SingularValueDecomposition( Matrix Arg ) { transpose = (Arg.RowCount < Arg.ColumnCount); // Derived from LINPACK code. // Initialize. double[][] A; if (transpose) { // copy of internal data, independent of Arg A = Matrix.Transpose(Arg).GetArray(); m = Arg.ColumnCount; n = Arg.RowCount; } else { A = Arg.CopyToJaggedArray(); m = Arg.RowCount; n = Arg.ColumnCount; } int nu = Math.Min(m, n); double[] s = new double[Math.Min(m + 1, n)]; double[][] U = Matrix.CreateMatrixData(m, nu); double[][] V = Matrix.CreateMatrixData(n, n); double[] e = new double[n]; double[] work = new double[m]; bool wantu = true; bool wantv = true; // 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] = Fn.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] = Fn.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 = Number.PositiveRelativeAccuracy; 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]) : 0.0) + (ks != k + 1 ? Math.Abs(e[ks - 1]) : 0.0); 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 = Fn.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 = Fn.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 = Fn.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 = Fn.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. 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 (wantu && (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; } } // (vermorel) transposing the results if needed if (transpose) { // swaping U and V double[][] T = V; V = U; U = T; } _u = new Matrix(U); _v = new Matrix(V); _singular = new Vector(s); InitOnDemandComputations(); }