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[mq]: eigensolver

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Gael Guennebaud 2009-09-01 16:20:56 +02:00
parent 67ccc6b851
commit 4d91229bdc
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2
Eigen/QR
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@ -42,6 +42,8 @@ namespace Eigen {
#include "src/QR/EigenSolver.h"
#include "src/QR/SelfAdjointEigenSolver.h"
#include "src/QR/HessenbergDecomposition.h"
#include "src/QR/ComplexSchur.h"
#include "src/QR/ComplexEigenSolver.h"
// declare all classes for a given matrix type
#define EIGEN_QR_MODULE_INSTANTIATE_TYPE(MATRIXTYPE,PREFIX) \

138
Eigen/src/QR/ComplexEigenSolver.h Normal file
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Claire Maurice
// Copyright (C) 2009 Gael Guennebaud <g.gael@free.fr>
//
// 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_COMPLEX_EIGEN_SOLVER_H
#define EIGEN_COMPLEX_EIGEN_SOLVER_H
#define MAXITER 30
template<typename _MatrixType> class ComplexEigenSolver
{
public:
typedef _MatrixType MatrixType;
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef std::complex<RealScalar> Complex;
typedef Matrix<Complex, MatrixType::ColsAtCompileTime,1> EigenvalueType;
typedef Matrix<Complex, MatrixType::RowsAtCompileTime,MatrixType::ColsAtCompileTime> EigenvectorType;
/**
* \brief Default Constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via ComplexEigenSolver::compute(const MatrixType&).
*/
ComplexEigenSolver() : m_eivec(), m_eivalues(), m_isInitialized(false)
{}
ComplexEigenSolver(const MatrixType& matrix)
: m_eivec(matrix.rows(),matrix.cols()),
m_eivalues(matrix.cols()),
m_isInitialized(false)
{
compute(matrix);
}
EigenvectorType eigenvectors(void) const
{
ei_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
return m_eivec;
}
EigenvalueType eigenvalues() const
{
ei_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
return m_eivalues;
}
void compute(const MatrixType& matrix);
protected:
MatrixType m_eivec;
EigenvalueType m_eivalues;
bool m_isInitialized;
};
template<typename MatrixType>
void ComplexEigenSolver<MatrixType>::compute(const MatrixType& matrix)
{
assert(matrix.cols() == matrix.rows());
int n = matrix.cols();
m_eivalues.resize(n,1);
RealScalar eps = epsilon<RealScalar>();
// Reduce to complex Schur form
ComplexSchur<MatrixType> schur(matrix);
m_eivalues = schur.matrixT().diagonal();
m_eivec.setZero();
Scalar d2, z;
RealScalar norm = matrix.norm();
// compute the (normalized) eigenvectors
for(int k=n-1 ; k>=0 ; k--)
{
d2 = schur.matrixT().coeff(k,k);
m_eivec.coeffRef(k,k) = Scalar(1.0,0.0);
for(int i=k-1 ; i>=0 ; i--)
{
m_eivec.coeffRef(i,k) = -schur.matrixT().coeff(i,k);
if(k-i-1>0)
m_eivec.coeffRef(i,k) -= (schur.matrixT().row(i).segment(i+1,k-i-1) * m_eivec.col(k).segment(i+1,k-i-1)).value();
z = schur.matrixT().coeff(i,i) - d2;
if(z==Scalar(0))
z.real() = eps * norm;
m_eivec.coeffRef(i,k) = m_eivec.coeff(i,k) / z;
}
m_eivec.col(k).normalize();
}
m_eivec = schur.matrixU() * m_eivec;
m_isInitialized = true;
// sort the eigenvalues
{
for (int i=0; i<n; i++)
{
int k;
m_eivalues.cwise().abs().end(n-i).minCoeff(&k);
if (k != 0)
{
k += i;
std::swap(m_eivalues[k],m_eivalues[i]);
m_eivec.col(i).swap(m_eivec.col(k));
}
}
}
}
#endif // EIGEN_COMPLEX_EIGEN_SOLVER_H

273
Eigen/src/QR/ComplexSchur.h Normal file
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Claire Maurice
// Copyright (C) 2009 Gael Guennebaud <g.gael@free.fr>
//
// 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_COMPLEX_SCHUR_H
#define EIGEN_COMPLEX_SCHUR_H
#define MAXITER 30
/** \ingroup QR
*
* \class ComplexShur
*
* \brief Performs a complex Shur decomposition of a real or complex square matrix
*
*/
template<typename _MatrixType> class ComplexSchur
{
public:
typedef _MatrixType MatrixType;
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef std::complex<RealScalar> Complex;
typedef Matrix<Complex, MatrixType::RowsAtCompileTime,MatrixType::ColsAtCompileTime> ComplexMatrixType;
/**
* \brief Default Constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via ComplexSchur::compute(const MatrixType&).
*/
ComplexSchur() : m_matT(), m_matU(), m_isInitialized(false)
{}
ComplexSchur(const MatrixType& matrix)
: m_matT(matrix.rows(),matrix.cols()),
m_matU(matrix.rows(),matrix.cols()),
m_isInitialized(false)
{
compute(matrix);
}
ComplexMatrixType matrixU() const
{
ei_assert(m_isInitialized && "ComplexSchur is not initialized.");
return m_matU;
}
ComplexMatrixType matrixT() const
{
ei_assert(m_isInitialized && "ComplexShur is not initialized.");
return m_matT;
}
void compute(const MatrixType& matrix);
protected:
ComplexMatrixType m_matT, m_matU;
bool m_isInitialized;
};
// computes the plane rotation G such that G' x |p| = | c s' |' |p| = |z|
// |q| |-s c' | |q| |0|
// and returns z if requested. Note that the returned c is real.
template<typename T> void ei_make_givens(const std::complex<T>& p, const std::complex<T>& q,
JacobiRotation<std::complex<T> >& rot, std::complex<T>* z=0)
{
typedef std::complex<T> Complex;
T scale, absx, absxy;
if(p==Complex(0))
{
// return identity
rot.c() = Complex(1,0);
rot.s() = Complex(0,0);
if(z) *z = p;
}
else
{
scale = cnorm1(p);
absx = scale * ei_sqrt(ei_abs2(p/scale));
scale = ei_abs(scale) + cnorm1(q);
absxy = scale * ei_sqrt((absx/scale)*(absx/scale) + ei_abs2(q/scale));
rot.c() = Complex(absx / absxy);
Complex np = p/absx;
rot.s() = -ei_conj(np) * q / absxy;
if(z) *z = np * absxy;
}
}
template<typename MatrixType>
void ComplexSchur<MatrixType>::compute(const MatrixType& matrix)
{
assert(matrix.cols() == matrix.rows());
int n = matrix.cols();
// Reduce to Hessenberg form
HessenbergDecomposition<MatrixType> hess(matrix);
m_matT = hess.matrixH();
m_matU = hess.matrixQ();
int iu = m_matT.cols() - 1;
int il;
RealScalar d,sd,sf;
Complex c,b,disc,r1,r2,kappa;
RealScalar eps = epsilon<RealScalar>();
int iter = 0;
while(true)
{
//locate the range in which to iterate
while(iu > 0)
{
d = cnorm1(m_matT.coeffRef(iu,iu)) + cnorm1(m_matT.coeffRef(iu-1,iu-1));
sd = cnorm1(m_matT.coeffRef(iu,iu-1));
if(sd >= eps * d) break; // FIXME : precision criterion ??
m_matT.coeffRef(iu,iu-1) = Complex(0);
iter = 0;
--iu;
}
if(iu==0) break;
iter++;
if(iter >= MAXITER)
{
// FIXME : what to do when iter==MAXITER ??
std::cerr << "MAXITER" << std::endl;
return;
}
il = iu-1;
while( il > 0 )
{
// check if the current 2x2 block on the diagonal is upper triangular
d = cnorm1(m_matT.coeffRef(il,il)) + cnorm1(m_matT.coeffRef(il-1,il-1));
sd = cnorm1(m_matT.coeffRef(il,il-1));
if(sd < eps * d) break; // FIXME : precision criterion ??
--il;
}
if( il != 0 ) m_matT.coeffRef(il,il-1) = Complex(0);
// compute the shift (the normalization by sf is to avoid under/overflow)
Matrix<Scalar,2,2> t = m_matT.template block<2,2>(iu-1,iu-1);
sf = t.cwise().abs().sum();
t /= sf;
c = t.determinant();
b = t.diagonal().sum();
disc = csqrt(b*b - RealScalar(4)*c);
r1 = (b+disc)/RealScalar(2);
r2 = (b-disc)/RealScalar(2);
if(cnorm1(r1) > cnorm1(r2))
r2 = c/r1;
else
r1 = c/r2;
if(cnorm1(r1-t.coeff(1,1)) < cnorm1(r2-t.coeff(1,1)))
kappa = sf * r1;
else
kappa = sf * r2;
// perform the QR step using Givens rotations
JacobiRotation<Complex> rot;
ei_make_givens(m_matT.coeff(il,il) - kappa, m_matT.coeff(il+1,il), rot);
for(int i=il ; i<iu ; i++)
{
m_matT.block(0,i,n,n-i).applyJacobiOnTheLeft(i, i+1, rot.adjoint());
m_matT.block(0,0,std::min(i+2,iu)+1,n).applyJacobiOnTheRight(i, i+1, rot);
m_matU.applyJacobiOnTheRight(i, i+1, rot);
if(i != iu-1)
{
int i1 = i+1;
int i2 = i+2;
ei_make_givens(m_matT.coeffRef(i1,i), m_matT.coeffRef(i2,i), rot, &m_matT.coeffRef(i1,i));
m_matT.coeffRef(i2,i) = Complex(0);
}
}
}
// FIXME : is it necessary ?
for(int i=0 ; i<n ; i++)
for(int j=0 ; j<n ; j++)
{
if(ei_abs(m_matT.coeff(i,j).real()) < eps)
m_matT.coeffRef(i,j).real() = 0;
if(ei_abs(m_matT.coeff(i,j).imag()) < eps)
m_matT.coeffRef(i,j).imag() = 0;
}
m_isInitialized = true;
}
// norm1 of complex numbers
template<typename T>
T cnorm1(const std::complex<T> &Z)
{
return(ei_abs(Z.real()) + ei_abs(Z.imag()));
}
/**
* Computes the principal value of the square root of the complex \a z.
*/
template<typename RealScalar>
std::complex<RealScalar> csqrt(const std::complex<RealScalar> &z)
{
RealScalar t, tre, tim;
t = ei_abs(z);
if (ei_abs(z.real()) <= ei_abs(z.imag()))
{
// No cancellation in these formulas
tre = ei_sqrt(0.5*(t+z.real()));
tim = ei_sqrt(0.5*(t-z.real()));
}
else
{
// Stable computation of the above formulas
if (z.real() > 0)
{
tre = t + z.real();
tim = ei_abs(z.imag())*ei_sqrt(0.5/tre);
tre = ei_sqrt(0.5*tre);
}
else
{
tim = t - z.real();
tre = ei_abs(z.imag())*ei_sqrt(0.5/tim);
tim = ei_sqrt(0.5*tim);
}
}
if(z.imag() < 0)
tim = -tim;
return (std::complex<RealScalar>(tre,tim));
}
#endif // EIGEN_COMPLEX_SCHUR_H

1
test/CMakeLists.txt
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@ -125,6 +125,7 @@ ei_add_test(qr_colpivoting)
ei_add_test(qr_fullpivoting)
ei_add_test(eigensolver_selfadjoint " " "${GSL_LIBRARIES}")
ei_add_test(eigensolver_generic " " "${GSL_LIBRARIES}")
ei_add_test(eigensolver_complex)
ei_add_test(svd)
ei_add_test(jacobisvd ${EI_OFLAG})
ei_add_test(geo_orthomethods)

70
test/eigensolver_complex.cpp Normal file
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@ -0,0 +1,70 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra. Eigen itself is part of the KDE project.
//
// Copyright (C) 2008-2009 Gael Guennebaud <g.gael@free.fr>
//
// 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/>.
#include "main.h"
#include <Eigen/QR>
#include <Eigen/LU>
template<typename MatrixType> void eigensolver(const MatrixType& m)
{
/* this test covers the following files:
ComplexEigenSolver.h
*/
int rows = m.rows();
int cols = m.cols();
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;
typedef Matrix<RealScalar, MatrixType::RowsAtCompileTime, 1> RealVectorType;
typedef typename std::complex<typename NumTraits<typename MatrixType::Scalar>::Real> Complex;
// RealScalar largerEps = 10*test_precision<RealScalar>();
MatrixType a = MatrixType::Random(rows,cols);
MatrixType a1 = MatrixType::Random(rows,cols);
MatrixType symmA = a.adjoint() * a + a1.adjoint() * a1;
// ComplexEigenSolver<MatrixType> ei0(symmA);
// VERIFY_IS_APPROX(symmA * ei0.eigenvectors(), ei0.eigenvectors() * ei0.eigenvalues().asDiagonal());
// a.imag().setZero();
// std::cerr << a << "\n\n";
ComplexEigenSolver<MatrixType> ei1(a);
// exit(1);
VERIFY_IS_APPROX(a * ei1.eigenvectors(), ei1.eigenvectors() * ei1.eigenvalues().asDiagonal());
}
void test_eigensolver_complex()
{
for(int i = 0; i < g_repeat; i++) {
// CALL_SUBTEST( eigensolver(Matrix4cf()) );
// CALL_SUBTEST( eigensolver(MatrixXcd(4,4)) );
CALL_SUBTEST( eigensolver(MatrixXcd(6,6)) );
// CALL_SUBTEST( eigensolver(MatrixXd(14,14)) );
}
}