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;
83 explicit RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
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;
229 ColumnVectorType m_workspaceVector;
232 bool m_isInitialized;
233 bool m_matUisUptodate;
238 Scalar computeNormOfT();
239 Index findSmallSubdiagEntry(Index iu);
240 void splitOffTwoRows(Index iu,
bool computeU,
const Scalar& exshift);
241 void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
242 void initFrancisQRStep(Index il, Index iu,
const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
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 eigen_assert(matrix.
cols() == matrix.
rows());
252 Index maxIters = m_maxIters;
264 template<
typename MatrixType>
265 template<
typename HessMatrixType,
typename OrthMatrixType>
272 Index maxIters = m_maxIters;
275 m_workspaceVector.
resize(m_matT.cols());
276 Scalar* workspace = &m_workspaceVector.coeffRef(0);
282 Index iu = m_matT.cols() - 1;
286 Scalar norm = computeNormOfT();
292 Index il = findSmallSubdiagEntry(iu);
297 m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
299 m_matT.coeffRef(iu, iu-1) = Scalar(0);
305 splitOffTwoRows(iu, computeU, exshift);
312 Vector3s firstHouseholderVector(0,0,0), shiftInfo;
313 computeShift(iu, iter, exshift, shiftInfo);
315 totalIter = totalIter + 1;
316 if (totalIter > maxIters)
break;
318 initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
319 performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
323 if(totalIter <= maxIters)
328 m_isInitialized =
true;
329 m_matUisUptodate = computeU;
334 template<
typename MatrixType>
337 const Index size = m_matT.cols();
342 for (
Index j = 0; j < size; ++j)
343 norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();
348 template<
typename MatrixType>
355 Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res));
364 template<
typename MatrixType>
369 const Index size = m_matT.cols();
373 Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
374 Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
375 m_matT.coeffRef(iu,iu) += exshift;
376 m_matT.coeffRef(iu-1,iu-1) += exshift;
380 Scalar z = sqrt(abs(q));
383 rot.
makeGivens(p + z, m_matT.coeff(iu, iu-1));
385 rot.
makeGivens(p - z, m_matT.coeff(iu, iu-1));
387 m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.
adjoint());
388 m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
389 m_matT.coeffRef(iu, iu-1) = Scalar(0);
391 m_matU.applyOnTheRight(iu-1, iu, rot);
395 m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
399 template<
typename MatrixType>
404 shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
405 shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
406 shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
411 exshift += shiftInfo.coeff(0);
412 for (
Index i = 0; i <= iu; ++i)
413 m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
414 Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2));
415 shiftInfo.coeffRef(0) = Scalar(0.75) * s;
416 shiftInfo.coeffRef(1) = Scalar(0.75) * s;
417 shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
423 Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
424 s = s * s + shiftInfo.coeff(2);
428 if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
430 s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
431 s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
433 for (
Index i = 0; i <= iu; ++i)
434 m_matT.coeffRef(i,i) -= s;
441 template<
typename MatrixType>
445 Vector3s& v = firstHouseholderVector;
447 for (im = iu-2; im >= il; --im)
449 const Scalar Tmm = m_matT.coeff(im,im);
450 const Scalar r = shiftInfo.coeff(0) - Tmm;
451 const Scalar s = shiftInfo.coeff(1) - Tmm;
452 v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
453 v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
454 v.coeffRef(2) = m_matT.coeff(im+2,im+1);
458 const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2)));
459 const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1)));
466 template<
typename MatrixType>
469 eigen_assert(im >= il);
470 eigen_assert(im <= iu-2);
472 const Index size = m_matT.cols();
474 for (
Index k = im; k <= iu-2; ++k)
476 bool firstIteration = (k == im);
480 v = firstHouseholderVector;
482 v = m_matT.template block<3,1>(k,k-1);
486 v.makeHouseholder(ess, tau, beta);
488 if (beta != Scalar(0))
490 if (firstIteration && k > il)
491 m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
492 else if (!firstIteration)
493 m_matT.coeffRef(k,k-1) = beta;
496 m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
497 m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
499 m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
508 if (beta != Scalar(0))
510 m_matT.coeffRef(iu-1, iu-2) = beta;
511 m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
512 m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
514 m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
518 for (
Index i = im+2; i <= iu; ++i)
520 m_matT.coeffRef(i,i-2) = Scalar(0);
522 m_matT.coeffRef(i,i-3) = Scalar(0);
528 #endif // EIGEN_REAL_SCHUR_H HouseholderSequenceType matrixQ() const
Reconstructs the orthogonal matrix Q in the decomposition.
Definition: HessenbergDecomposition.h:234
MatrixHReturnType matrixH() const
Constructs the Hessenberg matrix H in the decomposition.
Definition: HessenbergDecomposition.h:262
Performs a real Schur decomposition of a square matrix.
Definition: RealSchur.h:54
void makeGivens(const Scalar &p, const Scalar &q, Scalar *z=0)
Definition: Jacobi.h:148
RealSchur & computeFromHessenberg(const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU)
Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T.
const MatrixType & matrixU() const
Returns the orthogonal matrix in the Schur decomposition.
Definition: RealSchur.h:127
Rotation given by a cosine-sine pair.
Definition: ForwardDeclarations.h:260
Holds information about the various numeric (i.e. scalar) types allowed by Eigen. ...
Definition: NumTraits.h:107
void makeHouseholder(EssentialPart &essential, Scalar &tau, RealScalar &beta) const
ComputationInfo info() const
Reports whether previous computation was successful.
Definition: RealSchur.h:195
Derived & derived()
Definition: EigenBase.h:44
void resize(Index rows, Index cols)
Definition: PlainObjectBase.h:252
Index rows() const
Definition: EigenBase.h:58
const MatrixType & matrixT() const
Returns the quasi-triangular matrix in the Schur decomposition.
Definition: RealSchur.h:144
Index getMaxIterations()
Returns the maximum number of iterations.
Definition: RealSchur.h:213
Definition: EigenBase.h:28
Eigen::Index Index
Definition: RealSchur.h:67
HessenbergDecomposition & compute(const EigenBase< InputType > &matrix)
Computes Hessenberg decomposition of given matrix.
Definition: HessenbergDecomposition.h:152
static const int m_maxIterationsPerRow
Maximum number of iterations per row.
Definition: RealSchur.h:223
JacobiRotation adjoint() const
Definition: Jacobi.h:62
RealSchur & compute(const EigenBase< InputType > &matrix, bool computeU=true)
Computes Schur decomposition of given matrix.
Definition: Constants.h:432
Index cols() const
Definition: EigenBase.h:61
RealSchur(Index size=RowsAtCompileTime==Dynamic?1:RowsAtCompileTime)
Default constructor.
Definition: RealSchur.h:83
RealSchur & setMaxIterations(Index maxIters)
Sets the maximum number of iterations allowed.
Definition: RealSchur.h:206
RealSchur(const EigenBase< InputType > &matrix, bool computeU=true)
Constructor; computes real Schur decomposition of given matrix.
Definition: RealSchur.h:104
ComputationInfo
Definition: Constants.h:430
Definition: Constants.h:436
Derived & setConstant(Index size, const Scalar &value)
Definition: CwiseNullaryOp.h:352