/// <summary>
/// Batch decomposition of the given matrix.
/// The decomposed matrices can be retrieved via instance methods.
/// </summary>
/// <param name="arg">
/// A rectangular matrix.
/// </param>
/// <param name="wantU">
/// Whether the matrix U is needed.
/// </param>
/// <param name="wantV">
/// Whether the matrix V is needed.
/// </param>
/// <param name="order">
/// whether the singular values must be ordered.
/// </param>
/// <exception cref="ArgumentException">
/// If
/// <code>
/// a.Rows < a.Columns
/// </code>
/// .
/// </exception>
private void BatchSVD(DoubleMatrix2D arg, bool wantU, bool wantV, bool order)
{
Property.DEFAULT.CheckRectangular(arg);
// Derived from LINPACK code.
// Initialize.
double[][] a = arg.ToArray();
_m = arg.Rows;
_n = arg.Columns;
int nu = Math.Min(_m, _n);
_s = new double[Math.Min(_m + 1, _n)];
_u = new DenseDoubleMatrix2D(_m, nu);
if (wantV)
{
_v = new DenseDoubleMatrix2D(_n, _n);
}
var e = new double[_n];
var work = new double[_m];
// 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] = Algebra.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] = Algebra.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 = 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) +
(ks != k + 1 ? Math.Abs(e[ks - 1]) : 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 = Algebra.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 = Algebra.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 =
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 = Algebra.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 = Algebra.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.
if (order)
{
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 (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;
}
}
}