levenbergmarquardtestimator.java

Apache的common math数学软件包

      for (double ratio = 0; ratio < 1.0e-4;) {

        // save the state
        for (int j = 0; j < solvedCols; ++j) {
          int pj = permutation[j];
          oldX[pj] = parameters[pj].getEstimate();
        }
        double previousCost = cost;
        double[] tmpVec = residuals;
        residuals = oldRes;
        oldRes    = tmpVec;
        
        // determine the Levenberg-Marquardt parameter
        determineLMParameter(oldRes, delta, diag, work1, work2, work3);

        // compute the new point and the norm of the evolution direction
        double lmNorm = 0;
        for (int j = 0; j < solvedCols; ++j) {
          int pj = permutation[j];
          lmDir[pj] = -lmDir[pj];
          parameters[pj].setEstimate(oldX[pj] + lmDir[pj]);
          double s = diag[pj] * lmDir[pj];
          lmNorm  += s * s;
        }
        lmNorm = Math.sqrt(lmNorm);

        // on the first iteration, adjust the initial step bound.
        if (firstIteration) {
          delta = Math.min(delta, lmNorm);
        }

        // evaluate the function at x + p and calculate its norm
        updateResidualsAndCost();

        // compute the scaled actual reduction
        double actRed = -1.0;
        if (0.1 * cost < previousCost) {
          double r = cost / previousCost;
          actRed = 1.0 - r * r;
        }

        // compute the scaled predicted reduction
        // and the scaled directional derivative
        for (int j = 0; j < solvedCols; ++j) {
          int pj = permutation[j];
          double dirJ = lmDir[pj];
          work1[j] = 0;
          for (int i = 0, index = pj; i <= j; ++i, index += cols) {
            work1[i] += jacobian[index] * dirJ;
          }
        }
        double coeff1 = 0;
        for (int j = 0; j < solvedCols; ++j) {
         coeff1 += work1[j] * work1[j];
        }
        double pc2 = previousCost * previousCost;
        coeff1 = coeff1 / pc2;
        double coeff2 = lmPar * lmNorm * lmNorm / pc2;
        double preRed = coeff1 + 2 * coeff2;
        double dirDer = -(coeff1 + coeff2);

        // ratio of the actual to the predicted reduction
        ratio = (preRed == 0) ? 0 : (actRed / preRed);

        // update the step bound
        if (ratio <= 0.25) {
          double tmp =
            (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
          if ((0.1 * cost >= previousCost) || (tmp < 0.1)) {
            tmp = 0.1;
          }
          delta = tmp * Math.min(delta, 10.0 * lmNorm);
          lmPar /= tmp;
        } else if ((lmPar == 0) || (ratio >= 0.75)) {
          delta = 2 * lmNorm;
          lmPar *= 0.5;
        }

        // test for successful iteration.
        if (ratio >= 1.0e-4) {
          // successful iteration, update the norm
          firstIteration = false;
          xNorm = 0;
          for (int k = 0; k < cols; ++k) {
            double xK = diag[k] * parameters[k].getEstimate();
            xNorm    += xK * xK;
          }
          xNorm = Math.sqrt(xNorm);
        } else {
          // failed iteration, reset the previous values
          cost = previousCost;
          for (int j = 0; j < solvedCols; ++j) {
            int pj = permutation[j];
            parameters[pj].setEstimate(oldX[pj]);
          }
          tmpVec    = residuals;
          residuals = oldRes;
          oldRes    = tmpVec;
        }
   
        // tests for convergence.
        if (((Math.abs(actRed) <= costRelativeTolerance) &&
             (preRed <= costRelativeTolerance) &&
             (ratio <= 2.0)) ||
             (delta <= parRelativeTolerance * xNorm)) {
          return;
        }

        // tests for termination and stringent tolerances
        // (2.2204e-16 is the machine epsilon for IEEE754)
        if ((Math.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
          throw new EstimationException("cost relative tolerance is too small ({0})," +
                                        " no further reduction in the" +
                                        " sum of squares is possible",
                                        new Object[] { new Double(costRelativeTolerance) });
        } else if (delta <= 2.2204e-16 * xNorm) {
          throw new EstimationException("parameters relative tolerance is too small" +
                                        " ({0}), no further improvement in" +
                                        " the approximate solution is possible",
                                        new Object[] { new Double(parRelativeTolerance) });
        } else if (maxCosine <= 2.2204e-16)  {
          throw new EstimationException("orthogonality tolerance is too small ({0})," +
                                        " solution is orthogonal to the jacobian",
                                        new Object[] { new Double(orthoTolerance) });
        }

      }

    }

  }

  /** 
   * Determine the Levenberg-Marquardt parameter.
   * <p>This implementation is a translation in Java of the MINPACK
   * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
   * routine.</p>
   * <p>This method sets the lmPar and lmDir attributes.</p>
   * <p>The authors of the original fortran function are:</p>
   * <ul>
   *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
   *   <li>Burton  S. Garbow</li>
   *   <li>Kenneth E. Hillstrom</li>
   *   <li>Jorge   J. More</li>
   * </ul>
   * <p>Luc Maisonobe did the Java translation.</p>
   * 
   * @param qy array containing qTy
   * @param delta upper bound on the euclidean norm of diagR * lmDir
   * @param diag diagonal matrix
   * @param work1 work array
   * @param work2 work array
   * @param work3 work array
   */
  private void determineLMParameter(double[] qy, double delta, double[] diag,
                                    double[] work1, double[] work2, double[] work3) {

    // compute and store in x the gauss-newton direction, if the
    // jacobian is rank-deficient, obtain a least squares solution
    for (int j = 0; j < rank; ++j) {
      lmDir[permutation[j]] = qy[j];
    }
    for (int j = rank; j < cols; ++j) {
      lmDir[permutation[j]] = 0;
    }
    for (int k = rank - 1; k >= 0; --k) {
      int pk = permutation[k];
      double ypk = lmDir[pk] / diagR[pk];
      for (int i = 0, index = pk; i < k; ++i, index += cols) {
        lmDir[permutation[i]] -= ypk * jacobian[index];
      }
      lmDir[pk] = ypk;
    }

    // evaluate the function at the origin, and test
    // for acceptance of the Gauss-Newton direction
    double dxNorm = 0;
    for (int j = 0; j < solvedCols; ++j) {
      int pj = permutation[j];
      double s = diag[pj] * lmDir[pj];
      work1[pj] = s;
      dxNorm += s * s;
    }
    dxNorm = Math.sqrt(dxNorm);
    double fp = dxNorm - delta;
    if (fp <= 0.1 * delta) {
      lmPar = 0;
      return;
    }

    // if the jacobian is not rank deficient, the Newton step provides
    // a lower bound, parl, for the zero of the function,
    // otherwise set this bound to zero
    double sum2, parl = 0;
    if (rank == solvedCols) {
      for (int j = 0; j < solvedCols; ++j) {
        int pj = permutation[j];
        work1[pj] *= diag[pj] / dxNorm; 
      }
      sum2 = 0;
      for (int j = 0; j < solvedCols; ++j) {
        int pj = permutation[j];
        double sum = 0;
        for (int i = 0, index = pj; i < j; ++i, index += cols) {
          sum += jacobian[index] * work1[permutation[i]];
        }
        double s = (work1[pj] - sum) / diagR[pj];
        work1[pj] = s;
        sum2 += s * s;
      }
      parl = fp / (delta * sum2);
    }

    // calculate an upper bound, paru, for the zero of the function
    sum2 = 0;
    for (int j = 0; j < solvedCols; ++j) {
      int pj = permutation[j];
      double sum = 0;
      for (int i = 0, index = pj; i <= j; ++i, index += cols) {
        sum += jacobian[index] * qy[i];
      }
      sum /= diag[pj];
      sum2 += sum * sum;
    }
    double gNorm = Math.sqrt(sum2);
    double paru = gNorm / delta;
    if (paru == 0) {
      // 2.2251e-308 is the smallest positive real for IEE754
      paru = 2.2251e-308 / Math.min(delta, 0.1);
    }

    // if the input par lies outside of the interval (parl,paru),
    // set par to the closer endpoint
    lmPar = Math.min(paru, Math.max(lmPar, parl));
    if (lmPar == 0) {
      lmPar = gNorm / dxNorm;
    }

    for (int countdown = 10; countdown >= 0; --countdown) {

      // evaluate the function at the current value of lmPar
      if (lmPar == 0) {
        lmPar = Math.max(2.2251e-308, 0.001 * paru);
      }
      double sPar = Math.sqrt(lmPar);
      for (int j = 0; j < solvedCols; ++j) {
        int pj = permutation[j];
        work1[pj] = sPar * diag[pj];
      }
      determineLMDirection(qy, work1, work2, work3);

      dxNorm = 0;
      for (int j = 0; j < solvedCols; ++j) {
        int pj = permutation[j];
        double s = diag[pj] * lmDir[pj];
        work3[pj] = s;
        dxNorm += s * s;
      }
      dxNorm = Math.sqrt(dxNorm);
      double previousFP = fp;
      fp = dxNorm - delta;

      // if the function is small enough, accept the current value
      // of lmPar, also test for the exceptional cases where parl is zero
      if ((Math.abs(fp) <= 0.1 * delta) ||
          ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
        return;
      }
 
      // compute the Newton correction
      for (int j = 0; j < solvedCols; ++j) {
       int pj = permutation[j];
        work1[pj] = work3[pj] * diag[pj] / dxNorm; 
      }
      for (int j = 0; j < solvedCols; ++j) {
        int pj = permutation[j];
        work1[pj] /= work2[j];
        double tmp = work1[pj];
        for (int i = j + 1; i < solvedCols; ++i) {
          work1[permutation[i]] -= jacobian[i * cols + pj] * tmp;
        }
      }
      sum2 = 0;
      for (int j = 0; j < solvedCols; ++j) {
        double s = work1[permutation[j]];
        sum2 += s * s;
      }
      double correction = fp / (delta * sum2);