RealSchur.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) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
00005 // Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
00006 //
00007 // Eigen is free software; you can redistribute it and/or
00008 // modify it under the terms of the GNU Lesser General Public
00009 // License as published by the Free Software Foundation; either
00010 // version 3 of the License, or (at your option) any later version.
00011 //
00012 // Alternatively, you can redistribute it and/or
00013 // modify it under the terms of the GNU General Public License as
00014 // published by the Free Software Foundation; either version 2 of
00015 // the License, or (at your option) any later version.
00016 //
00017 // Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
00018 // WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
00019 // FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
00020 // GNU General Public License for more details.
00021 //
00022 // You should have received a copy of the GNU Lesser General Public
00023 // License and a copy of the GNU General Public License along with
00024 // Eigen. If not, see <http://www.gnu.org/licenses/>.
00025 
00026 #ifndef EIGEN_REAL_SCHUR_H
00027 #define EIGEN_REAL_SCHUR_H
00028 
00029 #include "./EigenvaluesCommon.h"
00030 #include "./HessenbergDecomposition.h"
00031 
00068 template<typename _MatrixType> class RealSchur
00069 {
00070   public:
00071     typedef _MatrixType MatrixType;
00072     enum {
00073       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00074       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00075       Options = MatrixType::Options,
00076       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00077       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
00078     };
00079     typedef typename MatrixType::Scalar Scalar;
00080     typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
00081     typedef typename MatrixType::Index Index;
00082 
00083     typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
00084     typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
00085 
00097     RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
00098             : m_matT(size, size),
00099               m_matU(size, size),
00100               m_workspaceVector(size),
00101               m_hess(size),
00102               m_isInitialized(false),
00103               m_matUisUptodate(false)
00104     { }
00105 
00116     RealSchur(const MatrixType& matrix, bool computeU = true)
00117             : m_matT(matrix.rows(),matrix.cols()),
00118               m_matU(matrix.rows(),matrix.cols()),
00119               m_workspaceVector(matrix.rows()),
00120               m_hess(matrix.rows()),
00121               m_isInitialized(false),
00122               m_matUisUptodate(false)
00123     {
00124       compute(matrix, computeU);
00125     }
00126 
00138     const MatrixType& matrixU() const
00139     {
00140       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
00141       eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
00142       return m_matU;
00143     }
00144 
00155     const MatrixType& matrixT() const
00156     {
00157       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
00158       return m_matT;
00159     }
00160   
00178     RealSchur& compute(const MatrixType& matrix, bool computeU = true);
00179 
00184     ComputationInfo info() const
00185     {
00186       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
00187       return m_info;
00188     }
00189 
00194     static const int m_maxIterations = 40;
00195 
00196   private:
00197     
00198     MatrixType m_matT;
00199     MatrixType m_matU;
00200     ColumnVectorType m_workspaceVector;
00201     HessenbergDecomposition<MatrixType> m_hess;
00202     ComputationInfo m_info;
00203     bool m_isInitialized;
00204     bool m_matUisUptodate;
00205 
00206     typedef Matrix<Scalar,3,1> Vector3s;
00207 
00208     Scalar computeNormOfT();
00209     Index findSmallSubdiagEntry(Index iu, Scalar norm);
00210     void splitOffTwoRows(Index iu, bool computeU, Scalar exshift);
00211     void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
00212     void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
00213     void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace);
00214 };
00215 
00216 
00217 template<typename MatrixType>
00218 RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU)
00219 {
00220   assert(matrix.cols() == matrix.rows());
00221 
00222   // Step 1. Reduce to Hessenberg form
00223   m_hess.compute(matrix);
00224   m_matT = m_hess.matrixH();
00225   if (computeU)
00226     m_matU = m_hess.matrixQ();
00227 
00228   // Step 2. Reduce to real Schur form  
00229   m_workspaceVector.resize(m_matT.cols());
00230   Scalar* workspace = &m_workspaceVector.coeffRef(0);
00231 
00232   // The matrix m_matT is divided in three parts. 
00233   // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero. 
00234   // Rows il,...,iu is the part we are working on (the active window).
00235   // Rows iu+1,...,end are already brought in triangular form.
00236   Index iu = m_matT.cols() - 1;
00237   Index iter = 0; // iteration count
00238   Scalar exshift = 0.0; // sum of exceptional shifts
00239   Scalar norm = computeNormOfT();
00240 
00241   while (iu >= 0)
00242   {
00243     Index il = findSmallSubdiagEntry(iu, norm);
00244 
00245     // Check for convergence
00246     if (il == iu) // One root found
00247     {
00248       m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
00249       if (iu > 0) 
00250         m_matT.coeffRef(iu, iu-1) = Scalar(0);
00251       iu--;
00252       iter = 0;
00253     }
00254     else if (il == iu-1) // Two roots found
00255     {
00256       splitOffTwoRows(iu, computeU, exshift);
00257       iu -= 2;
00258       iter = 0;
00259     }
00260     else // No convergence yet
00261     {
00262       // The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )
00263       Vector3s firstHouseholderVector(0,0,0), shiftInfo;
00264       computeShift(iu, iter, exshift, shiftInfo);
00265       iter = iter + 1; 
00266       if (iter > m_maxIterations) break;
00267       Index im;
00268       initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
00269       performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
00270     }
00271   } 
00272 
00273   if(iter <= m_maxIterations) 
00274     m_info = Success;
00275   else
00276     m_info = NoConvergence;
00277 
00278   m_isInitialized = true;
00279   m_matUisUptodate = computeU;
00280   return *this;
00281 }
00282 
00284 template<typename MatrixType>
00285 inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
00286 {
00287   const Index size = m_matT.cols();
00288   // FIXME to be efficient the following would requires a triangular reduxion code
00289   // Scalar norm = m_matT.upper().cwiseAbs().sum() 
00290   //               + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
00291   Scalar norm = 0.0;
00292   for (Index j = 0; j < size; ++j)
00293     norm += m_matT.row(j).segment((std::max)(j-1,Index(0)), size-(std::max)(j-1,Index(0))).cwiseAbs().sum();
00294   return norm;
00295 }
00296 
00298 template<typename MatrixType>
00299 inline typename MatrixType::Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu, Scalar norm)
00300 {
00301   Index res = iu;
00302   while (res > 0)
00303   {
00304     Scalar s = internal::abs(m_matT.coeff(res-1,res-1)) + internal::abs(m_matT.coeff(res,res));
00305     if (s == 0.0)
00306       s = norm;
00307     if (internal::abs(m_matT.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s)
00308       break;
00309     res--;
00310   }
00311   return res;
00312 }
00313 
00315 template<typename MatrixType>
00316 inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, Scalar exshift)
00317 {
00318   const Index size = m_matT.cols();
00319 
00320   // The eigenvalues of the 2x2 matrix [a b; c d] are 
00321   // trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
00322   Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
00323   Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);   // q = tr^2 / 4 - det = discr/4
00324   m_matT.coeffRef(iu,iu) += exshift;
00325   m_matT.coeffRef(iu-1,iu-1) += exshift;
00326 
00327   if (q >= Scalar(0)) // Two real eigenvalues
00328   {
00329     Scalar z = internal::sqrt(internal::abs(q));
00330     JacobiRotation<Scalar> rot;
00331     if (p >= Scalar(0))
00332       rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
00333     else
00334       rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
00335 
00336     m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
00337     m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
00338     m_matT.coeffRef(iu, iu-1) = Scalar(0); 
00339     if (computeU)
00340       m_matU.applyOnTheRight(iu-1, iu, rot);
00341   }
00342 
00343   if (iu > 1) 
00344     m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
00345 }
00346 
00348 template<typename MatrixType>
00349 inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
00350 {
00351   shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
00352   shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
00353   shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
00354 
00355   // Wilkinson's original ad hoc shift
00356   if (iter == 10)
00357   {
00358     exshift += shiftInfo.coeff(0);
00359     for (Index i = 0; i <= iu; ++i)
00360       m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
00361     Scalar s = internal::abs(m_matT.coeff(iu,iu-1)) + internal::abs(m_matT.coeff(iu-1,iu-2));
00362     shiftInfo.coeffRef(0) = Scalar(0.75) * s;
00363     shiftInfo.coeffRef(1) = Scalar(0.75) * s;
00364     shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
00365   }
00366 
00367   // MATLAB's new ad hoc shift
00368   if (iter == 30)
00369   {
00370     Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
00371     s = s * s + shiftInfo.coeff(2);
00372     if (s > Scalar(0))
00373     {
00374       s = internal::sqrt(s);
00375       if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
00376         s = -s;
00377       s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
00378       s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
00379       exshift += s;
00380       for (Index i = 0; i <= iu; ++i)
00381         m_matT.coeffRef(i,i) -= s;
00382       shiftInfo.setConstant(Scalar(0.964));
00383     }
00384   }
00385 }
00386 
00388 template<typename MatrixType>
00389 inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
00390 {
00391   Vector3s& v = firstHouseholderVector; // alias to save typing
00392 
00393   for (im = iu-2; im >= il; --im)
00394   {
00395     const Scalar Tmm = m_matT.coeff(im,im);
00396     const Scalar r = shiftInfo.coeff(0) - Tmm;
00397     const Scalar s = shiftInfo.coeff(1) - Tmm;
00398     v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
00399     v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
00400     v.coeffRef(2) = m_matT.coeff(im+2,im+1);
00401     if (im == il) {
00402       break;
00403     }
00404     const Scalar lhs = m_matT.coeff(im,im-1) * (internal::abs(v.coeff(1)) + internal::abs(v.coeff(2)));
00405     const Scalar rhs = v.coeff(0) * (internal::abs(m_matT.coeff(im-1,im-1)) + internal::abs(Tmm) + internal::abs(m_matT.coeff(im+1,im+1)));
00406     if (internal::abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
00407     {
00408       break;
00409     }
00410   }
00411 }
00412 
00414 template<typename MatrixType>
00415 inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
00416 {
00417   assert(im >= il);
00418   assert(im <= iu-2);
00419 
00420   const Index size = m_matT.cols();
00421 
00422   for (Index k = im; k <= iu-2; ++k)
00423   {
00424     bool firstIteration = (k == im);
00425 
00426     Vector3s v;
00427     if (firstIteration)
00428       v = firstHouseholderVector;
00429     else
00430       v = m_matT.template block<3,1>(k,k-1);
00431 
00432     Scalar tau, beta;
00433     Matrix<Scalar, 2, 1> ess;
00434     v.makeHouseholder(ess, tau, beta);
00435     
00436     if (beta != Scalar(0)) // if v is not zero
00437     {
00438       if (firstIteration && k > il)
00439         m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
00440       else if (!firstIteration)
00441         m_matT.coeffRef(k,k-1) = beta;
00442 
00443       // These Householder transformations form the O(n^3) part of the algorithm
00444       m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
00445       m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
00446       if (computeU)
00447         m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
00448     }
00449   }
00450 
00451   Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);
00452   Scalar tau, beta;
00453   Matrix<Scalar, 1, 1> ess;
00454   v.makeHouseholder(ess, tau, beta);
00455 
00456   if (beta != Scalar(0)) // if v is not zero
00457   {
00458     m_matT.coeffRef(iu-1, iu-2) = beta;
00459     m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
00460     m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
00461     if (computeU)
00462       m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
00463   }
00464 
00465   // clean up pollution due to round-off errors
00466   for (Index i = im+2; i <= iu; ++i)
00467   {
00468     m_matT.coeffRef(i,i-2) = Scalar(0);
00469     if (i > im+2)
00470       m_matT.coeffRef(i,i-3) = Scalar(0);
00471   }
00472 }
00473 
00474 #endif // EIGEN_REAL_SCHUR_H


libicr
Author(s): Robert Krug
autogenerated on Mon Jan 6 2014 11:33:17