#region Math.NET Iridium (LGPL) by Vermorel, Ruegg + Contributors
// Math.NET Iridium, part of the Math.NET Project
// http://mathnet.opensourcedotnet.info
//
// Copyright (c) 2004-2008, Joannes Vermorel, http://www.vermorel.com
//                          Christoph Rüegg, http://christoph.ruegg.name
//
// Contribution: The MathWorks and NIST [2000]
//
// This program 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 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 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 program; if not, write to the Free Software
// Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
#endregion

using System;

using MathNet.Numerics;

namespace MathNet.Numerics.LinearAlgebra
{

    /// <summary>
    /// Eigenvalues and eigenvectors of a real matrix.
    /// </summary>
    /// <remarks>
    /// 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.Multiply(D.Multiply(V.Transpose())) and 
    /// V.Multiply(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.Multiply(V) equals V.Multiply(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().
    /// </remarks>
    [Serializable]
    public class EigenvalueDecomposition
    {
        /// <summary>Row and column dimension (square matrix).</summary>
        int n;

        /// <summary>Symmetry flag.</summary>
        bool isSymmetric;

        /// <summary>Arrays for internal storage of eigenvalues.</summary>
        double[] d, e;

        /// <summary>Array for internal storage of eigenvectors.</summary>
        double[][] V;

        /// <summary>Array for internal storage of nonsymmetric Hessenberg form.</summary>
        double[][] H;

        ComplexVector _eigenValues;
        Vector _eigenValuesReal;
        Vector _eigenValuesImag;

        [NonSerialized()]
        double cdivr, cdivi;

        OnDemandComputation<Matrix> _blockDiagonalOnDemand;
        OnDemandComputation<Matrix> _eigenVectorsOnDemand;
        OnDemandComputation<ComplexVector> _eigenValuesOnDemand;

        /// <summary>
        /// Check for symmetry, then construct the eigenvalue decomposition
        /// </summary>
        /// <remarks>Provides access to D and V</remarks>
        /// <param name="Arg">Square matrix</param>
        public
        EigenvalueDecomposition(
            Matrix Arg
            )
        {
            double[][] A = Arg;
            n = Arg.ColumnCount;
            V = Matrix.CreateMatrixData(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];
                    }
                }

                SymmetricTridiagonalize();

                SymmetricDiagonalize();
            }
            else
            {
                H = new double[n][];
                for(int i = 0; i < n; i++)
                {
                    double[] Hi = new double[n];
                    double[] Ai = A[i];

                    for(int j = 0; j < n; j++)
                    {
                        Hi[j] = Ai[j];
                    }

                    H[i] = Hi;
                }

                NonsymmetricReduceToHessenberg();

                NonsymmetricReduceHessenberToRealSchur();
            }

            _eigenValuesReal = new Vector(d);
            _eigenValuesImag = new Vector(e);
            _eigenValues = ComplexVector.Create(d, e);

            InitOnDemandComputations();
        }

        /// <summary>
        /// Constructs the eigenvalue decomposition from a symmetrical, 
        /// tridiagonal matrix.
        /// </summary>
        public
        EigenvalueDecomposition(
            double[] d,
            double[] e
            )
        {
            // TODO: unit test missing for EigenvalueDecomposition constructor.

            n = d.Length;
            V = Matrix.CreateMatrixData(n, n);

            this.d = new double[n];
            Array.Copy(d, 0, this.d, 0, n);

            this.e = new double[n];
            Array.Copy(e, 0, this.e, 1, n - 1);

            for(int i = 0; i < n; i++)
            {
                V[i][i] = 1;
            }

            SymmetricDiagonalize();

            _eigenValuesReal = new Vector(this.d);
            _eigenValuesImag = new Vector(this.e);
            _eigenValues = ComplexVector.Create(this.d, this.e);

            InitOnDemandComputations();
        }

        /// <summary>
        /// Gets the eigenvalues.
        /// </summary>
        /// <returns>diag(D)</returns>
        public ComplexVector EigenValues
        {
            get { return _eigenValues; }
        }

        /// <summary>
        /// Gets the real part of the eigenvalues.
        /// </summary>
        /// <returns>real(diag(D))</returns>
        public Vector RealEigenvalues
        {
            get { return _eigenValuesReal; }
        }

        /// <summary>
        /// Gets the imaginary part of the eigenvalues.
        /// </summary>
        /// <returns>imag(diag(D))</returns>
        public Vector ImagEigenvalues
        {
            get { return _eigenValuesImag; }
        }

        /// <summary>
        /// Gets the block diagonal eigenvalue matrix
        /// </summary>
        public Matrix BlockDiagonal
        {
            get { return _blockDiagonalOnDemand.Compute(); }
        }

        /// <summary>
        /// Returns the eigenvector matrix
        /// </summary>
        public Matrix EigenVectors
        {
            get { return _eigenVectorsOnDemand.Compute(); }
        }

        void
        InitOnDemandComputations()
        {
            _blockDiagonalOnDemand = new OnDemandComputation<Matrix>(ComputeBlockDiagonalMatrix);
            _eigenValuesOnDemand = new OnDemandComputation<ComplexVector>(ComputeEigenValues);
            _eigenVectorsOnDemand = new OnDemandComputation<Matrix>(ComputeEigentVectors);
        }

        ComplexVector
        ComputeEigenValues()
        {
            Complex[] eigenvalues = new Complex[n];

            for(int i = 0; i < eigenvalues.Length; i++)
            {
                eigenvalues[i] = new Complex(d[i], e[i]);
            }

            return new ComplexVector(eigenvalues);
        }

        Matrix
        ComputeBlockDiagonalMatrix()
        {
            double[][] D = Matrix.CreateMatrixData(n, n);
            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 new Matrix(D);
        }

        Matrix
        ComputeEigentVectors()
        {
            return new Matrix(V);
        }

        /// <summary>
        /// Symmetric Householder reduction to tridiagonal form.
        /// </summary>
        void
        SymmetricTridiagonalize()
        {
            // 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;
        }

        /// <summary>
        /// Symmetric tridiagonal QL algorithm.
        /// </summary>
        void
        SymmetricDiagonalize()
        {
            // 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 = Number.PositiveRelativeAccuracy;
            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 = Fn.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 = Fn.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;
                    }
                }
            }
        }

        /// <summary>
        /// Nonsymmetric reduction to Hessenberg form.
        /// </summary>
        void
        NonsymmetricReduceToHessenberg()
        {
            // 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;

            double[] ort = new double[n];

            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];
                        }
                    }
                }
            }
        }

        /// <summary>
        /// Nonsymmetric reduction from Hessenberg to real Schur form.
        /// </summary>
        void
        NonsymmetricReduceHessenberToRealSchur()
        {
            // 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 = Number.PositiveRelativeAccuracy;
            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++)
                    {
                        bool 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 != 0.0)
                        {
                            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++)
                            {
                                double[] Hi = H[i];
                                p = x * Hi[k] + y * Hi[k + 1];

                                if(notlast)
                                {
                                    p = p + z * Hi[k + 2];
                                    Hi[k + 2] = Hi[k + 2] - p * r;
                                }

                                Hi[k] = Hi[k] - p;
                                Hi[k + 1] = Hi[k + 1] - p * q;
                            }

                            // Accumulate transformations

                            for(int i = low; i <= high; i++)
                            {
                                double[] Vi = V[i];
                                p = x * Vi[k] + y * Vi[k + 1];

                                if(notlast)
                                {
                                    p = p + z * Vi[k + 2];
                                    Vi[k + 2] = Vi[k + 2] - p * r;
                                }

                                Vi[k] = Vi[k] - p;
                                Vi[k + 1] = Vi[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 == 0.0)
                {
                    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] == 0.0)
                            {
                                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;
                }
            }
        }

        /// <summary>
        /// Complex scalar division.
        /// </summary>
        void
        cdiv(
            double xr,
            double xi,
            double yr,
            double yi
            )
        {
            // TODO (cdr, 2008-03-11): Refactor

            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;
            }
        }
    }
}
