-/* $RCSfile$\r
- * $Author: egonw $\r
- * $Date: 2005-11-10 09:52:44f -0600 (Thu, 10 Nov 2005) $\r
- * $Revision: 4255 $\r
- *\r
- * Copyright (C) 2003-2005 Miguel, Jmol Development, www.jmol.org\r
- *\r
- * Contact: jmol-developers@lists.sf.net\r
- *\r
- * This library is free software; you can redistribute it and/or\r
- * modify it under the terms of the GNU Lesser General Public\r
- * License as published by the Free Software Foundation; either\r
- * version 2.1 of the License, or (at your option) any later version.\r
- *\r
- * This library is distributed in the hope that it will be useful,\r
- * but WITHOUT ANY WARRANTY; without even the implied warranty of\r
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU\r
- * Lesser General Public License for more details.\r
- *\r
- * You should have received a copy of the GNU Lesser General Public\r
- * License along with this library; if not, write to the Free Software\r
- * Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.\r
- */\r
-\r
-package javajs.util;\r
-\r
-import javajs.api.EigenInterface;\r
-\r
-\r
-/**\r
- * Eigenvalues and eigenvectors of a real matrix.\r
- * See javajs.api.EigenInterface() as well.\r
- * \r
- * adapted by Bob Hanson from http://math.nist.gov/javanumerics/jama/ (public\r
- * domain); adding quaternion superimposition capability; removing\r
- * nonsymmetric reduction to Hessenberg form, which we do not need in Jmol.\r
- * \r
- * Output is as a set of double[n] columns, but for the EigenInterface\r
- * we return them as V3[3] and float[3] (or double[3]) values.\r
- * \r
- * Eigenvalues and eigenvectors are sorted from smallest to largest eigenvalue.\r
- * \r
- * <P>\r
- * If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal\r
- * and the eigenvector matrix V is orthogonal. I.e. A =\r
- * V.times(D.times(V.transpose())) and V.times(V.transpose()) equals the\r
- * identity matrix.\r
- * <P>\r
- * If A is not symmetric, then the eigenvalue matrix D is block diagonal with\r
- * the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda +\r
- * i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The columns of V represent\r
- * the eigenvectors in the sense that A*V = V*D, i.e. A.times(V) equals\r
- * V.times(D). The matrix V may be badly conditioned, or even singular, so the\r
- * validity of the equation A = V*D*inverse(V) depends upon V.cond().\r
- **/\r
-\r
-public class Eigen implements EigenInterface {\r
-\r
- /* ------------------------\r
- Public Methods\r
- * ------------------------ */\r
-\r
- public Eigen() {}\r
- \r
- public Eigen set(int n) {\r
- this.n = n;\r
- V = new double[n][n];\r
- d = new double[n];\r
- e = new double[n];\r
- return this;\r
- }\r
-\r
- @Override\r
- public Eigen setM(double[][] m) {\r
- set(m.length);\r
- calc(m);\r
- return this;\r
- }\r
-\r
- /**\r
- * return values sorted from smallest to largest value.\r
- */\r
- @Override\r
- public double[] getEigenvalues() {\r
- return d;\r
- }\r
-\r
- /**\r
- * Specifically for 3x3 systems, returns eigenVectors as V3[3]\r
- * and values as float[3]; sorted from smallest to largest value.\r
- * \r
- * @param eigenVectors returned vectors\r
- * @param eigenValues returned values\r
- * \r
- */\r
- @Override\r
- public void fillFloatArrays(V3[] eigenVectors, float[] eigenValues) {\r
- for (int i = 0; i < 3; i++) {\r
- if (eigenVectors != null) {\r
- if (eigenVectors[i] == null)\r
- eigenVectors[i] = new V3();\r
- eigenVectors[i].set((float) V[0][i], (float) V[1][i], (float) V[2][i]);\r
- }\r
- if (eigenValues != null)\r
- eigenValues[i] = (float) d[i];\r
- }\r
- }\r
-\r
- /**\r
- * Transpose V and turn into floats; sorted from smallest to largest value.\r
- * \r
- * @return ROWS of eigenvectors f[0], f[1], f[2], etc.\r
- */\r
- @Override\r
- public float[][] getEigenvectorsFloatTransposed() {\r
- float[][] f = new float[n][n];\r
- for (int i = n; --i >= 0;)\r
- for (int j = n; --j >= 0;)\r
- f[j][i] = (float) V[i][j];\r
- return f;\r
- }\r
-\r
-\r
- /**\r
- * Check for symmetry, then construct the eigenvalue decomposition\r
- * \r
- * @param A\r
- * Square matrix\r
- */\r
-\r
- public void calc(double[][] A) {\r
-\r
- /* Jmol only has need of symmetric solutions \r
- * \r
- issymmetric = true;\r
- \r
- for (int j = 0; (j < n) & issymmetric; j++) {\r
- for (int i = 0; (i < n) & issymmetric; i++) {\r
- issymmetric = (A[i][j] == A[j][i]);\r
- }\r
- }\r
-\r
- if (issymmetric) {\r
- */\r
- for (int i = 0; i < n; i++) {\r
- for (int j = 0; j < n; j++) {\r
- V[i][j] = A[i][j];\r
- }\r
- }\r
-\r
- // Tridiagonalize.\r
- tred2();\r
-\r
- // Diagonalize.\r
- tql2();\r
- /*\r
- } else {\r
- H = new double[n][n];\r
- ort = new double[n];\r
-\r
- for (int j = 0; j < n; j++) {\r
- for (int i = 0; i < n; i++) {\r
- H[i][j] = A[i][j];\r
- }\r
- }\r
-\r
- // Reduce to Hessenberg form.\r
- orthes();\r
-\r
- // Reduce Hessenberg to real Schur form.\r
- hqr2();\r
- }\r
- */\r
-\r
- }\r
-\r
- /**\r
- * Return the real parts of the eigenvalues\r
- * \r
- * @return real(diag(D))\r
- */\r
-\r
- public double[] getRealEigenvalues() {\r
- return d;\r
- }\r
-\r
- /**\r
- * Return the imaginary parts of the eigenvalues\r
- * \r
- * @return imag(diag(D))\r
- */\r
-\r
- public double[] getImagEigenvalues() {\r
- return e;\r
- }\r
-\r
- /* ------------------------\r
- Class variables\r
- * ------------------------ */\r
-\r
- /**\r
- * Row and column dimension (square matrix).\r
- * \r
- * @serial matrix dimension.\r
- */\r
- private int n = 3;\r
-\r
- /**\r
- * Symmetry flag.\r
- * \r
- * @serial internal symmetry flag.\r
- */\r
- //private boolean issymmetric = true;\r
-\r
- /**\r
- * Arrays for internal storage of eigenvalues.\r
- * \r
- * @serial internal storage of eigenvalues.\r
- */\r
- private double[] d, e;\r
-\r
- /**\r
- * Array for internal storage of eigenvectors.\r
- * \r
- * @serial internal storage of eigenvectors.\r
- */\r
- private double[][] V;\r
-\r
- /**\r
- * Array for internal storage of nonsymmetric Hessenberg form.\r
- * \r
- * @serial internal storage of nonsymmetric Hessenberg form.\r
- */\r
- //private double[][] H;\r
-\r
- /**\r
- * Working storage for nonsymmetric algorithm.\r
- * \r
- * @serial working storage for nonsymmetric algorithm.\r
- */\r
- //private double[] ort;\r
-\r
- /* ------------------------\r
- Private Methods\r
- * ------------------------ */\r
-\r
- // Symmetric Householder reduction to tridiagonal form.\r
-\r
- private void tred2() {\r
-\r
- // This is derived from the Algol procedures tred2 by\r
- // Bowdler, Martin, Reinsch, and Wilkinson, Handbook for\r
- // Auto. Comp., Vol.ii-Linear Algebra, and the corresponding\r
- // Fortran subroutine in EISPACK.\r
-\r
- for (int j = 0; j < n; j++) {\r
- d[j] = V[n - 1][j];\r
- }\r
-\r
- // Householder reduction to tridiagonal form.\r
-\r
- for (int i = n - 1; i > 0; i--) {\r
-\r
- // Scale to avoid under/overflow.\r
-\r
- double scale = 0.0;\r
- double h = 0.0;\r
- for (int k = 0; k < i; k++) {\r
- scale = scale + Math.abs(d[k]);\r
- }\r
- if (scale == 0.0) {\r
- e[i] = d[i - 1];\r
- for (int j = 0; j < i; j++) {\r
- d[j] = V[i - 1][j];\r
- V[i][j] = 0.0;\r
- V[j][i] = 0.0;\r
- }\r
- } else {\r
-\r
- // Generate Householder vector.\r
-\r
- for (int k = 0; k < i; k++) {\r
- d[k] /= scale;\r
- h += d[k] * d[k];\r
- }\r
- double f = d[i - 1];\r
- double g = Math.sqrt(h);\r
- if (f > 0) {\r
- g = -g;\r
- }\r
- e[i] = scale * g;\r
- h = h - f * g;\r
- d[i - 1] = f - g;\r
- for (int j = 0; j < i; j++) {\r
- e[j] = 0.0;\r
- }\r
-\r
- // Apply similarity transformation to remaining columns.\r
-\r
- for (int j = 0; j < i; j++) {\r
- f = d[j];\r
- V[j][i] = f;\r
- g = e[j] + V[j][j] * f;\r
- for (int k = j + 1; k <= i - 1; k++) {\r
- g += V[k][j] * d[k];\r
- e[k] += V[k][j] * f;\r
- }\r
- e[j] = g;\r
- }\r
- f = 0.0;\r
- for (int j = 0; j < i; j++) {\r
- e[j] /= h;\r
- f += e[j] * d[j];\r
- }\r
- double hh = f / (h + h);\r
- for (int j = 0; j < i; j++) {\r
- e[j] -= hh * d[j];\r
- }\r
- for (int j = 0; j < i; j++) {\r
- f = d[j];\r
- g = e[j];\r
- for (int k = j; k <= i - 1; k++) {\r
- V[k][j] -= (f * e[k] + g * d[k]);\r
- }\r
- d[j] = V[i - 1][j];\r
- V[i][j] = 0.0;\r
- }\r
- }\r
- d[i] = h;\r
- }\r
-\r
- // Accumulate transformations.\r
-\r
- for (int i = 0; i < n - 1; i++) {\r
- V[n - 1][i] = V[i][i];\r
- V[i][i] = 1.0;\r
- double h = d[i + 1];\r
- if (h != 0.0) {\r
- for (int k = 0; k <= i; k++) {\r
- d[k] = V[k][i + 1] / h;\r
- }\r
- for (int j = 0; j <= i; j++) {\r
- double g = 0.0;\r
- for (int k = 0; k <= i; k++) {\r
- g += V[k][i + 1] * V[k][j];\r
- }\r
- for (int k = 0; k <= i; k++) {\r
- V[k][j] -= g * d[k];\r
- }\r
- }\r
- }\r
- for (int k = 0; k <= i; k++) {\r
- V[k][i + 1] = 0.0;\r
- }\r
- }\r
- for (int j = 0; j < n; j++) {\r
- d[j] = V[n - 1][j];\r
- V[n - 1][j] = 0.0;\r
- }\r
- V[n - 1][n - 1] = 1.0;\r
- e[0] = 0.0;\r
- }\r
-\r
- // Symmetric tridiagonal QL algorithm.\r
-\r
- private void tql2() {\r
-\r
- // This is derived from the Algol procedures tql2, by\r
- // Bowdler, Martin, Reinsch, and Wilkinson, Handbook for\r
- // Auto. Comp., Vol.ii-Linear Algebra, and the corresponding\r
- // Fortran subroutine in EISPACK.\r
-\r
- for (int i = 1; i < n; i++) {\r
- e[i - 1] = e[i];\r
- }\r
- e[n - 1] = 0.0;\r
-\r
- double f = 0.0;\r
- double tst1 = 0.0;\r
- double eps = Math.pow(2.0, -52.0);\r
- for (int l = 0; l < n; l++) {\r
-\r
- // Find small subdiagonal element\r
-\r
- tst1 = Math.max(tst1, Math.abs(d[l]) + Math.abs(e[l]));\r
- int m = l;\r
- while (m < n) {\r
- if (Math.abs(e[m]) <= eps * tst1) {\r
- break;\r
- }\r
- m++;\r
- }\r
-\r
- // If m == l, d[l] is an eigenvalue,\r
- // otherwise, iterate.\r
-\r
- if (m > l) {\r
- int iter = 0;\r
- do {\r
- iter = iter + 1; // (Could check iteration count here.)\r
-\r
- // Compute implicit shift\r
-\r
- double g = d[l];\r
- double p = (d[l + 1] - g) / (2.0 * e[l]);\r
- double r = hypot(p, 1.0);\r
- if (p < 0) {\r
- r = -r;\r
- }\r
- d[l] = e[l] / (p + r);\r
- d[l + 1] = e[l] * (p + r);\r
- double dl1 = d[l + 1];\r
- double h = g - d[l];\r
- for (int i = l + 2; i < n; i++) {\r
- d[i] -= h;\r
- }\r
- f = f + h;\r
-\r
- // Implicit QL transformation.\r
-\r
- p = d[m];\r
- double c = 1.0;\r
- double c2 = c;\r
- double c3 = c;\r
- double el1 = e[l + 1];\r
- double s = 0.0;\r
- double s2 = 0.0;\r
- for (int i = m - 1; i >= l; i--) {\r
- c3 = c2;\r
- c2 = c;\r
- s2 = s;\r
- g = c * e[i];\r
- h = c * p;\r
- r = hypot(p, e[i]);\r
- e[i + 1] = s * r;\r
- s = e[i] / r;\r
- c = p / r;\r
- p = c * d[i] - s * g;\r
- d[i + 1] = h + s * (c * g + s * d[i]);\r
-\r
- // Accumulate transformation.\r
-\r
- for (int k = 0; k < n; k++) {\r
- h = V[k][i + 1];\r
- V[k][i + 1] = s * V[k][i] + c * h;\r
- V[k][i] = c * V[k][i] - s * h;\r
- }\r
- }\r
- p = -s * s2 * c3 * el1 * e[l] / dl1;\r
- e[l] = s * p;\r
- d[l] = c * p;\r
-\r
- // Check for convergence.\r
-\r
- } while (Math.abs(e[l]) > eps * tst1);\r
- }\r
- d[l] = d[l] + f;\r
- e[l] = 0.0;\r
- }\r
-\r
- // Sort eigenvalues and corresponding vectors.\r
-\r
- for (int i = 0; i < n - 1; i++) {\r
- int k = i;\r
- double p = d[i];\r
- for (int j = i + 1; j < n; j++) {\r
- if (d[j] < p) {\r
- k = j;\r
- p = d[j];\r
- }\r
- }\r
- if (k != i) {\r
- d[k] = d[i];\r
- d[i] = p;\r
- for (int j = 0; j < n; j++) {\r
- p = V[j][i];\r
- V[j][i] = V[j][k];\r
- V[j][k] = p;\r
- }\r
- }\r
- }\r
- }\r
-\r
- private static double hypot(double a, double b) {\r
-\r
- // sqrt(a^2 + b^2) without under/overflow. \r
-\r
- double r;\r
- if (Math.abs(a) > Math.abs(b)) {\r
- r = b / a;\r
- r = Math.abs(a) * Math.sqrt(1 + r * r);\r
- } else if (b != 0) {\r
- r = a / b;\r
- r = Math.abs(b) * Math.sqrt(1 + r * r);\r
- } else {\r
- r = 0.0;\r
- }\r
- return r;\r
- }\r
-\r
- // Nonsymmetric reduction to Hessenberg form.\r
-\r
- /*\r
- private void orthes() {\r
-\r
- // This is derived from the Algol procedures orthes and ortran,\r
- // by Martin and Wilkinson, Handbook for Auto. Comp.,\r
- // Vol.ii-Linear Algebra, and the corresponding\r
- // Fortran subroutines in EISPACK.\r
-\r
- int low = 0;\r
- int high = n - 1;\r
-\r
- for (int m = low + 1; m <= high - 1; m++) {\r
-\r
- // Scale column.\r
-\r
- double scale = 0.0;\r
- for (int i = m; i <= high; i++) {\r
- scale = scale + Math.abs(H[i][m - 1]);\r
- }\r
- if (scale != 0.0) {\r
-\r
- // Compute Householder transformation.\r
-\r
- double h = 0.0;\r
- for (int i = high; i >= m; i--) {\r
- ort[i] = H[i][m - 1] / scale;\r
- h += ort[i] * ort[i];\r
- }\r
- double g = Math.sqrt(h);\r
- if (ort[m] > 0) {\r
- g = -g;\r
- }\r
- h = h - ort[m] * g;\r
- ort[m] = ort[m] - g;\r
-\r
- // Apply Householder similarity transformation\r
- // H = (I-u*u'/h)*H*(I-u*u')/h)\r
-\r
- for (int j = m; j < n; j++) {\r
- double f = 0.0;\r
- for (int i = high; i >= m; i--) {\r
- f += ort[i] * H[i][j];\r
- }\r
- f = f / h;\r
- for (int i = m; i <= high; i++) {\r
- H[i][j] -= f * ort[i];\r
- }\r
- }\r
-\r
- for (int i = 0; i <= high; i++) {\r
- double f = 0.0;\r
- for (int j = high; j >= m; j--) {\r
- f += ort[j] * H[i][j];\r
- }\r
- f = f / h;\r
- for (int j = m; j <= high; j++) {\r
- H[i][j] -= f * ort[j];\r
- }\r
- }\r
- ort[m] = scale * ort[m];\r
- H[m][m - 1] = scale * g;\r
- }\r
- }\r
-\r
- // Accumulate transformations (Algol's ortran).\r
-\r
- for (int i = 0; i < n; i++) {\r
- for (int j = 0; j < n; j++) {\r
- V[i][j] = (i == j ? 1.0 : 0.0);\r
- }\r
- }\r
-\r
- for (int m = high - 1; m >= low + 1; m--) {\r
- if (H[m][m - 1] != 0.0) {\r
- for (int i = m + 1; i <= high; i++) {\r
- ort[i] = H[i][m - 1];\r
- }\r
- for (int j = m; j <= high; j++) {\r
- double g = 0.0;\r
- for (int i = m; i <= high; i++) {\r
- g += ort[i] * V[i][j];\r
- }\r
- // Double division avoids possible underflow\r
- g = (g / ort[m]) / H[m][m - 1];\r
- for (int i = m; i <= high; i++) {\r
- V[i][j] += g * ort[i];\r
- }\r
- }\r
- }\r
- }\r
- }\r
-\r
- // Complex scalar division.\r
-\r
- private transient double cdivr, cdivi;\r
-\r
- private void cdiv(double xr, double xi, double yr, double yi) {\r
- double r, d;\r
- if (Math.abs(yr) > Math.abs(yi)) {\r
- r = yi / yr;\r
- d = yr + r * yi;\r
- cdivr = (xr + r * xi) / d;\r
- cdivi = (xi - r * xr) / d;\r
- } else {\r
- r = yr / yi;\r
- d = yi + r * yr;\r
- cdivr = (r * xr + xi) / d;\r
- cdivi = (r * xi - xr) / d;\r
- }\r
- }\r
-\r
- // Nonsymmetric reduction from Hessenberg to real Schur form.\r
-\r
- private void hqr2() {\r
-\r
- // This is derived from the Algol procedure hqr2,\r
- // by Martin and Wilkinson, Handbook for Auto. Comp.,\r
- // Vol.ii-Linear Algebra, and the corresponding\r
- // Fortran subroutine in EISPACK.\r
-\r
- // Initialize\r
-\r
- int nn = this.n;\r
- int n = nn - 1;\r
- int low = 0;\r
- int high = nn - 1;\r
- double eps = Math.pow(2.0, -52.0);\r
- double exshift = 0.0;\r
- double p = 0, q = 0, r = 0, s = 0, z = 0, t, w, x, y;\r
-\r
- // Store roots isolated by balanc and compute matrix norm\r
-\r
- double norm = 0.0;\r
- for (int i = 0; i < nn; i++) {\r
- if (i < low || i > high) {\r
- d[i] = H[i][i];\r
- e[i] = 0.0;\r
- }\r
- for (int j = Math.max(i - 1, 0); j < nn; j++) {\r
- norm = norm + Math.abs(H[i][j]);\r
- }\r
- }\r
-\r
- // Outer loop over eigenvalue index\r
-\r
- int iter = 0;\r
- while (n >= low) {\r
-\r
- // Look for single small sub-diagonal element\r
-\r
- int l = n;\r
- while (l > low) {\r
- s = Math.abs(H[l - 1][l - 1]) + Math.abs(H[l][l]);\r
- if (s == 0.0) {\r
- s = norm;\r
- }\r
- if (Math.abs(H[l][l - 1]) < eps * s) {\r
- break;\r
- }\r
- l--;\r
- }\r
-\r
- // Check for convergence\r
- // One root found\r
-\r
- if (l == n) {\r
- H[n][n] = H[n][n] + exshift;\r
- d[n] = H[n][n];\r
- e[n] = 0.0;\r
- n--;\r
- iter = 0;\r
-\r
- // Two roots found\r
-\r
- } else if (l == n - 1) {\r
- w = H[n][n - 1] * H[n - 1][n];\r
- p = (H[n - 1][n - 1] - H[n][n]) / 2.0;\r
- q = p * p + w;\r
- z = Math.sqrt(Math.abs(q));\r
- H[n][n] = H[n][n] + exshift;\r
- H[n - 1][n - 1] = H[n - 1][n - 1] + exshift;\r
- x = H[n][n];\r
-\r
- // Real pair\r
-\r
- if (q >= 0) {\r
- if (p >= 0) {\r
- z = p + z;\r
- } else {\r
- z = p - z;\r
- }\r
- d[n - 1] = x + z;\r
- d[n] = d[n - 1];\r
- if (z != 0.0) {\r
- d[n] = x - w / z;\r
- }\r
- e[n - 1] = 0.0;\r
- e[n] = 0.0;\r
- x = H[n][n - 1];\r
- s = Math.abs(x) + Math.abs(z);\r
- p = x / s;\r
- q = z / s;\r
- r = Math.sqrt(p * p + q * q);\r
- p = p / r;\r
- q = q / r;\r
-\r
- // Row modification\r
-\r
- for (int j = n - 1; j < nn; j++) {\r
- z = H[n - 1][j];\r
- H[n - 1][j] = q * z + p * H[n][j];\r
- H[n][j] = q * H[n][j] - p * z;\r
- }\r
-\r
- // Column modification\r
-\r
- for (int i = 0; i <= n; i++) {\r
- z = H[i][n - 1];\r
- H[i][n - 1] = q * z + p * H[i][n];\r
- H[i][n] = q * H[i][n] - p * z;\r
- }\r
-\r
- // Accumulate transformations\r
-\r
- for (int i = low; i <= high; i++) {\r
- z = V[i][n - 1];\r
- V[i][n - 1] = q * z + p * V[i][n];\r
- V[i][n] = q * V[i][n] - p * z;\r
- }\r
-\r
- // Complex pair\r
-\r
- } else {\r
- d[n - 1] = x + p;\r
- d[n] = x + p;\r
- e[n - 1] = z;\r
- e[n] = -z;\r
- }\r
- n = n - 2;\r
- iter = 0;\r
-\r
- // No convergence yet\r
-\r
- } else {\r
-\r
- // Form shift\r
-\r
- x = H[n][n];\r
- y = 0.0;\r
- w = 0.0;\r
- if (l < n) {\r
- y = H[n - 1][n - 1];\r
- w = H[n][n - 1] * H[n - 1][n];\r
- }\r
-\r
- // Wilkinson's original ad hoc shift\r
-\r
- if (iter == 10) {\r
- exshift += x;\r
- for (int i = low; i <= n; i++) {\r
- H[i][i] -= x;\r
- }\r
- s = Math.abs(H[n][n - 1]) + Math.abs(H[n - 1][n - 2]);\r
- x = y = 0.75 * s;\r
- w = -0.4375 * s * s;\r
- }\r
-\r
- // MATLAB's new ad hoc shift\r
-\r
- if (iter == 30) {\r
- s = (y - x) / 2.0;\r
- s = s * s + w;\r
- if (s > 0) {\r
- s = Math.sqrt(s);\r
- if (y < x) {\r
- s = -s;\r
- }\r
- s = x - w / ((y - x) / 2.0 + s);\r
- for (int i = low; i <= n; i++) {\r
- H[i][i] -= s;\r
- }\r
- exshift += s;\r
- x = y = w = 0.964;\r
- }\r
- }\r
-\r
- iter = iter + 1; // (Could check iteration count here.)\r
-\r
- // Look for two consecutive small sub-diagonal elements\r
-\r
- int m = n - 2;\r
- while (m >= l) {\r
- z = H[m][m];\r
- r = x - z;\r
- s = y - z;\r
- p = (r * s - w) / H[m + 1][m] + H[m][m + 1];\r
- q = H[m + 1][m + 1] - z - r - s;\r
- r = H[m + 2][m + 1];\r
- s = Math.abs(p) + Math.abs(q) + Math.abs(r);\r
- p = p / s;\r
- q = q / s;\r
- r = r / s;\r
- if (m == l) {\r
- break;\r
- }\r
- if (Math.abs(H[m][m - 1]) * (Math.abs(q) + Math.abs(r)) < eps\r
- * (Math.abs(p) * (Math.abs(H[m - 1][m - 1]) + Math.abs(z) + Math\r
- .abs(H[m + 1][m + 1])))) {\r
- break;\r
- }\r
- m--;\r
- }\r
-\r
- for (int i = m + 2; i <= n; i++) {\r
- H[i][i - 2] = 0.0;\r
- if (i > m + 2) {\r
- H[i][i - 3] = 0.0;\r
- }\r
- }\r
-\r
- // Double QR step involving rows l:n and columns m:n\r
-\r
- for (int k = m; k <= n - 1; k++) {\r
- boolean notlast = (k != n - 1);\r
- if (k != m) {\r
- p = H[k][k - 1];\r
- q = H[k + 1][k - 1];\r
- r = (notlast ? H[k + 2][k - 1] : 0.0);\r
- x = Math.abs(p) + Math.abs(q) + Math.abs(r);\r
- if (x != 0.0) {\r
- p = p / x;\r
- q = q / x;\r
- r = r / x;\r
- }\r
- }\r
- if (x == 0.0) {\r
- break;\r
- }\r
- s = Math.sqrt(p * p + q * q + r * r);\r
- if (p < 0) {\r
- s = -s;\r
- }\r
- if (s != 0) {\r
- if (k != m) {\r
- H[k][k - 1] = -s * x;\r
- } else if (l != m) {\r
- H[k][k - 1] = -H[k][k - 1];\r
- }\r
- p = p + s;\r
- x = p / s;\r
- y = q / s;\r
- z = r / s;\r
- q = q / p;\r
- r = r / p;\r
-\r
- // Row modification\r
-\r
- for (int j = k; j < nn; j++) {\r
- p = H[k][j] + q * H[k + 1][j];\r
- if (notlast) {\r
- p = p + r * H[k + 2][j];\r
- H[k + 2][j] = H[k + 2][j] - p * z;\r
- }\r
- H[k][j] = H[k][j] - p * x;\r
- H[k + 1][j] = H[k + 1][j] - p * y;\r
- }\r
-\r
- // Column modification\r
-\r
- for (int i = 0; i <= Math.min(n, k + 3); i++) {\r
- p = x * H[i][k] + y * H[i][k + 1];\r
- if (notlast) {\r
- p = p + z * H[i][k + 2];\r
- H[i][k + 2] = H[i][k + 2] - p * r;\r
- }\r
- H[i][k] = H[i][k] - p;\r
- H[i][k + 1] = H[i][k + 1] - p * q;\r
- }\r
-\r
- // Accumulate transformations\r
-\r
- for (int i = low; i <= high; i++) {\r
- p = x * V[i][k] + y * V[i][k + 1];\r
- if (notlast) {\r
- p = p + z * V[i][k + 2];\r
- V[i][k + 2] = V[i][k + 2] - p * r;\r
- }\r
- V[i][k] = V[i][k] - p;\r
- V[i][k + 1] = V[i][k + 1] - p * q;\r
- }\r
- } // (s != 0)\r
- } // k loop\r
- } // check convergence\r
- } // while (n >= low)\r
-\r
- // Backsubstitute to find vectors of upper triangular form\r
-\r
- if (norm == 0.0) {\r
- return;\r
- }\r
-\r
- for (n = nn - 1; n >= 0; n--) {\r
- p = d[n];\r
- q = e[n];\r
-\r
- // Real vector\r
-\r
- if (q == 0) {\r
- int l = n;\r
- H[n][n] = 1.0;\r
- for (int i = n - 1; i >= 0; i--) {\r
- w = H[i][i] - p;\r
- r = 0.0;\r
- for (int j = l; j <= n; j++) {\r
- r = r + H[i][j] * H[j][n];\r
- }\r
- if (e[i] < 0.0) {\r
- z = w;\r
- s = r;\r
- } else {\r
- l = i;\r
- if (e[i] == 0.0) {\r
- if (w != 0.0) {\r
- H[i][n] = -r / w;\r
- } else {\r
- H[i][n] = -r / (eps * norm);\r
- }\r
-\r
- // Solve real equations\r
-\r
- } else {\r
- x = H[i][i + 1];\r
- y = H[i + 1][i];\r
- q = (d[i] - p) * (d[i] - p) + e[i] * e[i];\r
- t = (x * s - z * r) / q;\r
- H[i][n] = t;\r
- if (Math.abs(x) > Math.abs(z)) {\r
- H[i + 1][n] = (-r - w * t) / x;\r
- } else {\r
- H[i + 1][n] = (-s - y * t) / z;\r
- }\r
- }\r
-\r
- // Overflow control\r
-\r
- t = Math.abs(H[i][n]);\r
- if ((eps * t) * t > 1) {\r
- for (int j = i; j <= n; j++) {\r
- H[j][n] = H[j][n] / t;\r
- }\r
- }\r
- }\r
- }\r
-\r
- // Complex vector\r
-\r
- } else if (q < 0) {\r
- int l = n - 1;\r
-\r
- // Last vector component imaginary so matrix is triangular\r
-\r
- if (Math.abs(H[n][n - 1]) > Math.abs(H[n - 1][n])) {\r
- H[n - 1][n - 1] = q / H[n][n - 1];\r
- H[n - 1][n] = -(H[n][n] - p) / H[n][n - 1];\r
- } else {\r
- cdiv(0.0, -H[n - 1][n], H[n - 1][n - 1] - p, q);\r
- H[n - 1][n - 1] = cdivr;\r
- H[n - 1][n] = cdivi;\r
- }\r
- H[n][n - 1] = 0.0;\r
- H[n][n] = 1.0;\r
- for (int i = n - 2; i >= 0; i--) {\r
- double ra, sa, vr, vi;\r
- ra = 0.0;\r
- sa = 0.0;\r
- for (int j = l; j <= n; j++) {\r
- ra = ra + H[i][j] * H[j][n - 1];\r
- sa = sa + H[i][j] * H[j][n];\r
- }\r
- w = H[i][i] - p;\r
-\r
- if (e[i] < 0.0) {\r
- z = w;\r
- r = ra;\r
- s = sa;\r
- } else {\r
- l = i;\r
- if (e[i] == 0) {\r
- cdiv(-ra, -sa, w, q);\r
- H[i][n - 1] = cdivr;\r
- H[i][n] = cdivi;\r
- } else {\r
-\r
- // Solve complex equations\r
-\r
- x = H[i][i + 1];\r
- y = H[i + 1][i];\r
- vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;\r
- vi = (d[i] - p) * 2.0 * q;\r
- if (vr == 0.0 & vi == 0.0) {\r
- vr = eps\r
- * norm\r
- * (Math.abs(w) + Math.abs(q) + Math.abs(x) + Math.abs(y) + Math\r
- .abs(z));\r
- }\r
- cdiv(x * r - z * ra + q * sa, x * s - z * sa - q * ra, vr, vi);\r
- H[i][n - 1] = cdivr;\r
- H[i][n] = cdivi;\r
- if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) {\r
- H[i + 1][n - 1] = (-ra - w * H[i][n - 1] + q * H[i][n]) / x;\r
- H[i + 1][n] = (-sa - w * H[i][n] - q * H[i][n - 1]) / x;\r
- } else {\r
- cdiv(-r - y * H[i][n - 1], -s - y * H[i][n], z, q);\r
- H[i + 1][n - 1] = cdivr;\r
- H[i + 1][n] = cdivi;\r
- }\r
- }\r
-\r
- // Overflow control\r
-\r
- t = Math.max(Math.abs(H[i][n - 1]), Math.abs(H[i][n]));\r
- if ((eps * t) * t > 1) {\r
- for (int j = i; j <= n; j++) {\r
- H[j][n - 1] = H[j][n - 1] / t;\r
- H[j][n] = H[j][n] / t;\r
- }\r
- }\r
- }\r
- }\r
- }\r
- }\r
-\r
- // Vectors of isolated roots\r
-\r
- for (int i = 0; i < nn; i++) {\r
- if (i < low || i > high) {\r
- for (int j = i; j < nn; j++) {\r
- V[i][j] = H[i][j];\r
- }\r
- }\r
- }\r
-\r
- // Back transformation to get eigenvectors of original matrix\r
-\r
- for (int j = nn - 1; j >= low; j--) {\r
- for (int i = low; i <= high; i++) {\r
- z = 0.0;\r
- for (int k = low; k <= Math.min(j, high); k++) {\r
- z = z + V[i][k] * H[k][j];\r
- }\r
- V[i][j] = z;\r
- }\r
- }\r
- }\r
- */\r
-\r
-\r
-}\r
+/* $RCSfile$
+ * $Author: egonw $
+ * $Date: 2005-11-10 09:52:44f -0600 (Thu, 10 Nov 2005) $
+ * $Revision: 4255 $
+ *
+ * Copyright (C) 2003-2005 Miguel, Jmol Development, www.jmol.org
+ *
+ * Contact: jmol-developers@lists.sf.net
+ *
+ * This library is free software; you can redistribute it and/or
+ * modify it under the terms of the GNU Lesser General Public
+ * License as published by the Free Software Foundation; either
+ * version 2.1 of the License, or (at your option) any later version.
+ *
+ * This library is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ * Lesser General Public License for more details.
+ *
+ * You should have received a copy of the GNU Lesser General Public
+ * License along with this library; if not, write to the Free Software
+ * Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
+ */
+
+package javajs.util;
+
+import javajs.api.EigenInterface;
+
+
+/**
+ * Eigenvalues and eigenvectors of a real matrix.
+ * See javajs.api.EigenInterface() as well.
+ *
+ * adapted by Bob Hanson from http://math.nist.gov/javanumerics/jama/ (public
+ * domain); adding quaternion superimposition capability; removing
+ * nonsymmetric reduction to Hessenberg form, which we do not need in Jmol.
+ *
+ * Output is as a set of double[n] columns, but for the EigenInterface
+ * we return them as V3[3] and float[3] (or double[3]) values.
+ *
+ * Eigenvalues and eigenvectors are sorted from smallest to largest eigenvalue.
+ *
+ * <P>
+ * 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.
+ * <P>
+ * 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;
+ }
+ }
+ }
+ */
+
+
+}
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