using System;
using YAMP;
namespace YAMP.Numerics
{
/// <summary>
/// Singular Value Decomposition.
/// For an m-by-n matrix A with m >= n, the singular value decomposition is
/// an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
/// an n-by-n orthogonal matrix V so that A = U*S*V'.
/// The singular values, sigma[k] = S[k][k], are ordered so that
/// sigma[0] >= sigma[1] >= ... >= sigma[n-1].
/// 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.
/// </summary>
public class SingularValueDecomposition
{
#region Members
/// <summary>
/// Arrays for internal storage of U and V.
/// </summary>
double[][] U, V;
/// <summary>
/// Array for internal storage of singular values.
/// </summary>
double[] s;
/// <summary>
/// Row and column dimensions.
/// </summary>
int m, n;
#endregion //Class variables
#region Constructor
/// <summary>
/// Construct the singular value decomposition
/// </summary>
/// <param name="Arg">Rectangular matrix</param>
/// <returns>Structure to access U, S and V.</returns>
public SingularValueDecomposition(MatrixValue Arg)
{
// Derived from LINPACK code.
// Initialize.
var A = Arg.GetRealArray();
m = Arg.DimensionY;
n = Arg.DimensionX;
var nu = Math.Min(m, n);
s = new double[Math.Min(m + 1, n)];
U = new double[m][];
for (int i = 0; i < m; i++)
U[i] = new double[nu];
V = new double[n][];
for (int i2 = 0; i2 < n; i2++)
V[i2] = new double[n];
var e = new double[n];
var work = new double[m];
var wantu = true;
var 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] = NumericHelpers.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] = NumericHelpers.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++)
{
var 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.
var 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++)
{
var t = 0.0;
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++)
{
var t = 0.0;
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.
var pp = p - 1;
var iter = 0;
var 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;
var 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:
{
var f = e[p - 2];
e[p - 2] = 0.0;
for (int j = p - 2; j >= k; j--)
{
var t = NumericHelpers.Hypot(s[j], f);
var cs = s[j] / t;
var 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:
{
var f = e[k - 1];
e[k - 1] = 0.0;
for (int j = k; j < p; j++)
{
var t = NumericHelpers.Hypot(s[j], f);
var cs = s[j] / t;
var 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.
var 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]));
var sp = s[p - 1] / scale;
var spm1 = s[p - 2] / scale;
var epm1 = e[p - 2] / scale;
var sk = s[k] / scale;
var ek = e[k] / scale;
var b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0;
var c = (sp * epm1) * (sp * epm1);
var 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);
}
var f = (sk + sp) * (sk - sp) + shift;
var g = sk * ek;
// Chase zeros.
for (int j = k; j < p - 1; j++)
{
var t = NumericHelpers.Hypot(f, g);
var cs = f / t;
var 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 = NumericHelpers.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;
var 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;
}
}
}
#endregion //Constructor
#region Public Properties
/// <summary>
/// Return the one-dimensional array of singular values
/// </summary>
/// <returns>diagonal of S.</returns>
virtual public double[] SingularValues
{
get
{
return s;
}
}
/// <summary>
/// Return the diagonal matrix of singular values
/// </summary>
/// <returns>S</returns>
virtual public MatrixValue S
{
get
{
var X = new MatrixValue(n, n);
for (int i = 1; i <= n; i++)
X[i, i].Value = s[i - 1];
return X;
}
}
#endregion // Public Properties
#region Public Methods
/// <summary>
/// Return the left singular vectors
/// </summary>
/// <returns>U</returns>
public virtual MatrixValue GetU()
{
return new MatrixValue(U, m, Math.Min(m + 1, n));
}
/// <summary>
/// Return the right singular vectors
/// </summary>
/// <returns>V</returns>
public virtual MatrixValue GetV()
{
return new MatrixValue(V, n, n);
}
/// <summary>
/// Two norm
/// </summary>
/// <returns>max(S)</returns>
public virtual double Norm2()
{
return s[0];
}
/// <summary>
/// Two norm condition number
/// </summary>
/// <returns>max(S)/min(S)</returns>
public virtual double Condition()
{
return s[0] / s[Math.Min(m, n) - 1];
}
/// <summary>
/// Effective numerical matrix rank
/// </summary>
/// <returns>Number of nonnegligible singular values.</returns>
public virtual int Rank()
{
var eps = Math.Pow(2.0, -52.0);
var tol = Math.Max(m, n) * s[0] * eps;
var r = 0;
for (int i = 0; i < s.Length; i++)
{
if (s[i] > tol)
r++;
}
return r;
}
#endregion //Public Methods
}
}
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