/// <summary>
/// Moore–Penrose pseudoinverse
/// If A = U • Σ • VT is the singular value decomposition of A, then A† = V • Σ† • UT.
/// For a diagonal matrix such as Σ, we get the pseudoinverse by taking the reciprocal of each non-zero element
/// on the diagonal, leaving the zeros in place, and transposing the resulting matrix.
/// In numerical computation, only elements larger than some small tolerance are taken to be nonzero,
/// and the others are replaced by zeros. For example, in the MATLAB or NumPy function pinv,
/// the tolerance is taken to be t = ε • max(m,n) • max(Σ), where ε is the machine epsilon. (Wikipedia)
/// </summary>
/// <param name="M">The matrix to pseudoinverse</param>
/// <returns>The pseudoinverse of this Matrix</returns>
public static Matrix PseudoInverse(this Matrix M)
{
Svd D = M.Svd(true);
Matrix W = (Matrix)D.W();
Vector s = (Vector)D.S();
// The first element of W has the maximum value.
double tolerance = Precision.EpsilonOf(2) * Math.Max(M.RowCount, M.ColumnCount) * W[0, 0];
for (int i = 0; i < s.Count; i++)
{
if (s[i] < tolerance)
{
s[i] = 0;
}
else
{
s[i] = 1 / s[i];
}
}
W.SetDiagonal(s);
// (U * W * VT)T is equivalent with V * WT * UT
return((Matrix)(D.U() * W * D.VT()).Transpose());
}
private static Matrix <double> PseudoInverse(Svd <double> svd)
{
Matrix <double> W = svd.W();
Vector <double> s = svd.S();
// The first element of W has the maximum value.
double tolerance = Precision.EpsilonOf(2) * Math.Max(svd.U().RowCount, svd.VT().ColumnCount) * W[0, 0];
for (int i = 0; i < s.Count; i++)
{
if (s[i] < tolerance)
{
s[i] = 0;
}
else
{
s[i] = 1 / s[i];
}
}
W.SetDiagonal(s);
// (U * W * VT)T is equivalent with V * WT * UT
return((svd.U() * W * svd.VT()).Transpose());
}
