00652     /** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A.
00653       *
00654       * \param b the right-hand-side of the equation to solve.
00655       *
00656       * \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.
00657       *
00658       * \note SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and least-squares solving.
00659       * In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$.
00660       */
00661     template<typename Rhs>
00662     inline const internal::solve_retval<JacobiSVD, Rhs>
00663     solve (const MatrixBase<Rhs>& b) const
00664     {
00665       eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
00666       eigen_assert(computeU () && computeV () && "JacobiSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
00667       return internal::solve_retval<JacobiSVD, Rhs>(*this, b.derived());
00668     }
00669 
00670     /** \returns the number of singular values that are not exactly 0 */
00671     Index nonzeroSingularValues () const
00672     {
00673       eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
00674       return m_nonzeroSingularValues;
00675     }
00676     
00677     /** \returns the rank of the matrix of which \c *this is the SVD.
00678       *
00679       * \note This method has to determine which singular values should be considered nonzero.
00680       *       For that, it uses the threshold value that you can control by calling
00681       *       setThreshold(const RealScalar&).
00682       */
00683     inline Index rank () const
00684     {
00685       using std::abs;
00686       eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
00687       if(m_singularValues.size()==0) return 0;
00688       RealScalar premultiplied_threshold = m_singularValues.coeff(0) * threshold();
00689       Index i = m_nonzeroSingularValues-1;
00690       while(i>=0 && m_singularValues.coeff(i) < premultiplied_threshold) --i;
00691       return i+1;
00692     }
00693     
00694     /** Allows to prescribe a threshold to be used by certain methods, such as rank() and solve(),
00695       * which need to determine when singular values are to be considered nonzero.
00696       * This is not used for the SVD decomposition itself.
00697       *
00698       * When it needs to get the threshold value, Eigen calls threshold().
00699       * The default is \c NumTraits<Scalar>::epsilon()
00700       *
00701       * \param threshold The new value to use as the threshold.
00702       *
00703       * A singular value will be considered nonzero if its value is strictly greater than
00704       *  \f$ \vert singular value \vert \leqslant threshold \times \vert max singular value \vert \f$.
00705       *
00706       * If you want to come back to the default behavior, call setThreshold(Default_t)
00707       */
00708     JacobiSVD& setThreshold(const RealScalar& threshold)
00709     {
00710       m_usePrescribedThreshold = true;
00711       m_prescribedThreshold = threshold;
00712       return *this;
00713     }
00714 
00715     /** Allows to come back to the default behavior, letting Eigen use its default formula for
00716       * determining the threshold.
00717       *
00718       * You should pass the special object Eigen::Default as parameter here.
00719       * \code svd.setThreshold(Eigen::Default); \endcode
00720       *
00721       * See the documentation of setThreshold(const RealScalar&).
00722       */
00723     JacobiSVD& setThreshold(Default_t)
00724     {
00725       m_usePrescribedThreshold = false;
00726       return *this;
00727     }
00728 
00729     /** Returns the threshold that will be used by certain methods such as rank().
00730       *
00731       * See the documentation of setThreshold(const RealScalar&).
00732       */
00733     RealScalar threshold() const
00734     {
00735       eigen_assert(m_isInitialized || m_usePrescribedThreshold);
00736       return m_usePrescribedThreshold ? m_prescribedThreshold
00737                                       : (std::max<Index>)(1,m_diagSize)*NumTraits<Scalar>::epsilon();
00738     }
00739 
00740     inline Index rows() const { return m_rows; }
00741     inline Index cols() const { return m_cols; }
00742 
00743   private:
00744     void allocate(Index rows, Index cols, unsigned int computationOptions);
00745     
00746     static void check_template_parameters()
00747     {
00748       EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
00749     }
00750 
00751   protected:
00752     MatrixUType m_matrixU;
00753     MatrixVType m_matrixV;
00754     SingularValuesType m_singularValues;
00755     WorkMatrixType m_workMatrix;
00756     bool m_isInitialized, m_isAllocated, m_usePrescribedThreshold;
00757     bool m_computeFullU, m_computeThinU;
00758     bool m_computeFullV, m_computeThinV;
00759     unsigned int m_computationOptions;
00760     Index m_nonzeroSingularValues, m_rows, m_cols, m_diagSize;
00761     RealScalar m_prescribedThreshold;
00762 
00763     template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex>
00764     friend struct internal::svd_precondition_2x2_block_to_be_real;
00765     template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything>
00766     friend struct internal::qr_preconditioner_impl;
00767 
00768     internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols;
00769     internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows;
00770     MatrixType m_scaledMatrix;
00771 };
00772 
00773 template<typename MatrixType, int QRPreconditioner>
00774 void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, unsigned int computationOptions)
00775 {
00776   eigen_assert(rows >= 0 && cols >= 0);
00777 
00778   if (m_isAllocated &&
00779       rows == m_rows &&
00780       cols == m_cols &&
00781       computationOptions == m_computationOptions)
00782   {
00783     return;
00784   }
00785 
00786   m_rows = rows;
00787   m_cols = cols;
00788   m_isInitialized = false;
00789   m_isAllocated = true;
00790   m_computationOptions = computationOptions;
00791   m_computeFullU = (computationOptions & ComputeFullU) != 0;
00792   m_computeThinU = (computationOptions & ComputeThinU) != 0;
00793   m_computeFullV = (computationOptions & ComputeFullV) != 0;
00794   m_computeThinV = (computationOptions & ComputeThinV) != 0;
00795   eigen_assert(!(m_computeFullU && m_computeThinU) && "JacobiSVD: you can't ask for both full and thin U");
00796   eigen_assert(!(m_computeFullV && m_computeThinV) && "JacobiSVD: you can't ask for both full and thin V");
00797   eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime==Dynamic) &&
00798               "JacobiSVD: thin U and V are only available when your matrix has a dynamic number of columns.");
00799   if (QRPreconditioner == FullPivHouseholderQRPreconditioner)
00800   {
00801       eigen_assert(!(m_computeThinU || m_computeThinV) &&
00802               "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
00803               "Use the ColPivHouseholderQR preconditioner instead.");
00804   }
00805   m_diagSize = (std::min)(m_rows, m_cols);
00806   m_singularValues.resize(m_diagSize);
00807   if(RowsAtCompileTime==Dynamic)
00808     m_matrixU.resize(m_rows, m_computeFullU ? m_rows
00809                             : m_computeThinU ? m_diagSize
00810                             : 0);
00811   if(ColsAtCompileTime==Dynamic)
00812     m_matrixV.resize(m_cols, m_computeFullV ? m_cols
00813                             : m_computeThinV ? m_diagSize
00814                             : 0);
00815   m_workMatrix.resize(m_diagSize, m_diagSize);
00816   
00817   if(m_cols>m_rows)   m_qr_precond_morecols.allocate(*this);
00818   if(m_rows>m_cols)   m_qr_precond_morerows.allocate(*this);
00819   if(m_rows!=m_cols)  m_scaledMatrix.resize(rows,cols);
00820 }
00821 
00822 template<typename MatrixType, int QRPreconditioner>
00823 JacobiSVD<MatrixType, QRPreconditioner>&
00824 JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions)
00825 {
00826   check_template_parameters();
00827   
00828   using std::abs;
00829   allocate(matrix.rows(), matrix.cols(), computationOptions);
00830 
00831   // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations,
00832   // only worsening the precision of U and V as we accumulate more rotations
00833   const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon();
00834 
00835   // limit for very small denormal numbers to be considered zero in order to avoid infinite loops (see bug 286)
00836   const RealScalar considerAsZero = RealScalar(2) * std::numeric_limits<RealScalar>::denorm_min();
00837 
00838   // Scaling factor to reduce over/under-flows
00839   RealScalar scale = matrix.cwiseAbs().maxCoeff();
00840   if(scale==RealScalar(0)) scale = RealScalar(1);
00841   
00842   /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
00843 
00844   if(m_rows!=m_cols)
00845   {
00846     m_scaledMatrix = matrix / scale;
00847     m_qr_precond_morecols.run(*this, m_scaledMatrix);
00848     m_qr_precond_morerows.run(*this, m_scaledMatrix);
00849   }
00850   else
00851   {
00852     m_workMatrix = matrix.block(0,0,m_diagSize,m_diagSize) / scale;
00853     if(m_computeFullU) m_matrixU.setIdentity(m_rows,m_rows);
00854     if(m_computeThinU) m_matrixU.setIdentity(m_rows,m_diagSize);
00855     if(m_computeFullV) m_matrixV.setIdentity(m_cols,m_cols);
00856     if(m_computeThinV) m_matrixV.setIdentity(m_cols, m_diagSize);
00857   }
00858 
00859   /*** step 2. The main Jacobi SVD iteration. ***/
00860 
00861   bool finished = false;
00862   while(!finished)
00863   {
00864     finished = true;
00865 
00866     // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix
00867 
00868     for(Index p = 1; p < m_diagSize; ++p)
00869     {
00870       for(Index q = 0; q < p; ++q)
00871       {
00872         // if this 2x2 sub-matrix is not diagonal already...
00873         // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't
00874         // keep us iterating forever. Similarly, small denormal numbers are considered zero.
00875         using std::max;
00876         RealScalar threshold = (max)(considerAsZero, precision * (max)(abs(m_workMatrix.coeff(p,p)),
00877                                                                        abs(m_workMatrix.coeff(q,q))));
00878         // We compare both values to threshold instead of calling max to be robust to NaN (See bug 791)
00879         if(abs(m_workMatrix.coeff(p,q))>threshold || abs(m_workMatrix.coeff(q,p)) > threshold)
00880         {
00881           finished = false;
00882 
00883           // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
00884           internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q);
00885           JacobiRotation<RealScalar>  j_left, j_right;
00886           internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right);
00887 
00888           // accumulate resulting Jacobi rotations
00889           m_workMatrix.applyOnTheLeft(p,q,j_left);
00890           if(computeU()) m_matrixU.applyOnTheRight(p,q,j_left.transpose());
00891 
00892           m_workMatrix.applyOnTheRight(p,q,j_right);
00893           if(computeV()) m_matrixV.applyOnTheRight(p,q,j_right);
00894         }
00895       }
00896     }
00897   }
00898 
00899   /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/
00900 
00901   for(Index i = 0; i < m_diagSize; ++i)
00902   {
00903     RealScalar a = abs(m_workMatrix.coeff(i,i));
00904     m_singularValues.coeffRef(i) = a;
00905     if(computeU() && (a!=RealScalar(0))) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a;
00906   }
00907 
00908   /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/
00909 
00910   m_nonzeroSingularValues = m_diagSize;
00911   for(Index i = 0; i < m_diagSize; i++)
00912   {
00913     Index pos;
00914     RealScalar maxRemainingSingularValue = m_singularValues.tail(m_diagSize-i).maxCoeff(&pos);
00915     if(maxRemainingSingularValue == RealScalar(0))
00916     {
00917       m_nonzeroSingularValues = i;
00918       break;
00919     }
00920     if(pos)
00921     {
00922       pos += i;
00923       std::swap(m_singularValues.coeffRef(i), m_singularValues.coeffRef(pos));
00924       if(computeU()) m_matrixU.col(pos).swap(m_matrixU.col(i));
00925       if(computeV()) m_matrixV.col(pos).swap(m_matrixV.col(i));
00926     }
00927   }
00928   
00929   m_singularValues *= scale;
00930 
00931   m_isInitialized = true;
00932   return *this;
00933 }
00934 
00935 namespace internal {
00936 template<typename _MatrixType, int QRPreconditioner, typename Rhs>
00937 struct solve_retval<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
00938   : solve_retval_base<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
00939 {
00940   typedef JacobiSVD<_MatrixType, QRPreconditioner> JacobiSVDType;
00941   EIGEN_MAKE_SOLVE_HELPERS(JacobiSVDType,Rhs)
00942 
00943   template<typename Dest> void evalTo(Dest& dst) const
00944   {
00945     eigen_assert(rhs().rows() == dec().rows());
00946 
00947     // A = U S V^*
00948     // So A^{-1} = V S^{-1} U^*
00949 
00950     Matrix<Scalar, Dynamic, Rhs::ColsAtCompileTime, 0, _MatrixType::MaxRowsAtCompileTime, Rhs::MaxColsAtCompileTime> tmp;
00951     Index rank = dec().rank();
00952     
00953     tmp.noalias() = dec().matrixU().leftCols(rank).adjoint() * rhs();
00954     tmp = dec().singularValues().head(rank).asDiagonal().inverse() * tmp;
00955     dst = dec().matrixV().leftCols(rank) * tmp;
00956   }
00957 };
00958 } // end namespace internal
00959 
00960 /** \svd_module
00961   *
00962   * \return the singular value decomposition of \c *this computed by two-sided
00963   * Jacobi transformations.
00964   *
00965   * \sa class JacobiSVD
00966   */
00967 template<typename Derived>
00968 JacobiSVD<typename MatrixBase<Derived>::PlainObject>
00969 MatrixBase<Derived>::jacobiSvd (unsigned int computationOptions) const
00970 {
00971   return JacobiSVD<PlainObject>(*this, computationOptions);
00972 }
00973 
00974 } // end namespace Eigen
00975 
00976 #endif // EIGEN_JACOBISVD_H