.NET Opensource

I've been incorporating Math.NET Iridium into a little hobby project of mine, and I seem to have run into a bug. In the code, I have several matrices, and compute the SVD for each. The first two compute just fine, but when I hit the third matrix, the call to myMatrix.SingularValueDecomposition just gets stuck in an infinite loop. Also, trying to compute the condition number of the same matrix also causes the program to hang.

Any ideas? I can provide the offending matrix if you'd like, but it is 120x120 so I didn't want to post it here without asking...

P.S. -- As a feature request...how about adding methods to the Matrix and Vector classes to parse/output a Matlab-format string (e.g. [[1 2 3; 4 5 6; 7 8 9;]])?

Sorry, I should have also mentioned that I know there's not a problem with the matrix itself, since I can put it into MATLAB and compute the SVD just fine. However, I'm trying to do the whole project in .NET, so that doesn't help me ;)

Seems there are two issues here:

• The algorithm doesn't converge (nicely or at all) as it should. Matlab apparently converges.
• The algorithm doesn't stop even though it does not converge.
  

I noticed that although iterations are counted, this count is never checked in the current revision (there's actually a comment where the check should be placed), that's why it hangs instead of throwing an exception. I'll have to check that.

Can you attach the failing matrix? So I can analyze the issue and build a unit test around it - if there are no data privacy/IP concerns.

It's happening on several different matrices. I'm using the SVD to do some analysis on graphs; the matrix I'm including below is the Laplace matrix of a disconnected graph (specifically, a weighted digraph with 120 vertices). I've also included the singular values of the matrix as calculated by Matlab. NOTE: If you want to put the matrix into Matlab yourself, the whole thing needs to be on one line (you may have to remove the newline characters in notepad or similar first).

Thanks for all your hard work! Perhaps one day I will be able to contribute some code back to the project...

>> testmatrix = [34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-29,0,0,0,0,0,0,0,0,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0; 0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-48,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0; 0,0,0,52,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0; 0,0,0,0,19,0,0,0,0,0,0,0,0,-31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0; 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0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,-53; 0,0,0,0,0,0,0,0,0,0,0,-26,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,-46,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,41,0,0,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,44,0,-31,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-33,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-44,0,31,0,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-32,0,0,0,0,46,0; 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-18,0,0,0,0,0,0,0,53; ];

>> svd(testmatrix)

ans =

  102.8591
   98.0000
   96.7471
   87.1436
   86.6487
   86.3134
   84.8646
   84.0238
   83.1986
   82.2192
   81.5843
   81.2158
   80.8084
   80.3243
   79.6994
   79.2465
   79.1581
   79.0569
   77.3692
   76.1183
   75.3127
   75.1665
   74.9667
   74.9667
   74.6726
   74.0945
   73.3348
   72.9932
   72.8011
   72.4431
   71.8471
   71.3863
   71.0634
   71.0634
   70.2282
   70.2140
   67.0075
   65.0077
   64.5600
   63.3877
   63.1981
   62.8013
   61.4003
   55.7136
   51.4198
   50.0400
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
         0
         0
         0
        
        
        


 

I have the same problem with the code, and I have changed the code so it is checking for numbers of iter and the hypot is also changed to a new function "givens" which is used for the four cases.

I have also changed the constructor interface so you should speify if you want U and Y , but this is just done to make the algorithm faster.

I have included the updated code below

#region Math.NET Iridium (LGPL) by Vermorel + Contributors
// Math.NET Iridium, part of the Math.NET Project
// http://mathnet.opensourcedotnet.info
//
// Copyright (c) 2004-2008, Joannes Vermorel, http://www.vermorel.com
//
// 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 System.Diagnostics;

namespace MathNet.Numerics.LinearAlgebra
{
  /// <summary>Singular Value Decomposition.</summary>
  /// <remarks>
  /// <p>For an m-by-n matrix A with m >= n, the singular value decomposition
  /// is an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
  /// an n-by-n orthogonal matrix V so that A = U*S*V'.</p>
  ///
  /// <p>The singular values, sigma[k] = S[k, k], are ordered so that
  /// sigma[0] >= sigma[1] >= ... >= sigma[n-1].</p>
  ///
  /// <p>The singular value decomposition always exists, so the constructor will
  /// never fail.  The matrix condition number and the effective numerical
  /// rank can be computed from this decomposition.</p>
  /// </remarks>
  [Serializable]
  public class SingularValueDecomposition
  {
  
    Matrix _u;
    Matrix _v;

    /// <summary>Array for internal storage of singular values.</summary>
    Vector _singular;

    /// <summary>Row dimensions.</summary>
    int m;

    /// <summary>Column dimensions.</summary>
    int n;

    /// <summary>Indicates whether all the results provided by the
    /// method or properties should be transposed.</summary>
    /// <remarks>
    /// (vermorel) The initial implementation was assuming that
    /// m &gt;= n, but in fact, it is easy to handle the case m &lt; n
    /// by transposing all the results.
    /// </remarks>
    bool transpose;

    OnDemandComputation<Matrix> _diagonalSingularValuesOnDemand;
    OnDemandComputation<int> _rankOnDemand;

    /// <summary>Construct the singular value decomposition.</summary>
    /// <remarks>Provides access to U, S and V.</remarks>
    /// <param name="Arg">Rectangular matrix</param>
    public
    SingularValueDecomposition(
        Matrix Arg,
        bool wantu,
        bool wantv
        )
    {
      transpose = (Arg.RowCount < Arg.ColumnCount);

      // Derived from LINPACK code.
      // Initialize.
      double[[ A;
      if (transpose)
      {
        // copy of internal data, independent of Arg
        A = Matrix.Transpose(Arg).GetArray();
        m = Arg.ColumnCount;
        n = Arg.RowCount;
      }
      else
      {
        A = Arg.CopyToJaggedArray();
        m = Arg.RowCount;
        n = Arg.ColumnCount;
      }

      int nu = Math.Min(m, n);
      double[ s = new double[Math.Min(m + 1, n)];
      double[[ U = Matrix.CreateMatrixData(m, nu);
      double[[ V = Matrix.CreateMatrixData(n, n);

      double[ e = new double[n];
      double[ work = new double[m];


      // Reduce A to bidiagonal form, storing the diagonal elements
      // in s and the super-diagonal elements in e.

      int nct = Math.Min(m - 1, n);
      int nrt = Math.Max(0, Math.Min(n - 2, m));
      for (int k = 0; k < Math.Max(nct, nrt); k++)
      {
        if (k < nct)
        {

          // Compute the transformation for the k-th column and
          // place the k-th diagonal in s[k].
          // Compute 2-norm of k-th column without under/overflow.
          s[k] = 0;

          for (int i = k; i < m; i++)
          {
            s[k] = Fn.Hypot(s[k], A[i][k]);
          }

          if (s[k] != 0.0)
          {
            if (A[k][k] < 0.0)
            {
              s[k] = -s[k];
            }

            for (int i = k; i < m; i++)
            {
              A[i][k] /= s[k];
            }

            A[k][k] += 1.0;
          }

          s[k] = -s[k];
        }

        for (int j = k + 1; j < n; j++)
        {
          if ((k < nct) & (s[k] != 0.0))
          {

            // Apply the transformation.

            double t = 0;
            for (int i = k; i < m; i++)
            {
              t += A[i][k] * A[i][j];
            }

            t = (-t) / A[k][k];
            for (int i = k; i < m; i++)
            {
              A[i][j] += t * A[i][k];
            }
          }

          // Place the k-th row of A into e for the
          // subsequent calculation of the row transformation.

          e[j] = A[k][j];
        }

        if (wantu & (k < nct))
        {

          // Place the transformation in U for subsequent back
          // multiplication.

          for (int i = k; i < m; i++)
          {
            U[i][k] = A[i][k];
          }
        }

        if (k < nrt)
        {

          // Compute the k-th row transformation and place the
          // k-th super-diagonal in e[k].
          // Compute 2-norm without under/overflow.
          e[k] = 0;

          for (int i = k + 1; i < n; i++)
          {
            e[k] = Fn.Hypot(e[k], e[i]);
          }

          if (e[k] != 0.0)
          {
            if (e[k + 1] < 0.0)
            {
              e[k] = -e[k];
            }

            for (int i = k + 1; i < n; i++)
            {
              e[i] /= e[k];
            }

            e[k + 1] += 1.0;
          }

          e[k] = -e[k];

          if ((k + 1 < m) & (e[k] != 0.0))
          {

            // Apply the transformation.

            for (int i = k + 1; i < m; i++)
            {
              work[i] = 0.0;
            }

            for (int j = k + 1; j < n; j++)
            {
              for (int i = k + 1; i < m; i++)
              {
                work[i] += e[j] * A[i][j];
              }
            }

            for (int j = k + 1; j < n; j++)
            {
              double t = (-e[j]) / e[k + 1];
              for (int i = k + 1; i < m; i++)
              {
                A[i][j] += t * work[i];
              }
            }
          }

          if (wantv)
          {

            // Place the transformation in V for subsequent
            // back multiplication.

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

      // Set up the final bidiagonal matrix or order p.

      int p = Math.Min(n, m + 1);

      if (nct < n)
      {
        s[nct] = A[nct][nct];
      }

      if (m < p)
      {
        s[p - 1] = 0.0;
      }

      if (nrt + 1 < p)
      {
        e[nrt] = A[nrt][p - 1];
      }

      e[p - 1] = 0.0;

      // If required, generate U.

      if (wantu)
      {
        for (int j = nct; j < nu; j++)
        {
          for (int i = 0; i < m; i++)
          {
            U[i][j] = 0.0;
          }

          U[j][j] = 1.0;
        }

        for (int k = nct - 1; k >= 0; k--)
        {
          if (s[k] != 0.0)
          {
            for (int j = k + 1; j < nu; j++)
            {
              double t = 0;
              for (int i = k; i < m; i++)
              {
                t += U[i][k] * U[i][j];
              }

              t = (-t) / U[k][k];
              for (int i = k; i < m; i++)
              {
                U[i][j] += t * U[i][k];
              }
            }

            for (int i = k; i < m; i++)
            {
              U[i][k] = -U[i][k];
            }

            U[k][k] = 1.0 + U[k][k];
            for (int i = 0; i < k - 1; i++)
            {
              U[i][k] = 0.0;
            }
          }
          else
          {
            for (int i = 0; i < m; i++)
            {
              U[i][k] = 0.0;
            }

            U[k][k] = 1.0;
          }
        }
      }

      // If required, generate V.

      if (wantv)
      {
        for (int k = n - 1; k >= 0; k--)
        {
          if ((k < nrt) & (e[k] != 0.0))
          {
            for (int j = k + 1; j < nu; j++)
            {
              double t = 0;
              for (int i = k + 1; i < n; i++)
              {
                t += V[i][k] * V[i][j];
              }

              t = (-t) / V[k + 1][k];
              for (int i = k + 1; i < n; i++)
              {
                V[i][j] += t * V[i][k];
              }
            }
          }

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

          V[k][k] = 1.0;
        }
      }

      // Main iteration loop for the singular values.

      int pp = p - 1;
      int iter = 0;
      double eps = Number.PositiveRelativeAccuracy;
      double tiny = double.Epsilon / eps;
      double snorm = 0;
      for (int ii = 0; ii < nu; ii++)
      {
        snorm = Math.Max(snorm, Math.Max(Math.Abs(s[ii]), Math.Abs(e[ii])));
      }
      while (p > 0)
      {
        int k, kase;

        // Here is where a test for too many iterations would go.

        // This section of the program inspects for
        // negligible elements in the s and e arrays.  On
        // completion the variables kase and k are set as follows.

        // kase = 1     if s(p) and e[k-1] are negligible and k<p
        // kase = 2     if s(k) is negligible and k<p
        // kase = 3     if e[k-1] is negligible, k<p, and
        //              s(k), ..., s(p) are not negligible (qr step).
        // kase = 4     if e(p-1) is negligible (convergence).

        for (k = p - 2; k >= -1; k--)
        {
          if (k == -1)
          {
            break;
          }
          double test0 = Math.Abs(s[k]) + Math.Abs(s[k + 1]);
          double ztest0 = Math.Abs(e[k]);         
          if ((ztest0 <= eps * test0) || (ztest0 <= tiny)
            || (iter > 20 && ztest0 <= eps*snorm))
          {
            e[k] = 0.0;
            break;
          }
        }

        if (k == p - 2)
        {
          kase = 4;
        }
        else
        {
          int ks;
          for (ks = p - 1; ks >= k; ks--)
          {
            if (ks == k)
            {
              break;
            }

            double t = (ks != p ? Math.Abs(e[ks]) : 0.0) + (ks != k + 1 ? Math.Abs(e[ks - 1]) : 0.0);
            if (Math.Abs(s[ks]) <= eps * t)
            {
              s[ks] = 0.0;
              break;
            }
          }

          if (ks == k)
          {
            kase = 3;
          }
          else if (ks == p - 1)
          {
            kase = 1;
          }
          else
          {
            kase = 2;
            k = ks;
          }
        }

        k++;

        // Perform the task indicated by kase.

        switch (kase)
        {
          // Deflate negligible s(p).
          case 1:
            {
              double f = e[p - 2];
              e[p - 2] = 0.0;
              for (int j = p - 2; j >= k; j--)
              {
                double t, cs, sn;
                givens(s[j], f, out t, out cs, out sn);
                s[j] = t;

                if (j != k)
                {
                  f = (-sn) * e[j - 1];
                  e[j - 1] = cs * e[j - 1];
                }

                if (wantv)
                {
                  for (int i = 0; i < n; i++)
                  {
                    t = cs * V[i][j] + sn * V[i][p - 1];
                    V[i][p - 1] = (-sn) * V[i][j] + cs * V[i][p - 1];
                    V[i][j] = t;
                  }
                }
              }
            }

            break;

          // Split at negligible s(k)
          case 2:
            {
              double f = e[k - 1];
              e[k - 1] = 0.0;
              for (int j = k; j < p; j++)
              {
                double t, cs, sn;
                givens(s[j], f, out t, out cs, out sn);
                s[j] = t;
                f = (-sn) * e[j];
                e[j] = cs * e[j];
                if (wantu)
                {
                  for (int i = 0; i < m; i++)
                  {
                    t = cs * U[i][j] + sn * U[i][k - 1];
                    U[i][k - 1] = (-sn) * U[i][j] + cs * U[i][k - 1];
                    U[i][j] = t;
                  }
                }
              }
            }

            break;

          // Perform one qr step.
          case 3:
            {
              // Calculate the shift.

              double scale = Math.Max(Math.Max(Math.Max(Math.Max(Math.Abs(s[p - 1]), Math.Abs(s[p - 2])), Math.Abs(e[p - 2])), Math.Abs(s[k])), Math.Abs(e[k]));
              double sp = s[p - 1] / scale;
              double spm1 = s[p - 2] / scale;
              double epm1 = e[p - 2] / scale;
              double sk = s[k] / scale;
              double ek = e[k] / scale;
              double b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0;
              double c = (sp * epm1) * (sp * epm1);
              double shift = 0.0;
              if ((b != 0.0) | (c != 0.0))
              {
                shift = Math.Sqrt(b * b + c);

                if (b < 0.0)
                {
                  shift = -shift;
                }

                shift = c / (b + shift);
              }

              double f = (sk + sp) * (sk - sp) + shift;
              double g = sk * ek;

              // Chase zeros.

              for (int j = k; j < p - 1; j++)
              {
                double t, cs, sn;
                givens( f,g, out t, out cs, out sn);

                if (j != k)
                {
                  e[j - 1] = t;
                }

                f = cs * s[j] + sn * e[j];
                e[j] = cs * e[j] - sn * s[j];
                g = sn * s[j + 1];
                s[j + 1] = cs * s[j + 1];

                if (wantv)
                {
                  for (int i = 0; i < n; i++)
                  {
                    t = cs * V[i][j] + sn * V[i][j + 1];
                    V[i][j + 1] = (-sn) * V[i][j] + cs * V[i][j + 1];
                    V[i][j] = t;
                  }
                }

                givens(f, g, out t, out cs, out sn);
                cs = f / t;
                sn = g / t;
                s[j] = t;
                f = cs * e[j] + sn * s[j + 1];
                s[j + 1] = (-sn) * e[j] + cs * s[j + 1];
                g = sn * e[j + 1];
                e[j + 1] = cs * e[j + 1];

                if (wantu && (j < m - 1))
                {
                  for (int i = 0; i < m; i++)
                  {
                    t = cs * U[i][j] + sn * U[i][j + 1];
                    U[i][j + 1] = (-sn) * U[i][j] + cs * U[i][j + 1];
                    U[i][j] = t;
                  }
                }
              }

              e[p - 2] = f;
              iter = iter + 1;
            }

            break;

          // Convergence.


          case 4:
            {
              // Make the singular values positive.

              if (s[k] <= 0.0)
              {
                s[k] = (s[k] < 0.0 ? -s[k] : 0.0);
                if (wantv)
                {
                  for (int i = 0; i <= pp; i++)
                  {
                    V[i][k] = -V[i][k];
                  }
                }
              }

              // Order the singular values.

              while (k < pp)
              {
                if (s[k] >= s[k + 1])
                {
                  break;
                }

                double t = s[k];
                s[k] = s[k + 1];
                s[k + 1] = t;

                if (wantv && (k < n - 1))
                {
                  for (int i = 0; i < n; i++)
                  {
                    t = V[i][k + 1];
                    V[i][k + 1] = V[i][k];
                    V[i][k] = t;
                  }
                }

                if (wantu && (k < m - 1))
                {
                  for (int i = 0; i < m; i++)
                  {
                    t = U[i][k + 1];
                    U[i][k + 1] = U[i][k];
                    U[i][k] = t;
                  }
                }

                k++;
              }

              iter = 0;
              p--;
            }

            break;
        }
      }

      // (vermorel) transposing the results if needed
      if (transpose)
      {
        // swaping U and V
        double[[ T = V;
        V = U;
        U = T;
      }

      _u = new Matrix(U);
      _v = new Matrix(V);
      _singular = new Vector(s);

      InitOnDemandComputations();
    }

    /// <summary>Gets the one-dimensional array of singular values.</summary>
    /// <returns>diagonal of S.</returns>
    public Vector SingularValues
    {
      get { return _singular; }
    }

    /// <summary>Get the diagonal matrix of singular values.</summary>
    public Matrix S
    {
      get
      {
        // TODO: bad name for this property
        return _diagonalSingularValuesOnDemand.Compute();
      }
    }

    /// <summary>Gets the left singular vectors (U matrix).</summary>
    public Matrix LeftSingularVectors
    {
      get { return _u; }
    }

    /// <summary>Gets the right singular vectors (V matrix).</summary>
    public Matrix RightSingularVectors
    {
      get { return _v; }
    }

    /// <summary>Two norm.</summary>
    /// <returns>max(S)</returns>
    public
    double
    Norm2()
    {
      // TODO (cdr, 2008-03-11): Change to property
      return _singular[0];
    }

    /// <summary>Two norm condition number.</summary>
    /// <returns>max(S)/min(S)</returns>
    public
    double
    Condition()
    {
      // TODO (cdr, 2008-03-11): Change to property
      return _singular[0] / _singular[Math.Min(m, n) - 1];
    }

    /// <summary>Effective numerical matrix rank - Number of nonnegligible singular values.</summary>
    public
    int
    Rank()
    {
      // TODO (cdr, 2008-03-11): Change to property
      return _rankOnDemand.Compute();
    }

    void
    InitOnDemandComputations()
    {
      _diagonalSingularValuesOnDemand = new OnDemandComputation<Matrix>(ComputeDiagonalSingularValues);
      _rankOnDemand = new OnDemandComputation<int>(ComputeRank);
    }

    Matrix
    ComputeDiagonalSingularValues()
    {
      return Matrix.Diagonal(_singular);
    }

    int
    ComputeRank()
    {
      double tol = Math.Max(m, n) * _singular[0] * Number.PositiveRelativeAccuracy;
      int r = 0;

      for (int i = 0; i < _singular.Length; i++)
      {
        if (_singular[i] > tol)
        {
          r++;
        }
      }

      return r;
    }
    private double dlapy2(double a, double b)
    {
      b = Math.Abs(b);
      a = Math.Abs(a);
      double y = a;
      if (b > a || double.IsNaN(b))
      {
        y = b;
        b = a;
      }
      if (!(Number.AlmostZero(y) || double.IsInfinity(y)))
      {
        double r = b / y;
        y = y * Math.Sqrt(1 + r * r);
      }
      return y;
    }

    private void givens(double a, double b, out double r, out double c, out double s)
    {
      r = dlapy2(a, b);
      if (r == 0)
      {
        c = 1;
        s = 0;
      }
      else
      {
        double rho;
        if (Math.Abs(a) > Math.Abs(b))
          rho = a;
        else
          rho = b;

        if (rho < 0)
          r = -r;

        c = a / r;
        s = b / r;
      }
    }
  }
}