/*
    * 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
    *
    *      https://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.
   */
  
  /*
   * This is not the original file distributed by the Apache Software Foundation
   * It has been modified by the Hipparchus project
   */
  package org.hipparchus.optim.nonlinear.vector.leastsquares;
  
  import java.util.Arrays;
  
  import org.hipparchus.exception.MathIllegalStateException;
  import org.hipparchus.linear.ArrayRealVector;
  import org.hipparchus.linear.RealMatrix;
  import org.hipparchus.optim.ConvergenceChecker;
  import org.hipparchus.optim.LocalizedOptimFormats;
  import org.hipparchus.optim.nonlinear.vector.leastsquares.LeastSquaresProblem.Evaluation;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.Incrementor;
  import org.hipparchus.util.Precision;
  
  
  /**
   * This class solves a least-squares problem using the Levenberg-Marquardt
   * algorithm.
   *
   * <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:</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>The redistribution policy for MINPACK is available <a
   * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
   * is reproduced below.</p>
   *
   * <blockquote>
   * <p>
   *    Minpack Copyright Notice (1999) University of Chicago.
   *    All rights reserved
   * </p>
   * <p>
   * Redistribution and use in source and binary forms, with or without
   * modification, are permitted provided that the following conditions
   * are met:</p>
   * <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>
  * </blockquote>
  *
  */
 public class LevenbergMarquardtOptimizer implements LeastSquaresOptimizer {
 
     /** Twice the "epsilon machine". */
     private static final double TWO_EPS = 2 * Precision.EPSILON;
 
     /* configuration parameters */
     /** 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;
 
     /** Default constructor.
      * <p>
      * 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);
     }
 
     /**
      * Construct an instance with all parameters specified.
      *
      * @param initialStepBoundFactor initial step bound factor
      * @param costRelativeTolerance  cost relative tolerance
      * @param parRelativeTolerance   parameters relative tolerance
      * @param orthoTolerance         orthogonality tolerance
      * @param qrRankingThreshold     threshold in the QR decomposition. Columns with a 2
      *                               norm less than this threshold are considered to be
      *                               all 0s.
      */
     public LevenbergMarquardtOptimizer(
             final double initialStepBoundFactor,
             final double costRelativeTolerance,
             final double parRelativeTolerance,
             final double orthoTolerance,
             final double qrRankingThreshold) {
         this.initialStepBoundFactor = initialStepBoundFactor;
         this.costRelativeTolerance = costRelativeTolerance;
         this.parRelativeTolerance = parRelativeTolerance;
         this.orthoTolerance = orthoTolerance;
         this.qrRankingThreshold = qrRankingThreshold;
     }
 
     /** Build new instance with initial step bound factor.
      * @param newInitialStepBoundFactor 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 newInitialStepBoundFactor}
      * itself. In most cases factor should lie in the interval
      * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
      * of the matrix is reduced.
      * @return a new instance.
      */
     public LevenbergMarquardtOptimizer withInitialStepBoundFactor(double newInitialStepBoundFactor) {
         return new LevenbergMarquardtOptimizer(
                 newInitialStepBoundFactor,
                 costRelativeTolerance,
                 parRelativeTolerance,
                 orthoTolerance,
                 qrRankingThreshold);
     }
 
     /** Build new instance with cost relative tolerance.
      * @param newCostRelativeTolerance Desired relative error in the sum of squares.
      * @return a new instance.
      */
     public LevenbergMarquardtOptimizer withCostRelativeTolerance(double newCostRelativeTolerance) {
         return new LevenbergMarquardtOptimizer(
                 initialStepBoundFactor,
                 newCostRelativeTolerance,
                 parRelativeTolerance,
                 orthoTolerance,
                 qrRankingThreshold);
     }
 
     /** Build new instance with parameter relative tolerance.
      * @param newParRelativeTolerance Desired relative error in the approximate solution
      * parameters.
      * @return a new instance.
      */
     public LevenbergMarquardtOptimizer withParameterRelativeTolerance(double newParRelativeTolerance) {
         return new LevenbergMarquardtOptimizer(
                 initialStepBoundFactor,
                 costRelativeTolerance,
                 newParRelativeTolerance,
                 orthoTolerance,
                 qrRankingThreshold);
     }
 
     /** Build new instance with ortho tolerance.
      * @param newOrthoTolerance Desired max cosine on the orthogonality between
      * the function vector and the columns of the Jacobian.
      * @return a new instance.
      */
     public LevenbergMarquardtOptimizer withOrthoTolerance(double newOrthoTolerance) {
         return new LevenbergMarquardtOptimizer(
                 initialStepBoundFactor,
                 costRelativeTolerance,
                 parRelativeTolerance,
                 newOrthoTolerance,
                 qrRankingThreshold);
     }
 
     /** Build new instance with ranking threshold.
      * @param newQRRankingThreshold 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.
      * @return a new instance.
      */
     public LevenbergMarquardtOptimizer withRankingThreshold(double newQRRankingThreshold) {
         return new LevenbergMarquardtOptimizer(
                 initialStepBoundFactor,
                 costRelativeTolerance,
                 parRelativeTolerance,
                 orthoTolerance,
                 newQRRankingThreshold);
     }
 
     /**
      * Gets the value of a tuning parameter.
      * @see #withInitialStepBoundFactor(double)
      *
      * @return the parameter's value.
      */
     public double getInitialStepBoundFactor() {
         return initialStepBoundFactor;
     }
 
     /**
      * Gets the value of a tuning parameter.
      * @see #withCostRelativeTolerance(double)
      *
      * @return the parameter's value.
      */
     public double getCostRelativeTolerance() {
         return costRelativeTolerance;
     }
 
     /**
      * Gets the value of a tuning parameter.
      * @see #withParameterRelativeTolerance(double)
      *
      * @return the parameter's value.
      */
     public double getParameterRelativeTolerance() {
         return parRelativeTolerance;
     }
 
     /**
      * Gets the value of a tuning parameter.
      * @see #withOrthoTolerance(double)
      *
      * @return the parameter's value.
      */
     public double getOrthoTolerance() {
         return orthoTolerance;
     }
 
     /**
      * Gets the value of a tuning parameter.
      * @see #withRankingThreshold(double)
      *
      * @return the parameter's value.
      */
     public double getRankingThreshold() {
         return qrRankingThreshold;
     }
 
     /** {@inheritDoc} */
     @Override
     public Optimum optimize(final LeastSquaresProblem problem) {
         // Pull in relevant data from the problem as locals.
         final int nR = problem.getObservationSize(); // Number of observed data.
         final int nC = problem.getParameterSize(); // Number of parameters.
         // Counters.
         final Incrementor iterationCounter = problem.getIterationCounter();
         final Incrementor evaluationCounter = problem.getEvaluationCounter();
         // Convergence criterion.
         final ConvergenceChecker<Evaluation> checker = problem.getConvergenceChecker();
 
         // arrays shared with the other private methods
         final int solvedCols  = FastMath.min(nR, nC);
         /* Parameters evolution direction associated with lmPar. */
         double[] lmDir = new double[nC];
         /* Levenberg-Marquardt parameter. */
         double lmPar = 0;
 
         // local point
         double   delta   = 0;
         double   xNorm   = 0;
         double[] diag    = new double[nC];
         double[] oldX    = new double[nC];
         double[] oldRes  = new double[nR];
         double[] qtf     = new double[nR];
         double[] work1   = new double[nC];
         double[] work2   = new double[nC];
         double[] work3   = new double[nC];
 
 
         // Evaluate the function at the starting point and calculate its norm.
         evaluationCounter.increment();
         //value will be reassigned in the loop
         Evaluation current = problem.evaluate(problem.getStart());
         double[] currentResiduals = current.getResiduals().toArray();
         double currentCost = current.getCost();
         double[] currentPoint = current.getPoint().toArray();
 
         // Outer loop.
         boolean firstIteration = true;
         while (true) {
             iterationCounter.increment();
 
             final Evaluation previous = current;
 
             // QR decomposition of the jacobian matrix
             final InternalData internalData = qrDecomposition(current.getJacobian(), solvedCols);
             final double[][] weightedJacobian = internalData.weightedJacobian;
             final int[] permutation = internalData.permutation;
             final double[] diagR = internalData.diagR;
             final double[] jacNorm = internalData.jacNorm;
 
             //residuals already have weights applied
             double[] weightedResidual = currentResiduals;
             for (int i = 0; i < nR; i++) {
                 qtf[i] = weightedResidual[i];
             }
 
             // compute Qt.res
             qTy(qtf, internalData);
 
             // 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.
                 return Optimum.of(
                         current,
                         evaluationCounter.getCount(),
                         iterationCounter.getCount());
             }
 
             // 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;
 
                 // determine the Levenberg-Marquardt parameter
                 lmPar = determineLMParameter(qtf, delta, diag,
                                      internalData, solvedCols,
                                      work1, work2, work3, lmDir, lmPar);
 
                 // 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.
                 evaluationCounter.increment();
                 current = problem.evaluate(new ArrayRealVector(currentPoint));
                 currentResiduals = current.getResiduals().toArray();
                 currentCost = current.getCost();
                 currentPoint = current.getPoint().toArray();
 
                 // 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 /= 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(iterationCounter.getCount(), previous, current)) {
                         return Optimum.of(current, evaluationCounter.getCount(), iterationCounter.getCount());
                     }
                 } 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;
                     // Reset "current" to previous values.
                     current = previous;
                 }
 
                 // Default convergence criteria.
                 if ((FastMath.abs(actRed) <= costRelativeTolerance &&
                      preRed <= costRelativeTolerance &&
                      ratio <= 2.0) ||
                     delta <= parRelativeTolerance * xNorm) {
                     return Optimum.of(current, evaluationCounter.getCount(), iterationCounter.getCount());
                 }
 
                 // tests for termination and stringent tolerances
                 if (FastMath.abs(actRed) <= TWO_EPS &&
                     preRed <= TWO_EPS &&
                     ratio <= 2.0) {
                     throw new MathIllegalStateException(LocalizedOptimFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
                                                         costRelativeTolerance);
                 } else if (delta <= TWO_EPS * xNorm) {
                     throw new MathIllegalStateException(LocalizedOptimFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
                                                         parRelativeTolerance);
                 } else if (maxCosine <= TWO_EPS) {
                     throw new MathIllegalStateException(LocalizedOptimFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
                                                         orthoTolerance);
                 }
             }
         }
     }
 
     /**
      * Holds internal data.
      * This structure was created so that all optimizer fields can be "final".
      * Code should be further refactored in order to not pass around arguments
      * that will modified in-place (cf. "work" arrays).
      */
     private static class InternalData {
         /** Weighted Jacobian. */
         private final double[][] weightedJacobian;
         /** Columns permutation array. */
         private final int[] permutation;
         /** Rank of the Jacobian matrix. */
         private final int rank;
         /** Diagonal elements of the R matrix in the QR decomposition. */
         private final double[] diagR;
         /** Norms of the columns of the jacobian matrix. */
         private final double[] jacNorm;
         /** Coefficients of the Householder transforms vectors. */
         private final double[] beta;
 
         /**
          * <p>
          * All arrays are stored by reference
          * </p>
          * @param weightedJacobian Weighted Jacobian.
          * @param permutation Columns permutation array.
          * @param rank Rank of the Jacobian matrix.
          * @param diagR Diagonal elements of the R matrix in the QR decomposition.
          * @param jacNorm Norms of the columns of the jacobian matrix.
          * @param beta Coefficients of the Householder transforms vectors.
          */
         InternalData(double[][] weightedJacobian,
                      int[] permutation,
                      int rank,
                      double[] diagR,
                      double[] jacNorm,
                      double[] beta) {
             this.weightedJacobian = weightedJacobian; // NOPMD - staring array references is intentional and documented here
             this.permutation = permutation;           // NOPMD - staring array references is intentional and documented here
             this.rank = rank;
             this.diagR = diagR;                       // NOPMD - staring array references is intentional and documented here
             this.jacNorm = jacNorm;                   // NOPMD - staring array references is intentional and documented here
             this.beta = beta;                         // NOPMD - staring array references is intentional and documented here
         }
     }
 
     /**
      * Determines 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 internalData Data (modified in-place in this method).
      * @param solvedCols Number of solved point.
      * @param work1 work array
      * @param work2 work array
      * @param work3 work array
      * @param lmDir the "returned" LM direction will be stored in this array.
      * @param lmPar the value of the LM parameter from the previous iteration.
      * @return the new LM parameter
      */
     private double determineLMParameter(double[] qy, double delta, double[] diag,
                                       InternalData internalData, int solvedCols,
                                       double[] work1, double[] work2, double[] work3,
                                       double[] lmDir, double lmPar) {
         final double[][] weightedJacobian = internalData.weightedJacobian;
         final int[] permutation = internalData.permutation;
         final int rank = internalData.rank;
         final double[] diagR = internalData.diagR;
 
         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 lmPar;
         }
 
         // 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, internalData, solvedCols, work3, lmDir);
 
             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 lmPar;
             }
 
             // 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);
         }
 
         return lmPar;
     }
 
     /**
      * 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 internalData Data (modified in-place in this method).
      * @param solvedCols Number of sloved point.
      * @param work work array
      * @param lmDir the "returned" LM direction is stored in this array
      */
     private void determineLMDirection(double[] qy, double[] diag,
                                       double[] lmDiag,
                                       InternalData internalData,
                                       int solvedCols,
                                       double[] work,
                                       double[] lmDir) {
         final int[] permutation = internalData.permutation;
         final double[][] weightedJacobian = internalData.weightedJacobian;
         final double[] diagR = internalData.diagR;
 
         // 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.
      * @param solvedCols Number of solved point.
      * @return data used in other methods of this class.
      * @throws MathIllegalStateException if the decomposition cannot be performed.
      */
     private InternalData qrDecomposition(RealMatrix jacobian, int solvedCols)
         throws MathIllegalStateException {
         // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
         // hence the multiplication by -1.
         final double[][] weightedJacobian = jacobian.scalarMultiply(-1).getData();
 
         final int nR = weightedJacobian.length;
         final int nC = weightedJacobian[0].length;
 
         final int[] permutation = new int[nC];
         final double[] diagR = new double[nC];
         final double[] jacNorm = new double[nC];
         final double[] beta = new double[nC];
 
         // 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 MathIllegalStateException(LocalizedOptimFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
                                                         nR, nC);
                 }
                 if (norm2 > ak2) {
                     nextColumn = i;
                     ak2        = norm2;
                 }
             }
             if (ak2 <= qrRankingThreshold) {
                 return new InternalData(weightedJacobian, permutation, k, diagR, jacNorm, beta);
             }
             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];
                 }
             }
         }
 
         return new InternalData(weightedJacobian, permutation, solvedCols, diagR, jacNorm, beta);
     }
 
     /**
      * Compute the product Qt.y for some Q.R. decomposition.
      *
      * @param y vector to multiply (will be overwritten with the result)
      * @param internalData Data.
      */
     private void qTy(double[] y,
                      InternalData internalData) {
         final double[][] weightedJacobian = internalData.weightedJacobian;
         final int[] permutation = internalData.permutation;
         final double[] beta = internalData.beta;
 
         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];
             }
         }
     }
 }