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Diffstat (limited to 'Eigen/src/Eigenvalues/RealSchur.h')
| -rw-r--r-- | Eigen/src/Eigenvalues/RealSchur.h | 388 |
1 files changed, 388 insertions, 0 deletions
diff --git a/Eigen/src/Eigenvalues/RealSchur.h b/Eigen/src/Eigenvalues/RealSchur.h new file mode 100644 index 000000000..395b80089 --- /dev/null +++ b/Eigen/src/Eigenvalues/RealSchur.h @@ -0,0 +1,388 @@ +// This file is part of Eigen, a lightweight C++ template library +// for linear algebra. +// +// Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr> +// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk> +// +// Eigen is free software; you can redistribute it and/or +// modify it under the terms of the GNU Lesser General Public +// License as published by the Free Software Foundation; either +// version 3 of the License, or (at your option) any later version. +// +// Alternatively, you can redistribute it and/or +// modify it under the terms of the GNU General Public License as +// published by the Free Software Foundation; either version 2 of +// the License, or (at your option) any later version. +// +// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY +// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS +// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the +// GNU General Public License for more details. +// +// You should have received a copy of the GNU Lesser General Public +// License and a copy of the GNU General Public License along with +// Eigen. If not, see <http://www.gnu.org/licenses/>. + +#ifndef EIGEN_REAL_SCHUR_H +#define EIGEN_REAL_SCHUR_H + +#include "./HessenbergDecomposition.h" + +/** \eigenvalues_module \ingroup Eigenvalues_Module + * \nonstableyet + * + * \class RealSchur + * + * \brief Performs a real Schur decomposition of a square matrix + */ +template<typename _MatrixType> class RealSchur +{ + public: + typedef _MatrixType MatrixType; + enum { + RowsAtCompileTime = MatrixType::RowsAtCompileTime, + ColsAtCompileTime = MatrixType::ColsAtCompileTime, + Options = MatrixType::Options, + MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, + MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime + }; + typedef typename MatrixType::Scalar Scalar; + typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar; + typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options, MaxColsAtCompileTime, 1> EigenvalueType; + + /** \brief Constructor; computes Schur decomposition of given matrix. */ + RealSchur(const MatrixType& matrix) + : m_matT(matrix.rows(),matrix.cols()), + m_matU(matrix.rows(),matrix.cols()), + m_eivalues(matrix.rows()), + m_isInitialized(false) + { + compute(matrix); + } + + /** \brief Returns the orthogonal matrix in the Schur decomposition. */ + const MatrixType& matrixU() const + { + ei_assert(m_isInitialized && "RealSchur is not initialized."); + return m_matU; + } + + /** \brief Returns the quasi-triangular matrix in the Schur decomposition. */ + const MatrixType& matrixT() const + { + ei_assert(m_isInitialized && "RealSchur is not initialized."); + return m_matT; + } + + /** \brief Returns vector of eigenvalues. + * + * This function will likely be removed. */ + const EigenvalueType& eigenvalues() const + { + ei_assert(m_isInitialized && "RealSchur is not initialized."); + return m_eivalues; + } + + /** \brief Computes Schur decomposition of given matrix. */ + void compute(const MatrixType& matrix); + + private: + + MatrixType m_matT; + MatrixType m_matU; + EigenvalueType m_eivalues; + bool m_isInitialized; + + void hqr2(); +}; + + +template<typename MatrixType> +void RealSchur<MatrixType>::compute(const MatrixType& matrix) +{ + assert(matrix.cols() == matrix.rows()); + + // Reduce to Hessenberg form + // TODO skip Q if skipU = true + HessenbergDecomposition<MatrixType> hess(matrix); + m_matT = hess.matrixH(); + m_matU = hess.matrixQ(); + + // Reduce to Real Schur form + hqr2(); + + m_isInitialized = true; +} + + +template<typename MatrixType> +void RealSchur<MatrixType>::hqr2() +{ + // This is derived from the Algol procedure hqr2, + // by Martin and Wilkinson, Handbook for Auto. Comp., + // Vol.ii-Linear Algebra, and the corresponding + // Fortran subroutine in EISPACK. + + // Initialize + const int size = m_matU.cols(); + int n = size-1; + const int low = 0; + const int high = size-1; + Scalar exshift = 0.0; + Scalar p=0,q=0,r=0,s=0,z=0,w,x,y; + + // Compute matrix norm + // FIXME to be efficient the following would requires a triangular reduxion code + // Scalar norm = m_matT.upper().cwiseAbs().sum() + m_matT.corner(BottomLeft,n,n).diagonal().cwiseAbs().sum(); + Scalar norm = 0.0; + for (int j = 0; j < size; ++j) + { + norm += m_matT.row(j).segment(std::max(j-1,0), size-std::max(j-1,0)).cwiseAbs().sum(); + } + + // Outer loop over eigenvalue index + int iter = 0; + while (n >= low) + { + // Look for single small sub-diagonal element + int l = n; + while (l > low) + { + s = ei_abs(m_matT.coeff(l-1,l-1)) + ei_abs(m_matT.coeff(l,l)); + if (s == 0.0) + s = norm; + if (ei_abs(m_matT.coeff(l,l-1)) < NumTraits<Scalar>::epsilon() * s) + break; + l--; + } + + // Check for convergence + // One root found + if (l == n) + { + m_matT.coeffRef(n,n) = m_matT.coeff(n,n) + exshift; + m_eivalues.coeffRef(n) = ComplexScalar(m_matT.coeff(n,n), 0.0); + n--; + iter = 0; + } + else if (l == n-1) // Two roots found + { + w = m_matT.coeff(n,n-1) * m_matT.coeff(n-1,n); + p = (m_matT.coeff(n-1,n-1) - m_matT.coeff(n,n)) * Scalar(0.5); + q = p * p + w; + z = ei_sqrt(ei_abs(q)); + m_matT.coeffRef(n,n) = m_matT.coeff(n,n) + exshift; + m_matT.coeffRef(n-1,n-1) = m_matT.coeff(n-1,n-1) + exshift; + x = m_matT.coeff(n,n); + + // Scalar pair + if (q >= 0) + { + if (p >= 0) + z = p + z; + else + z = p - z; + + m_eivalues.coeffRef(n-1) = ComplexScalar(x + z, 0.0); + m_eivalues.coeffRef(n) = ComplexScalar(z!=0.0 ? x - w / z : m_eivalues.coeff(n-1).real(), 0.0); + + x = m_matT.coeff(n,n-1); + s = ei_abs(x) + ei_abs(z); + p = x / s; + q = z / s; + r = ei_sqrt(p * p+q * q); + p = p / r; + q = q / r; + + // Row modification + for (int j = n-1; j < size; ++j) + { + z = m_matT.coeff(n-1,j); + m_matT.coeffRef(n-1,j) = q * z + p * m_matT.coeff(n,j); + m_matT.coeffRef(n,j) = q * m_matT.coeff(n,j) - p * z; + } + + // Column modification + for (int i = 0; i <= n; ++i) + { + z = m_matT.coeff(i,n-1); + m_matT.coeffRef(i,n-1) = q * z + p * m_matT.coeff(i,n); + m_matT.coeffRef(i,n) = q * m_matT.coeff(i,n) - p * z; + } + + // Accumulate transformations + for (int i = low; i <= high; ++i) + { + z = m_matU.coeff(i,n-1); + m_matU.coeffRef(i,n-1) = q * z + p * m_matU.coeff(i,n); + m_matU.coeffRef(i,n) = q * m_matU.coeff(i,n) - p * z; + } + } + else // Complex pair + { + m_eivalues.coeffRef(n-1) = ComplexScalar(x + p, z); + m_eivalues.coeffRef(n) = ComplexScalar(x + p, -z); + } + n = n - 2; + iter = 0; + } + else // No convergence yet + { + // Form shift + x = m_matT.coeff(n,n); + y = 0.0; + w = 0.0; + if (l < n) + { + y = m_matT.coeff(n-1,n-1); + w = m_matT.coeff(n,n-1) * m_matT.coeff(n-1,n); + } + + // Wilkinson's original ad hoc shift + if (iter == 10) + { + exshift += x; + for (int i = low; i <= n; ++i) + m_matT.coeffRef(i,i) -= x; + s = ei_abs(m_matT.coeff(n,n-1)) + ei_abs(m_matT.coeff(n-1,n-2)); + x = y = Scalar(0.75) * s; + w = Scalar(-0.4375) * s * s; + } + + // MATLAB's new ad hoc shift + if (iter == 30) + { + s = Scalar((y - x) / 2.0); + s = s * s + w; + if (s > 0) + { + s = ei_sqrt(s); + if (y < x) + s = -s; + s = Scalar(x - w / ((y - x) / 2.0 + s)); + for (int i = low; i <= n; ++i) + m_matT.coeffRef(i,i) -= s; + exshift += s; + x = y = w = Scalar(0.964); + } + } + + iter = iter + 1; // (Could check iteration count here.) + + // Look for two consecutive small sub-diagonal elements + int m = n-2; + while (m >= l) + { + z = m_matT.coeff(m,m); + r = x - z; + s = y - z; + p = (r * s - w) / m_matT.coeff(m+1,m) + m_matT.coeff(m,m+1); + q = m_matT.coeff(m+1,m+1) - z - r - s; + r = m_matT.coeff(m+2,m+1); + s = ei_abs(p) + ei_abs(q) + ei_abs(r); + p = p / s; + q = q / s; + r = r / s; + if (m == l) { + break; + } + if (ei_abs(m_matT.coeff(m,m-1)) * (ei_abs(q) + ei_abs(r)) < + NumTraits<Scalar>::epsilon() * (ei_abs(p) * (ei_abs(m_matT.coeff(m-1,m-1)) + ei_abs(z) + + ei_abs(m_matT.coeff(m+1,m+1))))) + { + break; + } + m--; + } + + for (int i = m+2; i <= n; ++i) + { + m_matT.coeffRef(i,i-2) = 0.0; + if (i > m+2) + m_matT.coeffRef(i,i-3) = 0.0; + } + + // Double QR step involving rows l:n and columns m:n + for (int k = m; k <= n-1; ++k) + { + int notlast = (k != n-1); + if (k != m) { + p = m_matT.coeff(k,k-1); + q = m_matT.coeff(k+1,k-1); + r = notlast ? m_matT.coeff(k+2,k-1) : Scalar(0); + x = ei_abs(p) + ei_abs(q) + ei_abs(r); + if (x != 0.0) + { + p = p / x; + q = q / x; + r = r / x; + } + } + + if (x == 0.0) + break; + + s = ei_sqrt(p * p + q * q + r * r); + + if (p < 0) + s = -s; + + if (s != 0) + { + if (k != m) + m_matT.coeffRef(k,k-1) = -s * x; + else if (l != m) + m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1); + + p = p + s; + x = p / s; + y = q / s; + z = r / s; + q = q / p; + r = r / p; + + // Row modification + for (int j = k; j < size; ++j) + { + p = m_matT.coeff(k,j) + q * m_matT.coeff(k+1,j); + if (notlast) + { + p = p + r * m_matT.coeff(k+2,j); + m_matT.coeffRef(k+2,j) = m_matT.coeff(k+2,j) - p * z; + } + m_matT.coeffRef(k,j) = m_matT.coeff(k,j) - p * x; + m_matT.coeffRef(k+1,j) = m_matT.coeff(k+1,j) - p * y; + } + + // Column modification + for (int i = 0; i <= std::min(n,k+3); ++i) + { + p = x * m_matT.coeff(i,k) + y * m_matT.coeff(i,k+1); + if (notlast) + { + p = p + z * m_matT.coeff(i,k+2); + m_matT.coeffRef(i,k+2) = m_matT.coeff(i,k+2) - p * r; + } + m_matT.coeffRef(i,k) = m_matT.coeff(i,k) - p; + m_matT.coeffRef(i,k+1) = m_matT.coeff(i,k+1) - p * q; + } + + // Accumulate transformations + for (int i = low; i <= high; ++i) + { + p = x * m_matU.coeff(i,k) + y * m_matU.coeff(i,k+1); + if (notlast) + { + p = p + z * m_matU.coeff(i,k+2); + m_matU.coeffRef(i,k+2) = m_matU.coeff(i,k+2) - p * r; + } + m_matU.coeffRef(i,k) = m_matU.coeff(i,k) - p; + m_matU.coeffRef(i,k+1) = m_matU.coeff(i,k+1) - p * q; + } + } // (s != 0) + } // k loop + } // check convergence + } // while (n >= low) +} + +#endif // EIGEN_REAL_SCHUR_H |