eigenvaluedecomposition.java

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      if (l == n) {
        H[n][n] = H[n][n] + exshift;
        d[n] = H[n][n];
        e[n] = 0.0;
        n--;
        iter = 0;

        // Two roots found

      } else if (l == n-1) {
        w = H[n][n-1] * H[n-1][n];
        p = (H[n-1][n-1] - H[n][n]) / 2.0;
        q = p * p + w;
        z = Math.sqrt(Math.abs(q));
        H[n][n] = H[n][n] + exshift;
        H[n-1][n-1] = H[n-1][n-1] + exshift;
        x = H[n][n];

        // Real pair

        if (q >= 0) {
          if (p >= 0) {
            z = p + z;
          } else {
            z = p - z;
          }
          d[n-1] = x + z;
          d[n] = d[n-1];
          if (z != 0.0) {
            d[n] = x - w / z;
          }
          e[n-1] = 0.0;
          e[n] = 0.0;
          x = H[n][n-1];
          s = Math.abs(x) + Math.abs(z);
          p = x / s;
          q = z / s;
          r = Math.sqrt(p * p+q * q);
          p = p / r;
          q = q / r;

          // Row modification

          for (int j = n-1; j < nn; j++) {
            z = H[n-1][j];
            H[n-1][j] = q * z + p * H[n][j];
            H[n][j] = q * H[n][j] - p * z;
          }

          // Column modification

          for (int i = 0; i <= n; i++) {
            z = H[i][n-1];
            H[i][n-1] = q * z + p * H[i][n];
            H[i][n] = q * H[i][n] - p * z;
          }

          // Accumulate transformations

          for (int i = low; i <= high; i++) {
            z = V[i][n-1];
            V[i][n-1] = q * z + p * V[i][n];
            V[i][n] = q * V[i][n] - p * z;
          }

          // Complex pair

        } else {
          d[n-1] = x + p;
          d[n] = x + p;
          e[n-1] = z;
          e[n] = -z;
        }
        n = n - 2;
        iter = 0;

        // No convergence yet

      } else {

        // Form shift

        x = H[n][n];
        y = 0.0;
        w = 0.0;
        if (l < n) {
          y = H[n-1][n-1];
          w = H[n][n-1] * H[n-1][n];
        }

        // Wilkinson's original ad hoc shift

        if (iter == 10) {
          exshift += x;
          for (int i = low; i <= n; i++) {
            H[i][i] -= x;
          }
          s = Math.abs(H[n][n-1]) + Math.abs(H[n-1][n-2]);
          x = y = 0.75 * s;
          w = -0.4375 * s * s;
        }

        // MATLAB's new ad hoc shift

        if (iter == 30) {
          s = (y - x) / 2.0;
          s = s * s + w;
          if (s > 0) {
            s = Math.sqrt(s);
            if (y < x) {
              s = -s;
            }
            s = x - w / ((y - x) / 2.0 + s);
            for (int i = low; i <= n; i++) {
              H[i][i] -= s;
            }
            exshift += s;
            x = y = w = 0.964;
          }
        }

        iter = iter + 1;   // (Could check iteration count here.)

        // Look for two consecutive small sub-diagonal elements

        int m = n-2;
        while (m >= l) {
          z = H[m][m];
          r = x - z;
          s = y - z;
          p = (r * s - w) / H[m+1][m] + H[m][m+1];
          q = H[m+1][m+1] - z - r - s;
          r = H[m+2][m+1];
          s = Math.abs(p) + Math.abs(q) + Math.abs(r);
          p = p / s;
          q = q / s;
          r = r / s;
          if (m == l) {
            break;
          }
          if (Math.abs(H[m][m-1]) * (Math.abs(q) + Math.abs(r)) <
              eps * (Math.abs(p) * (Math.abs(H[m-1][m-1]) + Math.abs(z) +
                  Math.abs(H[m+1][m+1])))) {
            break;
                  }
          m--;
        }

        for (int i = m+2; i <= n; i++) {
          H[i][i-2] = 0.0;
          if (i > m+2) {
            H[i][i-3] = 0.0;
          }
        }

        // Double QR step involving rows l:n and columns m:n

        for (int k = m; k <= n-1; k++) {
          boolean notlast = (k != n-1);
          if (k != m) {
            p = H[k][k-1];
            q = H[k+1][k-1];
            r = (notlast ? H[k+2][k-1] : 0.0);
            x = Math.abs(p) + Math.abs(q) + Math.abs(r);
            if (x != 0.0) {
              p = p / x;
              q = q / x;
              r = r / x;
            }
          }
          if (x == 0.0) {
            break;
          }
          s = Math.sqrt(p * p + q * q + r * r);
          if (p < 0) {
            s = -s;
          }
          if (s != 0) {
            if (k != m) {
              H[k][k-1] = -s * x;
            } else if (l != m) {
              H[k][k-1] = -H[k][k-1];
            }
            p = p + s;
            x = p / s;
            y = q / s;
            z = r / s;
            q = q / p;
            r = r / p;

            // Row modification

            for (int j = k; j < nn; j++) {
              p = H[k][j] + q * H[k+1][j];
              if (notlast) {
                p = p + r * H[k+2][j];
                H[k+2][j] = H[k+2][j] - p * z;
              }
              H[k][j] = H[k][j] - p * x;
              H[k+1][j] = H[k+1][j] - p * y;
            }

            // Column modification

            for (int i = 0; i <= Math.min(n,k+3); i++) {
              p = x * H[i][k] + y * H[i][k+1];
              if (notlast) {
                p = p + z * H[i][k+2];
                H[i][k+2] = H[i][k+2] - p * r;
              }
              H[i][k] = H[i][k] - p;
              H[i][k+1] = H[i][k+1] - p * q;
            }

            // Accumulate transformations

            for (int i = low; i <= high; i++) {
              p = x * V[i][k] + y * V[i][k+1];
              if (notlast) {
                p = p + z * V[i][k+2];
                V[i][k+2] = V[i][k+2] - p * r;
              }
              V[i][k] = V[i][k] - p;
              V[i][k+1] = V[i][k+1] - p * q;
            }
          }  // (s != 0)
        }  // k loop
      }  // check convergence
    }  // while (n >= low)

    // Backsubstitute to find vectors of upper triangular form

    if (norm == 0.0) {
      return;
    }

    for (n = nn-1; n >= 0; n--) {
      p = d[n];
      q = e[n];

      // Real vector

      if (q == 0) {
        int l = n;
        H[n][n] = 1.0;
        for (int i = n-1; i >= 0; i--) {
          w = H[i][i] - p;
          r = 0.0;
          for (int j = l; j <= n; j++) {
            r = r + H[i][j] * H[j][n];
          }
          if (e[i] < 0.0) {
            z = w;
            s = r;
          } else {
            l = i;
            if (e[i] == 0.0) {
              if (w != 0.0) {
                H[i][n] = -r / w;
              } else {
                H[i][n] = -r / (eps * norm);
              }

              // Solve real equations

            } else {
              x = H[i][i+1];
              y = H[i+1][i];
              q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
              t = (x * s - z * r) / q;
              H[i][n] = t;
              if (Math.abs(x) > Math.abs(z)) {
                H[i+1][n] = (-r - w * t) / x;
              } else {
                H[i+1][n] = (-s - y * t) / z;
              }
            }

            // Overflow control

            t = Math.abs(H[i][n]);
            if ((eps * t) * t > 1) {
              for (int j = i; j <= n; j++) {
                H[j][n] = H[j][n] / t;
              }
            }
          }
        }

        // Complex vector

      } else if (q < 0) {
        int l = n-1;

        // Last vector component imaginary so matrix is triangular

        if (Math.abs(H[n][n-1]) > Math.abs(H[n-1][n])) {
          H[n-1][n-1] = q / H[n][n-1];
          H[n-1][n] = -(H[n][n] - p) / H[n][n-1];
        } else {
          cdiv(0.0,-H[n-1][n],H[n-1][n-1]-p,q);
          H[n-1][n-1] = cdivr;
          H[n-1][n] = cdivi;
        }
        H[n][n-1] = 0.0;
        H[n][n] = 1.0;
        for (int i = n-2; i >= 0; i--) {
          double ra,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) {
              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;
      }
    }
  }


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

    issymmetric = true;
    for (int j = 0; (j < n) & issymmetric; j++) {
      for (int i = 0; (i < n) & issymmetric; i++) {
        issymmetric = (A[i][j] == A[j][i]);
      }
    }

    if (issymmetric) {
      for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
          V[i][j] = A[i][j];
        }
      }

      // Tridiagonalize.
      tred2();

      // Diagonalize.
      tql2();

    } else {
      H = new double[n][n];
      ort = new double[n];

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

      // Reduce to Hessenberg form.
      orthes();

      // Reduce Hessenberg to real Schur form.
      hqr2();
    }
  }

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

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

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

  /** 
   * Return the block diagonal eigenvalue matrix
   * @return     D
   */
  public Matrix getD() {
    Matrix X = new Matrix(n,n);
    double[][] D = X.getArray();
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < n; j++) {
        D[i][j] = 0.0;
      }
      D[i][i] = d[i];
      if (e[i] > 0) {
        D[i][i+1] = e[i];
      } else if (e[i] < 0) {
        D[i][i-1] = e[i];
      }
    }
    return X;
  }
}