/// <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;
}
}
}
/// <summary>
/// Constructs and returns a new QR decomposition object; computed by Householder reflections;
/// The decomposed matrices can be retrieved via instance methods of the returned decomposition object.
/// Return a decomposition object to access <i>R</i> and the Householder vectors <i>H</i>, and to compute <i>Q</i>.
/// </summary>
/// <param name="A">A rectangular matrix.</param>
/// <exception cref="ArgumentException">if <i>A.Rows < A.Columns</i>.</exception>
public QRDecomposition(DoubleMatrix2D A)
{
Property.DEFAULT.CheckRectangular(A);
Functions F = Functions.functions;
// Initialize.
QR = A.Copy();
m = A.Rows;
n = A.Columns;
Rdiag = A.Like1D(n);
//Rdiag = new double[n];
DoubleDoubleFunction hypot = Algebra.HypotFunction();
// precompute and cache some views to avoid regenerating them time and again
DoubleMatrix1D[] QRcolumns = new DoubleMatrix1D[n];
DoubleMatrix1D[] QRcolumnsPart = new DoubleMatrix1D[n];
for (int k = 0; k < n; k++)
{
QRcolumns[k] = QR.ViewColumn(k);
QRcolumnsPart[k] = QR.ViewColumn(k).ViewPart(k, m - k);
}
// Main loop.
for (int k = 0; k < n; k++)
{
//DoubleMatrix1D QRcolk = QR.ViewColumn(k).ViewPart(k,m-k);
// Compute 2-norm of k-th column without under/overflow.
double nrm = 0;
//if (k<m) nrm = QRcolumnsPart[k].aggregate(hypot,F.identity);
for (int i = k; i < m; i++)
{ // fixes bug reported by [email protected]
nrm = Algebra.Hypot(nrm, QR[i, k]);
}
if (nrm != 0.0)
{
// Form k-th Householder vector.
if (QR[k, k] < 0)
{
nrm = -nrm;
}
QRcolumnsPart[k].Assign(F2.Div(nrm));
/*
* for (int i = k; i < m; i++) {
* QR[i][k] /= nrm;
* }
*/
QR[k, k] = QR[k, k] + 1;
// Apply transformation to remaining columns.
for (int j = k + 1; j < n; j++)
{
DoubleMatrix1D QRcolj = QR.ViewColumn(j).ViewPart(k, m - k);
double s = QRcolumnsPart[k].ZDotProduct(QRcolj);
/*
* // fixes bug reported by John Chambers
* DoubleMatrix1D QRcolj = QR.ViewColumn(j).ViewPart(k,m-k);
* double s = QRcolumnsPart[k].ZDotProduct(QRcolumns[j]);
* double s = 0.0;
* for (int i = k; i < m; i++) {
* s += QR[i][k]*QR[i][j];
* }
*/
s = -s / QR[k, k];
//QRcolumnsPart[j].Assign(QRcolumns[k], F.PlusMult(s));
for (int i = k; i < m; i++)
{
QR[i, j] = QR[i, j] + s * QR[i, k];
}
}
}
Rdiag[k] = -nrm;
}
}
/// <summary>
/// Symmetric tridiagonal QL algorithm.
/// </summary>
private void Tql2()
{
// This is derived from the Algol procedures tql2, by
// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
// Autod Compd, Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutine in EISPACK.
for (int i = 1; i < n; i++)
{
e[i - 1] = e[i];
}
e[n - 1] = 0.0;
double f = 0.0;
double tst1 = 0.0;
double eps = System.Math.Pow(2.0, -52.0);
for (int l = 0; l < n; l++)
{
// Find small subdiagonal element
tst1 = System.Math.Max(tst1, System.Math.Abs(d[l]) + System.Math.Abs(e[l]));
int m = l;
while (m < n)
{
if (System.Math.Abs(e[m]) <= eps * tst1)
{
break;
}
m++;
}
// If m == l, d[l] is an eigenvalue,
// otherwise, iterate.
if (m > l)
{
int iter = 0;
do
{
iter = iter + 1; // (Could check iteration count hered)
// Compute implicit shift
double g = d[l];
double p = (d[l + 1] - g) / (2.0 * e[l]);
double r = Algebra.Hypot(p, 1.0);
if (p < 0)
{
r = -r;
}
d[l] = e[l] / (p + r);
d[l + 1] = e[l] * (p + r);
double dl1 = d[l + 1];
double h = g - d[l];
for (int i = l + 2; i < n; i++)
{
d[i] -= h;
}
f = f + h;
// Implicit QL transformation.
p = d[m];
double c = 1.0;
double c2 = c;
double c3 = c;
double el1 = e[l + 1];
double s = 0.0;
double s2 = 0.0;
for (int i = m - 1; i >= l; i--)
{
c3 = c2;
c2 = c;
s2 = s;
g = c * e[i];
h = c * p;
r = Algebra.Hypot(p, e[i]);
e[i + 1] = s * r;
s = e[i] / r;
c = p / r;
p = c * d[i] - s * g;
d[i + 1] = h + s * (c * g + s * d[i]);
// Accumulate transformation.
for (int k = 0; k < n; k++)
{
h = v[k][i + 1];
v[k][i + 1] = s * v[k][i] + c * h;
v[k][i] = c * v[k][i] - s * h;
}
}
p = -s * s2 * c3 * el1 * e[l] / dl1;
e[l] = s * p;
d[l] = c * p;
// Check for convergence.
} while (System.Math.Abs(e[l]) > eps * tst1);
}
d[l] = d[l] + f;
e[l] = 0.0;
}
// Sort eigenvalues and corresponding vectors.
for (int i = 0; i < n - 1; i++)
{
int k = i;
double p = d[i];
for (int j = i + 1; j < n; j++)
{
if (d[j] < p)
{
k = j;
p = d[j];
}
}
if (k != i)
{
d[k] = d[i];
d[i] = p;
for (int j = 0; j < n; j++)
{
p = v[j][i];
v[j][i] = v[j][k];
v[j][k] = p;
}
}
}
}
public double Dnrm2(DoubleMatrix1D x)
{
return(System.Math.Sqrt(Algebra.Norm2(x)));
}
public void Solve(DoubleMatrix2D B)
{
int CUT_OFF = 10;
//algebra.property().checkRectangular(LU);
int m = M;
int n = N;
if (B.Rows != m)
{
throw new ArgumentException("Matrix row dimensions must agree.");
}
if (!this.IsNonsingular)
{
throw new ArgumentException("Matrix is singular.");
}
// right hand side with pivoting
// Matrix Xmat = B.getMatrix(piv,0,nx-1);
if (this.work1 == null || this.work1.Length < m)
{
this.work1 = new int[m];
}
//if (this.work2 == null || this.work2.Length < m) this.work2 = new int[m];
Algebra.PermuteRows(B, this.piv, this.work1);
if (m * n == 0)
{
return; // nothing to do
}
int nx = B.Columns;
//precompute and cache some views to avoid regenerating them time and again
DoubleMatrix1D[] Brows = new DoubleMatrix1D[n];
for (int k = 0; k < n; k++)
{
Brows[k] = B.ViewRow(k);
}
// transformations
Cern.Jet.Math.Mult div = Cern.Jet.Math.Mult.Div(0);
Cern.Jet.Math.PlusMult minusMult = Cern.Jet.Math.PlusMult.MinusMult(0);
IntArrayList nonZeroIndexes = new IntArrayList(); // sparsity
DoubleMatrix1D Browk = Cern.Colt.Matrix.DoubleFactory1D.Dense.Make(nx); // blocked row k
// Solve L*Y = B(piv,:)
for (int k = 0; k < n; k++)
{
// blocking (make copy of k-th row to localize references)
Browk.Assign(Brows[k]);
// sparsity detection
int maxCardinality = nx / CUT_OFF; // == heuristic depending on speedup
Browk.GetNonZeros(nonZeroIndexes, null, maxCardinality);
int cardinality = nonZeroIndexes.Count;
Boolean sparse = (cardinality < maxCardinality);
for (int i = k + 1; i < n; i++)
{
//for (int j = 0; j < nx; j++) B[i][j] -= B[k][j]*LU[i][k];
//for (int j = 0; j < nx; j++) B.set(i,j, B.Get(i,j) - B.Get(k,j)*LU.Get(i,k));
minusMult.Multiplicator = -LU[i, k];
if (minusMult.Multiplicator != 0)
{
if (sparse)
{
Brows[i].Assign(Browk, minusMult, nonZeroIndexes);
}
else
{
Brows[i].Assign(Browk, minusMult);
}
}
}
}
// Solve U*B = Y;
for (int k = n - 1; k >= 0; k--)
{
// for (int j = 0; j < nx; j++) B[k][j] /= LU[k][k];
// for (int j = 0; j < nx; j++) B.set(k,j, B.Get(k,j) / LU.Get(k,k));
div.Multiplicator = 1 / LU[k, k];
Brows[k].Assign(div);
// blocking
if (Browk == null)
{
Browk = Cern.Colt.Matrix.DoubleFactory1D.Dense.Make(B.Columns);
}
Browk.Assign(Brows[k]);
// sparsity detection
int maxCardinality = nx / CUT_OFF; // == heuristic depending on speedup
Browk.GetNonZeros(nonZeroIndexes, null, maxCardinality);
int cardinality = nonZeroIndexes.Count;
Boolean sparse = (cardinality < maxCardinality);
//Browk.GetNonZeros(nonZeroIndexes,null);
//Boolean sparse = nonZeroIndexes.Count < nx/10;
for (int i = 0; i < k; i++)
{
// for (int j = 0; j < nx; j++) B[i][j] -= B[k][j]*LU[i][k];
// for (int j = 0; j < nx; j++) B.set(i,j, B.Get(i,j) - B.Get(k,j)*LU.Get(i,k));
minusMult.Multiplicator = -LU[i, k];
if (minusMult.Multiplicator != 0)
{
if (sparse)
{
Brows[i].Assign(Browk, minusMult, nonZeroIndexes);
}
else
{
Brows[i].Assign(Browk, minusMult);
}
}
}
}
}
public void Solve(DoubleMatrix1D B)
{
//algebra.property().checkRectangular(LU);
int m = M;
int n = N;
if (B.Count() != m)
{
throw new ArgumentException("Matrix dimensions must agree.");
}
if (!this.IsNonsingular)
{
throw new ArgumentException("Matrix is singular.");
}
// right hand side with pivoting
// Matrix Xmat = B.getMatrix(piv,0,nx-1);
if (this.workDouble == null || this.workDouble.Length < m)
{
this.workDouble = new double[m];
}
Algebra.Permute(B, this.piv, this.workDouble);
if (m * n == 0)
{
return; // nothing to do
}
// Solve L*Y = B(piv,:)
for (int k = 0; k < n; k++)
{
double f = B[k];
if (f != 0)
{
for (int i = k + 1; i < n; i++)
{
// B[i] -= B[k]*LU[i][k];
double v = LU[i, k];
if (v != 0)
{
B[i] = B[i] - f * v;
}
}
}
}
// Solve U*B = Y;
for (int k = n - 1; k >= 0; k--)
{
// B[k] /= LU[k,k]
B[k] = B[k] / LU[k, k];
double f = B[k];
if (f != 0)
{
for (int i = 0; i < k; i++)
{
// B[i] -= B[k]*LU[i][k];
double v = LU[i, k];
if (v != 0)
{
B[i] = B[i] - f * v;
}
}
}
}
}