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 = 2 * Constants.DoubleEpsilon;
Double exshift = 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;
for (int i = 0; i < nn; i++)
{
if (i < low | i > high)
{
d[i] = H[i, i];
e[i] = 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)
s = norm;
if (Double.IsNaN(s))
break;
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;
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;
q = p * p + w;
z = (Double)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)
d[n] = x - w / z;
e[n - 1] = 0;
e[n] = 0;
x = H[n, n - 1];
s = System.Math.Abs(x) + System.Math.Abs(z);
p = x / s;
q = z / s;
r = (Double)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;
w = 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 = (Double)0.75 * s;
w = (Double)(-0.4375) * s * s;
}
// MATLAB's new ad hoc shift
if (iter == 30)
{
s = (y - x) / 2;
s = s * s + w;
if (s > 0)
{
s = (Double)System.Math.Sqrt(s);
if (y < x) s = -s;
s = x - w / ((y - x) / 2 + s);
for (int i = low; i <= n; i++)
H[i, i] -= s;
exshift += s;
x = y = w = (Double)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;
if (i > m + 2)
H[i, i - 3] = 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);
x = System.Math.Abs(p) + System.Math.Abs(q) + System.Math.Abs(r);
if (x != 0)
{
p = p / x;
q = q / x;
r = r / x;
}
}
if (x == 0) break;
s = (Double)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)
{
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;
for (int i = n - 1; i >= 0; i--)
{
w = H[i, i] - p;
r = 0;
for (int j = l; j <= n; j++)
r = r + H[i, j] * H[j, n];
if (e[i] < 0)
{
z = w;
s = r;
}
else
{
l = i;
if (e[i] == 0)
{
H[i, n] = (w != 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, -H[n - 1, n], H[n - 1, n - 1] - p, q, out H[n - 1, n - 1], out H[n - 1, n]);
}
H[n, n - 1] = 0;
H[n, n] = 1;
for (int i = n - 2; i >= 0; i--)
{
Double ra, sa, vr, vi;
ra = 0;
sa = 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)
{
z = w;
r = ra;
s = sa;
}
else
{
l = i;
if (e[i] == 0)
{
cdiv(-ra, -sa, w, q, out H[i, n - 1], out H[i, n]);
}
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 * q;
if (vr == 0 & vi == 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, out H[i, n - 1], out H[i, n]);
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, out H[i + 1, n - 1], out H[i + 1, n]);
}
}
// 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;
for (int k = low; k <= System.Math.Min(j, high); k++)
z = z + V[i, k] * H[k, j];
V[i, j] = z;
}
}
}
#endregion