private void Compute(Matrix symmetricMatrix)
{
var tmpDiag = new SparseVector(size);
Blas.Default.Copy(_diagonal, tmpDiag);
var tmpAccumulate = new SparseVector(size);
Matrix s = symmetricMatrix.Clone();
for (int ite = 1; ite <= MaxIterations; ite++)
{
// abs sum of upper triangel
double sum = 0.0;
for (int j = 0; j < size - 1; j++)
{
for (int k = j + 1; k < size; k++)
{
sum += Math.Abs(s[j, k]);
}
}
if (sum == 0.0)
{
return;
}
// To speed up computation a threshold is introduced to
// make sure it is worthy to perform the Jacobi rotation
double threshold;
if (ite < 5)
{
threshold = 0.2 * sum / (size * size);
}
else
{
threshold = 0.0;
}
// sweep upper triangle
for (int j = 0; j < size - 1; j++)
{
for (int k = j + 1; k < size; k++)
{
double smll = Math.Abs(s[j, k]);
if (ite > 5 &&
smll < EpsPrec * Math.Abs(_diagonal.Data[j]) &&
smll < EpsPrec * Math.Abs(_diagonal.Data[k]))
{
s[j, k] = 0.0;
}
else if (Math.Abs(s[j, k]) > threshold)
{
double heig = _diagonal.Data[k] - _diagonal.Data[j];
double tang;
if (smll < EpsPrec * Math.Abs(heig))
{
tang = s[j, k] / heig;
}
else
{
double beta = 0.5 * heig / s[j, k];
tang = 1.0 / (Math.Abs(beta) + Math.Sqrt(1.0 + beta * beta));
if (beta < 0.0)
{
tang = -tang;
}
}
double cosin = 1.0 / Math.Sqrt(1.0 + tang * tang);
double sine = tang * cosin;
double rho = sine / (1.0 + cosin);
heig = tang * s[j, k];
tmpAccumulate.Data[j] -= heig;
tmpAccumulate.Data[k] += heig;
_diagonal.Data[j] -= heig;
_diagonal.Data[k] += heig;
s[j, k] = 0.0;
// perform rotation on upper triangle
int l = 0;
for ( ; l < j; l++)
{
JacobiRotate(s, rho, sine, l, j, l, k);
}
for (++l; l < k; l++)
{
JacobiRotate(s, rho, sine, j, l, l, k);
}
for (++l; l < size; l++)
{
JacobiRotate(s, rho, sine, j, l, k, l);
}
for (l = 0; l < size; l++)
{
JacobiRotate(_eigenVectors, rho, sine, l, j, l, k);
}
}
}
}
// tmpDiag.Add(tmpAccumulate);
// diagonal = tmpDiag.Clone();
// tmpAccumulate.Fill(0.0);
for (int j = 0; j < size; j++)
{
tmpDiag.Data[j] += tmpAccumulate.Data[j];
_diagonal.Data[j] = tmpDiag.Data[j];
tmpAccumulate.Data[j] = 0.0;
}
}
throw new ApplicationException("TODO: Too many iterations reached");
}
/// <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)
{
transpose = (Arg.RowCount < Arg.ColumnCount);
// Derived from LINPACK code.
// Initialize.
var A = Arg.Clone();
if (transpose)
{
A.Transpose();
}
m = A.RowCount;
n = A.ColumnCount;
int nu = Math.Min(m, n);
s = new double[Math.Min(m + 1, n)];
U = new Matrix(m, nu);
V = new Matrix(n, n);
var e = new double[n];
var work = new double[m];
const bool wantu = true;
const bool wantv = true;
// Reduce A to bidiagonal form, storing the diagonal elements
// in s and the super-diagonal elements in e.
int nct = Math.Min(m - 1, n);
int nrt = Math.Max(0, Math.Min(n - 2, m));
for (int k = 0; k < 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] = Fn.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)
{
continue;
}
// 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] = Fn.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 (Math.Abs(e[k]) > eps * (Math.Abs(s[k]) + Math.Abs(s[k + 1])))
{
continue;
}
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?Math.Abs(e[ks]):0.0) + (ks != k + 1?Math.Abs(e[ks - 1]):0.0);
if (Math.Abs(s[ks]) > eps * t)
{
continue;
}
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 = Fn.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 = Fn.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 = Fn.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 = Fn.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))
{
continue;
}
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;
}
}
// (vermorel) transposing the results if needed
if (!transpose)
{
return;
}
// swaping U and V
Matrix T = V;
V = U;
U = T;
}
/// <summary>LU Decomposition</summary>
/// <param name="A"> Rectangular matrix
/// </param>
/// <returns> Structure to access L, U and piv.
/// </returns>
public LUDecomposition(Matrix A)
{
// Use a "left-looking", dot-product, Crout/Doolittle algorithm.
_lu = A.Clone();
_piv = new int[m];
for (int i = 0; i < m; i++)
{
_piv[i] = i;
}
_pivsign = 1;
//double[] LUrowi;
var LUcolj = new double[m];
// Outer loop.
for (int j = 0; j < n; j++)
{
// Make a copy of the j-th column to localize references.
for (int i = 0; i < m; i++)
{
LUcolj[i] = _lu[i, j];
}
// Apply previous transformations.
for (int i = 0; i < m; i++)
{
//LUrowi = LU[i];
// Most of the time is spent in the following dot product.
int kmax = Math.Min(i, j);
double s = 0.0;
for (int k = 0; k < kmax; k++)
{
s += _lu[i, k] * LUcolj[k];
}
_lu[i, j] = LUcolj[i] -= s;
}
// Find pivot and exchange if necessary.
int p = j;
for (int i = j + 1; i < m; i++)
{
if (Math.Abs(LUcolj[i]) > Math.Abs(LUcolj[p]))
{
p = i;
}
}
if (p != j)
{
for (int k = 0; k < n; k++)
{
double t = _lu[p, k]; _lu[p, k] = _lu[j, k]; _lu[j, k] = t;
}
int k2 = _piv[p]; _piv[p] = _piv[j]; _piv[j] = k2;
_pivsign = -_pivsign;
}
// Compute multipliers.
if (j < m & _lu[j, j] != 0.0)
{
for (int i = j + 1; i < m; i++)
{
_lu[i, j] /= _lu[j, j];
}
}
}
}