/// <summary>
/// Solves a system of linear equations, <b>Ax = b</b>, with A SVD factorized.
/// </summary>
/// <param name="input">The right hand side vector, <b>b</b>.</param>
/// <param name="result">The left hand side <see cref="Matrix{T}"/>, <b>x</b>.</param>
public override void Solve(Vector <Complex32> input, Vector <Complex32> result)
{
if (input == null)
{
throw new ArgumentNullException("input");
}
if (result == null)
{
throw new ArgumentNullException("result");
}
if (!ComputeVectors)
{
throw new InvalidOperationException(Resources.SingularVectorsNotComputed);
}
// Ax=b where A is an m x n matrix
// Check that b is a column vector with m entries
if (MatrixU.RowCount != input.Count)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
// Check that x is a column vector with n entries
if (MatrixVT.ColumnCount != result.Count)
{
throw Matrix.DimensionsDontMatch <ArgumentException>(MatrixVT, result);
}
var mn = Math.Min(MatrixU.RowCount, MatrixVT.ColumnCount);
var tmp = new Complex32[MatrixVT.ColumnCount];
for (var j = 0; j < MatrixVT.ColumnCount; j++)
{
var value = Complex32.Zero;
if (j < mn)
{
for (var i = 0; i < MatrixU.RowCount; i++)
{
value += MatrixU.At(i, j).Conjugate() * input[i];
}
value /= VectorS[j];
}
tmp[j] = value;
}
for (var j = 0; j < MatrixVT.ColumnCount; j++)
{
var value = Complex32.Zero;
for (var i = 0; i < MatrixVT.ColumnCount; i++)
{
value += MatrixVT.At(i, j).Conjugate() * tmp[i];
}
result[j] = value;
}
}
/// <summary>
/// Solves a system of linear equations, <b>Ax = b</b>, with A SVD factorized.
/// </summary>
/// <param name="input">The right hand side vector, <b>b</b>.</param>
/// <returns>The left hand side <see cref="Vector{T}"/>, <b>x</b>.</returns>
public virtual Vector <T> Solve(Vector <T> input)
{
// Check for proper arguments.
if (input == null)
{
throw new ArgumentNullException("input");
}
if (!ComputeVectors)
{
throw new InvalidOperationException(Resources.SingularVectorsNotComputed);
}
var x = MatrixU.CreateVector(MatrixVT.ColumnCount);
Solve(input, x);
return(x);
}
/// <summary>Returns the singular values as a diagonal <see cref="Matrix{T}"/>.</summary>
/// <returns>The singular values as a diagonal <see cref="Matrix{T}"/>.</returns>
public Matrix <T> W()
{
var rows = MatrixU.RowCount;
var columns = MatrixVT.ColumnCount;
var result = MatrixU.CreateMatrix(rows, columns);
for (var i = 0; i < rows; i++)
{
for (var j = 0; j < columns; j++)
{
if (i == j)
{
result.At(i, i, VectorS[i]);
}
}
}
return(result);
}
/// <summary>
/// Solves a system of linear equations, <b>AX = B</b>, with A SVD factorized.
/// </summary>
/// <param name="input">The right hand side <see cref="Matrix{T}"/>, <b>B</b>.</param>
/// <param name="result">The left hand side <see cref="Matrix{T}"/>, <b>X</b>.</param>
public override void Solve(Matrix <float> input, Matrix <float> result)
{
// Check for proper arguments.
if (input == null)
{
throw new ArgumentNullException("input");
}
if (result == null)
{
throw new ArgumentNullException("result");
}
if (!ComputeVectors)
{
throw new InvalidOperationException(Resources.SingularVectorsNotComputed);
}
// The solution X should have the same number of columns as B
if (input.ColumnCount != result.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSameColumnDimension);
}
// The dimension compatibility conditions for X = A\B require the two matrices A and B to have the same number of rows
if (MatrixU.RowCount != input.RowCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSameRowDimension);
}
// The solution X row dimension is equal to the column dimension of A
if (MatrixVT.ColumnCount != result.RowCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSameColumnDimension);
}
var mn = Math.Min(MatrixU.RowCount, MatrixVT.ColumnCount);
var bn = input.ColumnCount;
var tmp = new float[MatrixVT.ColumnCount];
for (var k = 0; k < bn; k++)
{
for (var j = 0; j < MatrixVT.ColumnCount; j++)
{
float value = 0;
if (j < mn)
{
for (var i = 0; i < MatrixU.RowCount; i++)
{
value += MatrixU.At(i, j) * input.At(i, k);
}
value /= VectorS[j];
}
tmp[j] = value;
}
for (var j = 0; j < MatrixVT.ColumnCount; j++)
{
float value = 0;
for (var i = 0; i < MatrixVT.ColumnCount; i++)
{
value += MatrixVT.At(i, j) * tmp[i];
}
result[j, k] = value;
}
}
}
/// <summary>
/// Initializes a new instance of the <see cref="UserSvd"/> class. This object will compute the
/// the singular value decomposition when the constructor is called and cache it's decomposition.
/// </summary>
/// <param name="matrix">The matrix to factor.</param>
/// <param name="computeVectors">Compute the singular U and VT vectors or not.</param>
/// <exception cref="ArgumentNullException">If <paramref name="matrix"/> is <c>null</c>.</exception>
/// <exception cref="ArgumentException">If SVD algorithm failed to converge with matrix <paramref name="matrix"/>.</exception>
public UserSvd(Matrix <float> matrix, bool computeVectors)
{
if (matrix == null)
{
throw new ArgumentNullException("matrix");
}
ComputeVectors = computeVectors;
var nm = Math.Min(matrix.RowCount + 1, matrix.ColumnCount);
var matrixCopy = matrix.Clone();
VectorS = matrixCopy.CreateVector(nm);
MatrixU = matrixCopy.CreateMatrix(matrixCopy.RowCount, matrixCopy.RowCount);
MatrixVT = matrixCopy.CreateMatrix(matrixCopy.ColumnCount, matrixCopy.ColumnCount);
const int Maxiter = 1000;
var e = new float[matrixCopy.ColumnCount];
var work = new float[matrixCopy.RowCount];
int i, j;
int l, lp1;
var cs = 0.0f;
var sn = 0.0f;
float t;
var ncu = matrixCopy.RowCount;
// Reduce matrixCopy to bidiagonal form, storing the diagonal elements
// In s and the super-diagonal elements in e.
var nct = Math.Min(matrixCopy.RowCount - 1, matrixCopy.ColumnCount);
var nrt = Math.Max(0, Math.Min(matrixCopy.ColumnCount - 2, matrixCopy.RowCount));
var lu = Math.Max(nct, nrt);
for (l = 0; l < lu; l++)
{
lp1 = l + 1;
if (l < nct)
{
// Compute the transformation for the l-th column and place the l-th diagonal in VectorS[l].
var xnorm = Dnrm2Column(matrixCopy, matrixCopy.RowCount, l, l);
VectorS[l] = xnorm;
if (VectorS[l] != 0.0)
{
if (matrixCopy.At(l, l) != 0.0)
{
VectorS[l] = Dsign(VectorS[l], matrixCopy.At(l, l));
}
DscalColumn(matrixCopy, matrixCopy.RowCount, l, l, 1.0f / VectorS[l]);
matrixCopy.At(l, l, (1.0f + matrixCopy.At(l, l)));
}
VectorS[l] = -VectorS[l];
}
for (j = lp1; j < matrixCopy.ColumnCount; j++)
{
if (l < nct)
{
if (VectorS[l] != 0.0)
{
// Apply the transformation.
t = -Ddot(matrixCopy, matrixCopy.RowCount, l, j, l) / matrixCopy.At(l, l);
for (var ii = l; ii < matrixCopy.RowCount; ii++)
{
matrixCopy.At(ii, j, matrixCopy.At(ii, j) + (t * matrixCopy.At(ii, l)));
}
}
}
// Place the l-th row of matrixCopy into e for the
// Subsequent calculation of the row transformation.
e[j] = matrixCopy.At(l, j);
}
if (ComputeVectors && l < nct)
{
// Place the transformation in u for subsequent back multiplication.
for (i = l; i < matrixCopy.RowCount; i++)
{
MatrixU.At(i, l, matrixCopy.At(i, l));
}
}
if (l >= nrt)
{
continue;
}
// Compute the l-th row transformation and place the l-th super-diagonal in e(l).
var enorm = Dnrm2Vector(e, lp1);
e[l] = enorm;
if (e[l] != 0.0)
{
if (e[lp1] != 0.0)
{
e[l] = Dsign(e[l], e[lp1]);
}
DscalVector(e, lp1, 1.0f / e[l]);
e[lp1] = 1.0f + e[lp1];
}
e[l] = -e[l];
if (lp1 < matrixCopy.RowCount && e[l] != 0.0)
{
// Apply the transformation.
for (i = lp1; i < matrixCopy.RowCount; i++)
{
work[i] = 0.0f;
}
for (j = lp1; j < matrixCopy.ColumnCount; j++)
{
for (var ii = lp1; ii < matrixCopy.RowCount; ii++)
{
work[ii] += e[j] * matrixCopy.At(ii, j);
}
}
for (j = lp1; j < matrixCopy.ColumnCount; j++)
{
var ww = -e[j] / e[lp1];
for (var ii = lp1; ii < matrixCopy.RowCount; ii++)
{
matrixCopy.At(ii, j, matrixCopy.At(ii, j) + (ww * work[ii]));
}
}
}
if (ComputeVectors)
{
// Place the transformation in v for subsequent back multiplication.
for (i = lp1; i < matrixCopy.ColumnCount; i++)
{
MatrixVT.At(i, l, e[i]);
}
}
}
// Set up the final bidiagonal matrixCopy or order m.
var m = Math.Min(matrixCopy.ColumnCount, matrixCopy.RowCount + 1);
var nctp1 = nct + 1;
var nrtp1 = nrt + 1;
if (nct < matrixCopy.ColumnCount)
{
VectorS[nctp1 - 1] = matrixCopy.At((nctp1 - 1), (nctp1 - 1));
}
if (matrixCopy.RowCount < m)
{
VectorS[m - 1] = 0.0f;
}
if (nrtp1 < m)
{
e[nrtp1 - 1] = matrixCopy.At((nrtp1 - 1), (m - 1));
}
e[m - 1] = 0.0f;
// If required, generate u.
if (ComputeVectors)
{
for (j = nctp1 - 1; j < ncu; j++)
{
for (i = 0; i < matrixCopy.RowCount; i++)
{
MatrixU.At(i, j, 0.0f);
}
MatrixU.At(j, j, 1.0f);
}
for (l = nct - 1; l >= 0; l--)
{
if (VectorS[l] != 0.0)
{
for (j = l + 1; j < ncu; j++)
{
t = -Ddot(MatrixU, matrixCopy.RowCount, l, j, l) / MatrixU.At(l, l);
for (var ii = l; ii < matrixCopy.RowCount; ii++)
{
MatrixU.At(ii, j, MatrixU.At(ii, j) + (t * MatrixU.At(ii, l)));
}
}
DscalColumn(MatrixU, matrixCopy.RowCount, l, l, -1.0f);
MatrixU.At(l, l, 1.0f + MatrixU.At(l, l));
for (i = 0; i < l; i++)
{
MatrixU.At(i, l, 0.0f);
}
}
else
{
for (i = 0; i < matrixCopy.RowCount; i++)
{
MatrixU.At(i, l, 0.0f);
}
MatrixU.At(l, l, 1.0f);
}
}
}
// If it is required, generate v.
if (ComputeVectors)
{
for (l = matrixCopy.ColumnCount - 1; l >= 0; l--)
{
lp1 = l + 1;
if (l < nrt)
{
if (e[l] != 0.0)
{
for (j = lp1; j < matrixCopy.ColumnCount; j++)
{
t = -Ddot(MatrixVT, matrixCopy.ColumnCount, l, j, lp1) / MatrixVT.At(lp1, l);
for (var ii = l; ii < matrixCopy.ColumnCount; ii++)
{
MatrixVT.At(ii, j, MatrixVT.At(ii, j) + (t * MatrixVT.At(ii, l)));
}
}
}
}
for (i = 0; i < matrixCopy.ColumnCount; i++)
{
MatrixVT.At(i, l, 0.0f);
}
MatrixVT.At(l, l, 1.0f);
}
}
// Transform s and e so that they are float .
for (i = 0; i < m; i++)
{
float r;
if (VectorS[i] != 0.0)
{
t = VectorS[i];
r = VectorS[i] / t;
VectorS[i] = t;
if (i < m - 1)
{
e[i] = e[i] / r;
}
if (ComputeVectors)
{
DscalColumn(MatrixU, matrixCopy.RowCount, i, 0, r);
}
}
// Exit
if (i == m - 1)
{
break;
}
if (e[i] != 0.0)
{
t = e[i];
r = t / e[i];
e[i] = t;
VectorS[i + 1] = VectorS[i + 1] * r;
if (ComputeVectors)
{
DscalColumn(MatrixVT, matrixCopy.ColumnCount, i + 1, 0, r);
}
}
}
// Main iteration loop for the singular values.
var mn = m;
var iter = 0;
while (m > 0)
{
// Quit if all the singular values have been found. If too many iterations have been performed,
// throw exception that Convergence Failed
if (iter >= Maxiter)
{
throw new ArgumentException(Resources.ConvergenceFailed);
}
// This section of the program inspects for negligible elements in the s and e arrays. On
// completion the variables kase and l are set as follows.
// Kase = 1 if VectorS[m] and e[l-1] are negligible and l < m
// Kase = 2 if VectorS[l] is negligible and l < m
// Kase = 3 if e[l-1] is negligible, l < m, and VectorS[l, ..., VectorS[m] are not negligible (qr step).
// Лase = 4 if e[m-1] is negligible (convergence).
float ztest;
float test;
for (l = m - 2; l >= 0; l--)
{
test = Math.Abs(VectorS[l]) + Math.Abs(VectorS[l + 1]);
ztest = test + Math.Abs(e[l]);
if (ztest.AlmostEqualInDecimalPlaces(test, 7))
{
e[l] = 0.0f;
break;
}
}
int kase;
if (l == m - 2)
{
kase = 4;
}
else
{
int ls;
for (ls = m - 1; ls > l; ls--)
{
test = 0.0f;
if (ls != m - 1)
{
test = test + Math.Abs(e[ls]);
}
if (ls != l + 1)
{
test = test + Math.Abs(e[ls - 1]);
}
ztest = test + Math.Abs(VectorS[ls]);
if (ztest.AlmostEqualInDecimalPlaces(test, 7))
{
VectorS[ls] = 0.0f;
break;
}
}
if (ls == l)
{
kase = 3;
}
else if (ls == m - 1)
{
kase = 1;
}
else
{
kase = 2;
l = ls;
}
}
l = l + 1;
// Perform the task indicated by kase.
int k;
float f;
switch (kase)
{
// Deflate negligible VectorS[m].
case 1:
f = e[m - 2];
e[m - 2] = 0.0f;
float t1;
for (var kk = l; kk < m - 1; kk++)
{
k = m - 2 - kk + l;
t1 = VectorS[k];
Drotg(ref t1, ref f, ref cs, ref sn);
VectorS[k] = t1;
if (k != l)
{
f = -sn * e[k - 1];
e[k - 1] = cs * e[k - 1];
}
if (ComputeVectors)
{
Drot(MatrixVT, matrixCopy.ColumnCount, k, m - 1, cs, sn);
}
}
break;
// Split at negligible VectorS[l].
case 2:
f = e[l - 1];
e[l - 1] = 0.0f;
for (k = l; k < m; k++)
{
t1 = VectorS[k];
Drotg(ref t1, ref f, ref cs, ref sn);
VectorS[k] = t1;
f = -sn * e[k];
e[k] = cs * e[k];
if (ComputeVectors)
{
Drot(MatrixU, matrixCopy.RowCount, k, l - 1, cs, sn);
}
}
break;
// Perform one qr step.
case 3:
// Calculate the shift.
var scale = 0.0f;
scale = Math.Max(scale, Math.Abs(VectorS[m - 1]));
scale = Math.Max(scale, Math.Abs(VectorS[m - 2]));
scale = Math.Max(scale, Math.Abs(e[m - 2]));
scale = Math.Max(scale, Math.Abs(VectorS[l]));
scale = Math.Max(scale, Math.Abs(e[l]));
var sm = VectorS[m - 1] / scale;
var smm1 = VectorS[m - 2] / scale;
var emm1 = e[m - 2] / scale;
var sl = VectorS[l] / scale;
var el = e[l] / scale;
var b = (((smm1 + sm) * (smm1 - sm)) + (emm1 * emm1)) / 2.0f;
var c = (sm * emm1) * (sm * emm1);
var shift = 0.0f;
if (b != 0.0 || c != 0.0)
{
shift = (float)Math.Sqrt((b * b) + c);
if (b < 0.0)
{
shift = -shift;
}
shift = c / (b + shift);
}
f = ((sl + sm) * (sl - sm)) + shift;
var g = sl * el;
// Chase zeros.
for (k = l; k < m - 1; k++)
{
Drotg(ref f, ref g, ref cs, ref sn);
if (k != l)
{
e[k - 1] = f;
}
f = (cs * VectorS[k]) + (sn * e[k]);
e[k] = (cs * e[k]) - (sn * VectorS[k]);
g = sn * VectorS[k + 1];
VectorS[k + 1] = cs * VectorS[k + 1];
if (ComputeVectors)
{
Drot(MatrixVT, matrixCopy.ColumnCount, k, k + 1, cs, sn);
}
Drotg(ref f, ref g, ref cs, ref sn);
VectorS[k] = f;
f = (cs * e[k]) + (sn * VectorS[k + 1]);
VectorS[k + 1] = (-sn * e[k]) + (cs * VectorS[k + 1]);
g = sn * e[k + 1];
e[k + 1] = cs * e[k + 1];
if (ComputeVectors && k < matrixCopy.RowCount)
{
Drot(MatrixU, matrixCopy.RowCount, k, k + 1, cs, sn);
}
}
e[m - 2] = f;
iter = iter + 1;
break;
// Convergence.
case 4:
// Make the singular value positive
if (VectorS[l] < 0.0)
{
VectorS[l] = -VectorS[l];
if (ComputeVectors)
{
DscalColumn(MatrixVT, matrixCopy.ColumnCount, l, 0, -1.0f);
}
}
// Order the singular value.
while (l != mn - 1)
{
if (VectorS[l] >= VectorS[l + 1])
{
break;
}
t = VectorS[l];
VectorS[l] = VectorS[l + 1];
VectorS[l + 1] = t;
if (ComputeVectors && l < matrixCopy.ColumnCount)
{
Dswap(MatrixVT, matrixCopy.ColumnCount, l, l + 1);
}
if (ComputeVectors && l < matrixCopy.RowCount)
{
Dswap(MatrixU, matrixCopy.RowCount, l, l + 1);
}
l = l + 1;
}
iter = 0;
m = m - 1;
break;
}
}
if (ComputeVectors)
{
MatrixVT = MatrixVT.Transpose();
}
// Adjust the size of s if rows < columns. We are using ported copy of linpack's svd code and it uses
// a singular vector of length mRows+1 when mRows < mColumns. The last element is not used and needs to be removed.
// we should port lapack's svd routine to remove this problem.
if (matrixCopy.RowCount < matrixCopy.ColumnCount)
{
nm--;
var tmp = matrixCopy.CreateVector(nm);
for (i = 0; i < nm; i++)
{
tmp[i] = VectorS[i];
}
VectorS = tmp;
}
}
/// <summary>Returns the left singular vectors as a <see cref="Matrix{T}"/>.</summary>
/// <returns>The left singular vectors. The matrix will be <c>null</c>, if <b>computeVectors</b> in the constructor is set to <c>false</c>.</returns>
public Matrix <T> U()
{
return(ComputeVectors ? MatrixU.Clone() : null);
}