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diff --git a/ml/dlib/dlib/matrix/matrix_eigenvalue.h b/ml/dlib/dlib/matrix/matrix_eigenvalue.h
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--- a/ml/dlib/dlib/matrix/matrix_eigenvalue.h
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@@ -1,1379 +0,0 @@
-// 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
-
-
-
-
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