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Dwsim / data /DWSIM.MathOps.Mapack /SingularValueDecomposition.cs
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// ----------------------------------------------
// Lutz Roeder's Mapack for .NET, September 2000
// Adapted from Mapack for COM and Jama routines.
// http://www.aisto.com/roeder/dotnet
// ----------------------------------------------
namespace Mapack
{
using System;
/// <summary>
/// Singular Value Decomposition for a rectangular matrix.
/// </summary>
/// <remarks>
/// For an m-by-n matrix <c>A</c> with <c>m >= n</c>, the singular value decomposition is
/// an m-by-n orthogonal matrix <c>U</c>, an n-by-n diagonal matrix <c>S</c>, and
/// an n-by-n orthogonal matrix <c>V</c> so that <c>A = U * S * V'</c>.
/// The singular values, <c>sigma[k] = S[k,k]</c>, are ordered so that
/// <c>sigma[0] >= sigma[1] >= ... >= sigma[n-1]</c>.
/// The singular value decompostion always exists, so the constructor will
/// never fail. The matrix condition number and the effective numerical
/// rank can be computed from this decomposition.
/// </remarks>
public class SingularValueDecomposition
{
private Matrix U;
private Matrix V;
private double[] s; // singular values
private int m;
private int n;
/// <summary>Construct singular value decomposition.</summary>
public SingularValueDecomposition(Matrix value)
{
if (value == null)
{
throw new ArgumentNullException("value");
}
Matrix copy = (Matrix) value.Clone();
double[][] a = copy.Array;
m = value.Rows;
n = value.Columns;
int nu = Math.Min(m,n);
s = new double [Math.Min(m+1,n)];
U = new Matrix(m, nu);
V = new Matrix(n, n);
double[][] u = U.Array;
double[][] v = V.Array;
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 = 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] = Hypotenuse(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] = Hypotenuse(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 = 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 = 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])))
{
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)
{
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 = Hypotenuse(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 = Hypotenuse(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 = Math.Max(Math.Max(Math.Max(Math.Max(Math.Abs(s[p-1]),Math.Abs(s[p-2])),Math.Abs(e[p-2])), Math.Abs(s[k])),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 = 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 = Hypotenuse(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 = Hypotenuse(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;
}
}
}
/// <summary>Returns the condition number <c>max(S) / min(S)</c>.</summary>
public double Condition
{
get
{
return s[0] / s[Math.Min(m, n) - 1];
}
}
/// <summary>Returns the Two norm.</summary>
public double Norm2
{
get
{
return s[0];
}
}
/// <summary>Returns the effective numerical matrix rank.</summary>
/// <value>Number of non-negligible singular values.</value>
public int Rank
{
get
{
double eps = Math.Pow(2.0,-52.0);
double tol = Math.Max(m, n) * s[0] * eps;
int r = 0;
for (int i = 0; i < s.Length; i++)
{
if (s[i] > tol)
{
r++;
}
}
return r;
}
}
/// <summary>Return the one-dimensional array of singular values.</summary>
public double[] Diagonal
{
get
{
return this.s;
}
}
private static double Hypotenuse(double a, double b)
{
if (Math.Abs(a) > Math.Abs(b))
{
double r = b / a;
return Math.Abs(a) * Math.Sqrt(1 + r * r);
}
if (b != 0)
{
double r = a / b;
return Math.Abs(b) * Math.Sqrt(1 + r * r);
}
return 0.0;
}
/// <summary>Returns the U matrix.</summary>
public Matrix UMatrix
{
get
{
return this.U;
}
}
/// <summary>Returns the U matrix.</summary>
public Matrix VMatrix
{
get
{
return this.V;
}
}
}
}