content: /*
   * This software is a cooperative product of The MathWorks and the National
   * Institute of Standards and Technology (NIST) which has been released to the
   * public domain. Neither The MathWorks nor NIST assumes any responsibility
   * whatsoever for its use by other parties, and makes no guarantees, expressed
   * or implied, about its quality, reliability, or any other characteristic.
   */
  
  /*
   * SingularValueDecomposition.java
   * Copyright (C) 1999 The Mathworks and NIST
   *
   */
  
  package weka.core.matrix;
  
  import weka.core.RevisionHandler;
  import weka.core.RevisionUtils;
  
  import java.io.Serializable;
  
  /** 
   * Singular Value Decomposition.
   * <P>
   * For an m-by-n matrix A with m &gt;= 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>
   * The singular values, sigma[k] = S[k][k], are ordered so that sigma[0] &gt;=
   * sigma[1] &gt;= ... &gt;= sigma[n-1].
   * <P>
   * The singular value decompostion always exists, so the constructor will never
   * fail.  The matrix condition number and the effective numerical rank can be
   * computed from this decomposition.
   * <p/>
   * Adapted from the <a href="http://math.nist.gov/javanumerics/jama/" target="_blank">JAMA</a> package.
   *
   * @author The Mathworks and NIST 
   * @author Fracpete (fracpete at waikato dot ac dot nz)
   * @version $Revision: 5953 $
   */
  public class SingularValueDecomposition 
    implements Serializable, RevisionHandler {
  
    /** for serialization */
    private static final long serialVersionUID = -8738089610999867951L;
    
    /** 
     * Arrays for internal storage of U and V.
     * @serial internal storage of U.
     * @serial internal storage of V.
     */
    private double[][] U, V;
  
    /** 
     * Array for internal storage of singular values.
     * @serial internal storage of singular values.
     */
    private double[] s;
  
    /** 
     * Row and column dimensions.
     * @serial row dimension.
     * @serial column dimension.
     */
    private int m, n;
  
    /** 
     * Construct the singular value decomposition
     * @param Arg    Rectangular matrix
     */
    public SingularValueDecomposition(Matrix Arg) {
  
      // Derived from LINPACK code.
      // Initialize.
      double[][] A = Arg.getArrayCopy();
      m = Arg.getRowDimension();
      n = Arg.getColumnDimension();
  
      /* Apparently the failing cases are only a proper subset of (m<n), 
  	 so let's not throw error.  Correct fix to come later?
      if (m<n) {
  	  throw new IllegalArgumentException("Jama SVD only works for m >= n"); }
      */
  
      int nu = Math.min(m,n);
      s = new double [Math.min(m+1,n)];
      U = new double [m][nu];
      V = new double [n][n];
      double[] e = new double [n];
      double[] work = new double [m];
      boolean wantu = true;
      boolean wantv = true;
  
      // 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] = Maths.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] = Maths.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 = Math.pow(2.0,-52.0);
      double tiny = Math.pow(2.0,-966.0);
      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;
          }
          if (Math.abs(e[k]) <=
            tiny + eps*(Math.abs(s[k]) + Math.abs(s[k+1]))) {
            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.) + 
                                                    (ks != k+1 ? Math.abs(e[ks-1]) : 0.);
            if (Math.abs(s[ks]) <= tiny + 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 = Maths.hypot(s[j],f);
                      double cs = s[j]/t;
                      double sn = f/t;
                      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 = Maths.hypot(s[j],f);
                      double cs = s[j]/t;
                      double sn = f/t;
                      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 = Maths.hypot(f,g);
                      double cs = f/t;
                      double sn = g/t;
                      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;
                        }
                      }
                      t = Maths.hypot(f,g);
                      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;
        }
      }
    }
  
    /** 
     * Return the left singular vectors
     * @return     U
     */
    public Matrix getU() {
      return new Matrix(U,m,Math.min(m+1,n));
    }
  
    /** 
     * Return the right singular vectors
     * @return     V
     */
    public Matrix getV() {
      return new Matrix(V,n,n);
    }
  
    /** 
     * Return the one-dimensional array of singular values
     * @return     diagonal of S.
     */
    public double[] getSingularValues() {
      return s;
    }
  
    /** 
     * Return the diagonal matrix of singular values
     * @return     S
     */
    public Matrix getS() {
      Matrix X = new Matrix(n,n);
      double[][] S = X.getArray();
      for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
          S[i][j] = 0.0;
        }
        S[i][i] = this.s[i];
      }
      return X;
    }
  
    /** 
     * Two norm
     * @return     max(S)
     */
    public double norm2() {
      return s[0];
    }
  
    /** 
     * Two norm condition number
     * @return     max(S)/min(S)
     */
    public double cond() {
      return s[0]/s[Math.min(m,n)-1];
    }
  
    /** 
     * Effective numerical matrix rank
     * @return     Number of nonnegligible singular values.
     */
    public int rank() {
      double eps = Math.pow(2.0,-52.0);
      double tol = Math.max(m,n)*s[0]*eps;
      int r = 0;
      for (int i = 0; i < s.length; i++) {
        if (s[i] > tol) {
          r++;
        }
      }
      return r;
    }
    
    /**
     * Returns the revision string.
     * 
     * @return		the revision
     */
    public String getRevision() {
      return RevisionUtils.extract("$Revision: 5953 $");
    }
  }
  