+ * If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal + * and the eigenvector matrix V is orthogonal. I.e. A = + * V.times(D.times(V.transpose())) and V.times(V.transpose()) equals the + * identity matrix. + *
+ * If A is not symmetric, then the eigenvalue matrix D is block diagonal with + * the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda + + * i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The columns of V represent + * the eigenvectors in the sense that A*V = V*D, i.e. A.times(V) equals + * V.times(D). The matrix V may be badly conditioned, or even singular, so the + * validity of the equation A = V*D*inverse(V) depends upon V.cond(). + **/ + +public class Eigen implements EigenInterface { + + /* ------------------------ + Public Methods + * ------------------------ */ + + public Eigen() {} + + public Eigen set(int n) { + this.n = n; + V = new double[n][n]; + d = new double[n]; + e = new double[n]; + return this; + } + + @Override + public Eigen setM(double[][] m) { + set(m.length); + calc(m); + return this; + } + + /** + * return values sorted from smallest to largest value. + */ + @Override + public double[] getEigenvalues() { + return d; + } + + /** + * Specifically for 3x3 systems, returns eigenVectors as V3[3] + * and values as float[3]; sorted from smallest to largest value. + * + * @param eigenVectors returned vectors + * @param eigenValues returned values + * + */ + @Override + public void fillFloatArrays(V3[] eigenVectors, float[] eigenValues) { + for (int i = 0; i < 3; i++) { + if (eigenVectors != null) { + if (eigenVectors[i] == null) + eigenVectors[i] = new V3(); + eigenVectors[i].set((float) V[0][i], (float) V[1][i], (float) V[2][i]); + } + if (eigenValues != null) + eigenValues[i] = (float) d[i]; + } + } + + /** + * Transpose V and turn into floats; sorted from smallest to largest value. + * + * @return ROWS of eigenvectors f[0], f[1], f[2], etc. + */ + @Override + public float[][] getEigenvectorsFloatTransposed() { + float[][] f = new float[n][n]; + for (int i = n; --i >= 0;) + for (int j = n; --j >= 0;) + f[j][i] = (float) V[i][j]; + return f; + } + + + /** + * Check for symmetry, then construct the eigenvalue decomposition + * + * @param A + * Square matrix + */ + + public void calc(double[][] A) { + + /* Jmol only has need of symmetric solutions + * + issymmetric = true; + + for (int j = 0; (j < n) & issymmetric; j++) { + for (int i = 0; (i < n) & issymmetric; i++) { + issymmetric = (A[i][j] == A[j][i]); + } + } + + if (issymmetric) { + */ + for (int i = 0; i < n; i++) { + for (int j = 0; j < n; j++) { + V[i][j] = A[i][j]; + } + } + + // Tridiagonalize. + tred2(); + + // Diagonalize. + tql2(); + /* + } else { + H = new double[n][n]; + ort = new double[n]; + + for (int j = 0; j < n; j++) { + for (int i = 0; i < n; i++) { + H[i][j] = A[i][j]; + } + } + + // Reduce to Hessenberg form. + orthes(); + + // Reduce Hessenberg to real Schur form. + hqr2(); + } + */ + + } + + /** + * Return the real parts of the eigenvalues + * + * @return real(diag(D)) + */ + + public double[] getRealEigenvalues() { + return d; + } + + /** + * Return the imaginary parts of the eigenvalues + * + * @return imag(diag(D)) + */ + + public double[] getImagEigenvalues() { + return e; + } + + /* ------------------------ + Class variables + * ------------------------ */ + + /** + * Row and column dimension (square matrix). + * + * @serial matrix dimension. + */ + private int n = 3; + + /** + * Symmetry flag. + * + * @serial internal symmetry flag. + */ + //private boolean issymmetric = true; + + /** + * Arrays for internal storage of eigenvalues. + * + * @serial internal storage of eigenvalues. + */ + private double[] d, e; + + /** + * Array for internal storage of eigenvectors. + * + * @serial internal storage of eigenvectors. + */ + private double[][] V; + + /** + * Array for internal storage of nonsymmetric Hessenberg form. + * + * @serial internal storage of nonsymmetric Hessenberg form. + */ + //private double[][] H; + + /** + * Working storage for nonsymmetric algorithm. + * + * @serial working storage for nonsymmetric algorithm. + */ + //private double[] ort; + + /* ------------------------ + Private Methods + * ------------------------ */ + + // Symmetric Householder reduction to tridiagonal form. + + private void tred2() { + + // This is derived from the Algol procedures tred2 by + // Bowdler, Martin, Reinsch, and Wilkinson, Handbook for + // Auto. Comp., Vol.ii-Linear Algebra, and the corresponding + // Fortran subroutine in EISPACK. + + for (int j = 0; j < n; j++) { + d[j] = V[n - 1][j]; + } + + // Householder reduction to tridiagonal form. + + for (int i = n - 1; i > 0; i--) { + + // Scale to avoid under/overflow. + + double scale = 0.0; + double h = 0.0; + for (int k = 0; k < i; k++) { + scale = scale + Math.abs(d[k]); + } + if (scale == 0.0) { + e[i] = d[i - 1]; + for (int j = 0; j < i; j++) { + d[j] = V[i - 1][j]; + V[i][j] = 0.0; + V[j][i] = 0.0; + } + } else { + + // Generate Householder vector. + + for (int k = 0; k < i; k++) { + d[k] /= scale; + h += d[k] * d[k]; + } + double f = d[i - 1]; + double g = Math.sqrt(h); + if (f > 0) { + g = -g; + } + e[i] = scale * g; + h = h - f * g; + d[i - 1] = f - g; + for (int j = 0; j < i; j++) { + e[j] = 0.0; + } + + // Apply similarity transformation to remaining columns. + + for (int j = 0; j < i; j++) { + f = d[j]; + V[j][i] = f; + g = e[j] + V[j][j] * f; + for (int k = j + 1; k <= i - 1; k++) { + g += V[k][j] * d[k]; + e[k] += V[k][j] * f; + } + e[j] = g; + } + f = 0.0; + for (int j = 0; j < i; j++) { + e[j] /= h; + f += e[j] * d[j]; + } + double hh = f / (h + h); + for (int j = 0; j < i; j++) { + e[j] -= hh * d[j]; + } + for (int j = 0; j < i; j++) { + f = d[j]; + g = e[j]; + for (int k = j; k <= i - 1; k++) { + V[k][j] -= (f * e[k] + g * d[k]); + } + d[j] = V[i - 1][j]; + V[i][j] = 0.0; + } + } + d[i] = h; + } + + // Accumulate transformations. + + for (int i = 0; i < n - 1; i++) { + V[n - 1][i] = V[i][i]; + V[i][i] = 1.0; + double h = d[i + 1]; + if (h != 0.0) { + for (int k = 0; k <= i; k++) { + d[k] = V[k][i + 1] / h; + } + for (int j = 0; j <= i; j++) { + double g = 0.0; + for (int k = 0; k <= i; k++) { + g += V[k][i + 1] * V[k][j]; + } + for (int k = 0; k <= i; k++) { + V[k][j] -= g * d[k]; + } + } + } + for (int k = 0; k <= i; k++) { + V[k][i + 1] = 0.0; + } + } + for (int j = 0; j < n; j++) { + d[j] = V[n - 1][j]; + V[n - 1][j] = 0.0; + } + V[n - 1][n - 1] = 1.0; + e[0] = 0.0; + } + + // Symmetric tridiagonal QL algorithm. + + private void tql2() { + + // This is derived from the Algol procedures tql2, by + // Bowdler, Martin, Reinsch, and Wilkinson, Handbook for + // Auto. Comp., 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 = Math.pow(2.0, -52.0); + for (int l = 0; l < n; l++) { + + // Find small subdiagonal element + + tst1 = Math.max(tst1, Math.abs(d[l]) + Math.abs(e[l])); + int m = l; + while (m < n) { + if (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 here.) + + // Compute implicit shift + + double g = d[l]; + double p = (d[l + 1] - g) / (2.0 * e[l]); + double r = 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 = 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 (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; + } + } + } + } + + private static double hypot(double a, double b) { + + // sqrt(a^2 + b^2) without under/overflow. + + double r; + if (Math.abs(a) > Math.abs(b)) { + r = b / a; + r = Math.abs(a) * Math.sqrt(1 + r * r); + } else if (b != 0) { + r = a / b; + r = Math.abs(b) * Math.sqrt(1 + r * r); + } else { + r = 0.0; + } + return r; + } + + // Nonsymmetric reduction to Hessenberg form. + + /* + private void orthes() { + + // This is derived from the Algol procedures orthes and ortran, + // by Martin and Wilkinson, Handbook for Auto. Comp., + // Vol.ii-Linear Algebra, and the corresponding + // Fortran subroutines in EISPACK. + + int low = 0; + int high = n - 1; + + for (int m = low + 1; m <= high - 1; m++) { + + // Scale column. + + double scale = 0.0; + for (int i = m; i <= high; i++) { + scale = scale + Math.abs(H[i][m - 1]); + } + if (scale != 0.0) { + + // Compute Householder transformation. + + double h = 0.0; + for (int i = high; i >= m; i--) { + ort[i] = H[i][m - 1] / scale; + h += ort[i] * ort[i]; + } + double g = Math.sqrt(h); + if (ort[m] > 0) { + g = -g; + } + h = h - ort[m] * g; + ort[m] = ort[m] - g; + + // Apply Householder similarity transformation + // H = (I-u*u'/h)*H*(I-u*u')/h) + + for (int j = m; j < n; j++) { + double f = 0.0; + for (int i = high; i >= m; i--) { + f += ort[i] * H[i][j]; + } + f = f / h; + for (int i = m; i <= high; i++) { + H[i][j] -= f * ort[i]; + } + } + + for (int i = 0; i <= high; i++) { + double f = 0.0; + for (int j = high; j >= m; j--) { + f += ort[j] * H[i][j]; + } + f = f / h; + for (int j = m; j <= high; j++) { + H[i][j] -= f * ort[j]; + } + } + ort[m] = scale * ort[m]; + H[m][m - 1] = scale * g; + } + } + + // Accumulate transformations (Algol's ortran). + + for (int i = 0; i < n; i++) { + for (int j = 0; j < n; j++) { + V[i][j] = (i == j ? 1.0 : 0.0); + } + } + + for (int m = high - 1; m >= low + 1; m--) { + if (H[m][m - 1] != 0.0) { + for (int i = m + 1; i <= high; i++) { + ort[i] = H[i][m - 1]; + } + for (int j = m; j <= high; j++) { + double g = 0.0; + for (int i = m; i <= high; i++) { + g += ort[i] * V[i][j]; + } + // Double division avoids possible underflow + g = (g / ort[m]) / H[m][m - 1]; + for (int i = m; i <= high; i++) { + V[i][j] += g * ort[i]; + } + } + } + } + } + + // Complex scalar division. + + private transient double cdivr, cdivi; + + private void cdiv(double xr, double xi, double yr, double yi) { + double r, d; + if (Math.abs(yr) > Math.abs(yi)) { + r = yi / yr; + d = yr + r * yi; + cdivr = (xr + r * xi) / d; + cdivi = (xi - r * xr) / d; + } else { + r = yr / yi; + d = yi + r * yr; + cdivr = (r * xr + xi) / d; + cdivi = (r * xi - xr) / d; + } + } + + // Nonsymmetric reduction from Hessenberg to real Schur form. + + private void hqr2() { + + // This is derived from the Algol procedure hqr2, + // by Martin and Wilkinson, Handbook for Auto. Comp., + // Vol.ii-Linear Algebra, and the corresponding + // Fortran subroutine in EISPACK. + + // Initialize + + int nn = this.n; + int n = nn - 1; + int low = 0; + int high = nn - 1; + double eps = Math.pow(2.0, -52.0); + double exshift = 0.0; + double p = 0, q = 0, r = 0, s = 0, z = 0, t, w, x, y; + + // Store roots isolated by balanc and compute matrix norm + + double norm = 0.0; + for (int i = 0; i < nn; i++) { + if (i < low || i > high) { + d[i] = H[i][i]; + e[i] = 0.0; + } + for (int j = Math.max(i - 1, 0); j < nn; j++) { + norm = norm + Math.abs(H[i][j]); + } + } + + // Outer loop over eigenvalue index + + int iter = 0; + while (n >= low) { + + // Look for single small sub-diagonal element + + int l = n; + while (l > low) { + s = Math.abs(H[l - 1][l - 1]) + Math.abs(H[l][l]); + if (s == 0.0) { + s = norm; + } + if (Math.abs(H[l][l - 1]) < eps * s) { + break; + } + l--; + } + + // Check for convergence + // One root found + + if (l == n) { + H[n][n] = H[n][n] + exshift; + d[n] = H[n][n]; + e[n] = 0.0; + n--; + iter = 0; + + // Two roots found + + } else if (l == n - 1) { + w = H[n][n - 1] * H[n - 1][n]; + p = (H[n - 1][n - 1] - H[n][n]) / 2.0; + q = p * p + w; + z = Math.sqrt(Math.abs(q)); + H[n][n] = H[n][n] + exshift; + H[n - 1][n - 1] = H[n - 1][n - 1] + exshift; + x = H[n][n]; + + // Real pair + + if (q >= 0) { + if (p >= 0) { + z = p + z; + } else { + z = p - z; + } + d[n - 1] = x + z; + d[n] = d[n - 1]; + if (z != 0.0) { + d[n] = x - w / z; + } + e[n - 1] = 0.0; + e[n] = 0.0; + x = H[n][n - 1]; + s = Math.abs(x) + Math.abs(z); + p = x / s; + q = z / s; + r = Math.sqrt(p * p + q * q); + p = p / r; + q = q / r; + + // Row modification + + for (int j = n - 1; j < nn; j++) { + z = H[n - 1][j]; + H[n - 1][j] = q * z + p * H[n][j]; + H[n][j] = q * H[n][j] - p * z; + } + + // Column modification + + for (int i = 0; i <= n; i++) { + z = H[i][n - 1]; + H[i][n - 1] = q * z + p * H[i][n]; + H[i][n] = q * H[i][n] - p * z; + } + + // Accumulate transformations + + for (int i = low; i <= high; i++) { + z = V[i][n - 1]; + V[i][n - 1] = q * z + p * V[i][n]; + V[i][n] = q * V[i][n] - p * z; + } + + // Complex pair + + } else { + d[n - 1] = x + p; + d[n] = x + p; + e[n - 1] = z; + e[n] = -z; + } + n = n - 2; + iter = 0; + + // No convergence yet + + } else { + + // Form shift + + x = H[n][n]; + y = 0.0; + w = 0.0; + if (l < n) { + y = H[n - 1][n - 1]; + w = H[n][n - 1] * H[n - 1][n]; + } + + // Wilkinson's original ad hoc shift + + if (iter == 10) { + exshift += x; + for (int i = low; i <= n; i++) { + H[i][i] -= x; + } + s = Math.abs(H[n][n - 1]) + Math.abs(H[n - 1][n - 2]); + x = y = 0.75 * s; + w = -0.4375 * s * s; + } + + // MATLAB's new ad hoc shift + + if (iter == 30) { + s = (y - x) / 2.0; + s = s * s + w; + if (s > 0) { + s = Math.sqrt(s); + if (y < x) { + s = -s; + } + s = x - w / ((y - x) / 2.0 + s); + for (int i = low; i <= n; i++) { + H[i][i] -= s; + } + exshift += s; + x = y = w = 0.964; + } + } + + iter = iter + 1; // (Could check iteration count here.) + + // Look for two consecutive small sub-diagonal elements + + int m = n - 2; + while (m >= l) { + z = H[m][m]; + r = x - z; + s = y - z; + p = (r * s - w) / H[m + 1][m] + H[m][m + 1]; + q = H[m + 1][m + 1] - z - r - s; + r = H[m + 2][m + 1]; + s = Math.abs(p) + Math.abs(q) + Math.abs(r); + p = p / s; + q = q / s; + r = r / s; + if (m == l) { + break; + } + if (Math.abs(H[m][m - 1]) * (Math.abs(q) + Math.abs(r)) < eps + * (Math.abs(p) * (Math.abs(H[m - 1][m - 1]) + Math.abs(z) + Math + .abs(H[m + 1][m + 1])))) { + break; + } + m--; + } + + for (int i = m + 2; i <= n; i++) { + H[i][i - 2] = 0.0; + if (i > m + 2) { + H[i][i - 3] = 0.0; + } + } + + // Double QR step involving rows l:n and columns m:n + + for (int k = m; k <= n - 1; k++) { + boolean notlast = (k != n - 1); + if (k != m) { + p = H[k][k - 1]; + q = H[k + 1][k - 1]; + r = (notlast ? H[k + 2][k - 1] : 0.0); + x = Math.abs(p) + Math.abs(q) + Math.abs(r); + if (x != 0.0) { + p = p / x; + q = q / x; + r = r / x; + } + } + if (x == 0.0) { + break; + } + s = Math.sqrt(p * p + q * q + r * r); + if (p < 0) { + s = -s; + } + if (s != 0) { + if (k != m) { + H[k][k - 1] = -s * x; + } else if (l != m) { + H[k][k - 1] = -H[k][k - 1]; + } + p = p + s; + x = p / s; + y = q / s; + z = r / s; + q = q / p; + r = r / p; + + // Row modification + + for (int j = k; j < nn; j++) { + p = H[k][j] + q * H[k + 1][j]; + if (notlast) { + p = p + r * H[k + 2][j]; + H[k + 2][j] = H[k + 2][j] - p * z; + } + H[k][j] = H[k][j] - p * x; + H[k + 1][j] = H[k + 1][j] - p * y; + } + + // Column modification + + for (int i = 0; i <= Math.min(n, k + 3); i++) { + p = x * H[i][k] + y * H[i][k + 1]; + if (notlast) { + p = p + z * H[i][k + 2]; + H[i][k + 2] = H[i][k + 2] - p * r; + } + H[i][k] = H[i][k] - p; + H[i][k + 1] = H[i][k + 1] - p * q; + } + + // Accumulate transformations + + for (int i = low; i <= high; i++) { + p = x * V[i][k] + y * V[i][k + 1]; + if (notlast) { + p = p + z * V[i][k + 2]; + V[i][k + 2] = V[i][k + 2] - p * r; + } + V[i][k] = V[i][k] - p; + V[i][k + 1] = V[i][k + 1] - p * q; + } + } // (s != 0) + } // k loop + } // check convergence + } // while (n >= low) + + // Backsubstitute to find vectors of upper triangular form + + if (norm == 0.0) { + return; + } + + for (n = nn - 1; n >= 0; n--) { + p = d[n]; + q = e[n]; + + // Real vector + + if (q == 0) { + int l = n; + H[n][n] = 1.0; + for (int i = n - 1; i >= 0; i--) { + w = H[i][i] - p; + r = 0.0; + for (int j = l; j <= n; j++) { + r = r + H[i][j] * H[j][n]; + } + if (e[i] < 0.0) { + z = w; + s = r; + } else { + l = i; + if (e[i] == 0.0) { + if (w != 0.0) { + H[i][n] = -r / w; + } else { + H[i][n] = -r / (eps * norm); + } + + // Solve real equations + + } else { + x = H[i][i + 1]; + y = H[i + 1][i]; + q = (d[i] - p) * (d[i] - p) + e[i] * e[i]; + t = (x * s - z * r) / q; + H[i][n] = t; + if (Math.abs(x) > Math.abs(z)) { + H[i + 1][n] = (-r - w * t) / x; + } else { + H[i + 1][n] = (-s - y * t) / z; + } + } + + // Overflow control + + t = Math.abs(H[i][n]); + if ((eps * t) * t > 1) { + for (int j = i; j <= n; j++) { + H[j][n] = H[j][n] / t; + } + } + } + } + + // Complex vector + + } else if (q < 0) { + int l = n - 1; + + // Last vector component imaginary so matrix is triangular + + if (Math.abs(H[n][n - 1]) > Math.abs(H[n - 1][n])) { + H[n - 1][n - 1] = q / H[n][n - 1]; + H[n - 1][n] = -(H[n][n] - p) / H[n][n - 1]; + } else { + cdiv(0.0, -H[n - 1][n], H[n - 1][n - 1] - p, q); + H[n - 1][n - 1] = cdivr; + H[n - 1][n] = cdivi; + } + H[n][n - 1] = 0.0; + H[n][n] = 1.0; + for (int i = n - 2; i >= 0; i--) { + double ra, sa, vr, vi; + ra = 0.0; + sa = 0.0; + for (int j = l; j <= n; j++) { + ra = ra + H[i][j] * H[j][n - 1]; + sa = sa + H[i][j] * H[j][n]; + } + w = H[i][i] - p; + + if (e[i] < 0.0) { + z = w; + r = ra; + s = sa; + } else { + l = i; + if (e[i] == 0) { + cdiv(-ra, -sa, w, q); + H[i][n - 1] = cdivr; + H[i][n] = cdivi; + } else { + + // Solve complex equations + + x = H[i][i + 1]; + y = H[i + 1][i]; + vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q; + vi = (d[i] - p) * 2.0 * q; + if (vr == 0.0 & vi == 0.0) { + vr = eps + * norm + * (Math.abs(w) + Math.abs(q) + Math.abs(x) + Math.abs(y) + Math + .abs(z)); + } + cdiv(x * r - z * ra + q * sa, x * s - z * sa - q * ra, vr, vi); + H[i][n - 1] = cdivr; + H[i][n] = cdivi; + if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) { + H[i + 1][n - 1] = (-ra - w * H[i][n - 1] + q * H[i][n]) / x; + H[i + 1][n] = (-sa - w * H[i][n] - q * H[i][n - 1]) / x; + } else { + cdiv(-r - y * H[i][n - 1], -s - y * H[i][n], z, q); + H[i + 1][n - 1] = cdivr; + H[i + 1][n] = cdivi; + } + } + + // Overflow control + + t = Math.max(Math.abs(H[i][n - 1]), Math.abs(H[i][n])); + if ((eps * t) * t > 1) { + for (int j = i; j <= n; j++) { + H[j][n - 1] = H[j][n - 1] / t; + H[j][n] = H[j][n] / t; + } + } + } + } + } + } + + // Vectors of isolated roots + + for (int i = 0; i < nn; i++) { + if (i < low || i > high) { + for (int j = i; j < nn; j++) { + V[i][j] = H[i][j]; + } + } + } + + // Back transformation to get eigenvectors of original matrix + + for (int j = nn - 1; j >= low; j--) { + for (int i = low; i <= high; i++) { + z = 0.0; + for (int k = low; k <= Math.min(j, high); k++) { + z = z + V[i][k] * H[k][j]; + } + V[i][j] = z; + } + } + } + */ + + +}