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+// 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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