// Accord Math Library
// Accord.NET framework
// http://www.crsouza.com
//
// Modifications copyright � C�sar Souza, 2009-2010
// cesarsouza at gmail.com
//
// Original work copyright � Lutz Roeder, 2000
// Adapted from Mapack for COM and Jama routines
//
namespace Accord.Math.Decompositions
{
using System;
using Accord.Math;
/// <summary>
/// Determines the eigenvalues and eigenvectors of a real square matrix.
/// </summary>
/// <remarks>
/// <para>
/// In the mathematical discipline of linear algebra, eigendecomposition
/// or sometimes spectral decomposition is the factorization of a matrix
/// into a canonical form, whereby the matrix is represented in terms of
/// its eigenvalues and eigenvectors.</para>
/// <para>
/// If <c>A</c> is symmetric, then <c>A = V * D * V'</c> and <c>A = V * V'</c>
/// where the eigenvalue matrix <c>D</c> is diagonal and the eigenvector matrix <c>V</c> is orthogonal.
/// If <c>A</c> is not symmetric, the eigenvalue matrix <c>D</c> is block diagonal
/// with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
/// <c>lambda + i*mu</c>, in 2-by-2 blocks, <c>[lambda, mu; -mu, lambda]</c>.
/// The columns of <c>V</c> represent the eigenvectors in the sense that <c>A * V = V * D</c>.
/// The matrix V may be badly conditioned, or even singular, so the validity of the equation
/// <c>A = V * D * inverse(V)</c> depends upon the condition of <c>V</c>.
/// </para>
/// </remarks>
public sealed class EigenvalueDecomposition
{
private int n; // matrix dimension
private double[] d, e; // storage of eigenvalues.
private double[,] V; // storage of eigenvectors.
private double[,] H; // storage of nonsymmetric Hessenberg form.
private double[] ort; // storage for nonsymmetric algorithm.
private double cdivr, cdivi;
private bool symmetric;
/// <summary>Construct an eigenvalue decomposition.</summary>
public EigenvalueDecomposition(double[,] value)
: this(value, value.IsSymmetric())
{
}
/// <summary>Construct an eigenvalue decomposition.</summary>
public EigenvalueDecomposition(double[,] value, bool assumeSymmetric)
{
if (value == null)
{
throw new ArgumentNullException("value", "Matrix cannot be null.");
}
if (value.GetLength(0) != value.GetLength(1))
{
throw new ArgumentException("Matrix is not a square matrix.", "value");
}
n = value.GetLength(1);
V = new double[n, n];
d = new double[n];
e = new double[n];
this.symmetric = assumeSymmetric;
if (this.symmetric)
{
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
V[i, j] = value[i, j];
// Tridiagonalize.
this.tred2();
// Diagonalize.
this.tql2();
}
else
{
H = new double[n, n];
ort = new double[n];
for (int j = 0; j < n; j++)
for (int i = 0; i < n; i++)
H[i, j] = value[i, j];
// Reduce to Hessenberg form.
this.orthes();
// Reduce Hessenberg to real Schur form.
this.hqr2();
}
}
private void tred2()
{
// Symmetric Householder reduction to tridiagonal form.
// 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 (int j = 0; j < n; j++)
d[j] = V[n - 1, j];
// Householder reduction to tridiagonal form.
for (int i = n - 1; i > 0; i--)
{
// Scale to avoid under/overflow.
double scale = 0.0;
double h = 0.0;
for (int k = 0; k < i; k++)
scale = scale + System.Math.Abs(d[k]);
if (scale == 0.0)
{
e[i] = d[i - 1];
for (int 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 (int k = 0; k < i; k++)
{
d[k] /= scale;
h += d[k] * d[k];
}
double f = d[i - 1];
double g = System.Math.Sqrt(h);
if (f > 0) g = -g;
e[i] = scale * g;
h = h - f * g;
d[i - 1] = f - g;
for (int j = 0; j < i; j++)
e[j] = 0.0;
// Apply similarity transformation to remaining columns.
for (int j = 0; j < i; j++)
{
f = d[j];
V[j, i] = f;
g = e[j] + V[j, j] * f;
for (int 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 (int j = 0; j < i; j++)
{
e[j] /= h;
f += e[j] * d[j];
}
double hh = f / (h + h);
for (int j = 0; j < i; j++)
e[j] -= hh * d[j];
for (int j = 0; j < i; j++)
{
f = d[j];
g = e[j];
for (int 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 (int i = 0; i < n - 1; i++)
{
V[n - 1, i] = V[i, i];
V[i, i] = 1.0;
double h = d[i + 1];
if (h != 0.0)
{
for (int k = 0; k <= i; k++)
d[k] = V[k, i + 1] / h;
for (int j = 0; j <= i; j++)
{
double g = 0.0;
for (int k = 0; k <= i; k++)
g += V[k, i + 1] * V[k, j];
for (int k = 0; k <= i; k++)
V[k, j] -= g * d[k];
}
}
for (int k = 0; k <= i; k++)
V[k, i + 1] = 0.0;
}
for (int 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;
}
private void tql2()
{
// Symmetric tridiagonal QL algorithm.
// 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 (int i = 1; i < n; i++)
e[i - 1] = e[i];
e[n - 1] = 0.0;
double f = 0.0;
double tst1 = 0.0;
double eps = System.Math.Pow(2.0, -52.0);
for (int l = 0; l < n; l++)
{
// Find small subdiagonal element.
tst1 = System.Math.Max(tst1, System.Math.Abs(d[l]) + System.Math.Abs(e[l]));
int m = l;
while (m < n)
{
if (System.Math.Abs(e[m]) <= eps * tst1)
break;
m++;
}
// If m == l, d[l] is an eigenvalue, otherwise, iterate.
if (m > l)
{
int iter = 0;
do
{
iter = iter + 1; // (Could check iteration count here.)
// Compute implicit shift
double g = d[l];
double p = (d[l + 1] - g) / (2.0 * e[l]);
double r = Accord.Math.Tools.Hypotenuse(p, 1.0);
if (p < 0)
{
r = -r;
}
d[l] = e[l] / (p + r);
d[l + 1] = e[l] * (p + r);
double dl1 = d[l + 1];
double h = g - d[l];
for (int i = l + 2; i < n; i++)
{
d[i] -= h;
}
f = f + h;
// Implicit QL transformation.
p = d[m];
double c = 1.0;
double c2 = c;
double c3 = c;
double el1 = e[l + 1];
double s = 0.0;
double s2 = 0.0;
for (int i = m - 1; i >= l; i--)
{
c3 = c2;
c2 = c;
s2 = s;
g = c * e[i];
h = c * p;
r = Accord.Math.Tools.Hypotenuse(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 (int 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 (System.Math.Abs(e[l]) > eps * tst1);
}
d[l] = d[l] + f;
e[l] = 0.0;
}
// Sort eigenvalues and corresponding vectors.
for (int i = 0; i < n - 1; i++)
{
int k = i;
double p = d[i];
for (int j = i + 1; j < n; j++)
{
if (d[j] < p)
{
k = j;
p = d[j];
}
}
if (k != i)
{
d[k] = d[i];
d[i] = p;
for (int j = 0; j < n; j++)
{
p = V[j, i];
V[j, i] = V[j, k];
V[j, k] = p;
}
}
}
}
private void orthes()
{
// Nonsymmetric reduction to Hessenberg form.
// 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 low = 0;
int high = n - 1;
for (int m = low + 1; m <= high - 1; m++)
{
// Scale column.
double scale = 0.0;
for (int i = m; i <= high; i++)
scale = scale + System.Math.Abs(H[i, m - 1]);
if (scale != 0.0)
{
// Compute Householder transformation.
double h = 0.0;
for (int i = high; i >= m; i--)
{
ort[i] = H[i, m - 1] / scale;
h += ort[i] * ort[i];
}
double g = System.Math.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 (int j = m; j < n; j++)
{
double f = 0.0;
for (int i = high; i >= m; i--)
f += ort[i] * H[i, j];
f = f / h;
for (int i = m; i <= high; i++)
H[i, j] -= f * ort[i];
}
for (int i = 0; i <= high; i++)
{
double f = 0.0;
for (int j = high; j >= m; j--)
f += ort[j] * H[i, j];
f = f / h;
for (int 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 (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
V[i, j] = (i == j ? 1.0 : 0.0);
for (int m = high - 1; m >= low + 1; m--)
{
if (H[m, m - 1] != 0.0)
{
for (int i = m + 1; i <= high; i++)
ort[i] = H[i, m - 1];
for (int j = m; j <= high; j++)
{
double g = 0.0;
for (int 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 (int i = m; i <= high; i++)
V[i, j] += g * ort[i];
}
}
}
}
private void cdiv(double xr, double xi, double yr, double yi)
{
// Complex scalar division.
double r;
double d;
if (System.Math.Abs(yr) > System.Math.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;
}
}
private void hqr2()
{
// Nonsymmetric reduction from Hessenberg to real Schur form.
// 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.
int nn = this.n;
int n = nn - 1;
int low = 0;
int high = nn - 1;
double eps = System.Math.Pow(2.0, -52.0);
double exshift = 0.0;
double p = 0;
double q = 0;
double r = 0;
double s = 0;
double z = 0;
double t;
double w;
double x;
double y;
// Store roots isolated by balanc and compute matrix norm
double norm = 0.0;
for (int i = 0; i < nn; i++)
{
if (i < low | i > high)
{
d[i] = H[i, i];
e[i] = 0.0;
}
for (int j = System.Math.Max(i - 1, 0); j < nn; j++)
norm = norm + System.Math.Abs(H[i, j]);
}
// 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 = System.Math.Abs(H[l - 1, l - 1]) + System.Math.Abs(H[l, l]);
if (s == 0.0) s = norm;
if (System.Math.Abs(H[l, l - 1]) < eps * s)
break;
l--;
}
// Check for convergence
if (l == n)
{
// One root found
H[n, n] = H[n, n] + exshift;
d[n] = H[n, n];
e[n] = 0.0;
n--;
iter = 0;
}
else if (l == n - 1)
{
// Two roots found
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 = System.Math.Sqrt(System.Math.Abs(q));
H[n, n] = H[n, n] + exshift;
H[n - 1, n - 1] = H[n - 1, n - 1] + exshift;
x = H[n, n];
if (q >= 0)
{
// Real pair
z = (p >= 0) ? (p + 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 = System.Math.Abs(x) + System.Math.Abs(z);
p = x / s;
q = z / s;
r = System.Math.Sqrt(p * p + q * q);
p = p / r;
q = q / r;
// Row modification
for (int 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 (int 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 (int 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;
}
}
else
{
// Complex pair
d[n - 1] = x + p;
d[n] = x + p;
e[n - 1] = z;
e[n] = -z;
}
n = n - 2;
iter = 0;
}
else
{
// No convergence yet
// 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 (int i = low; i <= n; i++)
H[i, i] -= x;
s = System.Math.Abs(H[n, n - 1]) + System.Math.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 = System.Math.Sqrt(s);
if (y < x) s = -s;
s = x - w / ((y - x) / 2.0 + s);
for (int i = low; i <= n; i++)
H[i, i] -= s;
exshift += s;
x = y = w = 0.964;
}
}
iter = iter + 1;
// Look for two consecutive small sub-diagonal elements
int 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 = System.Math.Abs(p) + System.Math.Abs(q) + System.Math.Abs(r);
p = p / s;
q = q / s;
r = r / s;
if (m == l)
break;
if (System.Math.Abs(H[m, m - 1]) * (System.Math.Abs(q) + System.Math.Abs(r)) < eps * (System.Math.Abs(p) * (System.Math.Abs(H[m - 1, m - 1]) + System.Math.Abs(z) + System.Math.Abs(H[m + 1, m + 1]))))
break;
m--;
}
for (int 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 (int k = m; k <= n - 1; k++)
{
bool 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 = System.Math.Abs(p) + System.Math.Abs(q) + System.Math.Abs(r);
if (x != 0.0)
{
p = p / x;
q = q / x;
r = r / x;
}
}
if (x == 0.0) break;
s = System.Math.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 (int 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 (int i = 0; i <= System.Math.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 (int 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;
}
}
}
}
}
// 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)
{
int l = n;
H[n, n] = 1.0;
for (int i = n - 1; i >= 0; i--)
{
w = H[i, i] - p;
r = 0.0;
for (int 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)
{
H[i, n] = (w != 0.0) ? (-r / w) : (-r / (eps * norm));
}
else
{
// Solve real equations
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;
H[i + 1, n] = (System.Math.Abs(x) > System.Math.Abs(z)) ? ((-r - w * t) / x) : ((-s - y * t) / z);
}
// Overflow control
t = System.Math.Abs(H[i, n]);
if ((eps * t) * t > 1)
for (int j = i; j <= n; j++)
H[j, n] = H[j, n] / t;
}
}
}
else if (q < 0)
{
// Complex vector
int l = n - 1;
// Last vector component imaginary so matrix is triangular
if (System.Math.Abs(H[n, n - 1]) > System.Math.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 (int i = n - 2; i >= 0; i--)
{
double ra, sa, vr, vi;
ra = 0.0;
sa = 0.0;
for (int 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 * (System.Math.Abs(w) + System.Math.Abs(q) + System.Math.Abs(x) + System.Math.Abs(y) + System.Math.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 (System.Math.Abs(x) > (System.Math.Abs(z) + System.Math.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 = System.Math.Max(System.Math.Abs(H[i, n - 1]), System.Math.Abs(H[i, n]));
if ((eps * t) * t > 1)
for (int 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 (int i = 0; i < nn; i++)
if (i < low | i > high)
for (int j = i; j < nn; j++)
V[i, j] = H[i, j];
// Back transformation to get eigenvectors of original matrix
for (int j = nn - 1; j >= low; j--)
for (int i = low; i <= high; i++)
{
z = 0.0;
for (int k = low; k <= System.Math.Min(j, high); k++)
z = z + V[i, k] * H[k, j];
V[i, j] = z;
}
}
/// <summary>Returns the real parts of the eigenvalues.</summary>
public double[] RealEigenvalues
{
get { return this.d; }
}
/// <summary>Returns the imaginary parts of the eigenvalues.</summary>
public double[] ImaginaryEigenvalues
{
get { return this.e; }
}
/// <summary>Returns the eigenvector matrix.</summary>
public double[,] Eigenvectors
{
get { return this.V; }
}
/// <summary>Returns the block diagonal eigenvalue matrix.</summary>
public double[,] DiagonalMatrix
{
get
{
double[,] x = new double[n, n];
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
x[i, j] = 0.0;
x[i, i] = d[i];
if (e[i] > 0)
{
x[i, i + 1] = e[i];
}
else if (e[i] < 0)
{
x[i, i - 1] = e[i];
}
}
return x;
}
}
}
}
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