mirror of
https://gitlab.com/libeigen/eigen.git
synced 2026-04-10 11:34:33 +08:00
* Draft of a eigenvalues solver
(does not support complex and does not re-use the QR decomposition) * Rewrite the cache friendly product to have only one instance per scalar type ! This significantly speeds up compilation time and reduces executable size. The current drawback is that some trivial expressions might be evaluated like conjugate or negate. * Renamed "cache optimal" to "cache friendly" * Added the ability to directly access matrix data of some expressions via: - the stride()/_stride() methods - DirectAccessBit flag (replace ReferencableBit)
This commit is contained in:
Gael Guennebaud
848
Eigen/src/QR/EigenSolver.h
Normal file
848
Eigen/src/QR/EigenSolver.h
Normal file
@@ -0,0 +1,848 @@
|
||||
// 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 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_EIGENSOLVER_H
|
||||
#define EIGEN_EIGENSOLVER_H
|
||||
|
||||
/** \class EigenSolver
|
||||
*
|
||||
* \brief Eigen values/vectors solver
|
||||
*
|
||||
* \param MatrixType the type of the matrix of which we are computing the eigen decomposition
|
||||
*
|
||||
* \note this code was adapted from JAMA (public domain)
|
||||
*
|
||||
* \sa MatrixBase::eigenvalues()
|
||||
*/
|
||||
template<typename _MatrixType> class EigenSolver
|
||||
{
|
||||
public:
|
||||
|
||||
typedef _MatrixType MatrixType;
|
||||
typedef typename MatrixType::Scalar Scalar;
|
||||
typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> VectorType;
|
||||
|
||||
EigenSolver(const MatrixType& matrix)
|
||||
: m_eivec(matrix.rows(), matrix.cols()),
|
||||
m_eivalr(matrix.cols()), m_eivali(matrix.cols()),
|
||||
m_H(matrix.rows(), matrix.cols()),
|
||||
m_ort(matrix.cols())
|
||||
{
|
||||
_compute(matrix);
|
||||
}
|
||||
|
||||
MatrixType eigenvectors(void) const { return m_eivec; }
|
||||
|
||||
VectorType eigenvalues(void) const { return m_eivalr; }
|
||||
|
||||
private:
|
||||
|
||||
void _compute(const MatrixType& matrix);
|
||||
|
||||
void tridiagonalization(void);
|
||||
void tql2(void);
|
||||
|
||||
void orthes(void);
|
||||
void hqr2(void);
|
||||
|
||||
protected:
|
||||
MatrixType m_eivec;
|
||||
VectorType m_eivalr, m_eivali;
|
||||
MatrixType m_H;
|
||||
VectorType m_ort;
|
||||
bool m_isSymmetric;
|
||||
};
|
||||
|
||||
template<typename MatrixType>
|
||||
void EigenSolver<MatrixType>::_compute(const MatrixType& matrix)
|
||||
{
|
||||
assert(matrix.cols() == matrix.rows());
|
||||
|
||||
m_isSymmetric = true;
|
||||
int n = matrix.cols();
|
||||
for (int j = 0; (j < n) && m_isSymmetric; j++) {
|
||||
for (int i = 0; (i < j) && m_isSymmetric; i++) {
|
||||
m_isSymmetric = (matrix(i,j) == matrix(j,i));
|
||||
}
|
||||
}
|
||||
|
||||
m_eivalr.resize(n,1);
|
||||
m_eivali.resize(n,1);
|
||||
|
||||
if (m_isSymmetric)
|
||||
{
|
||||
m_eivec = matrix;
|
||||
|
||||
// Tridiagonalize.
|
||||
tridiagonalization();
|
||||
|
||||
// Diagonalize.
|
||||
tql2();
|
||||
}
|
||||
else
|
||||
{
|
||||
m_H = matrix;
|
||||
m_ort.resize(n, 1);
|
||||
|
||||
// Reduce to Hessenberg form.
|
||||
orthes();
|
||||
|
||||
// Reduce Hessenberg to real Schur form.
|
||||
hqr2();
|
||||
}
|
||||
std::cout << m_eivali.transpose() << "\n";
|
||||
}
|
||||
|
||||
|
||||
// Symmetric Householder reduction to tridiagonal form.
|
||||
template<typename MatrixType>
|
||||
void EigenSolver<MatrixType>::tridiagonalization(void)
|
||||
{
|
||||
|
||||
// This is derived from the Algol procedures tred2 by
|
||||
// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
|
||||
// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
|
||||
// Fortran subroutine in EISPACK.
|
||||
|
||||
int n = m_eivec.cols();
|
||||
m_eivalr = m_eivec.row(m_eivalr.size()-1);
|
||||
|
||||
// Householder reduction to tridiagonal form.
|
||||
for (int i = n-1; i > 0; i--)
|
||||
{
|
||||
// Scale to avoid under/overflow.
|
||||
Scalar scale = 0.0;
|
||||
Scalar h = 0.0;
|
||||
scale = m_eivalr.start(i).cwiseAbs().sum();
|
||||
|
||||
if (scale == 0.0)
|
||||
{
|
||||
m_eivali[i] = m_eivalr[i-1];
|
||||
m_eivalr.start(i) = m_eivec.row(i-1).start(i);
|
||||
m_eivec.corner(TopLeft, i, i) = m_eivec.corner(TopLeft, i, i).diagonal().asDiagonal();
|
||||
}
|
||||
else
|
||||
{
|
||||
// Generate Householder vector.
|
||||
m_eivalr.start(i) /= scale;
|
||||
h = m_eivalr.start(i).cwiseAbs2().sum();
|
||||
|
||||
Scalar f = m_eivalr[i-1];
|
||||
Scalar g = ei_sqrt(h);
|
||||
if (f > 0)
|
||||
g = -g;
|
||||
m_eivali[i] = scale * g;
|
||||
h = h - f * g;
|
||||
m_eivalr[i-1] = f - g;
|
||||
m_eivali.start(i).setZero();
|
||||
|
||||
// Apply similarity transformation to remaining columns.
|
||||
for (int j = 0; j < i; j++)
|
||||
{
|
||||
f = m_eivalr[j];
|
||||
m_eivec(j,i) = f;
|
||||
g = m_eivali[j] + m_eivec(j,j) * f;
|
||||
int bSize = i-j-1;
|
||||
if (bSize>0)
|
||||
{
|
||||
g += (m_eivec.col(j).block(j+1, bSize).transpose() * m_eivalr.block(j+1, bSize))(0,0);
|
||||
m_eivali.block(j+1, bSize) += m_eivec.col(j).block(j+1, bSize) * f;
|
||||
}
|
||||
m_eivali[j] = g;
|
||||
}
|
||||
|
||||
f = (m_eivali.start(i).transpose() * m_eivalr.start(i))(0,0);
|
||||
m_eivali.start(i) = (m_eivali.start(i) - (f / (h + h)) * m_eivalr.start(i))/h;
|
||||
|
||||
m_eivec.corner(TopLeft, i, i).lower() -=
|
||||
( (m_eivali.start(i) * m_eivalr.start(i).transpose()).lazy()
|
||||
+ (m_eivalr.start(i) * m_eivali.start(i).transpose()).lazy());
|
||||
|
||||
m_eivalr.start(i) = m_eivec.row(i-1).start(i);
|
||||
m_eivec.row(i).start(i).setZero();
|
||||
}
|
||||
m_eivalr[i] = h;
|
||||
}
|
||||
|
||||
// Accumulate transformations.
|
||||
for (int i = 0; i < n-1; i++)
|
||||
{
|
||||
m_eivec(n-1,i) = m_eivec(i,i);
|
||||
m_eivec(i,i) = 1.0;
|
||||
Scalar h = m_eivalr[i+1];
|
||||
// FIXME this does not looks very stable ;)
|
||||
if (h != 0.0)
|
||||
{
|
||||
m_eivalr.start(i+1) = m_eivec.col(i+1).start(i+1) / h;
|
||||
m_eivec.corner(TopLeft, i+1, i+1) -= m_eivalr.start(i+1)
|
||||
* ( m_eivec.col(i+1).start(i+1).transpose() * m_eivec.corner(TopLeft, i+1, i+1) );
|
||||
}
|
||||
m_eivec.col(i+1).start(i+1).setZero();
|
||||
}
|
||||
m_eivalr = m_eivec.row(m_eivalr.size()-1);
|
||||
m_eivec.row(m_eivalr.size()-1).setZero();
|
||||
m_eivec(n-1,n-1) = 1.0;
|
||||
m_eivali[0] = 0.0;
|
||||
}
|
||||
|
||||
|
||||
// Symmetric tridiagonal QL algorithm.
|
||||
template<typename MatrixType>
|
||||
void EigenSolver<MatrixType>::tql2(void)
|
||||
{
|
||||
|
||||
// This is derived from the Algol procedures tql2, by
|
||||
// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
|
||||
// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
|
||||
// Fortran subroutine in EISPACK.
|
||||
|
||||
int n = m_eivalr.size();
|
||||
|
||||
for (int i = 1; i < n; i++) {
|
||||
m_eivali[i-1] = m_eivali[i];
|
||||
}
|
||||
m_eivali[n-1] = 0.0;
|
||||
|
||||
Scalar f = 0.0;
|
||||
Scalar tst1 = 0.0;
|
||||
Scalar eps = std::pow(2.0,-52.0);
|
||||
for (int l = 0; l < n; l++)
|
||||
{
|
||||
// Find small subdiagonal element
|
||||
tst1 = std::max(tst1,ei_abs(m_eivalr[l]) + ei_abs(m_eivali[l]));
|
||||
int m = l;
|
||||
|
||||
while ( (m < n) && (ei_abs(m_eivali[m]) > eps*tst1) )
|
||||
m++;
|
||||
|
||||
// If m == l, m_eivalr[l] is an eigenvalue,
|
||||
// otherwise, iterate.
|
||||
if (m > l)
|
||||
{
|
||||
int iter = 0;
|
||||
do
|
||||
{
|
||||
iter = iter + 1;
|
||||
|
||||
// Compute implicit shift
|
||||
Scalar g = m_eivalr[l];
|
||||
Scalar p = (m_eivalr[l+1] - g) / (2.0 * m_eivali[l]);
|
||||
Scalar r = hypot(p,1.0);
|
||||
if (p < 0)
|
||||
r = -r;
|
||||
|
||||
m_eivalr[l] = m_eivali[l] / (p + r);
|
||||
m_eivalr[l+1] = m_eivali[l] * (p + r);
|
||||
Scalar dl1 = m_eivalr[l+1];
|
||||
Scalar h = g - m_eivalr[l];
|
||||
if (l+2<n)
|
||||
m_eivalr.end(n-l-2) -= VectorType::constant(n-l-2, h);
|
||||
f = f + h;
|
||||
|
||||
// Implicit QL transformation.
|
||||
p = m_eivalr[m];
|
||||
Scalar c = 1.0;
|
||||
Scalar c2 = c;
|
||||
Scalar c3 = c;
|
||||
Scalar el1 = m_eivali[l+1];
|
||||
Scalar s = 0.0;
|
||||
Scalar s2 = 0.0;
|
||||
for (int i = m-1; i >= l; i--)
|
||||
{
|
||||
c3 = c2;
|
||||
c2 = c;
|
||||
s2 = s;
|
||||
g = c * m_eivali[i];
|
||||
h = c * p;
|
||||
r = hypot(p,m_eivali[i]);
|
||||
m_eivali[i+1] = s * r;
|
||||
s = m_eivali[i] / r;
|
||||
c = p / r;
|
||||
p = c * m_eivalr[i] - s * g;
|
||||
m_eivalr[i+1] = h + s * (c * g + s * m_eivalr[i]);
|
||||
|
||||
// Accumulate transformation.
|
||||
for (int k = 0; k < n; k++)
|
||||
{
|
||||
h = m_eivec(k,i+1);
|
||||
m_eivec(k,i+1) = s * m_eivec(k,i) + c * h;
|
||||
m_eivec(k,i) = c * m_eivec(k,i) - s * h;
|
||||
}
|
||||
}
|
||||
p = -s * s2 * c3 * el1 * m_eivali[l] / dl1;
|
||||
m_eivali[l] = s * p;
|
||||
m_eivalr[l] = c * p;
|
||||
|
||||
// Check for convergence.
|
||||
} while (ei_abs(m_eivali[l]) > eps*tst1);
|
||||
}
|
||||
m_eivalr[l] = m_eivalr[l] + f;
|
||||
m_eivali[l] = 0.0;
|
||||
}
|
||||
|
||||
// Sort eigenvalues and corresponding vectors.
|
||||
// TODO use a better sort algorithm !!
|
||||
for (int i = 0; i < n-1; i++)
|
||||
{
|
||||
int k = i;
|
||||
Scalar minValue = m_eivalr[i];
|
||||
for (int j = i+1; j < n; j++)
|
||||
{
|
||||
if (m_eivalr[j] < minValue)
|
||||
{
|
||||
k = j;
|
||||
minValue = m_eivalr[j];
|
||||
}
|
||||
}
|
||||
if (k != i)
|
||||
{
|
||||
std::swap(m_eivalr[i], m_eivalr[k]);
|
||||
m_eivec.col(i).swap(m_eivec.col(k));
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
// Nonsymmetric reduction to Hessenberg form.
|
||||
template<typename MatrixType>
|
||||
void EigenSolver<MatrixType>::orthes(void)
|
||||
{
|
||||
// This is derived from the Algol procedures orthes and ortran,
|
||||
// by Martin and Wilkinson, Handbook for Auto. Comp.,
|
||||
// Vol.ii-Linear Algebra, and the corresponding
|
||||
// Fortran subroutines in EISPACK.
|
||||
|
||||
int n = m_eivec.cols();
|
||||
int low = 0;
|
||||
int high = n-1;
|
||||
|
||||
for (int m = low+1; m <= high-1; m++)
|
||||
{
|
||||
// Scale column.
|
||||
Scalar scale = m_H.block(m, m-1, high-m+1, 1).cwiseAbs().sum();
|
||||
if (scale != 0.0)
|
||||
{
|
||||
// Compute Householder transformation.
|
||||
Scalar h = 0.0;
|
||||
// FIXME could be rewritten, but this one looks better wrt cache
|
||||
for (int i = high; i >= m; i--)
|
||||
{
|
||||
m_ort[i] = m_H(i,m-1)/scale;
|
||||
h += m_ort[i] * m_ort[i];
|
||||
}
|
||||
Scalar g = ei_sqrt(h);
|
||||
if (m_ort[m] > 0)
|
||||
g = -g;
|
||||
h = h - m_ort[m] * g;
|
||||
m_ort[m] = m_ort[m] - g;
|
||||
|
||||
// Apply Householder similarity transformation
|
||||
// H = (I-u*u'/h)*H*(I-u*u')/h)
|
||||
int bSize = high-m+1;
|
||||
m_H.block(m, m, bSize, n-m) -= ((m_ort.block(m, bSize)/h)
|
||||
* (m_ort.block(m, bSize).transpose() * m_H.block(m, m, bSize, n-m)).lazy()).lazy();
|
||||
|
||||
m_H.block(0, m, high+1, bSize) -= ((m_H.block(0, m, high+1, bSize) * m_ort.block(m, bSize)).lazy()
|
||||
* (m_ort.block(m, bSize)/h).transpose()).lazy();
|
||||
|
||||
m_ort[m] = scale*m_ort[m];
|
||||
m_H(m,m-1) = scale*g;
|
||||
}
|
||||
}
|
||||
|
||||
// Accumulate transformations (Algol's ortran).
|
||||
m_eivec.setIdentity();
|
||||
|
||||
for (int m = high-1; m >= low+1; m--)
|
||||
{
|
||||
if (m_H(m,m-1) != 0.0)
|
||||
{
|
||||
m_ort.block(m+1, high-m) = m_H.col(m-1).block(m+1, high-m);
|
||||
|
||||
int bSize = high-m+1;
|
||||
m_eivec.block(m, m, bSize, bSize) += ( (m_ort.block(m, bSize) / (m_H(m,m-1) * m_ort[m] ) )
|
||||
* (m_ort.block(m, bSize).transpose() * m_eivec.block(m, m, bSize, bSize)).lazy());
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
// Complex scalar division.
|
||||
template<typename Scalar>
|
||||
std::complex<Scalar> cdiv(Scalar xr, Scalar xi, Scalar yr, Scalar yi)
|
||||
{
|
||||
Scalar r,d;
|
||||
if (ei_abs(yr) > ei_abs(yi))
|
||||
{
|
||||
r = yi/yr;
|
||||
d = yr + r*yi;
|
||||
return std::complex<Scalar>((xr + r*xi)/d, (xi - r*xr)/d);
|
||||
}
|
||||
else
|
||||
{
|
||||
r = yr/yi;
|
||||
d = yi + r*yr;
|
||||
return std::complex<Scalar>((r*xr + xi)/d, (r*xi - xr)/d);
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
// Nonsymmetric reduction from Hessenberg to real Schur form.
|
||||
template<typename MatrixType>
|
||||
void EigenSolver<MatrixType>::hqr2(void)
|
||||
{
|
||||
// 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
|
||||
int nn = m_eivec.cols();
|
||||
int n = nn-1;
|
||||
int low = 0;
|
||||
int high = nn-1;
|
||||
Scalar eps = pow(2.0,-52.0);
|
||||
Scalar exshift = 0.0;
|
||||
Scalar p=0,q=0,r=0,s=0,z=0,t,w,x,y;
|
||||
|
||||
// Store roots isolated by balanc and compute matrix norm
|
||||
// FIXME to be efficient the following would requires a triangular reduxion code
|
||||
// Scalar norm = m_H.upper().cwiseAbs().sum() + m_H.corner(BottomLeft,n,n).diagonal().cwiseAbs().sum();
|
||||
Scalar norm = 0.0;
|
||||
for (int j = 0; j < nn; j++)
|
||||
{
|
||||
// FIXME what's the purpose of the following since the condition is always false
|
||||
if ((j < low) || (j > high))
|
||||
{
|
||||
m_eivalr[j] = m_H(j,j);
|
||||
m_eivali[j] = 0.0;
|
||||
}
|
||||
norm += m_H.col(j).start(std::min(j+1,nn)).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_H(l-1,l-1)) + ei_abs(m_H(l,l));
|
||||
if (s == 0.0)
|
||||
s = norm;
|
||||
if (ei_abs(m_H(l,l-1)) < eps * s)
|
||||
break;
|
||||
l--;
|
||||
}
|
||||
|
||||
// Check for convergence
|
||||
// One root found
|
||||
if (l == n)
|
||||
{
|
||||
m_H(n,n) = m_H(n,n) + exshift;
|
||||
m_eivalr[n] = m_H(n,n);
|
||||
m_eivali[n] = 0.0;
|
||||
n--;
|
||||
iter = 0;
|
||||
}
|
||||
else if (l == n-1) // Two roots found
|
||||
{
|
||||
w = m_H(n,n-1) * m_H(n-1,n);
|
||||
p = (m_H(n-1,n-1) - m_H(n,n)) / 2.0;
|
||||
q = p * p + w;
|
||||
z = ei_sqrt(ei_abs(q));
|
||||
m_H(n,n) = m_H(n,n) + exshift;
|
||||
m_H(n-1,n-1) = m_H(n-1,n-1) + exshift;
|
||||
x = m_H(n,n);
|
||||
|
||||
// Scalar pair
|
||||
if (q >= 0)
|
||||
{
|
||||
if (p >= 0)
|
||||
z = p + z;
|
||||
else
|
||||
z = p - z;
|
||||
|
||||
m_eivalr[n-1] = x + z;
|
||||
m_eivalr[n] = m_eivalr[n-1];
|
||||
if (z != 0.0)
|
||||
m_eivalr[n] = x - w / z;
|
||||
|
||||
m_eivali[n-1] = 0.0;
|
||||
m_eivali[n] = 0.0;
|
||||
x = m_H(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 < nn; j++)
|
||||
{
|
||||
z = m_H(n-1,j);
|
||||
m_H(n-1,j) = q * z + p * m_H(n,j);
|
||||
m_H(n,j) = q * m_H(n,j) - p * z;
|
||||
}
|
||||
|
||||
// Column modification
|
||||
for (int i = 0; i <= n; i++)
|
||||
{
|
||||
z = m_H(i,n-1);
|
||||
m_H(i,n-1) = q * z + p * m_H(i,n);
|
||||
m_H(i,n) = q * m_H(i,n) - p * z;
|
||||
}
|
||||
|
||||
// Accumulate transformations
|
||||
for (int i = low; i <= high; i++)
|
||||
{
|
||||
z = m_eivec(i,n-1);
|
||||
m_eivec(i,n-1) = q * z + p * m_eivec(i,n);
|
||||
m_eivec(i,n) = q * m_eivec(i,n) - p * z;
|
||||
}
|
||||
}
|
||||
else // Complex pair
|
||||
{
|
||||
m_eivalr[n-1] = x + p;
|
||||
m_eivalr[n] = x + p;
|
||||
m_eivali[n-1] = z;
|
||||
m_eivali[n] = -z;
|
||||
}
|
||||
n = n - 2;
|
||||
iter = 0;
|
||||
}
|
||||
else // No convergence yet
|
||||
{
|
||||
// Form shift
|
||||
x = m_H(n,n);
|
||||
y = 0.0;
|
||||
w = 0.0;
|
||||
if (l < n)
|
||||
{
|
||||
y = m_H(n-1,n-1);
|
||||
w = m_H(n,n-1) * m_H(n-1,n);
|
||||
}
|
||||
|
||||
// Wilkinson's original ad hoc shift
|
||||
if (iter == 10)
|
||||
{
|
||||
exshift += x;
|
||||
for (int i = low; i <= n; i++)
|
||||
m_H(i,i) -= x;
|
||||
s = ei_abs(m_H(n,n-1)) + ei_abs(m_H(n-1,n-2));
|
||||
x = y = 0.75 * s;
|
||||
w = -0.4375 * s * s;
|
||||
}
|
||||
|
||||
// MATLAB's new ad hoc shift
|
||||
if (iter == 30)
|
||||
{
|
||||
s = (y - x) / 2.0;
|
||||
s = s * s + w;
|
||||
if (s > 0)
|
||||
{
|
||||
s = ei_sqrt(s);
|
||||
if (y < x)
|
||||
s = -s;
|
||||
s = x - w / ((y - x) / 2.0 + s);
|
||||
for (int i = low; i <= n; i++)
|
||||
m_H(i,i) -= s;
|
||||
exshift += s;
|
||||
x = y = w = 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_H(m,m);
|
||||
r = x - z;
|
||||
s = y - z;
|
||||
p = (r * s - w) / m_H(m+1,m) + m_H(m,m+1);
|
||||
q = m_H(m+1,m+1) - z - r - s;
|
||||
r = m_H(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_H(m,m-1)) * (ei_abs(q) + ei_abs(r)) <
|
||||
eps * (ei_abs(p) * (ei_abs(m_H(m-1,m-1)) + ei_abs(z) +
|
||||
ei_abs(m_H(m+1,m+1)))))
|
||||
{
|
||||
break;
|
||||
}
|
||||
m--;
|
||||
}
|
||||
|
||||
for (int i = m+2; i <= n; i++)
|
||||
{
|
||||
m_H(i,i-2) = 0.0;
|
||||
if (i > m+2)
|
||||
m_H(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_H(k,k-1);
|
||||
q = m_H(k+1,k-1);
|
||||
r = (notlast ? m_H(k+2,k-1) : 0.0);
|
||||
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_H(k,k-1) = -s * x;
|
||||
else if (l != m)
|
||||
m_H(k,k-1) = -m_H(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 < nn; j++)
|
||||
{
|
||||
p = m_H(k,j) + q * m_H(k+1,j);
|
||||
if (notlast)
|
||||
{
|
||||
p = p + r * m_H(k+2,j);
|
||||
m_H(k+2,j) = m_H(k+2,j) - p * z;
|
||||
}
|
||||
m_H(k,j) = m_H(k,j) - p * x;
|
||||
m_H(k+1,j) = m_H(k+1,j) - p * y;
|
||||
}
|
||||
|
||||
// Column modification
|
||||
for (int i = 0; i <= std::min(n,k+3); i++)
|
||||
{
|
||||
p = x * m_H(i,k) + y * m_H(i,k+1);
|
||||
if (notlast)
|
||||
{
|
||||
p = p + z * m_H(i,k+2);
|
||||
m_H(i,k+2) = m_H(i,k+2) - p * r;
|
||||
}
|
||||
m_H(i,k) = m_H(i,k) - p;
|
||||
m_H(i,k+1) = m_H(i,k+1) - p * q;
|
||||
}
|
||||
|
||||
// Accumulate transformations
|
||||
for (int i = low; i <= high; i++)
|
||||
{
|
||||
p = x * m_eivec(i,k) + y * m_eivec(i,k+1);
|
||||
if (notlast)
|
||||
{
|
||||
p = p + z * m_eivec(i,k+2);
|
||||
m_eivec(i,k+2) = m_eivec(i,k+2) - p * r;
|
||||
}
|
||||
m_eivec(i,k) = m_eivec(i,k) - p;
|
||||
m_eivec(i,k+1) = m_eivec(i,k+1) - p * q;
|
||||
}
|
||||
} // (s != 0)
|
||||
} // k loop
|
||||
} // check convergence
|
||||
} // while (n >= low)
|
||||
|
||||
// Backsubstitute to find vectors of upper triangular form
|
||||
if (norm == 0.0)
|
||||
{
|
||||
return;
|
||||
}
|
||||
|
||||
for (n = nn-1; n >= 0; n--)
|
||||
{
|
||||
p = m_eivalr[n];
|
||||
q = m_eivali[n];
|
||||
|
||||
// Scalar vector
|
||||
if (q == 0)
|
||||
{
|
||||
int l = n;
|
||||
m_H(n,n) = 1.0;
|
||||
for (int i = n-1; i >= 0; i--)
|
||||
{
|
||||
w = m_H(i,i) - p;
|
||||
r = (m_H.row(i).end(nn-l) * m_H.col(n).end(nn-l))(0,0);
|
||||
|
||||
if (m_eivali[i] < 0.0)
|
||||
{
|
||||
z = w;
|
||||
s = r;
|
||||
}
|
||||
else
|
||||
{
|
||||
l = i;
|
||||
if (m_eivali[i] == 0.0)
|
||||
{
|
||||
if (w != 0.0)
|
||||
m_H(i,n) = -r / w;
|
||||
else
|
||||
m_H(i,n) = -r / (eps * norm);
|
||||
}
|
||||
else // Solve real equations
|
||||
{
|
||||
x = m_H(i,i+1);
|
||||
y = m_H(i+1,i);
|
||||
q = (m_eivalr[i] - p) * (m_eivalr[i] - p) + m_eivali[i] * m_eivali[i];
|
||||
t = (x * s - z * r) / q;
|
||||
m_H(i,n) = t;
|
||||
if (ei_abs(x) > ei_abs(z))
|
||||
m_H(i+1,n) = (-r - w * t) / x;
|
||||
else
|
||||
m_H(i+1,n) = (-s - y * t) / z;
|
||||
}
|
||||
|
||||
// Overflow control
|
||||
t = ei_abs(m_H(i,n));
|
||||
if ((eps * t) * t > 1)
|
||||
m_H.col(n).end(nn-i) /= t;
|
||||
}
|
||||
}
|
||||
}
|
||||
else if (q < 0) // Complex vector
|
||||
{
|
||||
std::complex<Scalar> cc;
|
||||
int l = n-1;
|
||||
|
||||
// Last vector component imaginary so matrix is triangular
|
||||
if (ei_abs(m_H(n,n-1)) > ei_abs(m_H(n-1,n)))
|
||||
{
|
||||
m_H(n-1,n-1) = q / m_H(n,n-1);
|
||||
m_H(n-1,n) = -(m_H(n,n) - p) / m_H(n,n-1);
|
||||
}
|
||||
else
|
||||
{
|
||||
cc = cdiv<Scalar>(0.0,-m_H(n-1,n),m_H(n-1,n-1)-p,q);
|
||||
m_H(n-1,n-1) = ei_real(cc);
|
||||
m_H(n-1,n) = ei_imag(cc);
|
||||
}
|
||||
m_H(n,n-1) = 0.0;
|
||||
m_H(n,n) = 1.0;
|
||||
for (int i = n-2; i >= 0; i--)
|
||||
{
|
||||
Scalar ra,sa,vr,vi;
|
||||
ra = (m_H.row(i).end(nn-l) * m_H.col(n-1).end(nn-l)).lazy()(0,0);
|
||||
sa = (m_H.row(i).end(nn-l) * m_H.col(n).end(nn-l)).lazy()(0,0);
|
||||
w = m_H(i,i) - p;
|
||||
|
||||
if (m_eivali[i] < 0.0)
|
||||
{
|
||||
z = w;
|
||||
r = ra;
|
||||
s = sa;
|
||||
}
|
||||
else
|
||||
{
|
||||
l = i;
|
||||
if (m_eivali[i] == 0)
|
||||
{
|
||||
cc = cdiv(-ra,-sa,w,q);
|
||||
m_H(i,n-1) = ei_real(cc);
|
||||
m_H(i,n) = ei_imag(cc);
|
||||
}
|
||||
else
|
||||
{
|
||||
// Solve complex equations
|
||||
x = m_H(i,i+1);
|
||||
y = m_H(i+1,i);
|
||||
vr = (m_eivalr[i] - p) * (m_eivalr[i] - p) + m_eivali[i] * m_eivali[i] - q * q;
|
||||
vi = (m_eivalr[i] - p) * 2.0 * q;
|
||||
if ((vr == 0.0) && (vi == 0.0))
|
||||
vr = eps * norm * (ei_abs(w) + ei_abs(q) + ei_abs(x) + ei_abs(y) + ei_abs(z));
|
||||
|
||||
cc= cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
|
||||
m_H(i,n-1) = ei_real(cc);
|
||||
m_H(i,n) = ei_imag(cc);
|
||||
if (ei_abs(x) > (ei_abs(z) + ei_abs(q)))
|
||||
{
|
||||
m_H(i+1,n-1) = (-ra - w * m_H(i,n-1) + q * m_H(i,n)) / x;
|
||||
m_H(i+1,n) = (-sa - w * m_H(i,n) - q * m_H(i,n-1)) / x;
|
||||
}
|
||||
else
|
||||
{
|
||||
cc = cdiv(-r-y*m_H(i,n-1),-s-y*m_H(i,n),z,q);
|
||||
m_H(i+1,n-1) = ei_real(cc);
|
||||
m_H(i+1,n) = ei_imag(cc);
|
||||
}
|
||||
}
|
||||
|
||||
// Overflow control
|
||||
t = std::max(ei_abs(m_H(i,n-1)),ei_abs(m_H(i,n)));
|
||||
if ((eps * t) * t > 1)
|
||||
m_H.block(i, n-1, nn-i, 2) /= t;
|
||||
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
// Vectors of isolated roots
|
||||
for (int i = 0; i < nn; i++)
|
||||
{
|
||||
// FIXME again what's the purpose of this test ?
|
||||
// in this algo low==0 and high==nn-1 !!
|
||||
if (i < low || i > high)
|
||||
{
|
||||
m_eivec.row(i).end(nn-i) = m_H.row(i).end(nn-i);
|
||||
}
|
||||
}
|
||||
|
||||
// Back transformation to get eigenvectors of original matrix
|
||||
int bRows = high-low+1;
|
||||
for (int j = nn-1; j >= low; j--)
|
||||
{
|
||||
int bSize = std::min(j,high)-low+1;
|
||||
m_eivec.col(j).block(low, bRows) = (m_eivec.block(low, low, bRows, bSize) * m_H.col(j).block(low, bSize));
|
||||
}
|
||||
}
|
||||
|
||||
#endif // EIGEN_EIGENSOLVER_H
|
||||
@@ -95,10 +95,11 @@ void QR<MatrixType>::_compute(const MatrixType& matrix)
|
||||
m_qr(k,k) += 1.0;
|
||||
|
||||
// apply transformation to remaining columns
|
||||
for (int j = k+1; j < cols; j++)
|
||||
int remainingCols = cols - k -1;
|
||||
if (remainingCols>0)
|
||||
{
|
||||
Scalar s = -(m_qr.col(k).end(remainingSize).transpose() * m_qr.col(j).end(remainingSize))(0,0) / m_qr(k,k);
|
||||
m_qr.col(j).end(remainingSize) += s * m_qr.col(k).end(remainingSize);
|
||||
m_qr.corner(BottomRight, remainingSize, remainingCols) -= (1./m_qr(k,k)) * m_qr.col(k).end(remainingSize)
|
||||
* (m_qr.col(k).end(remainingSize).transpose() * m_qr.corner(BottomRight, remainingSize, remainingCols));
|
||||
}
|
||||
}
|
||||
m_norms[k] = -nrm;
|
||||
|
||||
Reference in New Issue
Block a user