public void EpsilonTest()
{
double x = 0.5;
double expected = 0.00000000000000011102230246251565;
double actual = Special.Epslon(x);
Assert.AreEqual(expected, actual);
x = 0.0;
expected = 0.0;
actual = Special.Epslon(x);
Assert.AreEqual(expected, actual);
x = 1.0;
expected = 0.00000000000000022204460492503131;
actual = Special.Epslon(x);
Assert.AreEqual(expected, actual);
}
/// <summary>
/// Adaptation of the original Fortran QZIT routine from EISPACK.
/// </summary>
/// <remarks>
/// This subroutine is the second step of the qz algorithm
/// for solving generalized matrix eigenvalue problems,
/// siam j. numer. anal. 10, 241-256(1973) by moler and stewart,
/// as modified in technical note nasa tn d-7305(1973) by ward.
///
/// This subroutine accepts a pair of real matrices, one of them
/// in upper hessenberg form and the other in upper triangular form.
/// it reduces the hessenberg matrix to quasi-triangular form using
/// orthogonal transformations while maintaining the triangular form
/// of the other matrix. it is usually preceded by qzhes and
/// followed by qzval and, possibly, qzvec.
///
/// For the full documentation, please check the original function.
/// </remarks>
private static int qzit(int n, double[,] a, double[,] b, double eps1, bool matz, double[,] z, ref int ierr)
{
int i, j, k, l = 0;
double r, s, t, a1, a2, a3 = 0;
int k1, k2, l1, ll;
double u1, u2, u3;
double v1, v2, v3;
double a11, a12, a21, a22, a33, a34, a43, a44;
double b11, b12, b22, b33, b34, b44;
int na, en, ld;
double ep;
double sh = 0;
int km1, lm1 = 0;
double ani, bni;
int ish, itn, its, enm2, lor1;
double epsa, epsb, anorm = 0, bnorm = 0;
int enorn;
bool notlas;
ierr = 0;
#region Compute epsa and epsb
for (i = 0; i < n; ++i)
{
ani = 0.0;
bni = 0.0;
if (i != 0)
{
ani = (Math.Abs(a[i, (i - 1)]));
}
for (j = i; j < n; ++j)
{
ani += Math.Abs(a[i, j]);
bni += Math.Abs(b[i, j]);
}
if (ani > anorm)
{
anorm = ani;
}
if (bni > bnorm)
{
bnorm = bni;
}
}
if (anorm == 0.0)
{
anorm = 1.0;
}
if (bnorm == 0.0)
{
bnorm = 1.0;
}
ep = eps1;
if (ep == 0.0)
{
// Use roundoff level if eps1 is zero
ep = Special.Epslon(1.0);
}
epsa = ep * anorm;
epsb = ep * bnorm;
#endregion
// Reduce a to quasi-triangular form, while keeping b triangular
lor1 = 0;
enorn = n;
en = n - 1;
itn = n * 30;
// Begin QZ step
L60:
if (en <= 1)
{
goto L1001;
}
if (!matz)
{
enorn = en + 1;
}
its = 0;
na = en - 1;
enm2 = na;
L70:
ish = 2;
// Check for convergence or reducibility.
for (ll = 0; ll <= en; ++ll)
{
lm1 = en - ll - 1;
l = lm1 + 1;
if (l + 1 == 1)
{
goto L95;
}
if ((Math.Abs(a[l, lm1])) <= epsa)
{
break;
}
}
L90:
a[l, lm1] = 0.0;
if (l < na)
{
goto L95;
}
// 1-by-1 or 2-by-2 block isolated
en = lm1;
goto L60;
// Check for small top of b
L95:
ld = l;
L100:
l1 = l + 1;
b11 = b[l, l];
if (Math.Abs(b11) > epsb)
{
goto L120;
}
b[l, l] = 0.0;
s = (Math.Abs(a[l, l]) + Math.Abs(a[l1, l]));
u1 = a[l, l] / s;
u2 = a[l1, l] / s;
r = Special.Sign(Math.Sqrt(u1 * u1 + u2 * u2), u1);
v1 = -(u1 + r) / r;
v2 = -u2 / r;
u2 = v2 / v1;
for (j = l; j < enorn; ++j)
{
t = a[l, j] + u2 * a[l1, j];
a[l, j] += t * v1;
a[l1, j] += t * v2;
t = b[l, j] + u2 * b[l1, j];
b[l, j] += t * v1;
b[l1, j] += t * v2;
}
if (l != 0)
{
a[l, lm1] = -a[l, lm1];
}
lm1 = l;
l = l1;
goto L90;
L120:
a11 = a[l, l] / b11;
a21 = a[l1, l] / b11;
if (ish == 1)
{
goto L140;
}
// Iteration strategy
if (itn == 0)
{
goto L1000;
}
if (its == 10)
{
goto L155;
}
// Determine type of shift
b22 = b[l1, l1];
if (Math.Abs(b22) < epsb)
{
b22 = epsb;
}
b33 = b[na, na];
if (Math.Abs(b33) < epsb)
{
b33 = epsb;
}
b44 = b[en, en];
if (Math.Abs(b44) < epsb)
{
b44 = epsb;
}
a33 = a[na, na] / b33;
a34 = a[na, en] / b44;
a43 = a[en, na] / b33;
a44 = a[en, en] / b44;
b34 = b[na, en] / b44;
t = (a43 * b34 - a33 - a44) * .5;
r = t * t + a34 * a43 - a33 * a44;
if (r < 0.0)
{
goto L150;
}
// Determine single shift zeroth column of a
ish = 1;
r = Math.Sqrt(r);
sh = -t + r;
s = -t - r;
if (Math.Abs(s - a44) < Math.Abs(sh - a44))
{
sh = s;
}
// Look for two consecutive small sub-diagonal elements of a.
for (ll = ld; ll + 1 <= enm2; ++ll)
{
l = enm2 + ld - ll - 1;
if (l == ld)
{
goto L140;
}
lm1 = l - 1;
l1 = l + 1;
t = a[l + 1, l + 1];
if (Math.Abs(b[l, l]) > epsb)
{
t -= sh * b[l, l];
}
if (Math.Abs(a[l, lm1]) <= (Math.Abs(t / a[l1, l])) * epsa)
{
goto L100;
}
}
L140:
a1 = a11 - sh;
a2 = a21;
if (l != ld)
{
a[l, lm1] = -a[l, lm1];
}
goto L160;
// Determine double shift zeroth column of a
L150:
a12 = a[l, l1] / b22;
a22 = a[l1, l1] / b22;
b12 = b[l, l1] / b22;
a1 = ((a33 - a11) * (a44 - a11) - a34 * a43 + a43 * b34 * a11) / a21 + a12 - a11 * b12;
a2 = a22 - a11 - a21 * b12 - (a33 - a11) - (a44 - a11) + a43 * b34;
a3 = a[l1 + 1, l1] / b22;
goto L160;
// Ad hoc shift
L155:
a1 = 0.0;
a2 = 1.0;
a3 = 1.1605;
L160:
++its;
--itn;
if (!matz)
{
lor1 = ld;
}
// Main loop
for (k = l; k <= na; ++k)
{
notlas = k != na && ish == 2;
k1 = k + 1;
k2 = k + 2;
km1 = Math.Max(k, l + 1) - 1; // Computing MAX
ll = Math.Min(en, k1 + ish); // Computing MIN
if (notlas)
{
goto L190;
}
// Zero a(k+1,k-1)
if (k == l)
{
goto L170;
}
a1 = a[k, km1];
a2 = a[k1, km1];
L170:
s = Math.Abs(a1) + Math.Abs(a2);
if (s == 0.0)
{
goto L70;
}
u1 = a1 / s;
u2 = a2 / s;
r = Special.Sign(Math.Sqrt(u1 * u1 + u2 * u2), u1);
v1 = -(u1 + r) / r;
v2 = -u2 / r;
u2 = v2 / v1;
for (j = km1; j < enorn; ++j)
{
t = a[k, j] + u2 * a[k1, j];
a[k, j] += t * v1;
a[k1, j] += t * v2;
t = b[k, j] + u2 * b[k1, j];
b[k, j] += t * v1;
b[k1, j] += t * v2;
}
if (k != l)
{
a[k1, km1] = 0.0;
}
goto L240;
// Zero a(k+1,k-1) and a(k+2,k-1)
L190:
if (k == l)
{
goto L200;
}
a1 = a[k, km1];
a2 = a[k1, km1];
a3 = a[k2, km1];
L200:
s = Math.Abs(a1) + Math.Abs(a2) + Math.Abs(a3);
if (s == 0.0)
{
goto L260;
}
u1 = a1 / s;
u2 = a2 / s;
u3 = a3 / s;
r = Special.Sign(Math.Sqrt(u1 * u1 + u2 * u2 + u3 * u3), u1);
v1 = -(u1 + r) / r;
v2 = -u2 / r;
v3 = -u3 / r;
u2 = v2 / v1;
u3 = v3 / v1;
for (j = km1; j < enorn; ++j)
{
t = a[k, j] + u2 * a[k1, j] + u3 * a[k2, j];
a[k, j] += t * v1;
a[k1, j] += t * v2;
a[k2, j] += t * v3;
t = b[k, j] + u2 * b[k1, j] + u3 * b[k2, j];
b[k, j] += t * v1;
b[k1, j] += t * v2;
b[k2, j] += t * v3;
}
if (k == l)
{
goto L220;
}
a[k1, km1] = 0.0;
a[k2, km1] = 0.0;
// Zero b(k+2,k+1) and b(k+2,k)
L220:
s = (Math.Abs(b[k2, k2])) + (Math.Abs(b[k2, k1])) + (Math.Abs(b[k2, k]));
if (s == 0.0)
{
goto L240;
}
u1 = b[k2, k2] / s;
u2 = b[k2, k1] / s;
u3 = b[k2, k] / s;
r = Special.Sign(Math.Sqrt(u1 * u1 + u2 * u2 + u3 * u3), u1);
v1 = -(u1 + r) / r;
v2 = -u2 / r;
v3 = -u3 / r;
u2 = v2 / v1;
u3 = v3 / v1;
for (i = lor1; i < ll + 1; ++i)
{
t = a[i, k2] + u2 * a[i, k1] + u3 * a[i, k];
a[i, k2] += t * v1;
a[i, k1] += t * v2;
a[i, k] += t * v3;
t = b[i, k2] + u2 * b[i, k1] + u3 * b[i, k];
b[i, k2] += t * v1;
b[i, k1] += t * v2;
b[i, k] += t * v3;
}
b[k2, k] = 0.0;
b[k2, k1] = 0.0;
if (matz)
{
for (i = 0; i < n; ++i)
{
t = z[i, k2] + u2 * z[i, k1] + u3 * z[i, k];
z[i, k2] += t * v1;
z[i, k1] += t * v2;
z[i, k] += t * v3;
}
}
// Zero b(k+1,k)
L240:
s = (Math.Abs(b[k1, k1])) + (Math.Abs(b[k1, k]));
if (s == 0.0)
{
goto L260;
}
u1 = b[k1, k1] / s;
u2 = b[k1, k] / s;
r = Special.Sign(Math.Sqrt(u1 * u1 + u2 * u2), u1);
v1 = -(u1 + r) / r;
v2 = -u2 / r;
u2 = v2 / v1;
for (i = lor1; i < ll + 1; ++i)
{
t = a[i, k1] + u2 * a[i, k];
a[i, k1] += t * v1;
a[i, k] += t * v2;
t = b[i, k1] + u2 * b[i, k];
b[i, k1] += t * v1;
b[i, k] += t * v2;
}
b[k1, k] = 0.0;
if (matz)
{
for (i = 0; i < n; ++i)
{
t = z[i, k1] + u2 * z[i, k];
z[i, k1] += t * v1;
z[i, k] += t * v2;
}
}
L260:
;
}
goto L70; // End QZ step
// Set error -- all eigenvalues have not converged after 30*n iterations
L1000:
ierr = en + 1;
// Save epsb for use by qzval and qzvec
L1001:
if (n > 1)
{
b[n - 1, 0] = epsb;
}
return(0);
}
/// <summary>
/// Single non-negative matrix factorization.
/// </summary>
private double nnmf(double[,] value,
ref double[,] w0, ref double[,] h0, NonnegativeFactorizationAlgorithm alg,
int maxIterations, double normChangeThreshold, double maxFactorChangeThreshold)
{
double[,] v = value;
double[,] h = h0;
double[,] w = w0;
double[,] z = null;
double dnorm0 = 0.0; // previous iteration norm
// Main loop
for (int iteration = 0; iteration < maxIterations; iteration++)
{
// Check which algorithm should be used
if (alg == NonnegativeFactorizationAlgorithm.MultiplicativeUpdate)
{
// Multiplicative update formula
h = new double[k, m];
w = new double[n, k];
if (z == null)
{
z = new double[k, k];
}
// Update H
for (int i = 0; i < k; i++)
{
for (int j = i; j < k; j++)
{
double s = 0.0;
for (int l = 0; l < n; l++)
{
s += w0[l, i] * w0[l, j];
}
z[i, j] = z[j, i] = s;
}
for (int j = 0; j < m; j++)
{
double d = 0.0;
for (int l = 0; l < k; l++)
{
d += z[i, l] * h0[l, j];
}
double s = 0.0;
for (int l = 0; l < n; l++)
{
s += w0[l, i] * v[l, j];
}
h[i, j] = h0[i, j] * s / (d + Special.Epslon(s));
}
}
// Update W
for (int j = 0; j < k; j++)
{
for (int i = j; i < k; i++)
{
double s = 0.0;
for (int l = 0; l < m; l++)
{
s += h[i, l] * h[j, l];
}
z[i, j] = z[j, i] = s;
}
for (int i = 0; i < n; i++)
{
double d = 0.0;
for (int l = 0; l < k; l++)
{
d += w0[i, l] * z[j, l];
}
double s = 0.0;
for (int l = 0; l < m; l++)
{
s += v[i, l] * h[j, l];
}
w[i, j] = w0[i, j] * s / (d + Special.Epslon(s));
}
}
}
else
{
// Alternating least squares update
h = w0.Solve(v); makepositive(h);
w = v.Divide(h); makepositive(w);
}
// Reconstruct matrix A using W and H
double[,] r = w.Multiply(h);
// Get norm of difference
double dnorm = normdiff(v, r);
// Get maximum change in factors
double dw = maxdiff(w0, w) / (sqrteps + maxabs(w0));
double dh = maxdiff(h0, h) / (sqrteps + maxabs(h0));
double delta = System.Math.Max(dw, dh);
if (iteration > 0) // Check for convergence
{
if (delta <= maxFactorChangeThreshold ||
dnorm <= normChangeThreshold * dnorm0)
{
break;
}
}
// Remember previous iteration results
dnorm0 = dnorm;
w0 = w; h0 = h;
}
return(dnorm0);
}