text: /*
   * Carrot2 project.
   *
   * Copyright (C) 2002-2020, Dawid Weiss, Stanisław Osiński.
   * All rights reserved.
   *
   * Refer to the full license file "carrot2.LICENSE"
   * in the root folder of the repository checkout or at:
   * https://www.carrot2.org/carrot2.LICENSE
   */
  package org.carrot2.math.mahout;
  
  public class SingularValueDecomposition {
  
    private final double[][] u;
    private final double[][] v;
  
    private final double[] s;
  
    private final int m;
    private final int n;
  
    private boolean transpositionNeeded = false;
  
    public SingularValueDecomposition(Matrix arg) {
      if (arg.numRows() < arg.numCols()) {
        transpositionNeeded = true;
      }
  
      // Derived from LINPACK code.
      // Initialize.
      double[][] a;
      if (transpositionNeeded) {
        // use the transpose Matrix
        m = arg.numCols();
        n = arg.numRows();
        a = new double[m][n];
        for (int i = 0; i < m; i++) {
          for (int j = 0; j < n; j++) {
            a[i][j] = arg.get(j, i);
          }
        }
      } else {
        m = arg.numRows();
        n = arg.numCols();
        a = new double[m][n];
        for (int i = 0; i < m; i++) {
          for (int j = 0; j < n; j++) {
            a[i][j] = arg.get(i, j);
          }
        }
      }
  
      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] = Algebra.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] = Algebra.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);
      while (p > 0) {
        int k;
  
        // 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]) <= eps * (Math.abs(s[k]) + Math.abs(s[k + 1]))) {
            e[k] = 0.0;
            break;
          }
        }
        int kase;
        if (k == p - 2) {
          kase = 4;
        } else {
          int ks;
          for (ks = p - 1; ks >= k; ks--) {
            if (ks == k) {
              break;
            }
            double t = (ks == p ? 0.0 : Math.abs(e[ks])) + (ks == k + 1 ? 0.0 : Math.abs(e[ks - 1]));
            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 = Algebra.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 = Algebra.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 = Algebra.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 = Algebra.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 += 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;
          default:
            throw new IllegalStateException();
        }
      }
    }
  
    public double cond() {
      return s[0] / s[Math.min(m, n) - 1];
    }
  
    public Matrix getS() {
      double[][] s = new double[n][n];
      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 new DenseMatrix(s);
    }
  
    public double[] getSingularValues() {
      return s;
    }
  
    public Matrix getU() {
      if (transpositionNeeded) { // case numRows() < numCols()
        return new DenseMatrix(v);
      } else {
        int numCols = Math.min(m + 1, n);
        Matrix r = new DenseMatrix(m, numCols);
        for (int i = 0; i < m; i++) {
          for (int j = 0; j < numCols; j++) {
            r.set(i, j, u[i][j]);
          }
        }
  
        return r;
      }
    }
  
    public Matrix getV() {
      if (transpositionNeeded) { // case numRows() < numCols()
        int numCols = Math.min(m + 1, n);
        Matrix r = new DenseMatrix(m, numCols);
        for (int i = 0; i < m; i++) {
          for (int j = 0; j < numCols; j++) {
            r.set(i, j, u[i][j]);
          }
        }
  
        return r;
      } else {
        return new DenseMatrix(v);
      }
    }
  
    public double norm2() {
      return s[0];
    }
  
    public int rank() {
      double eps = Math.pow(2.0, -52.0);
      double tol = Math.max(m, n) * s[0] * eps;
      int r = 0;
      for (double value : s) {
        if (value > tol) {
          r++;
        }
      }
      return r;
    }
  
    Matrix getCovariance(double minSingularValue) {
      Matrix j = new DenseMatrix(s.length, s.length);
      Matrix vMat = new DenseMatrix(this.v);
      for (int i = 0; i < s.length; i++) {
        j.set(i, i, s[i] >= minSingularValue ? 1 / (s[i] * s[i]) : 0.0);
      }
      return vMat.times(j).times(vMat.transpose());
    }
  
    @Override
    public String toString() {
      StringBuilder buf = new StringBuilder();
      buf.append("---------------------------------------------------------------------\n");
      buf.append("SingularValueDecomposition(A) --> cond(A), rank(A), norm2(A), U, S, V\n");
      buf.append("---------------------------------------------------------------------\n");
  
      buf.append("cond = ");
      String unknown = "Illegal operation or error: ";
      try {
        buf.append(String.valueOf(this.cond()));
      } catch (IllegalArgumentException exc) {
        buf.append(unknown).append(exc.getMessage());
      }
  
      buf.append("\nrank = ");
      try {
        buf.append(String.valueOf(this.rank()));
      } catch (IllegalArgumentException exc) {
        buf.append(unknown).append(exc.getMessage());
      }
  
      buf.append("\nnorm2 = ");
      try {
        buf.append(String.valueOf(this.norm2()));
      } catch (IllegalArgumentException exc) {
        buf.append(unknown).append(exc.getMessage());
      }
  
      buf.append("\n\nU = ");
      try {
        buf.append(String.valueOf(this.getU()));
      } catch (IllegalArgumentException exc) {
        buf.append(unknown).append(exc.getMessage());
      }
  
      buf.append("\n\nS = ");
      try {
        buf.append(String.valueOf(this.getS()));
      } catch (IllegalArgumentException exc) {
        buf.append(unknown).append(exc.getMessage());
      }
  
      buf.append("\n\nV = ");
      try {
        buf.append(String.valueOf(this.getV()));
      } catch (IllegalArgumentException exc) {
        buf.append(unknown).append(exc.getMessage());
      }
  
      return buf.toString();
    }
  }
  