coco
1380 lines
40 KiB
C++
1380 lines
40 KiB
C++
// Copyright (C) 2009 Davis E. King (davis@dlib.net)
|
|
// License: Boost Software License See LICENSE.txt for the full license.
|
|
// This code was adapted from code from the JAMA part of NIST's TNT library.
|
|
// See: http://math.nist.gov/tnt/
|
|
#ifndef DLIB_MATRIX_EIGENVALUE_DECOMPOSITION_H
|
|
#define DLIB_MATRIX_EIGENVALUE_DECOMPOSITION_H
|
|
|
|
#include "matrix.h"
|
|
#include "matrix_utilities.h"
|
|
#include "matrix_subexp.h"
|
|
#include <algorithm>
|
|
#include <complex>
|
|
#include <cmath>
|
|
|
|
#ifdef DLIB_USE_LAPACK
|
|
#include "lapack/geev.h"
|
|
#include "lapack/syev.h"
|
|
#include "lapack/syevr.h"
|
|
#endif
|
|
|
|
#define DLIB_LAPACK_EIGENVALUE_DECOMP_SIZE_THRESH 4
|
|
|
|
namespace dlib
|
|
{
|
|
|
|
template <
|
|
typename matrix_exp_type
|
|
>
|
|
class eigenvalue_decomposition
|
|
{
|
|
|
|
public:
|
|
|
|
const static long NR = matrix_exp_type::NR;
|
|
const static long NC = matrix_exp_type::NC;
|
|
typedef typename matrix_exp_type::type type;
|
|
typedef typename matrix_exp_type::mem_manager_type mem_manager_type;
|
|
typedef typename matrix_exp_type::layout_type layout_type;
|
|
|
|
typedef typename matrix_exp_type::matrix_type matrix_type;
|
|
typedef matrix<type,NR,1,mem_manager_type,layout_type> column_vector_type;
|
|
|
|
typedef matrix<std::complex<type>,0,0,mem_manager_type,layout_type> complex_matrix_type;
|
|
typedef matrix<std::complex<type>,NR,1,mem_manager_type,layout_type> complex_column_vector_type;
|
|
|
|
|
|
// You have supplied an invalid type of matrix_exp_type. You have
|
|
// to use this object with matrices that contain float or double type data.
|
|
COMPILE_TIME_ASSERT((is_same_type<float, type>::value ||
|
|
is_same_type<double, type>::value ));
|
|
|
|
|
|
template <typename EXP>
|
|
eigenvalue_decomposition(
|
|
const matrix_exp<EXP>& A
|
|
);
|
|
|
|
template <typename EXP>
|
|
eigenvalue_decomposition(
|
|
const matrix_op<op_make_symmetric<EXP> >& A
|
|
);
|
|
|
|
long dim (
|
|
) const;
|
|
|
|
const complex_column_vector_type get_eigenvalues (
|
|
) const;
|
|
|
|
const column_vector_type& get_real_eigenvalues (
|
|
) const;
|
|
|
|
const column_vector_type& get_imag_eigenvalues (
|
|
) const;
|
|
|
|
const complex_matrix_type get_v (
|
|
) const;
|
|
|
|
const complex_matrix_type get_d (
|
|
) const;
|
|
|
|
const matrix_type& get_pseudo_v (
|
|
) const;
|
|
|
|
const matrix_type get_pseudo_d (
|
|
) const;
|
|
|
|
private:
|
|
|
|
/** Row and column dimension (square matrix). */
|
|
long n;
|
|
|
|
bool issymmetric;
|
|
|
|
/** Arrays for internal storage of eigenvalues. */
|
|
|
|
column_vector_type d; /* real part */
|
|
column_vector_type e; /* img part */
|
|
|
|
/** Array for internal storage of eigenvectors. */
|
|
matrix_type V;
|
|
|
|
/** Array for internal storage of nonsymmetric Hessenberg form.
|
|
@serial internal storage of nonsymmetric Hessenberg form.
|
|
*/
|
|
matrix_type H;
|
|
|
|
|
|
/** Working storage for nonsymmetric algorithm.
|
|
@serial working storage for nonsymmetric algorithm.
|
|
*/
|
|
column_vector_type ort;
|
|
|
|
// Symmetric Householder reduction to tridiagonal form.
|
|
void tred2();
|
|
|
|
|
|
// Symmetric tridiagonal QL algorithm.
|
|
void tql2 ();
|
|
|
|
|
|
// Nonsymmetric reduction to Hessenberg form.
|
|
void orthes ();
|
|
|
|
|
|
// Complex scalar division.
|
|
type cdivr, cdivi;
|
|
void cdiv_(type xr, type xi, type yr, type yi);
|
|
|
|
|
|
// Nonsymmetric reduction from Hessenberg to real Schur form.
|
|
void hqr2 ();
|
|
};
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
// ----------------------------------------------------------------------------------------
|
|
// Public member functions
|
|
// ----------------------------------------------------------------------------------------
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
template <typename EXP>
|
|
eigenvalue_decomposition<matrix_exp_type>::
|
|
eigenvalue_decomposition(
|
|
const matrix_exp<EXP>& A_
|
|
)
|
|
{
|
|
COMPILE_TIME_ASSERT((is_same_type<type, typename EXP::type>::value));
|
|
|
|
|
|
const_temp_matrix<EXP> A(A_);
|
|
|
|
// make sure requires clause is not broken
|
|
DLIB_ASSERT(A.nr() == A.nc() && A.size() > 0,
|
|
"\teigenvalue_decomposition::eigenvalue_decomposition(A)"
|
|
<< "\n\tYou can only use this on square matrices"
|
|
<< "\n\tA.nr(): " << A.nr()
|
|
<< "\n\tA.nc(): " << A.nc()
|
|
<< "\n\tA.size(): " << A.size()
|
|
<< "\n\tthis: " << this
|
|
);
|
|
|
|
|
|
n = A.nc();
|
|
V.set_size(n,n);
|
|
d.set_size(n);
|
|
e.set_size(n);
|
|
|
|
|
|
issymmetric = true;
|
|
for (long j = 0; (j < n) && issymmetric; j++)
|
|
{
|
|
for (long i = 0; (i < n) && issymmetric; i++)
|
|
{
|
|
issymmetric = (A(i,j) == A(j,i));
|
|
}
|
|
}
|
|
|
|
if (issymmetric)
|
|
{
|
|
V = A;
|
|
|
|
#ifdef DLIB_USE_LAPACK
|
|
if (A.nr() > DLIB_LAPACK_EIGENVALUE_DECOMP_SIZE_THRESH)
|
|
{
|
|
e = 0;
|
|
|
|
// We could compute the result using syev()
|
|
//lapack::syev('V', 'L', V, d);
|
|
|
|
// Instead, we use syevr because its faster and maybe more stable.
|
|
matrix_type tempA(A);
|
|
matrix<lapack::integer,0,0,mem_manager_type,layout_type> isupz;
|
|
|
|
lapack::integer temp;
|
|
lapack::syevr('V','A','L',tempA,0,0,0,0,-1,temp,d,V,isupz);
|
|
return;
|
|
}
|
|
#endif
|
|
// Tridiagonalize.
|
|
tred2();
|
|
|
|
// Diagonalize.
|
|
tql2();
|
|
|
|
}
|
|
else
|
|
{
|
|
|
|
#ifdef DLIB_USE_LAPACK
|
|
if (A.nr() > DLIB_LAPACK_EIGENVALUE_DECOMP_SIZE_THRESH)
|
|
{
|
|
matrix<type,0,0,mem_manager_type, column_major_layout> temp, vl, vr;
|
|
temp = A;
|
|
lapack::geev('N', 'V', temp, d, e, vl, vr);
|
|
V = vr;
|
|
return;
|
|
}
|
|
#endif
|
|
H = A;
|
|
|
|
ort.set_size(n);
|
|
|
|
// Reduce to Hessenberg form.
|
|
orthes();
|
|
|
|
// Reduce Hessenberg to real Schur form.
|
|
hqr2();
|
|
}
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
template <typename EXP>
|
|
eigenvalue_decomposition<matrix_exp_type>::
|
|
eigenvalue_decomposition(
|
|
const matrix_op<op_make_symmetric<EXP> >& A
|
|
)
|
|
{
|
|
COMPILE_TIME_ASSERT((is_same_type<type, typename EXP::type>::value));
|
|
|
|
|
|
// make sure requires clause is not broken
|
|
DLIB_ASSERT(A.nr() == A.nc() && A.size() > 0,
|
|
"\teigenvalue_decomposition::eigenvalue_decomposition(A)"
|
|
<< "\n\tYou can only use this on square matrices"
|
|
<< "\n\tA.nr(): " << A.nr()
|
|
<< "\n\tA.nc(): " << A.nc()
|
|
<< "\n\tA.size(): " << A.size()
|
|
<< "\n\tthis: " << this
|
|
);
|
|
|
|
|
|
n = A.nc();
|
|
V.set_size(n,n);
|
|
d.set_size(n);
|
|
e.set_size(n);
|
|
|
|
|
|
V = A;
|
|
|
|
#ifdef DLIB_USE_LAPACK
|
|
if (A.nr() > DLIB_LAPACK_EIGENVALUE_DECOMP_SIZE_THRESH)
|
|
{
|
|
e = 0;
|
|
|
|
// We could compute the result using syev()
|
|
//lapack::syev('V', 'L', V, d);
|
|
|
|
// Instead, we use syevr because its faster and maybe more stable.
|
|
matrix_type tempA(A);
|
|
matrix<lapack::integer,0,0,mem_manager_type,layout_type> isupz;
|
|
|
|
lapack::integer temp;
|
|
lapack::syevr('V','A','L',tempA,0,0,0,0,-1,temp,d,V,isupz);
|
|
return;
|
|
}
|
|
#endif
|
|
// Tridiagonalize.
|
|
tred2();
|
|
|
|
// Diagonalize.
|
|
tql2();
|
|
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
const typename eigenvalue_decomposition<matrix_exp_type>::matrix_type& eigenvalue_decomposition<matrix_exp_type>::
|
|
get_pseudo_v (
|
|
) const
|
|
{
|
|
return V;
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
long eigenvalue_decomposition<matrix_exp_type>::
|
|
dim (
|
|
) const
|
|
{
|
|
return V.nr();
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
const typename eigenvalue_decomposition<matrix_exp_type>::complex_column_vector_type eigenvalue_decomposition<matrix_exp_type>::
|
|
get_eigenvalues (
|
|
) const
|
|
{
|
|
return complex_matrix(get_real_eigenvalues(), get_imag_eigenvalues());
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
const typename eigenvalue_decomposition<matrix_exp_type>::column_vector_type& eigenvalue_decomposition<matrix_exp_type>::
|
|
get_real_eigenvalues (
|
|
) const
|
|
{
|
|
return d;
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
const typename eigenvalue_decomposition<matrix_exp_type>::column_vector_type& eigenvalue_decomposition<matrix_exp_type>::
|
|
get_imag_eigenvalues (
|
|
) const
|
|
{
|
|
return e;
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
const typename eigenvalue_decomposition<matrix_exp_type>::complex_matrix_type eigenvalue_decomposition<matrix_exp_type>::
|
|
get_d (
|
|
) const
|
|
{
|
|
return diagm(complex_matrix(get_real_eigenvalues(), get_imag_eigenvalues()));
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
const typename eigenvalue_decomposition<matrix_exp_type>::complex_matrix_type eigenvalue_decomposition<matrix_exp_type>::
|
|
get_v (
|
|
) const
|
|
{
|
|
complex_matrix_type CV(n,n);
|
|
|
|
for (long i = 0; i < n; i++)
|
|
{
|
|
if (e(i) > 0)
|
|
{
|
|
set_colm(CV,i) = complex_matrix(colm(V,i), colm(V,i+1));
|
|
}
|
|
else if (e(i) < 0)
|
|
{
|
|
set_colm(CV,i) = complex_matrix(colm(V,i), colm(V,i-1));
|
|
}
|
|
else
|
|
{
|
|
set_colm(CV,i) = complex_matrix(colm(V,i), uniform_matrix<type>(n,1,0));
|
|
}
|
|
}
|
|
|
|
return CV;
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
const typename eigenvalue_decomposition<matrix_exp_type>::matrix_type eigenvalue_decomposition<matrix_exp_type>::
|
|
get_pseudo_d (
|
|
) const
|
|
{
|
|
matrix_type D(n,n);
|
|
|
|
for (long i = 0; i < n; i++)
|
|
{
|
|
for (long j = 0; j < n; j++)
|
|
{
|
|
D(i,j) = 0.0;
|
|
}
|
|
D(i,i) = d(i);
|
|
if (e(i) > 0)
|
|
{
|
|
D(i,i+1) = e(i);
|
|
}
|
|
else if (e(i) < 0)
|
|
{
|
|
D(i,i-1) = e(i);
|
|
}
|
|
}
|
|
|
|
return D;
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
// ----------------------------------------------------------------------------------------
|
|
// Private member functions
|
|
// ----------------------------------------------------------------------------------------
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
// Symmetric Householder reduction to tridiagonal form.
|
|
template <typename matrix_exp_type>
|
|
void eigenvalue_decomposition<matrix_exp_type>::
|
|
tred2()
|
|
{
|
|
using std::abs;
|
|
using std::sqrt;
|
|
|
|
// 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.
|
|
|
|
for (long j = 0; j < n; j++)
|
|
{
|
|
d(j) = V(n-1,j);
|
|
}
|
|
|
|
// Householder reduction to tridiagonal form.
|
|
|
|
for (long i = n-1; i > 0; i--)
|
|
{
|
|
|
|
// Scale to avoid under/overflow.
|
|
|
|
type scale = 0.0;
|
|
type h = 0.0;
|
|
for (long k = 0; k < i; k++)
|
|
{
|
|
scale = scale + abs(d(k));
|
|
}
|
|
if (scale == 0.0)
|
|
{
|
|
e(i) = d(i-1);
|
|
for (long j = 0; j < i; j++)
|
|
{
|
|
d(j) = V(i-1,j);
|
|
V(i,j) = 0.0;
|
|
V(j,i) = 0.0;
|
|
}
|
|
}
|
|
else
|
|
{
|
|
|
|
// Generate Householder vector.
|
|
|
|
for (long k = 0; k < i; k++)
|
|
{
|
|
d(k) /= scale;
|
|
h += d(k) * d(k);
|
|
}
|
|
type f = d(i-1);
|
|
type g = sqrt(h);
|
|
if (f > 0)
|
|
{
|
|
g = -g;
|
|
}
|
|
e(i) = scale * g;
|
|
h = h - f * g;
|
|
d(i-1) = f - g;
|
|
for (long j = 0; j < i; j++)
|
|
{
|
|
e(j) = 0.0;
|
|
}
|
|
|
|
// Apply similarity transformation to remaining columns.
|
|
|
|
for (long j = 0; j < i; j++)
|
|
{
|
|
f = d(j);
|
|
V(j,i) = f;
|
|
g = e(j) + V(j,j) * f;
|
|
for (long k = j+1; k <= i-1; k++)
|
|
{
|
|
g += V(k,j) * d(k);
|
|
e(k) += V(k,j) * f;
|
|
}
|
|
e(j) = g;
|
|
}
|
|
f = 0.0;
|
|
for (long j = 0; j < i; j++)
|
|
{
|
|
e(j) /= h;
|
|
f += e(j) * d(j);
|
|
}
|
|
type hh = f / (h + h);
|
|
for (long j = 0; j < i; j++)
|
|
{
|
|
e(j) -= hh * d(j);
|
|
}
|
|
for (long j = 0; j < i; j++)
|
|
{
|
|
f = d(j);
|
|
g = e(j);
|
|
for (long k = j; k <= i-1; k++)
|
|
{
|
|
V(k,j) -= (f * e(k) + g * d(k));
|
|
}
|
|
d(j) = V(i-1,j);
|
|
V(i,j) = 0.0;
|
|
}
|
|
}
|
|
d(i) = h;
|
|
}
|
|
|
|
// Accumulate transformations.
|
|
|
|
for (long i = 0; i < n-1; i++)
|
|
{
|
|
V(n-1,i) = V(i,i);
|
|
V(i,i) = 1.0;
|
|
type h = d(i+1);
|
|
if (h != 0.0)
|
|
{
|
|
for (long k = 0; k <= i; k++)
|
|
{
|
|
d(k) = V(k,i+1) / h;
|
|
}
|
|
for (long j = 0; j <= i; j++)
|
|
{
|
|
type g = 0.0;
|
|
for (long k = 0; k <= i; k++)
|
|
{
|
|
g += V(k,i+1) * V(k,j);
|
|
}
|
|
for (long k = 0; k <= i; k++)
|
|
{
|
|
V(k,j) -= g * d(k);
|
|
}
|
|
}
|
|
}
|
|
for (long k = 0; k <= i; k++)
|
|
{
|
|
V(k,i+1) = 0.0;
|
|
}
|
|
}
|
|
for (long j = 0; j < n; j++)
|
|
{
|
|
d(j) = V(n-1,j);
|
|
V(n-1,j) = 0.0;
|
|
}
|
|
V(n-1,n-1) = 1.0;
|
|
e(0) = 0.0;
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
void eigenvalue_decomposition<matrix_exp_type>::
|
|
tql2 ()
|
|
{
|
|
using std::pow;
|
|
using std::min;
|
|
using std::max;
|
|
using std::abs;
|
|
|
|
// 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.
|
|
|
|
for (long i = 1; i < n; i++)
|
|
{
|
|
e(i-1) = e(i);
|
|
}
|
|
e(n-1) = 0.0;
|
|
|
|
type f = 0.0;
|
|
type tst1 = 0.0;
|
|
const type eps = std::numeric_limits<type>::epsilon();
|
|
for (long l = 0; l < n; l++)
|
|
{
|
|
|
|
// Find small subdiagonal element
|
|
|
|
tst1 = max(tst1,abs(d(l)) + abs(e(l)));
|
|
long m = l;
|
|
|
|
// Original while-loop from Java code
|
|
while (m < n)
|
|
{
|
|
if (abs(e(m)) <= eps*tst1)
|
|
{
|
|
break;
|
|
}
|
|
m++;
|
|
}
|
|
if (m == n)
|
|
--m;
|
|
|
|
|
|
// If m == l, d(l) is an eigenvalue,
|
|
// otherwise, iterate.
|
|
|
|
if (m > l)
|
|
{
|
|
long iter = 0;
|
|
do
|
|
{
|
|
iter = iter + 1; // (Could check iteration count here.)
|
|
|
|
// Compute implicit shift
|
|
|
|
type g = d(l);
|
|
type p = (d(l+1) - g) / (2.0 * e(l));
|
|
type r = hypot(p,(type)1.0);
|
|
if (p < 0)
|
|
{
|
|
r = -r;
|
|
}
|
|
d(l) = e(l) / (p + r);
|
|
d(l+1) = e(l) * (p + r);
|
|
type dl1 = d(l+1);
|
|
type h = g - d(l);
|
|
for (long i = l+2; i < n; i++)
|
|
{
|
|
d(i) -= h;
|
|
}
|
|
f = f + h;
|
|
|
|
// Implicit QL transformation.
|
|
|
|
p = d(m);
|
|
type c = 1.0;
|
|
type c2 = c;
|
|
type c3 = c;
|
|
type el1 = e(l+1);
|
|
type s = 0.0;
|
|
type s2 = 0.0;
|
|
for (long i = m-1; i >= l; i--)
|
|
{
|
|
c3 = c2;
|
|
c2 = c;
|
|
s2 = s;
|
|
g = c * e(i);
|
|
h = c * p;
|
|
r = hypot(p,e(i));
|
|
e(i+1) = s * r;
|
|
s = e(i) / r;
|
|
c = p / r;
|
|
p = c * d(i) - s * g;
|
|
d(i+1) = h + s * (c * g + s * d(i));
|
|
|
|
// Accumulate transformation.
|
|
|
|
for (long k = 0; k < n; k++)
|
|
{
|
|
h = V(k,i+1);
|
|
V(k,i+1) = s * V(k,i) + c * h;
|
|
V(k,i) = c * V(k,i) - s * h;
|
|
}
|
|
}
|
|
p = -s * s2 * c3 * el1 * e(l) / dl1;
|
|
e(l) = s * p;
|
|
d(l) = c * p;
|
|
|
|
// Check for convergence.
|
|
|
|
} while (abs(e(l)) > eps*tst1);
|
|
}
|
|
d(l) = d(l) + f;
|
|
e(l) = 0.0;
|
|
}
|
|
|
|
/*
|
|
The code to sort the eigenvalues and eigenvectors
|
|
has been removed from here since, in the non-symmetric case,
|
|
we can't sort the eigenvalues in a meaningful way. If we left this
|
|
code in here then the user might supply what they thought was a symmetric
|
|
matrix but was actually slightly non-symmetric due to rounding error
|
|
and then they would end up in the non-symmetric eigenvalue solver
|
|
where the eigenvalues don't end up getting sorted. So to avoid
|
|
any possible user confusion I'm just removing this.
|
|
*/
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
void eigenvalue_decomposition<matrix_exp_type>::
|
|
orthes ()
|
|
{
|
|
using std::abs;
|
|
using std::sqrt;
|
|
|
|
// 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.
|
|
|
|
long low = 0;
|
|
long high = n-1;
|
|
|
|
for (long m = low+1; m <= high-1; m++)
|
|
{
|
|
|
|
// Scale column.
|
|
|
|
type scale = 0.0;
|
|
for (long i = m; i <= high; i++)
|
|
{
|
|
scale = scale + abs(H(i,m-1));
|
|
}
|
|
if (scale != 0.0)
|
|
{
|
|
|
|
// Compute Householder transformation.
|
|
|
|
type h = 0.0;
|
|
for (long i = high; i >= m; i--)
|
|
{
|
|
ort(i) = H(i,m-1)/scale;
|
|
h += ort(i) * ort(i);
|
|
}
|
|
type g = sqrt(h);
|
|
if (ort(m) > 0)
|
|
{
|
|
g = -g;
|
|
}
|
|
h = h - ort(m) * g;
|
|
ort(m) = ort(m) - g;
|
|
|
|
// Apply Householder similarity transformation
|
|
// H = (I-u*u'/h)*H*(I-u*u')/h)
|
|
|
|
for (long j = m; j < n; j++)
|
|
{
|
|
type f = 0.0;
|
|
for (long i = high; i >= m; i--)
|
|
{
|
|
f += ort(i)*H(i,j);
|
|
}
|
|
f = f/h;
|
|
for (long i = m; i <= high; i++)
|
|
{
|
|
H(i,j) -= f*ort(i);
|
|
}
|
|
}
|
|
|
|
for (long i = 0; i <= high; i++)
|
|
{
|
|
type f = 0.0;
|
|
for (long j = high; j >= m; j--)
|
|
{
|
|
f += ort(j)*H(i,j);
|
|
}
|
|
f = f/h;
|
|
for (long j = m; j <= high; j++)
|
|
{
|
|
H(i,j) -= f*ort(j);
|
|
}
|
|
}
|
|
ort(m) = scale*ort(m);
|
|
H(m,m-1) = scale*g;
|
|
}
|
|
}
|
|
|
|
// Accumulate transformations (Algol's ortran).
|
|
|
|
for (long i = 0; i < n; i++)
|
|
{
|
|
for (long j = 0; j < n; j++)
|
|
{
|
|
V(i,j) = (i == j ? 1.0 : 0.0);
|
|
}
|
|
}
|
|
|
|
for (long m = high-1; m >= low+1; m--)
|
|
{
|
|
if (H(m,m-1) != 0.0)
|
|
{
|
|
for (long i = m+1; i <= high; i++)
|
|
{
|
|
ort(i) = H(i,m-1);
|
|
}
|
|
for (long j = m; j <= high; j++)
|
|
{
|
|
type g = 0.0;
|
|
for (long i = m; i <= high; i++)
|
|
{
|
|
g += ort(i) * V(i,j);
|
|
}
|
|
// Double division avoids possible underflow
|
|
g = (g / ort(m)) / H(m,m-1);
|
|
for (long i = m; i <= high; i++)
|
|
{
|
|
V(i,j) += g * ort(i);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
void eigenvalue_decomposition<matrix_exp_type>::
|
|
cdiv_(type xr, type xi, type yr, type yi)
|
|
{
|
|
using std::abs;
|
|
type r,d;
|
|
if (abs(yr) > abs(yi))
|
|
{
|
|
r = yi/yr;
|
|
d = yr + r*yi;
|
|
cdivr = (xr + r*xi)/d;
|
|
cdivi = (xi - r*xr)/d;
|
|
}
|
|
else
|
|
{
|
|
r = yr/yi;
|
|
d = yi + r*yr;
|
|
cdivr = (r*xr + xi)/d;
|
|
cdivi = (r*xi - xr)/d;
|
|
}
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
template <typename matrix_exp_type>
|
|
void eigenvalue_decomposition<matrix_exp_type>::
|
|
hqr2 ()
|
|
{
|
|
using std::pow;
|
|
using std::min;
|
|
using std::max;
|
|
using std::abs;
|
|
using std::sqrt;
|
|
|
|
// 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
|
|
|
|
long nn = this->n;
|
|
long n = nn-1;
|
|
long low = 0;
|
|
long high = nn-1;
|
|
const type eps = std::numeric_limits<type>::epsilon();
|
|
type exshift = 0.0;
|
|
type p=0,q=0,r=0,s=0,z=0,t,w,x,y;
|
|
|
|
// Store roots isolated by balanc and compute matrix norm
|
|
|
|
type norm = 0.0;
|
|
for (long i = 0; i < nn; i++)
|
|
{
|
|
if ((i < low) || (i > high))
|
|
{
|
|
d(i) = H(i,i);
|
|
e(i) = 0.0;
|
|
}
|
|
for (long j = max(i-1,0L); j < nn; j++)
|
|
{
|
|
norm = norm + abs(H(i,j));
|
|
}
|
|
}
|
|
|
|
// Outer loop over eigenvalue index
|
|
|
|
long iter = 0;
|
|
while (n >= low)
|
|
{
|
|
|
|
// Look for single small sub-diagonal element
|
|
|
|
long l = n;
|
|
while (l > low)
|
|
{
|
|
s = abs(H(l-1,l-1)) + abs(H(l,l));
|
|
if (s == 0.0)
|
|
{
|
|
s = norm;
|
|
}
|
|
if (abs(H(l,l-1)) < eps * s)
|
|
{
|
|
break;
|
|
}
|
|
l--;
|
|
}
|
|
|
|
// Check for convergence
|
|
// One root found
|
|
|
|
if (l == n)
|
|
{
|
|
H(n,n) = H(n,n) + exshift;
|
|
d(n) = H(n,n);
|
|
e(n) = 0.0;
|
|
n--;
|
|
iter = 0;
|
|
|
|
// Two roots found
|
|
|
|
}
|
|
else if (l == n-1)
|
|
{
|
|
w = H(n,n-1) * H(n-1,n);
|
|
p = (H(n-1,n-1) - H(n,n)) / 2.0;
|
|
q = p * p + w;
|
|
z = sqrt(abs(q));
|
|
H(n,n) = H(n,n) + exshift;
|
|
H(n-1,n-1) = H(n-1,n-1) + exshift;
|
|
x = H(n,n);
|
|
|
|
// type pair
|
|
|
|
if (q >= 0)
|
|
{
|
|
if (p >= 0)
|
|
{
|
|
z = p + z;
|
|
}
|
|
else
|
|
{
|
|
z = p - z;
|
|
}
|
|
d(n-1) = x + z;
|
|
d(n) = d(n-1);
|
|
if (z != 0.0)
|
|
{
|
|
d(n) = x - w / z;
|
|
}
|
|
e(n-1) = 0.0;
|
|
e(n) = 0.0;
|
|
x = H(n,n-1);
|
|
s = abs(x) + abs(z);
|
|
p = x / s;
|
|
q = z / s;
|
|
r = sqrt(p * p+q * q);
|
|
p = p / r;
|
|
q = q / r;
|
|
|
|
// Row modification
|
|
|
|
for (long j = n-1; j < nn; j++)
|
|
{
|
|
z = H(n-1,j);
|
|
H(n-1,j) = q * z + p * H(n,j);
|
|
H(n,j) = q * H(n,j) - p * z;
|
|
}
|
|
|
|
// Column modification
|
|
|
|
for (long i = 0; i <= n; i++)
|
|
{
|
|
z = H(i,n-1);
|
|
H(i,n-1) = q * z + p * H(i,n);
|
|
H(i,n) = q * H(i,n) - p * z;
|
|
}
|
|
|
|
// Accumulate transformations
|
|
|
|
for (long i = low; i <= high; i++)
|
|
{
|
|
z = V(i,n-1);
|
|
V(i,n-1) = q * z + p * V(i,n);
|
|
V(i,n) = q * V(i,n) - p * z;
|
|
}
|
|
|
|
// Complex pair
|
|
|
|
}
|
|
else
|
|
{
|
|
d(n-1) = x + p;
|
|
d(n) = x + p;
|
|
e(n-1) = z;
|
|
e(n) = -z;
|
|
}
|
|
n = n - 2;
|
|
iter = 0;
|
|
|
|
// No convergence yet
|
|
|
|
}
|
|
else
|
|
{
|
|
|
|
// Form shift
|
|
|
|
x = H(n,n);
|
|
y = 0.0;
|
|
w = 0.0;
|
|
if (l < n)
|
|
{
|
|
y = H(n-1,n-1);
|
|
w = H(n,n-1) * H(n-1,n);
|
|
}
|
|
|
|
// Wilkinson's original ad hoc shift
|
|
|
|
if (iter == 10)
|
|
{
|
|
exshift += x;
|
|
for (long i = low; i <= n; i++)
|
|
{
|
|
H(i,i) -= x;
|
|
}
|
|
s = abs(H(n,n-1)) + abs(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 = sqrt(s);
|
|
if (y < x)
|
|
{
|
|
s = -s;
|
|
}
|
|
s = x - w / ((y - x) / 2.0 + s);
|
|
for (long i = low; i <= n; i++)
|
|
{
|
|
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
|
|
|
|
long m = n-2;
|
|
while (m >= l)
|
|
{
|
|
z = H(m,m);
|
|
r = x - z;
|
|
s = y - z;
|
|
p = (r * s - w) / H(m+1,m) + H(m,m+1);
|
|
q = H(m+1,m+1) - z - r - s;
|
|
r = H(m+2,m+1);
|
|
s = abs(p) + abs(q) + abs(r);
|
|
p = p / s;
|
|
q = q / s;
|
|
r = r / s;
|
|
if (m == l)
|
|
{
|
|
break;
|
|
}
|
|
if (abs(H(m,m-1)) * (abs(q) + abs(r)) <
|
|
eps * (abs(p) * (abs(H(m-1,m-1)) + abs(z) +
|
|
abs(H(m+1,m+1)))))
|
|
{
|
|
break;
|
|
}
|
|
m--;
|
|
}
|
|
|
|
for (long i = m+2; i <= n; i++)
|
|
{
|
|
H(i,i-2) = 0.0;
|
|
if (i > m+2)
|
|
{
|
|
H(i,i-3) = 0.0;
|
|
}
|
|
}
|
|
|
|
// Double QR step involving rows l:n and columns m:n
|
|
|
|
for (long k = m; k <= n-1; k++)
|
|
{
|
|
long notlast = (k != n-1);
|
|
if (k != m)
|
|
{
|
|
p = H(k,k-1);
|
|
q = H(k+1,k-1);
|
|
r = (notlast ? H(k+2,k-1) : 0.0);
|
|
x = abs(p) + abs(q) + abs(r);
|
|
if (x != 0.0)
|
|
{
|
|
p = p / x;
|
|
q = q / x;
|
|
r = r / x;
|
|
}
|
|
}
|
|
if (x == 0.0)
|
|
{
|
|
break;
|
|
}
|
|
s = sqrt(p * p + q * q + r * r);
|
|
if (p < 0)
|
|
{
|
|
s = -s;
|
|
}
|
|
if (s != 0)
|
|
{
|
|
if (k != m)
|
|
{
|
|
H(k,k-1) = -s * x;
|
|
}
|
|
else if (l != m)
|
|
{
|
|
H(k,k-1) = -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 (long j = k; j < nn; j++)
|
|
{
|
|
p = H(k,j) + q * H(k+1,j);
|
|
if (notlast)
|
|
{
|
|
p = p + r * H(k+2,j);
|
|
H(k+2,j) = H(k+2,j) - p * z;
|
|
}
|
|
H(k,j) = H(k,j) - p * x;
|
|
H(k+1,j) = H(k+1,j) - p * y;
|
|
}
|
|
|
|
// Column modification
|
|
|
|
for (long i = 0; i <= min(n,k+3); i++)
|
|
{
|
|
p = x * H(i,k) + y * H(i,k+1);
|
|
if (notlast)
|
|
{
|
|
p = p + z * H(i,k+2);
|
|
H(i,k+2) = H(i,k+2) - p * r;
|
|
}
|
|
H(i,k) = H(i,k) - p;
|
|
H(i,k+1) = H(i,k+1) - p * q;
|
|
}
|
|
|
|
// Accumulate transformations
|
|
|
|
for (long i = low; i <= high; i++)
|
|
{
|
|
p = x * V(i,k) + y * V(i,k+1);
|
|
if (notlast)
|
|
{
|
|
p = p + z * V(i,k+2);
|
|
V(i,k+2) = V(i,k+2) - p * r;
|
|
}
|
|
V(i,k) = V(i,k) - p;
|
|
V(i,k+1) = V(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 = d(n);
|
|
q = e(n);
|
|
|
|
// Real vector
|
|
|
|
if (q == 0)
|
|
{
|
|
long l = n;
|
|
H(n,n) = 1.0;
|
|
for (long i = n-1; i >= 0; i--)
|
|
{
|
|
w = H(i,i) - p;
|
|
r = 0.0;
|
|
for (long j = l; j <= n; j++)
|
|
{
|
|
r = r + H(i,j) * H(j,n);
|
|
}
|
|
if (e(i) < 0.0)
|
|
{
|
|
z = w;
|
|
s = r;
|
|
}
|
|
else
|
|
{
|
|
l = i;
|
|
if (e(i) == 0.0)
|
|
{
|
|
if (w != 0.0)
|
|
{
|
|
H(i,n) = -r / w;
|
|
}
|
|
else
|
|
{
|
|
H(i,n) = -r / (eps * norm);
|
|
}
|
|
|
|
// Solve real equations
|
|
|
|
}
|
|
else
|
|
{
|
|
x = H(i,i+1);
|
|
y = H(i+1,i);
|
|
q = (d(i) - p) * (d(i) - p) + e(i) * e(i);
|
|
t = (x * s - z * r) / q;
|
|
H(i,n) = t;
|
|
if (abs(x) > abs(z))
|
|
{
|
|
H(i+1,n) = (-r - w * t) / x;
|
|
}
|
|
else
|
|
{
|
|
H(i+1,n) = (-s - y * t) / z;
|
|
}
|
|
}
|
|
|
|
// Overflow control
|
|
|
|
t = abs(H(i,n));
|
|
if ((eps * t) * t > 1)
|
|
{
|
|
for (long j = i; j <= n; j++)
|
|
{
|
|
H(j,n) = H(j,n) / t;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// Complex vector
|
|
|
|
}
|
|
else if (q < 0)
|
|
{
|
|
long l = n-1;
|
|
|
|
// Last vector component imaginary so matrix is triangular
|
|
|
|
if (abs(H(n,n-1)) > abs(H(n-1,n)))
|
|
{
|
|
H(n-1,n-1) = q / H(n,n-1);
|
|
H(n-1,n) = -(H(n,n) - p) / H(n,n-1);
|
|
}
|
|
else
|
|
{
|
|
cdiv_(0.0,-H(n-1,n),H(n-1,n-1)-p,q);
|
|
H(n-1,n-1) = cdivr;
|
|
H(n-1,n) = cdivi;
|
|
}
|
|
H(n,n-1) = 0.0;
|
|
H(n,n) = 1.0;
|
|
for (long i = n-2; i >= 0; i--)
|
|
{
|
|
type ra,sa,vr,vi;
|
|
ra = 0.0;
|
|
sa = 0.0;
|
|
for (long j = l; j <= n; j++)
|
|
{
|
|
ra = ra + H(i,j) * H(j,n-1);
|
|
sa = sa + H(i,j) * H(j,n);
|
|
}
|
|
w = H(i,i) - p;
|
|
|
|
if (e(i) < 0.0)
|
|
{
|
|
z = w;
|
|
r = ra;
|
|
s = sa;
|
|
}
|
|
else
|
|
{
|
|
l = i;
|
|
if (e(i) == 0)
|
|
{
|
|
cdiv_(-ra,-sa,w,q);
|
|
H(i,n-1) = cdivr;
|
|
H(i,n) = cdivi;
|
|
}
|
|
else
|
|
{
|
|
|
|
// Solve complex equations
|
|
|
|
x = H(i,i+1);
|
|
y = H(i+1,i);
|
|
vr = (d(i) - p) * (d(i) - p) + e(i) * e(i) - q * q;
|
|
vi = (d(i) - p) * 2.0 * q;
|
|
if ((vr == 0.0) && (vi == 0.0))
|
|
{
|
|
vr = eps * norm * (abs(w) + abs(q) +
|
|
abs(x) + abs(y) + abs(z));
|
|
}
|
|
cdiv_(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
|
|
H(i,n-1) = cdivr;
|
|
H(i,n) = cdivi;
|
|
if (abs(x) > (abs(z) + abs(q)))
|
|
{
|
|
H(i+1,n-1) = (-ra - w * H(i,n-1) + q * H(i,n)) / x;
|
|
H(i+1,n) = (-sa - w * H(i,n) - q * H(i,n-1)) / x;
|
|
}
|
|
else
|
|
{
|
|
cdiv_(-r-y*H(i,n-1),-s-y*H(i,n),z,q);
|
|
H(i+1,n-1) = cdivr;
|
|
H(i+1,n) = cdivi;
|
|
}
|
|
}
|
|
|
|
// Overflow control
|
|
|
|
t = max(abs(H(i,n-1)),abs(H(i,n)));
|
|
if ((eps * t) * t > 1)
|
|
{
|
|
for (long j = i; j <= n; j++)
|
|
{
|
|
H(j,n-1) = H(j,n-1) / t;
|
|
H(j,n) = H(j,n) / t;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// Vectors of isolated roots
|
|
|
|
for (long i = 0; i < nn; i++)
|
|
{
|
|
if (i < low || i > high)
|
|
{
|
|
for (long j = i; j < nn; j++)
|
|
{
|
|
V(i,j) = H(i,j);
|
|
}
|
|
}
|
|
}
|
|
|
|
// Back transformation to get eigenvectors of original matrix
|
|
|
|
for (long j = nn-1; j >= low; j--)
|
|
{
|
|
for (long i = low; i <= high; i++)
|
|
{
|
|
z = 0.0;
|
|
for (long k = low; k <= min(j,high); k++)
|
|
{
|
|
z = z + V(i,k) * H(k,j);
|
|
}
|
|
V(i,j) = z;
|
|
}
|
|
}
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
|
|
}
|
|
|
|
#endif // DLIB_MATRIX_EIGENVALUE_DECOMPOSITION_H
|
|
|
|
|
|
|
|
|