/*
    * Licensed to the Apache Software Foundation (ASF) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * The ASF licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
    * the License.  You may obtain a copy of the License at
    *
    *      http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  package org.apache.commons.math3.optim.nonlinear.vector.jacobian;
  
  import java.util.Arrays;
  import org.apache.commons.math3.exception.ConvergenceException;
  import org.apache.commons.math3.exception.MathUnsupportedOperationException;
  import org.apache.commons.math3.exception.util.LocalizedFormats;
  import org.apache.commons.math3.optim.PointVectorValuePair;
  import org.apache.commons.math3.optim.ConvergenceChecker;
  import org.apache.commons.math3.linear.RealMatrix;
  import org.apache.commons.math3.util.Precision;
  import org.apache.commons.math3.util.FastMath;
  
  
  /**
   * This class solves a least-squares problem using the Levenberg-Marquardt
   * algorithm.
   * <br/>
   * Constraints are not supported: the call to
   * {@link #optimize(OptimizationData[]) optimize} will throw
   * {@link MathUnsupportedOperationException} if bounds are passed to it.
   *
   * <p>This implementation <em>should</em> work even for over-determined systems
   * (i.e. systems having more point than equations). Over-determined systems
   * are solved by ignoring the point which have the smallest impact according
   * to their jacobian column norm. Only the rank of the matrix and some loop bounds
   * are changed to implement this.</p>
   *
   * <p>The resolution engine is a simple translation of the MINPACK <a
   * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
   * changes. The changes include the over-determined resolution, the use of
   * inherited convergence checker and the Q.R. decomposition which has been
   * rewritten following the algorithm described in the
   * P. Lascaux and R. Theodor book <i>Analyse num&eacute;rique matricielle
   * appliqu&eacute;e &agrave; l'art de l'ing&eacute;nieur</i>, Masson 1986.</p>
   * <p>The authors of the original fortran version are:
   * <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>
   * The redistribution policy for MINPACK is available <a
   * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
   * is reproduced below.</p>
   *
   * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
   * <tr><td>
   *    Minpack Copyright Notice (1999) University of Chicago.
   *    All rights reserved
   * </td></tr>
   * <tr><td>
   * Redistribution and use in source and binary forms, with or without
   * modification, are permitted provided that the following conditions
   * are met:
   * <ol>
   *  <li>Redistributions of source code must retain the above copyright
   *      notice, this list of conditions and the following disclaimer.</li>
   * <li>Redistributions in binary form must reproduce the above
   *     copyright notice, this list of conditions and the following
   *     disclaimer in the documentation and/or other materials provided
   *     with the distribution.</li>
   * <li>The end-user documentation included with the redistribution, if any,
   *     must include the following acknowledgment:
   *     <code>This product includes software developed by the University of
   *           Chicago, as Operator of Argonne National Laboratory.</code>
   *     Alternately, this acknowledgment may appear in the software itself,
   *     if and wherever such third-party acknowledgments normally appear.</li>
   * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
   *     WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
   *     UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
   *     THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
   *     IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
   *     OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
   *     OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
   *     OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
   *     USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
   *     THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
   *     DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
   *     UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
   *     BE CORRECTED.</strong></li>
   * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
   *     HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
   *     ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
  *     INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
  *     ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
  *     PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
  *     SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
  *     (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
  *     EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
  *     POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
  * <ol></td></tr>
  * </table>
  *
  * @version $Id: LevenbergMarquardtOptimizer.java 1462503 2013-03-29 15:48:27Z luc $
  * @since 2.0
  */
 public class LevenbergMarquardtOptimizer
     extends AbstractLeastSquaresOptimizer {
     /** Twice the "epsilon machine". */
     private static final double TWO_EPS = 2 * Precision.EPSILON;
     /** Number of solved point. */
     private int solvedCols;
     /** Diagonal elements of the R matrix in the Q.R. decomposition. */
     private double[] diagR;
     /** Norms of the columns of the jacobian matrix. */
     private double[] jacNorm;
     /** Coefficients of the Householder transforms vectors. */
     private double[] beta;
     /** Columns permutation array. */
     private int[] permutation;
     /** Rank of the jacobian matrix. */
     private int rank;
     /** Levenberg-Marquardt parameter. */
     private double lmPar;
     /** Parameters evolution direction associated with lmPar. */
     private double[] lmDir;
     /** Positive input variable used in determining the initial step bound. */
     private final double initialStepBoundFactor;
     /** Desired relative error in the sum of squares. */
     private final double costRelativeTolerance;
     /**  Desired relative error in the approximate solution parameters. */
     private final double parRelativeTolerance;
     /** Desired max cosine on the orthogonality between the function vector
      * and the columns of the jacobian. */
     private final double orthoTolerance;
     /** Threshold for QR ranking. */
     private final double qrRankingThreshold;
     /** Weighted residuals. */
     private double[] weightedResidual;
     /** Weighted Jacobian. */
     private double[][] weightedJacobian;
 
     /**
      * Build an optimizer for least squares problems with default values
      * for all the tuning parameters (see the {@link
      * #LevenbergMarquardtOptimizer(double,double,double,double,double)
      * other contructor}.
      * The default values for the algorithm settings are:
      * <ul>
      *  <li>Initial step bound factor: 100</li>
      *  <li>Cost relative tolerance: 1e-10</li>
      *  <li>Parameters relative tolerance: 1e-10</li>
      *  <li>Orthogonality tolerance: 1e-10</li>
      *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
      * </ul>
      */
     public LevenbergMarquardtOptimizer() {
         this(100, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
     }
 
     /**
      * Constructor that allows the specification of a custom convergence
      * checker.
      * Note that all the usual convergence checks will be <em>disabled</em>.
      * The default values for the algorithm settings are:
      * <ul>
      *  <li>Initial step bound factor: 100</li>
      *  <li>Cost relative tolerance: 1e-10</li>
      *  <li>Parameters relative tolerance: 1e-10</li>
      *  <li>Orthogonality tolerance: 1e-10</li>
      *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
      * </ul>
      *
      * @param checker Convergence checker.
      */
     public LevenbergMarquardtOptimizer(ConvergenceChecker<PointVectorValuePair> checker) {
         this(100, checker, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
     }
 
     /**
      * Constructor that allows the specification of a custom convergence
      * checker, in addition to the standard ones.
      *
      * @param initialStepBoundFactor Positive input variable used in
      * determining the initial step bound. This bound is set to the
      * product of initialStepBoundFactor and the euclidean norm of
      * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
      * itself. In most cases factor should lie in the interval
      * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
      * @param checker Convergence checker.
      * @param costRelativeTolerance Desired relative error in the sum of
      * squares.
      * @param parRelativeTolerance Desired relative error in the approximate
      * solution parameters.
      * @param orthoTolerance Desired max cosine on the orthogonality between
      * the function vector and the columns of the Jacobian.
      * @param threshold Desired threshold for QR ranking. If the squared norm
      * of a column vector is smaller or equal to this threshold during QR
      * decomposition, it is considered to be a zero vector and hence the rank
      * of the matrix is reduced.
      */
     public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
                                        ConvergenceChecker<PointVectorValuePair> checker,
                                        double costRelativeTolerance,
                                        double parRelativeTolerance,
                                        double orthoTolerance,
                                        double threshold) {
         super(checker);
         this.initialStepBoundFactor = initialStepBoundFactor;
         this.costRelativeTolerance = costRelativeTolerance;
         this.parRelativeTolerance = parRelativeTolerance;
         this.orthoTolerance = orthoTolerance;
         this.qrRankingThreshold = threshold;
     }
 
     /**
      * Build an optimizer for least squares problems with default values
      * for some of the tuning parameters (see the {@link
      * #LevenbergMarquardtOptimizer(double,double,double,double,double)
      * other contructor}.
      * The default values for the algorithm settings are:
      * <ul>
      *  <li>Initial step bound factor}: 100</li>
      *  <li>QR ranking threshold}: {@link Precision#SAFE_MIN}</li>
      * </ul>
      *
      * @param costRelativeTolerance Desired relative error in the sum of
      * squares.
      * @param parRelativeTolerance Desired relative error in the approximate
      * solution parameters.
      * @param orthoTolerance Desired max cosine on the orthogonality between
      * the function vector and the columns of the Jacobian.
      */
     public LevenbergMarquardtOptimizer(double costRelativeTolerance,
                                        double parRelativeTolerance,
                                        double orthoTolerance) {
         this(100,
              costRelativeTolerance, parRelativeTolerance, orthoTolerance,
              Precision.SAFE_MIN);
     }
 
     /**
      * The arguments control the behaviour of the default convergence checking
      * procedure.
      * Additional criteria can defined through the setting of a {@link
      * ConvergenceChecker}.
      *
      * @param initialStepBoundFactor Positive input variable used in
      * determining the initial step bound. This bound is set to the
      * product of initialStepBoundFactor and the euclidean norm of
      * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
      * itself. In most cases factor should lie in the interval
      * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
      * @param costRelativeTolerance Desired relative error in the sum of
      * squares.
      * @param parRelativeTolerance Desired relative error in the approximate
      * solution parameters.
      * @param orthoTolerance Desired max cosine on the orthogonality between
      * the function vector and the columns of the Jacobian.
      * @param threshold Desired threshold for QR ranking. If the squared norm
      * of a column vector is smaller or equal to this threshold during QR
      * decomposition, it is considered to be a zero vector and hence the rank
      * of the matrix is reduced.
      */
     public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
                                        double costRelativeTolerance,
                                        double parRelativeTolerance,
                                        double orthoTolerance,
                                        double threshold) {
         super(null); // No custom convergence criterion.
         this.initialStepBoundFactor = initialStepBoundFactor;
         this.costRelativeTolerance = costRelativeTolerance;
         this.parRelativeTolerance = parRelativeTolerance;
         this.orthoTolerance = orthoTolerance;
         this.qrRankingThreshold = threshold;
     }
 
     /** {@inheritDoc} */
     @Override
     protected PointVectorValuePair doOptimize() {
         checkParameters();
 
         final int nR = getTarget().length; // Number of observed data.
         final double[] currentPoint = getStartPoint();
         final int nC = currentPoint.length; // Number of parameters.
 
         // arrays shared with the other private methods
         solvedCols  = FastMath.min(nR, nC);
         diagR       = new double[nC];
         jacNorm     = new double[nC];
         beta        = new double[nC];
         permutation = new int[nC];
         lmDir       = new double[nC];
 
         // local point
         double   delta   = 0;
         double   xNorm   = 0;
         double[] diag    = new double[nC];
         double[] oldX    = new double[nC];
         double[] oldRes  = new double[nR];
         double[] oldObj  = new double[nR];
         double[] qtf     = new double[nR];
         double[] work1   = new double[nC];
         double[] work2   = new double[nC];
         double[] work3   = new double[nC];
 
         final RealMatrix weightMatrixSqrt = getWeightSquareRoot();
 
         // Evaluate the function at the starting point and calculate its norm.
         double[] currentObjective = computeObjectiveValue(currentPoint);
         double[] currentResiduals = computeResiduals(currentObjective);
         PointVectorValuePair current = new PointVectorValuePair(currentPoint, currentObjective);
         double currentCost = computeCost(currentResiduals);
 
         // Outer loop.
         lmPar = 0;
         boolean firstIteration = true;
         final ConvergenceChecker<PointVectorValuePair> checker = getConvergenceChecker();
         while (true) {
             incrementIterationCount();
 
             final PointVectorValuePair previous = current;
 
             // QR decomposition of the jacobian matrix
             qrDecomposition(computeWeightedJacobian(currentPoint));
 
             weightedResidual = weightMatrixSqrt.operate(currentResiduals);
             for (int i = 0; i < nR; i++) {
                 qtf[i] = weightedResidual[i];
             }
 
             // compute Qt.res
             qTy(qtf);
 
             // now we don't need Q anymore,
             // so let jacobian contain the R matrix with its diagonal elements
             for (int k = 0; k < solvedCols; ++k) {
                 int pk = permutation[k];
                 weightedJacobian[k][pk] = diagR[pk];
             }
 
             if (firstIteration) {
                 // scale the point according to the norms of the columns
                 // of the initial jacobian
                 xNorm = 0;
                 for (int k = 0; k < nC; ++k) {
                     double dk = jacNorm[k];
                     if (dk == 0) {
                         dk = 1.0;
                     }
                     double xk = dk * currentPoint[k];
                     xNorm  += xk * xk;
                     diag[k] = dk;
                 }
                 xNorm = FastMath.sqrt(xNorm);
 
                 // initialize the step bound delta
                 delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
             }
 
             // check orthogonality between function vector and jacobian columns
             double maxCosine = 0;
             if (currentCost != 0) {
                 for (int j = 0; j < solvedCols; ++j) {
                     int    pj = permutation[j];
                     double s  = jacNorm[pj];
                     if (s != 0) {
                         double sum = 0;
                         for (int i = 0; i <= j; ++i) {
                             sum += weightedJacobian[i][pj] * qtf[i];
                         }
                         maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * currentCost));
                     }
                 }
             }
             if (maxCosine <= orthoTolerance) {
                 // Convergence has been reached.
                 setCost(currentCost);
                 return current;
             }
 
             // rescale if necessary
             for (int j = 0; j < nC; ++j) {
                 diag[j] = FastMath.max(diag[j], jacNorm[j]);
             }
 
             // Inner loop.
             for (double ratio = 0; ratio < 1.0e-4;) {
 
                 // save the state
                 for (int j = 0; j < solvedCols; ++j) {
                     int pj = permutation[j];
                     oldX[pj] = currentPoint[pj];
                 }
                 final double previousCost = currentCost;
                 double[] tmpVec = weightedResidual;
                 weightedResidual = oldRes;
                 oldRes    = tmpVec;
                 tmpVec    = currentObjective;
                 currentObjective = oldObj;
                 oldObj    = tmpVec;
 
                 // determine the Levenberg-Marquardt parameter
                 determineLMParameter(qtf, 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];
                     currentPoint[pj] = oldX[pj] + lmDir[pj];
                     double s = diag[pj] * lmDir[pj];
                     lmNorm  += s * s;
                 }
                 lmNorm = FastMath.sqrt(lmNorm);
                 // on the first iteration, adjust the initial step bound.
                 if (firstIteration) {
                     delta = FastMath.min(delta, lmNorm);
                 }
 
                 // Evaluate the function at x + p and calculate its norm.
                 currentObjective = computeObjectiveValue(currentPoint);
                 currentResiduals = computeResiduals(currentObjective);
                 current = new PointVectorValuePair(currentPoint, currentObjective);
                 currentCost = computeCost(currentResiduals);
 
                 // compute the scaled actual reduction
                 double actRed = -1.0;
                 if (0.1 * currentCost < previousCost) {
                     double r = currentCost / 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; i <= j; ++i) {
                         work1[i] += weightedJacobian[i][pj] * 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 * currentCost >= previousCost) || (tmp < 0.1)) {
                             tmp = 0.1;
                         }
                         delta = tmp * FastMath.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 < nC; ++k) {
                         double xK = diag[k] * currentPoint[k];
                         xNorm += xK * xK;
                     }
                     xNorm = FastMath.sqrt(xNorm);
 
                     // tests for convergence.
                     if (checker != null && checker.converged(getIterations(), previous, current)) {
                         setCost(currentCost);
                         return current;
                     }
                 } else {
                     // failed iteration, reset the previous values
                     currentCost = previousCost;
                     for (int j = 0; j < solvedCols; ++j) {
                         int pj = permutation[j];
                         currentPoint[pj] = oldX[pj];
                     }
                     tmpVec    = weightedResidual;
                     weightedResidual = oldRes;
                     oldRes    = tmpVec;
                     tmpVec    = currentObjective;
                     currentObjective = oldObj;
                     oldObj    = tmpVec;
                     // Reset "current" to previous values.
                     current = new PointVectorValuePair(currentPoint, currentObjective);
                 }
 
                 // Default convergence criteria.
                 if ((FastMath.abs(actRed) <= costRelativeTolerance &&
                      preRed <= costRelativeTolerance &&
                      ratio <= 2.0) ||
                     delta <= parRelativeTolerance * xNorm) {
                     setCost(currentCost);
                     return current;
                 }
 
                 // tests for termination and stringent tolerances
                 if (FastMath.abs(actRed) <= TWO_EPS &&
                     preRed <= TWO_EPS &&
                     ratio <= 2.0) {
                     throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
                                                    costRelativeTolerance);
                 } else if (delta <= TWO_EPS * xNorm) {
                     throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
                                                    parRelativeTolerance);
                 } else if (maxCosine <= TWO_EPS) {
                     throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
                                                    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) {
         final int nC = weightedJacobian[0].length;
 
         // 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 < nC; ++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; i < k; ++i) {
                 lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk];
             }
             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 = FastMath.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;
         double 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; i < j; ++i) {
                     sum += weightedJacobian[i][pj] * 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; i <= j; ++i) {
                 sum += weightedJacobian[i][pj] * qy[i];
             }
             sum /= diag[pj];
             sum2 += sum * sum;
         }
         double gNorm = FastMath.sqrt(sum2);
         double paru = gNorm / delta;
         if (paru == 0) {
             paru = Precision.SAFE_MIN / FastMath.min(delta, 0.1);
         }
 
         // if the input par lies outside of the interval (parl,paru),
         // set par to the closer endpoint
         lmPar = FastMath.min(paru, FastMath.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 = FastMath.max(Precision.SAFE_MIN, 0.001 * paru);
             }
             double sPar = FastMath.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 = FastMath.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 ((FastMath.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]] -= weightedJacobian[i][pj] * tmp;
                 }
             }
             sum2 = 0;
             for (int j = 0; j < solvedCols; ++j) {
                 double s = work1[permutation[j]];
                 sum2 += s * s;
             }
             double correction = fp / (delta * sum2);
 
             // depending on the sign of the function, update parl or paru.
             if (fp > 0) {
                 parl = FastMath.max(parl, lmPar);
             } else if (fp < 0) {
                 paru = FastMath.min(paru, lmPar);
             }
 
             // compute an improved estimate for lmPar
             lmPar = FastMath.max(parl, lmPar + correction);
 
         }
     }
 
     /**
      * Solve a*x = b and d*x = 0 in the least squares sense.
      * <p>This implementation is a translation in Java of the MINPACK
      * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
      * routine.</p>
      * <p>This method sets the lmDir and lmDiag 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 diag diagonal matrix
      * @param lmDiag diagonal elements associated with lmDir
      * @param work work array
      */
     private void determineLMDirection(double[] qy, double[] diag,
                                       double[] lmDiag, double[] work) {
 
         // copy R and Qty to preserve input and initialize s
         //  in particular, save the diagonal elements of R in lmDir
         for (int j = 0; j < solvedCols; ++j) {
             int pj = permutation[j];
             for (int i = j + 1; i < solvedCols; ++i) {
                 weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]];
             }
             lmDir[j] = diagR[pj];
             work[j]  = qy[j];
         }
 
         // eliminate the diagonal matrix d using a Givens rotation
         for (int j = 0; j < solvedCols; ++j) {
 
             // prepare the row of d to be eliminated, locating the
             // diagonal element using p from the Q.R. factorization
             int pj = permutation[j];
             double dpj = diag[pj];
             if (dpj != 0) {
                 Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
             }
             lmDiag[j] = dpj;
 
             //  the transformations to eliminate the row of d
             // modify only a single element of Qty
             // beyond the first n, which is initially zero.
             double qtbpj = 0;
             for (int k = j; k < solvedCols; ++k) {
                 int pk = permutation[k];
 
                 // determine a Givens rotation which eliminates the
                 // appropriate element in the current row of d
                 if (lmDiag[k] != 0) {
 
                     final double sin;
                     final double cos;
                     double rkk = weightedJacobian[k][pk];
                     if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
                         final double cotan = rkk / lmDiag[k];
                         sin   = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
                         cos   = sin * cotan;
                     } else {
                         final double tan = lmDiag[k] / rkk;
                         cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
                         sin = cos * tan;
                     }
 
                     // compute the modified diagonal element of R and
                     // the modified element of (Qty,0)
                     weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
                     final double temp = cos * work[k] + sin * qtbpj;
                     qtbpj = -sin * work[k] + cos * qtbpj;
                     work[k] = temp;
 
                     // accumulate the tranformation in the row of s
                     for (int i = k + 1; i < solvedCols; ++i) {
                         double rik = weightedJacobian[i][pk];
                         final double temp2 = cos * rik + sin * lmDiag[i];
                         lmDiag[i] = -sin * rik + cos * lmDiag[i];
                         weightedJacobian[i][pk] = temp2;
                     }
                 }
             }
 
             // store the diagonal element of s and restore
             // the corresponding diagonal element of R
             lmDiag[j] = weightedJacobian[j][permutation[j]];
             weightedJacobian[j][permutation[j]] = lmDir[j];
         }
 
         // solve the triangular system for z, if the system is
         // singular, then obtain a least squares solution
         int nSing = solvedCols;
         for (int j = 0; j < solvedCols; ++j) {
             if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
                 nSing = j;
             }
             if (nSing < solvedCols) {
                 work[j] = 0;
             }
         }
         if (nSing > 0) {
             for (int j = nSing - 1; j >= 0; --j) {
                 int pj = permutation[j];
                 double sum = 0;
                 for (int i = j + 1; i < nSing; ++i) {
                     sum += weightedJacobian[i][pj] * work[i];
                 }
                 work[j] = (work[j] - sum) / lmDiag[j];
             }
         }
 
         // permute the components of z back to components of lmDir
         for (int j = 0; j < lmDir.length; ++j) {
             lmDir[permutation[j]] = work[j];
         }
     }
 
     /**
      * Decompose a matrix A as A.P = Q.R using Householder transforms.
      * <p>As suggested in the P. Lascaux and R. Theodor book
      * <i>Analyse num&eacute;rique matricielle appliqu&eacute;e &agrave;
      * l'art de l'ing&eacute;nieur</i> (Masson, 1986), instead of representing
      * the Householder transforms with u<sub>k</sub> unit vectors such that:
      * <pre>
      * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
      * </pre>
      * we use <sub>k</sub> non-unit vectors such that:
      * <pre>
      * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
      * </pre>
      * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
      * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
      * them from the v<sub>k</sub> vectors would be costly.</p>
      * <p>This decomposition handles rank deficient cases since the tranformations
      * are performed in non-increasing columns norms order thanks to columns
      * pivoting. The diagonal elements of the R matrix are therefore also in
      * non-increasing absolute values order.</p>
      *
      * @param jacobian Weighted Jacobian matrix at the current point.
      * @exception ConvergenceException if the decomposition cannot be performed
      */
     private void qrDecomposition(RealMatrix jacobian) throws ConvergenceException {
         // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
         // hence the multiplication by -1.
         weightedJacobian = jacobian.scalarMultiply(-1).getData();
 
         final int nR = weightedJacobian.length;
         final int nC = weightedJacobian[0].length;
 
         // initializations
         for (int k = 0; k < nC; ++k) {
             permutation[k] = k;
             double norm2 = 0;
             for (int i = 0; i < nR; ++i) {
                 double akk = weightedJacobian[i][k];
                 norm2 += akk * akk;
             }
             jacNorm[k] = FastMath.sqrt(norm2);
         }
 
         // transform the matrix column after column
         for (int k = 0; k < nC; ++k) {
 
             // select the column with the greatest norm on active components
             int nextColumn = -1;
             double ak2 = Double.NEGATIVE_INFINITY;
             for (int i = k; i < nC; ++i) {
                 double norm2 = 0;
                 for (int j = k; j < nR; ++j) {
                     double aki = weightedJacobian[j][permutation[i]];
                     norm2 += aki * aki;
                 }
                 if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
                     throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
                                                    nR, nC);
                 }
                 if (norm2 > ak2) {
                     nextColumn = i;
                     ak2        = norm2;
                 }
             }
             if (ak2 <= qrRankingThreshold) {
                 rank = k;
                 return;
             }
             int pk                  = permutation[nextColumn];
             permutation[nextColumn] = permutation[k];
             permutation[k]          = pk;
 
             // choose alpha such that Hk.u = alpha ek
             double akk   = weightedJacobian[k][pk];
             double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
             double betak = 1.0 / (ak2 - akk * alpha);
             beta[pk]     = betak;
 
             // transform the current column
             diagR[pk]        = alpha;
             weightedJacobian[k][pk] -= alpha;
 
             // transform the remaining columns
             for (int dk = nC - 1 - k; dk > 0; --dk) {
                 double gamma = 0;
                 for (int j = k; j < nR; ++j) {
                     gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]];
                 }
                 gamma *= betak;
                 for (int j = k; j < nR; ++j) {
                     weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk];
                 }
             }
         }
         rank = solvedCols;
     }
 
     /**
      * Compute the product Qt.y for some Q.R. decomposition.
      *
      * @param y vector to multiply (will be overwritten with the result)
      */
     private void qTy(double[] y) {
         final int nR = weightedJacobian.length;
         final int nC = weightedJacobian[0].length;
 
         for (int k = 0; k < nC; ++k) {
             int pk = permutation[k];
             double gamma = 0;
             for (int i = k; i < nR; ++i) {
                 gamma += weightedJacobian[i][pk] * y[i];
             }
             gamma *= beta[pk];
             for (int i = k; i < nR; ++i) {
                 y[i] -= gamma * weightedJacobian[i][pk];
             }
         }
     }
 
     /**
      * @throws MathUnsupportedOperationException if bounds were passed to the
      * {@link #optimize(OptimizationData[]) optimize} method.
      */
     private void checkParameters() {
         if (getLowerBound() != null ||
             getUpperBound() != null) {
             throw new MathUnsupportedOperationException(LocalizedFormats.CONSTRAINT);
         }
     }
 }