11 #ifndef EIGEN_REAL_SCHUR_H 12 #define EIGEN_REAL_SCHUR_H 14 #include "./HessenbergDecomposition.h" 57 typedef _MatrixType MatrixType;
59 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
60 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
61 Options = MatrixType::Options,
62 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
63 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
65 typedef typename MatrixType::Scalar Scalar;
66 typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
86 m_workspaceVector(size),
88 m_isInitialized(false),
89 m_matUisUptodate(false),
103 template<
typename InputType>
105 : m_matT(matrix.rows(),matrix.cols()),
106 m_matU(matrix.rows(),matrix.cols()),
107 m_workspaceVector(matrix.rows()),
108 m_hess(matrix.rows()),
109 m_isInitialized(false),
110 m_matUisUptodate(false),
129 eigen_assert(m_isInitialized &&
"RealSchur is not initialized.");
130 eigen_assert(m_matUisUptodate &&
"The matrix U has not been computed during the RealSchur decomposition.");
146 eigen_assert(m_isInitialized &&
"RealSchur is not initialized.");
169 template<
typename InputType>
189 template<
typename HessMatrixType,
typename OrthMatrixType>
197 eigen_assert(m_isInitialized &&
"RealSchur is not initialized.");
208 m_maxIters = maxIters;
232 bool m_isInitialized;
233 bool m_matUisUptodate;
238 Scalar computeNormOfT();
239 Index findSmallSubdiagEntry(
Index iu,
const Scalar& considerAsZero);
240 void splitOffTwoRows(
Index iu,
bool computeU,
const Scalar& exshift);
243 void performFrancisQRStep(
Index il,
Index im,
Index iu,
bool computeU,
const Vector3s& firstHouseholderVector, Scalar* workspace);
247 template<
typename MatrixType>
248 template<
typename InputType>
251 const Scalar considerAsZero = (std::numeric_limits<Scalar>::min)();
253 eigen_assert(matrix.
cols() == matrix.
rows());
254 Index maxIters = m_maxIters;
256 maxIters = m_maxIterationsPerRow * matrix.
rows();
258 Scalar scale = matrix.
derived().cwiseAbs().maxCoeff();
259 if(scale<considerAsZero)
261 m_matT.setZero(matrix.
rows(),matrix.
cols());
263 m_matU.setIdentity(matrix.
rows(),matrix.
cols());
265 m_isInitialized =
true;
266 m_matUisUptodate = computeU;
271 m_hess.compute(matrix.
derived()/scale);
274 computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU);
280 template<
typename MatrixType>
281 template<
typename HessMatrixType,
typename OrthMatrixType>
290 Index maxIters = m_maxIters;
292 maxIters = m_maxIterationsPerRow * matrixH.rows();
293 m_workspaceVector.resize(m_matT.cols());
294 Scalar* workspace = &m_workspaceVector.coeffRef(0);
300 Index iu = m_matT.cols() - 1;
304 Scalar norm = computeNormOfT();
307 Scalar considerAsZero = numext::maxi<Scalar>( norm * numext::abs2(NumTraits<Scalar>::epsilon()),
308 (std::numeric_limits<Scalar>::min)() );
314 Index il = findSmallSubdiagEntry(iu,considerAsZero);
319 m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
321 m_matT.coeffRef(iu, iu-1) = Scalar(0);
327 splitOffTwoRows(iu, computeU, exshift);
334 Vector3s firstHouseholderVector = Vector3s::Zero(), shiftInfo;
335 computeShift(iu, iter, exshift, shiftInfo);
337 totalIter = totalIter + 1;
338 if (totalIter > maxIters)
break;
340 initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
341 performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
345 if(totalIter <= maxIters)
350 m_isInitialized =
true;
351 m_matUisUptodate = computeU;
356 template<
typename MatrixType>
357 inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
359 const Index size = m_matT.cols();
364 for (Index j = 0; j < size; ++j)
365 norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();
370 template<
typename MatrixType>
371 inline Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu,
const Scalar& considerAsZero)
377 Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res));
379 s = numext::maxi<Scalar>(s * NumTraits<Scalar>::epsilon(), considerAsZero);
381 if (abs(m_matT.coeff(res,res-1)) <= s)
389 template<
typename MatrixType>
390 inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu,
bool computeU,
const Scalar& exshift)
394 const Index size = m_matT.cols();
398 Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
399 Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
400 m_matT.coeffRef(iu,iu) += exshift;
401 m_matT.coeffRef(iu-1,iu-1) += exshift;
405 Scalar z = sqrt(abs(q));
406 JacobiRotation<Scalar> rot;
408 rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
410 rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
412 m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
413 m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
414 m_matT.coeffRef(iu, iu-1) = Scalar(0);
416 m_matU.applyOnTheRight(iu-1, iu, rot);
420 m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
424 template<
typename MatrixType>
425 inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
429 shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
430 shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
431 shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
436 exshift += shiftInfo.coeff(0);
437 for (Index i = 0; i <= iu; ++i)
438 m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
439 Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2));
440 shiftInfo.coeffRef(0) = Scalar(0.75) * s;
441 shiftInfo.coeffRef(1) = Scalar(0.75) * s;
442 shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
448 Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
449 s = s * s + shiftInfo.coeff(2);
453 if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
455 s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
456 s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
458 for (Index i = 0; i <= iu; ++i)
459 m_matT.coeffRef(i,i) -= s;
460 shiftInfo.setConstant(Scalar(0.964));
466 template<
typename MatrixType>
467 inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu,
const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
470 Vector3s& v = firstHouseholderVector;
472 for (im = iu-2; im >= il; --im)
474 const Scalar Tmm = m_matT.coeff(im,im);
475 const Scalar r = shiftInfo.coeff(0) - Tmm;
476 const Scalar s = shiftInfo.coeff(1) - Tmm;
477 v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
478 v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
479 v.coeffRef(2) = m_matT.coeff(im+2,im+1);
483 const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2)));
484 const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1)));
485 if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
491 template<
typename MatrixType>
492 inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu,
bool computeU,
const Vector3s& firstHouseholderVector, Scalar* workspace)
494 eigen_assert(im >= il);
495 eigen_assert(im <= iu-2);
497 const Index size = m_matT.cols();
499 for (Index k = im; k <= iu-2; ++k)
501 bool firstIteration = (k == im);
505 v = firstHouseholderVector;
507 v = m_matT.template block<3,1>(k,k-1);
510 Matrix<Scalar, 2, 1> ess;
511 v.makeHouseholder(ess, tau, beta);
513 if (beta != Scalar(0))
515 if (firstIteration && k > il)
516 m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
517 else if (!firstIteration)
518 m_matT.coeffRef(k,k-1) = beta;
521 m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
522 m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
524 m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
528 Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);
530 Matrix<Scalar, 1, 1> ess;
531 v.makeHouseholder(ess, tau, beta);
533 if (beta != Scalar(0))
535 m_matT.coeffRef(iu-1, iu-2) = beta;
536 m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
537 m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
539 m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
543 for (Index i = im+2; i <= iu; ++i)
545 m_matT.coeffRef(i,i-2) = Scalar(0);
547 m_matT.coeffRef(i,i-3) = Scalar(0);
553 #endif // EIGEN_REAL_SCHUR_H Performs a real Schur decomposition of a square matrix.
Definition: RealSchur.h:54
RealSchur & computeFromHessenberg(const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU)
Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T.
Namespace containing all symbols from the Eigen library.
Definition: Core:309
Derived & derived()
Definition: EigenBase.h:45
RealSchur(Index size=RowsAtCompileTime==Dynamic ? 1 :RowsAtCompileTime)
Default constructor.
Definition: RealSchur.h:83
Index getMaxIterations()
Returns the maximum number of iterations.
Definition: RealSchur.h:213
Definition: EigenBase.h:29
Eigen::Index Index
Definition: RealSchur.h:67
Index rows() const
Definition: EigenBase.h:59
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
The Index type as used for the API.
Definition: Meta.h:33
static const int m_maxIterationsPerRow
Maximum number of iterations per row.
Definition: RealSchur.h:223
RealSchur & compute(const EigenBase< InputType > &matrix, bool computeU=true)
Computes Schur decomposition of given matrix.
const MatrixType & matrixT() const
Returns the quasi-triangular matrix in the Schur decomposition.
Definition: RealSchur.h:144
Definition: Constants.h:432
const MatrixType & matrixU() const
Returns the orthogonal matrix in the Schur decomposition.
Definition: RealSchur.h:127
RealSchur & setMaxIterations(Index maxIters)
Sets the maximum number of iterations allowed.
Definition: RealSchur.h:206
const int Dynamic
Definition: Constants.h:21
ComputationInfo info() const
Reports whether previous computation was successful.
Definition: RealSchur.h:195
RealSchur(const EigenBase< InputType > &matrix, bool computeU=true)
Constructor; computes real Schur decomposition of given matrix.
Definition: RealSchur.h:104
Index cols() const
Definition: EigenBase.h:62
ComputationInfo
Definition: Constants.h:430
Definition: Constants.h:436