/*
 * This program is free software; you can redistribute it and/or
 * modify it under the terms of the GNU General Public License as
 * published by the Free Software Foundation; either version 2 of the
 * License, or (at your option) any later version.
 *
 * This program 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 PURPOUSE. See the GNU
 * General Public License for more details.
 *
 * You should have recieved a copy of the GNU General Public License
 * along with this program; if not write to the Free Software
 * Foundation, inc., 59 Temple Place, Suite 330, Boston MA 02111-1307
 * USA
 */

package Jama;

import Jama.util.*;


/**
 * Eigenvalues and eigenvectors of a real matrix.
 * 
 * <P>
 * If A is symmetric, then A = VDV' 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>
 * 
 * <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 +
 * imu, in 2-by-2 blocks, [lambda, mu; -mu, lambda].  The     columns of V
 * represent the eigenvectors in the sense that AV = VD,     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 = VDinverse(V) depends upon
 * V.cond().
 * </p>
 */
public class EigenvalueDecomposition implements java.io.Serializable {
    /* ------------------------
       Class variables
     * ------------------------ */

    /**
     * Row and column dimension (square matrix).
     *
     * @serial matrix dimension.
     */
    private int n;

    /**
     * Symmetry flag.
     *
     * @serial internal symmetry flag.
     */
    private boolean issymmetric;

    /**
     * Arrays for internal storage of eigenvalues.
     *
     * @serial internal storage of eigenvalues.
     */
    private double[] d;

    /**
     * Arrays for internal storage of eigenvalues.
     *
     * @serial internal storage of eigenvalues.
     */
    private double[] 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;

    // Complex scalar division.
    private transient double cdivr;

    // Complex scalar division.
    private transient double cdivi;

    /* ------------------------
       Constructor
     * ------------------------ */

    /**
     * Check for symmetry, then construct the eigenvalue decomposition
     *
     * @param A Square matrix
     */
    public EigenvalueDecomposition(Matrix Arg) {
        double[][] A = Arg.getArray();
        n = Arg.getColumnDimension();
        V = new double[n][n];
        d = new double[n];
        e = new double[n];

        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();
        }
    }

    private void cdiv(double xr, double xi, double yr, double yi) {
        double r;
        double 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;
        }
    }

    /**
     * Return the block diagonal eigenvalue matrix
     *
     * @return D
     */
    public Matrix getD() {
        Matrix X = new Matrix(n, n);
        double[][] D = X.getArray();

        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                D[i][j] = 0.0;
            }

            D[i][i] = d[i];

            if (e[i] > 0) {
                D[i][i + 1] = e[i];
            } else if (e[i] < 0) {
                D[i][i - 1] = e[i];
            }
        }

        return X;
    }

    /**
     * Return the imaginary parts of the eigenvalues
     *
     * @return imag(diag(D))
     */
    public double[] getImagEigenvalues() {
        return e;
    }

    /**
     * Return the real parts of the eigenvalues
     *
     * @return real(diag(D))
     */
    public double[] getRealEigenvalues() {
        return d;
    }

    /* ------------------------
       Public Methods
     * ------------------------ */

    /**
     * Return the eigenvector matrix
     *
     * @return V
     */
    public Matrix getV() {
        return new Matrix(V, n, n);
    }

    // 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;
        double q = 0;
        double r = 0;
        double s = 0;
        double z = 0;
        double t;
        double w;
        double x;
        double 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;
                    double sa;
                    double vr;
                    double 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;
            }
        }
    }

    // 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]);
                    }
                }
            }
        }
    }

    // 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 = Maths.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 = Maths.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 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;
    }
}

