JacobiSVD.h
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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     if(n==0)
00379     {
00380       z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
00381       work_matrix.row(p) *= z;
00382       if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z);
00383       z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
00384       work_matrix.row(q) *= z;
00385       if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
00386     }
00387     else
00388     {
00389       rot.c() = conj(work_matrix.coeff(p,p)) / n;
00390       rot.s() = work_matrix.coeff(q,p) / n;
00391       work_matrix.applyOnTheLeft(p,q,rot);
00392       if(svd.computeU()) svd.m_matrixU.applyOnTheRight(p,q,rot.adjoint());
00393       if(work_matrix.coeff(p,q) != Scalar(0))
00394       {
00395         Scalar z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
00396         work_matrix.col(q) *= z;
00397         if(svd.computeV()) svd.m_matrixV.col(q) *= z;
00398       }
00399       if(work_matrix.coeff(q,q) != Scalar(0))
00400       {
00401         z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
00402         work_matrix.row(q) *= z;
00403         if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
00404       }
00405     }
00406   }
00407 };
00408 
00409 template<typename MatrixType, typename RealScalar, typename Index>
00410 void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q,
00411                             JacobiRotation<RealScalar> *j_left,
00412                             JacobiRotation<RealScalar> *j_right)
00413 {
00414   using std::sqrt;
00415   Matrix<RealScalar,2,2> m;
00416   m << numext::real(matrix.coeff(p,p)), numext::real(matrix.coeff(p,q)),
00417        numext::real(matrix.coeff(q,p)), numext::real(matrix.coeff(q,q));
00418   JacobiRotation<RealScalar> rot1;
00419   RealScalar t = m.coeff(0,0) + m.coeff(1,1);
00420   RealScalar d = m.coeff(1,0) - m.coeff(0,1);
00421   if(t == RealScalar(0))
00422   {
00423     rot1.c() = RealScalar(0);
00424     rot1.s() = d > RealScalar(0) ? RealScalar(1) : RealScalar(-1);
00425   }
00426   else
00427   {
00428     RealScalar u = d / t;
00429     rot1.c() = RealScalar(1) / sqrt(RealScalar(1) + numext::abs2(u));
00430     rot1.s() = rot1.c() * u;
00431   }
00432   m.applyOnTheLeft(0,1,rot1);
00433   j_right->makeJacobi(m,0,1);
00434   *j_left  = rot1 * j_right->transpose();
00435 }
00436 
00437 } // end namespace internal
00438 
00492 template<typename _MatrixType, int QRPreconditioner> class JacobiSVD
00493 {
00494   public:
00495 
00496     typedef _MatrixType MatrixType;
00497     typedef typename MatrixType::Scalar Scalar;
00498     typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
00499     typedef typename MatrixType::Index Index;
00500     enum {
00501       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00502       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00503       DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime),
00504       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00505       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
00506       MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime),
00507       MatrixOptions = MatrixType::Options
00508     };
00509 
00510     typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime,
00511                    MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime>
00512             MatrixUType;
00513     typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime,
00514                    MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime>
00515             MatrixVType;
00516     typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;
00517     typedef typename internal::plain_row_type<MatrixType>::type RowType;
00518     typedef typename internal::plain_col_type<MatrixType>::type ColType;
00519     typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime,
00520                    MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime>
00521             WorkMatrixType;
00522 
00528     JacobiSVD()
00529       : m_isInitialized(false),
00530         m_isAllocated(false),
00531         m_computationOptions(0),
00532         m_rows(-1), m_cols(-1)
00533     {}
00534 
00535 
00542     JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0)
00543       : m_isInitialized(false),
00544         m_isAllocated(false),
00545         m_computationOptions(0),
00546         m_rows(-1), m_cols(-1)
00547     {
00548       allocate(rows, cols, computationOptions);
00549     }
00550 
00561     JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
00562       : m_isInitialized(false),
00563         m_isAllocated(false),
00564         m_computationOptions(0),
00565         m_rows(-1), m_cols(-1)
00566     {
00567       compute(matrix, computationOptions);
00568     }
00569 
00580     JacobiSVD& compute(const MatrixType& matrix, unsigned int computationOptions);
00581 
00588     JacobiSVD& compute(const MatrixType& matrix)
00589     {
00590       return compute(matrix, m_computationOptions);
00591     }
00592 
00602     const MatrixUType& matrixU() const
00603     {
00604       eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
00605       eigen_assert(computeU() && "This JacobiSVD decomposition didn't compute U. Did you ask for it?");
00606       return m_matrixU;
00607     }
00608 
00618     const MatrixVType& matrixV() const
00619     {
00620       eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
00621       eigen_assert(computeV() && "This JacobiSVD decomposition didn't compute V. Did you ask for it?");
00622       return m_matrixV;
00623     }
00624 
00630     const SingularValuesType& singularValues() const
00631     {
00632       eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
00633       return m_singularValues;
00634     }
00635 
00637     inline bool computeU() const { return m_computeFullU || m_computeThinU; }
00639     inline bool computeV() const { return m_computeFullV || m_computeThinV; }
00640 
00650     template<typename Rhs>
00651     inline const internal::solve_retval<JacobiSVD, Rhs>
00652     solve(const MatrixBase<Rhs>& b) const
00653     {
00654       eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
00655       eigen_assert(computeU() && computeV() && "JacobiSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
00656       return internal::solve_retval<JacobiSVD, Rhs>(*this, b.derived());
00657     }
00658 
00660     Index nonzeroSingularValues() const
00661     {
00662       eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
00663       return m_nonzeroSingularValues;
00664     }
00665 
00666     inline Index rows() const { return m_rows; }
00667     inline Index cols() const { return m_cols; }
00668 
00669   private:
00670     void allocate(Index rows, Index cols, unsigned int computationOptions);
00671 
00672   protected:
00673     MatrixUType m_matrixU;
00674     MatrixVType m_matrixV;
00675     SingularValuesType m_singularValues;
00676     WorkMatrixType m_workMatrix;
00677     bool m_isInitialized, m_isAllocated;
00678     bool m_computeFullU, m_computeThinU;
00679     bool m_computeFullV, m_computeThinV;
00680     unsigned int m_computationOptions;
00681     Index m_nonzeroSingularValues, m_rows, m_cols, m_diagSize;
00682 
00683     template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex>
00684     friend struct internal::svd_precondition_2x2_block_to_be_real;
00685     template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything>
00686     friend struct internal::qr_preconditioner_impl;
00687 
00688     internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols;
00689     internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows;
00690 };
00691 
00692 template<typename MatrixType, int QRPreconditioner>
00693 void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, unsigned int computationOptions)
00694 {
00695   eigen_assert(rows >= 0 && cols >= 0);
00696 
00697   if (m_isAllocated &&
00698       rows == m_rows &&
00699       cols == m_cols &&
00700       computationOptions == m_computationOptions)
00701   {
00702     return;
00703   }
00704 
00705   m_rows = rows;
00706   m_cols = cols;
00707   m_isInitialized = false;
00708   m_isAllocated = true;
00709   m_computationOptions = computationOptions;
00710   m_computeFullU = (computationOptions & ComputeFullU) != 0;
00711   m_computeThinU = (computationOptions & ComputeThinU) != 0;
00712   m_computeFullV = (computationOptions & ComputeFullV) != 0;
00713   m_computeThinV = (computationOptions & ComputeThinV) != 0;
00714   eigen_assert(!(m_computeFullU && m_computeThinU) && "JacobiSVD: you can't ask for both full and thin U");
00715   eigen_assert(!(m_computeFullV && m_computeThinV) && "JacobiSVD: you can't ask for both full and thin V");
00716   eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime==Dynamic) &&
00717               "JacobiSVD: thin U and V are only available when your matrix has a dynamic number of columns.");
00718   if (QRPreconditioner == FullPivHouseholderQRPreconditioner)
00719   {
00720       eigen_assert(!(m_computeThinU || m_computeThinV) &&
00721               "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
00722               "Use the ColPivHouseholderQR preconditioner instead.");
00723   }
00724   m_diagSize = (std::min)(m_rows, m_cols);
00725   m_singularValues.resize(m_diagSize);
00726   if(RowsAtCompileTime==Dynamic)
00727     m_matrixU.resize(m_rows, m_computeFullU ? m_rows
00728                             : m_computeThinU ? m_diagSize
00729                             : 0);
00730   if(ColsAtCompileTime==Dynamic)
00731     m_matrixV.resize(m_cols, m_computeFullV ? m_cols
00732                             : m_computeThinV ? m_diagSize
00733                             : 0);
00734   m_workMatrix.resize(m_diagSize, m_diagSize);
00735   
00736   if(m_cols>m_rows) m_qr_precond_morecols.allocate(*this);
00737   if(m_rows>m_cols) m_qr_precond_morerows.allocate(*this);
00738 }
00739 
00740 template<typename MatrixType, int QRPreconditioner>
00741 JacobiSVD<MatrixType, QRPreconditioner>&
00742 JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions)
00743 {
00744   using std::abs;
00745   allocate(matrix.rows(), matrix.cols(), computationOptions);
00746 
00747   // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations,
00748   // only worsening the precision of U and V as we accumulate more rotations
00749   const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon();
00750 
00751   // limit for very small denormal numbers to be considered zero in order to avoid infinite loops (see bug 286)
00752   const RealScalar considerAsZero = RealScalar(2) * std::numeric_limits<RealScalar>::denorm_min();
00753 
00754   /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
00755 
00756   if(!m_qr_precond_morecols.run(*this, matrix) && !m_qr_precond_morerows.run(*this, matrix))
00757   {
00758     m_workMatrix = matrix.block(0,0,m_diagSize,m_diagSize);
00759     if(m_computeFullU) m_matrixU.setIdentity(m_rows,m_rows);
00760     if(m_computeThinU) m_matrixU.setIdentity(m_rows,m_diagSize);
00761     if(m_computeFullV) m_matrixV.setIdentity(m_cols,m_cols);
00762     if(m_computeThinV) m_matrixV.setIdentity(m_cols, m_diagSize);
00763   }
00764 
00765   /*** step 2. The main Jacobi SVD iteration. ***/
00766 
00767   bool finished = false;
00768   while(!finished)
00769   {
00770     finished = true;
00771 
00772     // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix
00773 
00774     for(Index p = 1; p < m_diagSize; ++p)
00775     {
00776       for(Index q = 0; q < p; ++q)
00777       {
00778         // if this 2x2 sub-matrix is not diagonal already...
00779         // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't
00780         // keep us iterating forever. Similarly, small denormal numbers are considered zero.
00781         using std::max;
00782         RealScalar threshold = (max)(considerAsZero, precision * (max)(abs(m_workMatrix.coeff(p,p)),
00783                                                                        abs(m_workMatrix.coeff(q,q))));
00784         if((max)(abs(m_workMatrix.coeff(p,q)),abs(m_workMatrix.coeff(q,p))) > threshold)
00785         {
00786           finished = false;
00787 
00788           // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
00789           internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q);
00790           JacobiRotation<RealScalar> j_left, j_right;
00791           internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right);
00792 
00793           // accumulate resulting Jacobi rotations
00794           m_workMatrix.applyOnTheLeft(p,q,j_left);
00795           if(computeU()) m_matrixU.applyOnTheRight(p,q,j_left.transpose());
00796 
00797           m_workMatrix.applyOnTheRight(p,q,j_right);
00798           if(computeV()) m_matrixV.applyOnTheRight(p,q,j_right);
00799         }
00800       }
00801     }
00802   }
00803 
00804   /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/
00805 
00806   for(Index i = 0; i < m_diagSize; ++i)
00807   {
00808     RealScalar a = abs(m_workMatrix.coeff(i,i));
00809     m_singularValues.coeffRef(i) = a;
00810     if(computeU() && (a!=RealScalar(0))) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a;
00811   }
00812 
00813   /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/
00814 
00815   m_nonzeroSingularValues = m_diagSize;
00816   for(Index i = 0; i < m_diagSize; i++)
00817   {
00818     Index pos;
00819     RealScalar maxRemainingSingularValue = m_singularValues.tail(m_diagSize-i).maxCoeff(&pos);
00820     if(maxRemainingSingularValue == RealScalar(0))
00821     {
00822       m_nonzeroSingularValues = i;
00823       break;
00824     }
00825     if(pos)
00826     {
00827       pos += i;
00828       std::swap(m_singularValues.coeffRef(i), m_singularValues.coeffRef(pos));
00829       if(computeU()) m_matrixU.col(pos).swap(m_matrixU.col(i));
00830       if(computeV()) m_matrixV.col(pos).swap(m_matrixV.col(i));
00831     }
00832   }
00833 
00834   m_isInitialized = true;
00835   return *this;
00836 }
00837 
00838 namespace internal {
00839 template<typename _MatrixType, int QRPreconditioner, typename Rhs>
00840 struct solve_retval<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
00841   : solve_retval_base<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
00842 {
00843   typedef JacobiSVD<_MatrixType, QRPreconditioner> JacobiSVDType;
00844   EIGEN_MAKE_SOLVE_HELPERS(JacobiSVDType,Rhs)
00845 
00846   template<typename Dest> void evalTo(Dest& dst) const
00847   {
00848     eigen_assert(rhs().rows() == dec().rows());
00849 
00850     // A = U S V^*
00851     // So A^{-1} = V S^{-1} U^*
00852 
00853     Matrix<Scalar, Dynamic, Rhs::ColsAtCompileTime, 0, _MatrixType::MaxRowsAtCompileTime, Rhs::MaxColsAtCompileTime> tmp;
00854     Index nonzeroSingVals = dec().nonzeroSingularValues();
00855     
00856     tmp.noalias() = dec().matrixU().leftCols(nonzeroSingVals).adjoint() * rhs();
00857     tmp = dec().singularValues().head(nonzeroSingVals).asDiagonal().inverse() * tmp;
00858     dst = dec().matrixV().leftCols(nonzeroSingVals) * tmp;
00859   }
00860 };
00861 } // end namespace internal
00862 
00870 template<typename Derived>
00871 JacobiSVD<typename MatrixBase<Derived>::PlainObject>
00872 MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const
00873 {
00874   return JacobiSVD<PlainObject>(*this, computationOptions);
00875 }
00876 
00877 } // end namespace Eigen
00878 
00879 #endif // EIGEN_JACOBISVD_H


acado
Author(s): Milan Vukov, Rien Quirynen
autogenerated on Sat Jun 8 2019 19:37:43