/// <summary>Frobenius norm</summary>
/// <returns>Sqrt of sum of squares of all elements.</returns>
public virtual double NormF()
{
double f = 0;
for (int i = 0; i < m; i++)
{
for (int j = 0; j < n; j++)
{
f = Maths.Hypot(f, A[i, j]);
}
}
return(f);
}
/// <summary>QR Decomposition, computed by Householder reflections.</summary>
/// <remarks>Provides access to R, the Householder vectors and computes Q.</remarks>
/// <param name="A">Rectangular matrix</param>
public QRDecomposition(Matrix A)
{
// Initialize.
QR = A.ArrayCopy;
Rdiag = new double[n];
// Main loop.
for (int k = 0; k < n; k++)
{
// Compute 2-norm of k-th column without under/overflow.
double nrm = 0;
for (int i = k; i < m; i++)
{
nrm = Maths.Hypot(nrm, QR[i, k]);
}
if (nrm != 0.0)
{
// Form k-th Householder vector.
if (QR[k, k] < 0)
{
nrm = -nrm;
}
for (int i = k; i < m; i++)
{
QR[i, k] /= nrm;
}
QR[k, k] += 1.0;
// Apply transformation to remaining columns.
for (int j = k + 1; j < n; j++)
{
double s = 0.0;
for (int i = k; i < m; i++)
{
s += QR[i, k] * QR[i, j];
}
s = (-s) / QR[k, k];
for (int i = k; i < m; i++)
{
QR[i, j] += s * QR[i, k];
}
}
}
Rdiag[k] = -nrm;
}
}
/// <summary>Construct the singular value decomposition.</summary>
/// <remarks>Provides access to U, S and V.</remarks>
/// <param name="Arg">Rectangular matrix</param>
public SingularValueDecomposition(Matrix Arg)
{
// Derived from LINPACK code.
// Initialize.
double[,] A = Arg.ArrayCopy;
m = Arg.RowDimension;
n = Arg.ColumnDimension;
int nu = System.Math.Min(m, n);
s = new double[System.Math.Min(m + 1, n)];
U = new double[m, nu];
V = new double[n, n];
double[] e = new double[n];
double[] work = new double[m];
bool wantu = true;
bool wantv = true;
// Reduce A to bidiagonal form, storing the diagonal elements
// in s and the super-diagonal elements in e.
int nct = System.Math.Min(m - 1, n);
int nrt = System.Math.Max(0, System.Math.Min(n - 2, m));
for (int k = 0; k < System.Math.Max(nct, nrt); k++)
{
if (k < nct)
{
// Compute the transformation for the k-th column and
// place the k-th diagonal in s[k].
// Compute 2-norm of k-th column without under/overflow.
s[k] = 0;
for (int i = k; i < m; i++)
{
s[k] = Maths.Hypot(s[k], A[i, k]);
}
if (s[k] != 0.0)
{
if (A[k, k] < 0.0)
{
s[k] = -s[k];
}
for (int i = k; i < m; i++)
{
A[i, k] /= s[k];
}
A[k, k] += 1.0;
}
s[k] = -s[k];
}
for (int j = k + 1; j < n; j++)
{
if ((k < nct) & (s[k] != 0.0))
{
// Apply the transformation.
double t = 0;
for (int i = k; i < m; i++)
{
t += A[i, k] * A[i, j];
}
t = (-t) / A[k, k];
for (int i = k; i < m; i++)
{
A[i, j] += t * A[i, k];
}
}
// Place the k-th row of A into e for the
// subsequent calculation of the row transformation.
e[j] = A[k, j];
}
if (wantu & (k < nct))
{
// Place the transformation in U for subsequent back
// multiplication.
for (int i = k; i < m; i++)
{
U[i, k] = A[i, k];
}
}
if (k < nrt)
{
// Compute the k-th row transformation and place the
// k-th super-diagonal in e[k].
// Compute 2-norm without under/overflow.
e[k] = 0;
for (int i = k + 1; i < n; i++)
{
e[k] = Maths.Hypot(e[k], e[i]);
}
if (e[k] != 0.0)
{
if (e[k + 1] < 0.0)
{
e[k] = -e[k];
}
for (int i = k + 1; i < n; i++)
{
e[i] /= e[k];
}
e[k + 1] += 1.0;
}
e[k] = -e[k];
if ((k + 1 < m) & (e[k] != 0.0))
{
// Apply the transformation.
for (int i = k + 1; i < m; i++)
{
work[i] = 0.0;
}
for (int j = k + 1; j < n; j++)
{
for (int i = k + 1; i < m; i++)
{
work[i] += e[j] * A[i, j];
}
}
for (int j = k + 1; j < n; j++)
{
double t = (-e[j]) / e[k + 1];
for (int i = k + 1; i < m; i++)
{
A[i, j] += t * work[i];
}
}
}
if (wantv)
{
// Place the transformation in V for subsequent
// back multiplication.
for (int i = k + 1; i < n; i++)
{
V[i, k] = e[i];
}
}
}
}
// Set up the final bidiagonal matrix or order p.
int p = System.Math.Min(n, m + 1);
if (nct < n)
{
s[nct] = A[nct, nct];
}
if (m < p)
{
s[p - 1] = 0.0;
}
if (nrt + 1 < p)
{
e[nrt] = A[nrt, p - 1];
}
e[p - 1] = 0.0;
// If required, generate U.
if (wantu)
{
for (int j = nct; j < nu; j++)
{
for (int i = 0; i < m; i++)
{
U[i, j] = 0.0;
}
U[j, j] = 1.0;
}
for (int k = nct - 1; k >= 0; k--)
{
if (s[k] != 0.0)
{
for (int j = k + 1; j < nu; j++)
{
double t = 0;
for (int i = k; i < m; i++)
{
t += U[i, k] * U[i, j];
}
t = (-t) / U[k, k];
for (int i = k; i < m; i++)
{
U[i, j] += t * U[i, k];
}
}
for (int i = k; i < m; i++)
{
U[i, k] = -U[i, k];
}
U[k, k] = 1.0 + U[k, k];
for (int i = 0; i < k - 1; i++)
{
U[i, k] = 0.0;
}
}
else
{
for (int i = 0; i < m; i++)
{
U[i, k] = 0.0;
}
U[k, k] = 1.0;
}
}
}
// If required, generate V.
if (wantv)
{
for (int k = n - 1; k >= 0; k--)
{
if ((k < nrt) & (e[k] != 0.0))
{
for (int j = k + 1; j < nu; j++)
{
double t = 0;
for (int i = k + 1; i < n; i++)
{
t += V[i, k] * V[i, j];
}
t = (-t) / V[k + 1, k];
for (int i = k + 1; i < n; i++)
{
V[i, j] += t * V[i, k];
}
}
}
for (int i = 0; i < n; i++)
{
V[i, k] = 0.0;
}
V[k, k] = 1.0;
}
}
// Main iteration loop for the singular values.
int pp = p - 1;
int iter = 0;
double eps = System.Math.Pow(2.0, -52.0);
while (p > 0)
{
int k, kase;
// Here is where a test for too many iterations would go.
// This section of the program inspects for
// negligible elements in the s and e arrays. On
// completion the variables kase and k are set as follows.
// kase = 1 if s(p) and e[k-1] are negligible and k<p
// kase = 2 if s(k) is negligible and k<p
// kase = 3 if e[k-1] is negligible, k<p, and
// s(k), ..., s(p) are not negligible (qr step).
// kase = 4 if e(p-1) is negligible (convergence).
for (k = p - 2; k >= -1; k--)
{
if (k == -1)
{
break;
}
if (System.Math.Abs(e[k]) <= eps * (System.Math.Abs(s[k]) + System.Math.Abs(s[k + 1])))
{
e[k] = 0.0;
break;
}
}
if (k == p - 2)
{
kase = 4;
}
else
{
int ks;
for (ks = p - 1; ks >= k; ks--)
{
if (ks == k)
{
break;
}
double t = (ks != p?System.Math.Abs(e[ks]):0.0) + (ks != k + 1?System.Math.Abs(e[ks - 1]):0.0);
if (System.Math.Abs(s[ks]) <= eps * t)
{
s[ks] = 0.0;
break;
}
}
if (ks == k)
{
kase = 3;
}
else if (ks == p - 1)
{
kase = 1;
}
else
{
kase = 2;
k = ks;
}
}
k++;
// Perform the task indicated by kase.
switch (kase)
{
// Deflate negligible s(p).
case 1:
{
double f = e[p - 2];
e[p - 2] = 0.0;
for (int j = p - 2; j >= k; j--)
{
double t = Maths.Hypot(s[j], f);
double cs = s[j] / t;
double sn = f / t;
s[j] = t;
if (j != k)
{
f = (-sn) * e[j - 1];
e[j - 1] = cs * e[j - 1];
}
if (wantv)
{
for (int i = 0; i < n; i++)
{
t = cs * V[i, j] + sn * V[i, p - 1];
V[i, p - 1] = (-sn) * V[i, j] + cs * V[i, p - 1];
V[i, j] = t;
}
}
}
}
break;
// Split at negligible s(k).
case 2:
{
double f = e[k - 1];
e[k - 1] = 0.0;
for (int j = k; j < p; j++)
{
double t = Maths.Hypot(s[j], f);
double cs = s[j] / t;
double sn = f / t;
s[j] = t;
f = (-sn) * e[j];
e[j] = cs * e[j];
if (wantu)
{
for (int i = 0; i < m; i++)
{
t = cs * U[i, j] + sn * U[i, k - 1];
U[i, k - 1] = (-sn) * U[i, j] + cs * U[i, k - 1];
U[i, j] = t;
}
}
}
}
break;
// Perform one qr step.
case 3:
{
// Calculate the shift.
double scale = System.Math.Max(System.Math.Max(System.Math.Max(System.Math.Max(System.Math.Abs(s[p - 1]), System.Math.Abs(s[p - 2])), System.Math.Abs(e[p - 2])), System.Math.Abs(s[k])), System.Math.Abs(e[k]));
double sp = s[p - 1] / scale;
double spm1 = s[p - 2] / scale;
double epm1 = e[p - 2] / scale;
double sk = s[k] / scale;
double ek = e[k] / scale;
double b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0;
double c = (sp * epm1) * (sp * epm1);
double shift = 0.0;
if ((b != 0.0) | (c != 0.0))
{
shift = System.Math.Sqrt(b * b + c);
if (b < 0.0)
{
shift = -shift;
}
shift = c / (b + shift);
}
double f = (sk + sp) * (sk - sp) + shift;
double g = sk * ek;
// Chase zeros.
for (int j = k; j < p - 1; j++)
{
double t = Maths.Hypot(f, g);
double cs = f / t;
double sn = g / t;
if (j != k)
{
e[j - 1] = t;
}
f = cs * s[j] + sn * e[j];
e[j] = cs * e[j] - sn * s[j];
g = sn * s[j + 1];
s[j + 1] = cs * s[j + 1];
if (wantv)
{
for (int i = 0; i < n; i++)
{
t = cs * V[i, j] + sn * V[i, j + 1];
V[i, j + 1] = (-sn) * V[i, j] + cs * V[i, j + 1];
V[i, j] = t;
}
}
t = Maths.Hypot(f, g);
cs = f / t;
sn = g / t;
s[j] = t;
f = cs * e[j] + sn * s[j + 1];
s[j + 1] = (-sn) * e[j] + cs * s[j + 1];
g = sn * e[j + 1];
e[j + 1] = cs * e[j + 1];
if (wantu && (j < m - 1))
{
for (int i = 0; i < m; i++)
{
t = cs * U[i, j] + sn * U[i, j + 1];
U[i, j + 1] = (-sn) * U[i, j] + cs * U[i, j + 1];
U[i, j] = t;
}
}
}
e[p - 2] = f;
iter = iter + 1;
}
break;
// Convergence.
case 4:
{
// Make the singular values positive.
if (s[k] <= 0.0)
{
s[k] = (s[k] < 0.0?-s[k]:0.0);
if (wantv)
{
for (int i = 0; i <= pp; i++)
{
V[i, k] = -V[i, k];
}
}
}
// Order the singular values.
while (k < pp)
{
if (s[k] >= s[k + 1])
{
break;
}
double t = s[k];
s[k] = s[k + 1];
s[k + 1] = t;
if (wantv && (k < n - 1))
{
for (int i = 0; i < n; i++)
{
t = V[i, k + 1]; V[i, k + 1] = V[i, k]; V[i, k] = t;
}
}
if (wantu && (k < m - 1))
{
for (int i = 0; i < m; i++)
{
t = U[i, k + 1]; U[i, k + 1] = U[i, k]; U[i, k] = t;
}
}
k++;
}
iter = 0;
p--;
}
break;
}
}
}
// Symmetric tridiagonal QL algorithm.
private void tql2()
{
// 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 = Maths.Hypot(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 = Maths.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 (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;
}
}
}
}