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00026 #ifndef EIGEN_REAL_SCHUR_H
00027 #define EIGEN_REAL_SCHUR_H
00028
00029 #include "./HessenbergDecomposition.h"
00030
00031 namespace Eigen {
00032
00069 template<typename _MatrixType> class RealSchur
00070 {
00071 public:
00072 typedef _MatrixType MatrixType;
00073 enum {
00074 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00075 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00076 Options = MatrixType::Options,
00077 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00078 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
00079 };
00080 typedef typename MatrixType::Scalar Scalar;
00081 typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
00082 typedef typename MatrixType::Index Index;
00083
00084 typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
00085 typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
00086
00098 RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
00099 : m_matT(size, size),
00100 m_matU(size, size),
00101 m_workspaceVector(size),
00102 m_hess(size),
00103 m_isInitialized(false),
00104 m_matUisUptodate(false)
00105 { }
00106
00117 RealSchur(const MatrixType& matrix, bool computeU = true)
00118 : m_matT(matrix.rows(),matrix.cols()),
00119 m_matU(matrix.rows(),matrix.cols()),
00120 m_workspaceVector(matrix.rows()),
00121 m_hess(matrix.rows()),
00122 m_isInitialized(false),
00123 m_matUisUptodate(false)
00124 {
00125 compute(matrix, computeU);
00126 }
00127
00139 const MatrixType& matrixU() const
00140 {
00141 eigen_assert(m_isInitialized && "RealSchur is not initialized.");
00142 eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
00143 return m_matU;
00144 }
00145
00156 const MatrixType& matrixT() const
00157 {
00158 eigen_assert(m_isInitialized && "RealSchur is not initialized.");
00159 return m_matT;
00160 }
00161
00179 RealSchur& compute(const MatrixType& matrix, bool computeU = true);
00180
00185 ComputationInfo info() const
00186 {
00187 eigen_assert(m_isInitialized && "RealSchur is not initialized.");
00188 return m_info;
00189 }
00190
00195 static const int m_maxIterations = 40;
00196
00197 private:
00198
00199 MatrixType m_matT;
00200 MatrixType m_matU;
00201 ColumnVectorType m_workspaceVector;
00202 HessenbergDecomposition<MatrixType> m_hess;
00203 ComputationInfo m_info;
00204 bool m_isInitialized;
00205 bool m_matUisUptodate;
00206
00207 typedef Matrix<Scalar,3,1> Vector3s;
00208
00209 Scalar computeNormOfT();
00210 Index findSmallSubdiagEntry(Index iu, Scalar norm);
00211 void splitOffTwoRows(Index iu, bool computeU, Scalar exshift);
00212 void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
00213 void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
00214 void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace);
00215 };
00216
00217
00218 template<typename MatrixType>
00219 RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU)
00220 {
00221 assert(matrix.cols() == matrix.rows());
00222
00223
00224 m_hess.compute(matrix);
00225 m_matT = m_hess.matrixH();
00226 if (computeU)
00227 m_matU = m_hess.matrixQ();
00228
00229
00230 m_workspaceVector.resize(m_matT.cols());
00231 Scalar* workspace = &m_workspaceVector.coeffRef(0);
00232
00233
00234
00235
00236
00237 Index iu = m_matT.cols() - 1;
00238 Index iter = 0;
00239 Scalar exshift(0);
00240 Scalar norm = computeNormOfT();
00241
00242 while (iu >= 0)
00243 {
00244 Index il = findSmallSubdiagEntry(iu, norm);
00245
00246
00247 if (il == iu)
00248 {
00249 m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
00250 if (iu > 0)
00251 m_matT.coeffRef(iu, iu-1) = Scalar(0);
00252 iu--;
00253 iter = 0;
00254 }
00255 else if (il == iu-1)
00256 {
00257 splitOffTwoRows(iu, computeU, exshift);
00258 iu -= 2;
00259 iter = 0;
00260 }
00261 else
00262 {
00263
00264 Vector3s firstHouseholderVector(0,0,0), shiftInfo;
00265 computeShift(iu, iter, exshift, shiftInfo);
00266 iter = iter + 1;
00267 if (iter > m_maxIterations) break;
00268 Index im;
00269 initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
00270 performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
00271 }
00272 }
00273
00274 if(iter <= m_maxIterations)
00275 m_info = Success;
00276 else
00277 m_info = NoConvergence;
00278
00279 m_isInitialized = true;
00280 m_matUisUptodate = computeU;
00281 return *this;
00282 }
00283
00285 template<typename MatrixType>
00286 inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
00287 {
00288 const Index size = m_matT.cols();
00289
00290
00291
00292 Scalar norm(0);
00293 for (Index j = 0; j < size; ++j)
00294 norm += m_matT.row(j).segment((std::max)(j-1,Index(0)), size-(std::max)(j-1,Index(0))).cwiseAbs().sum();
00295 return norm;
00296 }
00297
00299 template<typename MatrixType>
00300 inline typename MatrixType::Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu, Scalar norm)
00301 {
00302 Index res = iu;
00303 while (res > 0)
00304 {
00305 Scalar s = internal::abs(m_matT.coeff(res-1,res-1)) + internal::abs(m_matT.coeff(res,res));
00306 if (s == 0.0)
00307 s = norm;
00308 if (internal::abs(m_matT.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s)
00309 break;
00310 res--;
00311 }
00312 return res;
00313 }
00314
00316 template<typename MatrixType>
00317 inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, Scalar exshift)
00318 {
00319 const Index size = m_matT.cols();
00320
00321
00322
00323 Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
00324 Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
00325 m_matT.coeffRef(iu,iu) += exshift;
00326 m_matT.coeffRef(iu-1,iu-1) += exshift;
00327
00328 if (q >= Scalar(0))
00329 {
00330 Scalar z = internal::sqrt(internal::abs(q));
00331 JacobiRotation<Scalar> rot;
00332 if (p >= Scalar(0))
00333 rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
00334 else
00335 rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
00336
00337 m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
00338 m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
00339 m_matT.coeffRef(iu, iu-1) = Scalar(0);
00340 if (computeU)
00341 m_matU.applyOnTheRight(iu-1, iu, rot);
00342 }
00343
00344 if (iu > 1)
00345 m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
00346 }
00347
00349 template<typename MatrixType>
00350 inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
00351 {
00352 shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
00353 shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
00354 shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
00355
00356
00357 if (iter == 10)
00358 {
00359 exshift += shiftInfo.coeff(0);
00360 for (Index i = 0; i <= iu; ++i)
00361 m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
00362 Scalar s = internal::abs(m_matT.coeff(iu,iu-1)) + internal::abs(m_matT.coeff(iu-1,iu-2));
00363 shiftInfo.coeffRef(0) = Scalar(0.75) * s;
00364 shiftInfo.coeffRef(1) = Scalar(0.75) * s;
00365 shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
00366 }
00367
00368
00369 if (iter == 30)
00370 {
00371 Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
00372 s = s * s + shiftInfo.coeff(2);
00373 if (s > Scalar(0))
00374 {
00375 s = internal::sqrt(s);
00376 if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
00377 s = -s;
00378 s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
00379 s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
00380 exshift += s;
00381 for (Index i = 0; i <= iu; ++i)
00382 m_matT.coeffRef(i,i) -= s;
00383 shiftInfo.setConstant(Scalar(0.964));
00384 }
00385 }
00386 }
00387
00389 template<typename MatrixType>
00390 inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
00391 {
00392 Vector3s& v = firstHouseholderVector;
00393
00394 for (im = iu-2; im >= il; --im)
00395 {
00396 const Scalar Tmm = m_matT.coeff(im,im);
00397 const Scalar r = shiftInfo.coeff(0) - Tmm;
00398 const Scalar s = shiftInfo.coeff(1) - Tmm;
00399 v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
00400 v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
00401 v.coeffRef(2) = m_matT.coeff(im+2,im+1);
00402 if (im == il) {
00403 break;
00404 }
00405 const Scalar lhs = m_matT.coeff(im,im-1) * (internal::abs(v.coeff(1)) + internal::abs(v.coeff(2)));
00406 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)));
00407 if (internal::abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
00408 {
00409 break;
00410 }
00411 }
00412 }
00413
00415 template<typename MatrixType>
00416 inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
00417 {
00418 assert(im >= il);
00419 assert(im <= iu-2);
00420
00421 const Index size = m_matT.cols();
00422
00423 for (Index k = im; k <= iu-2; ++k)
00424 {
00425 bool firstIteration = (k == im);
00426
00427 Vector3s v;
00428 if (firstIteration)
00429 v = firstHouseholderVector;
00430 else
00431 v = m_matT.template block<3,1>(k,k-1);
00432
00433 Scalar tau, beta;
00434 Matrix<Scalar, 2, 1> ess;
00435 v.makeHouseholder(ess, tau, beta);
00436
00437 if (beta != Scalar(0))
00438 {
00439 if (firstIteration && k > il)
00440 m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
00441 else if (!firstIteration)
00442 m_matT.coeffRef(k,k-1) = beta;
00443
00444
00445 m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
00446 m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
00447 if (computeU)
00448 m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
00449 }
00450 }
00451
00452 Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);
00453 Scalar tau, beta;
00454 Matrix<Scalar, 1, 1> ess;
00455 v.makeHouseholder(ess, tau, beta);
00456
00457 if (beta != Scalar(0))
00458 {
00459 m_matT.coeffRef(iu-1, iu-2) = beta;
00460 m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
00461 m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
00462 if (computeU)
00463 m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
00464 }
00465
00466
00467 for (Index i = im+2; i <= iu; ++i)
00468 {
00469 m_matT.coeffRef(i,i-2) = Scalar(0);
00470 if (i > im+2)
00471 m_matT.coeffRef(i,i-3) = Scalar(0);
00472 }
00473 }
00474
00475 }
00476
00477 #endif // EIGEN_REAL_SCHUR_H