/// <summary>
/// Use Gobul Reinsch to decompose the matrix.
/// http://people.duke.edu/~hpgavin/SystemID/References/Golub+Reinsch-NM-1970.pdf
/// </summary>
/// <param name="a"></param>
/// <returns></returns>
public SingularValueDecomposition DecomposeGolubReinsch(double[][] a)
{
// http://people.duke.edu/~hpgavin/SystemID/References/Golub+Reinsch-NM-1970.pdf
int maxIteration = 50;
double s = 0.0, h = 0.0, f = 0.0, c = 0.0, z = 0.0, y = 0.0;
int l = 0;
var eps = double.Epsilon;
var tol = 1.11022302462516E-49;
int m = a.Length;
int n = a[0].Length;
var u = a.ToArray();
double[] e = new double[n];
double[] q = new double[n];
double[][] v = new double[n][];
for (int k = 0; k < n; k++)
{
v[k] = new double[n];
}
double g = 0.0, x = 0.0;
for (int i = 0; i < n; i++)
{
s = 0.0;
e[i] = g;
l = i + 1;
for (int j = i; j < m; j++)
{
s += System.Math.Pow(u[j][i], 2);
}
if (s <= tol)
{
g = 0.0;
}
else
{
f = u[i][i];
if (f < 0.0)
{
g = System.Math.Sqrt(s);
}
else
{
g = -System.Math.Sqrt(s);
}
h = f * g - s;
u[i][i] = f - g;
for (int j = l; j < n; j++)
{
s = 0.0;
for (int k = i; k < m; k++)
{
s += u[k][i] * u[k][j];
}
f = s / h;
for (int k = i; k < m; k++)
{
u[k][j] = u[k][j] + f * u[k][i];
}
}
}
q[i] = g;
s = 0.0;
for (int j = l; j < n; j++)
{
s = s + u[i][j] * u[i][j];
}
if (s <= tol)
{
g = 0.0;
}
else
{
f = u[i][i + 1];
if (f < 0.0)
{
g = System.Math.Sqrt(s);
}
else
{
g = -System.Math.Sqrt(s);
}
h = f * g - s;
u[i][i + 1] = f - g;
for (int j = l; j < n; j++)
{
e[j] = u[i][j] / h;
}
for (int j = l; j < m; j++)
{
s = 0.0;
for (int k = l; k < n; k++)
{
s = s + (u[j][k] * u[i][k]);
}
for (int k = l; k < n; k++)
{
u[j][k] = u[j][k] + (s * e[k]);
}
}
}
y = System.Math.Abs(q[i]) + System.Math.Abs(e[i]);
if (y > x)
{
x = y;
}
}
// accumulation of right hand transformations.
for (int i = n - 1; i >= 0; i--)
{
if (g != 0.0)
{
h = g * u[i][i + 1];
for (int j = l; j < n; j++)
{
v[j][i] = u[i][j] / h;
}
for (int j = l; j < n; j++)
{
s = 0.0;
for (int k = l; k < n; k++)
{
s += u[i][k] * v[k][j];
}
for (int k = l; k < n; k++)
{
v[k][j] += (s * v[k][i]);
}
}
}
for (int j = l; j < n; j++)
{
v[i][j] = 0.0;
v[j][i] = 0.0;
}
v[i][i] = 1.0;
g = e[i];
l = i;
}
// accumulation of the left hand transformations.
for (int i = n - 1; i >= 0; i--)
{
l = i + 1;
g = q[i];
for (int j = l; j < n; j++)
{
u[i][j] = 0.0;
}
if (g != 0.0)
{
h = u[i][i] * g;
for (int j = l; j < n; j++)
{
s = 0.0;
for (int k = l; k < m; k++)
{
s += (u[k][i] * u[k][j]);
}
f = s / h;
for (int k = i; k < m; k++)
{
u[k][j] = u[k][j] + f * u[k][i];
}
}
for (int j = i; j < m; j++)
{
u[j][i] = u[j][i] / g;
}
}
else
{
for (int j = i; j < m; j++)
{
u[j][i] = 0.0;
}
}
u[i][i] = u[i][i] + 1.0;
}
eps = eps * x;
// diagonalization of the bidiagonal form.
for (int k = n - 1; k >= 0; k--)
{
for (int iteration = 0; iteration < maxIteration; iteration++)
{
bool testFConvergence = false;
// test f splitting.
for (l = k; l >= 0; l--)
{
if (System.Math.Abs(e[l]) <= eps)
{
testFConvergence = true;
break;
}
if (System.Math.Abs(q[l - 1]) <= eps)
{
break;
}
}
if (!testFConvergence)
{
c = 0.0;
s = 0.0;
var l1 = l - 1;
for (int i = l; i < k + 1; i++)
{
f = s * e[i];
e[i] = c * e[i];
if (System.Math.Abs(f) <= eps)
{
break;
}
g = q[i];
h = Hypotenuse(f, g);
q[i] = h;
c = g / h;
s = -f / h;
for (int j = 0; j < m; j++)
{
y = u[j][l1];
z = u[j][i];
u[j][l1] = y * c + z * s;
u[j][i] = -y * s + z * c;
}
}
}
// test f convergence.
z = q[k];
if (l == k)
{
if (z < 0.0)
{
q[k] = -z;
for (int j = 0; j < n; j++)
{
v[j][k] = -v[j][k];
}
}
break;
}
if (iteration >= maxIteration - 1)
{
break;
}
// shift from bottom 2x2 minor.
x = q[l];
y = q[k - 1];
g = e[k - 1];
h = e[k];
f = ((y - z) * (y + z) + (g - h) * (g + h)) / (2.0 * h * y);
g = Hypotenuse(f, 1.0);
if (f < 0)
{
f = ((x - z) * (x + z) + h * (y / (f - g) - h)) / x;
}
else
{
f = ((x - z) * (x + z) + h * (y / (f + g) - h)) / x;
}
// next QR transformation.
c = 1.0;
s = 1.0;
for (int i = l + 1; i < k + 1; i++)
{
g = e[i];
y = q[i];
h = s * g;
g = c * g;
z = Hypotenuse(f, h);
e[i - 1] = z;
c = f / z;
s = h / z;
f = x * c + g * s;
g = -x * s + g * c;
h = y * s;
y = y * c;
for (int j = 0; j < n; j++)
{
x = v[j][i - 1];
z = v[j][i];
v[j][i - 1] = x * c + z * s;
v[j][i] = -x * s + z * c;
}
z = Hypotenuse(f, h);
q[i - 1] = z;
c = f / z;
s = h / z;
f = c * g + s * y;
x = -s * g + c * y;
for (int j = 0; j < m; j++)
{
y = u[j][i - 1];
z = u[j][i];
u[j][i - 1] = y * c + z * s;
u[j][i] = -y * s + z * c;
}
}
e[l] = 0.0;
e[k] = f;
q[k] = x;
}
}
Result = new SingularValueDecompositionResult
{
S = Matrix.BuildIdentityMatrix(q),
U = u,
V = v,
Input = a
};
return(this);
}
public SingularValueDecomposition DecomposeNaive(Matrix matrix)
{
var ata = matrix.Transpose().Multiply(matrix).Evaluate();
var ataNormalizedVectors = DecomposeMatrix(ata);
var sum = Matrix.BuildEmptyMatrix(matrix.NbRows, matrix.NbColumns);
for (int i = 0; i < ataNormalizedVectors.Count; i++)
{
if (sum.NbRows <= i || sum.NbColumns <= i)
{
break;
}
sum.SetValue(i, i, MathEntity.Sqrt(ataNormalizedVectors[i].EingenValue).Eval());
}
var v = Matrix.BuildEmptyMatrix(ataNormalizedVectors[0].Vector.Length, ataNormalizedVectors.Count);
var nbColumns = ataNormalizedVectors.Where(_ =>
{
if (_.EingenValue == 0)
{
return(false);
}
return(true);
}).Count();
var u = Matrix.BuildEmptyMatrix(sum.NbRows, nbColumns);
for (int column = 0; column < ataNormalizedVectors.Count; column++)
{
var eingenValue = (NumberEntity)ataNormalizedVectors[column].EingenValue;
var vector = ataNormalizedVectors[column].Vector;
var vectorMatrix = Matrix.BuildEmptyMatrix(vector.Length, 1);
for (int row = 0; row < vector.Length; row++)
{
v.SetValue(row, column, vector[row]);
vectorMatrix.SetValue(row, 0, vector[row]);
}
if (eingenValue.Number.Value == 0)
{
continue;
}
var val = (Number.Create(1) / MathEntity.Sqrt(ataNormalizedVectors[column].EingenValue)).Eval();
var uColumn = matrix.Multiply(val).Multiply(vectorMatrix);
for (int row = 0; row < uColumn.NbRows; row++)
{
u.SetValue(row, column, uColumn.GetValue(row, 0).Eval());
}
}
Result = new SingularValueDecompositionResult
{
U = u,
S = sum,
V = v,
Input = matrix
};
return(this);
}