public Matrix <double> PseudoInverse(Matrix <double> M) { var svd = M.Svd(true); var W = svd.W; var s = svd.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((svd.U * W * svd.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()); }
/// <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()); }
public override double EpsilonOf(double a) { return(Precision.EpsilonOf(a)); }