LevenbergMarquardtOptimizer.java

  1. /*
  2.  * Licensed to the Apache Software Foundation (ASF) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * The ASF licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *      http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.apache.commons.math3.optimization.general;

  19. import java.util.Arrays;

  21. import org.apache.commons.math3.exception.ConvergenceException;
  22. import org.apache.commons.math3.exception.util.LocalizedFormats;
  23. import org.apache.commons.math3.optimization.PointVectorValuePair;
  24. import org.apache.commons.math3.optimization.ConvergenceChecker;
  25. import org.apache.commons.math3.linear.RealMatrix;
  26. import org.apache.commons.math3.util.Precision;
  27. import org.apache.commons.math3.util.FastMath;


  30. /**
  31.  * This class solves a least squares problem using the Levenberg-Marquardt algorithm.
  32.  *
  33.  * <p>This implementation <em>should</em> work even for over-determined systems
  34.  * (i.e. systems having more point than equations). Over-determined systems
  35.  * are solved by ignoring the point which have the smallest impact according
  36.  * to their jacobian column norm. Only the rank of the matrix and some loop bounds
  37.  * are changed to implement this.</p>
  38.  *
  39.  * <p>The resolution engine is a simple translation of the MINPACK <a
  40.  * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
  41.  * changes. The changes include the over-determined resolution, the use of
  42.  * inherited convergence checker and the Q.R. decomposition which has been
  43.  * rewritten following the algorithm described in the
  44.  * P. Lascaux and R. Theodor book <i>Analyse num&eacute;rique matricielle
  45.  * appliqu&eacute;e &agrave; l'art de l'ing&eacute;nieur</i>, Masson 1986.</p>
  46.  * <p>The authors of the original fortran version are:
  47.  * <ul>
  48.  * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
  49.  * <li>Burton S. Garbow</li>
  50.  * <li>Kenneth E. Hillstrom</li>
  51.  * <li>Jorge J. More</li>
  52.  * </ul>
  53.  * The redistribution policy for MINPACK is available <a
  54.  * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
  55.  * is reproduced below.</p>
  56.  *
  57.  * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
  58.  * <tr><td>
  59.  *    Minpack Copyright Notice (1999) University of Chicago.
  60.  *    All rights reserved
  61.  * </td></tr>
  62.  * <tr><td>
  63.  * Redistribution and use in source and binary forms, with or without
  64.  * modification, are permitted provided that the following conditions
  65.  * are met:
  66.  * <ol>
  67.  *  <li>Redistributions of source code must retain the above copyright
  68.  *      notice, this list of conditions and the following disclaimer.</li>
  69.  * <li>Redistributions in binary form must reproduce the above
  70.  *     copyright notice, this list of conditions and the following
  71.  *     disclaimer in the documentation and/or other materials provided
  72.  *     with the distribution.</li>
  73.  * <li>The end-user documentation included with the redistribution, if any,
  74.  *     must include the following acknowledgment:
  75.  *     <code>This product includes software developed by the University of
  76.  *           Chicago, as Operator of Argonne National Laboratory.</code>
  77.  *     Alternately, this acknowledgment may appear in the software itself,
  78.  *     if and wherever such third-party acknowledgments normally appear.</li>
  79.  * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
  80.  *     WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
  81.  *     UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
  82.  *     THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
  83.  *     IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
  84.  *     OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
  85.  *     OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
  86.  *     OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
  87.  *     USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
  88.  *     THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
  89.  *     DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
  90.  *     UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
  91.  *     BE CORRECTED.</strong></li>
  92.  * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
  93.  *     HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
  94.  *     ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
  95.  *     INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
  96.  *     ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
  97.  *     PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
  98.  *     SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
  99.  *     (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
  100.  *     EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
  101.  *     POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
  102.  * <ol></td></tr>
  103.  * </table>
  104.  * @deprecated As of 3.1 (to be removed in 4.0).
  105.  * @since 2.0
  106.  *
  107.  */
  108. @Deprecated
  109. public class LevenbergMarquardtOptimizer extends AbstractLeastSquaresOptimizer {
  110.     /** Number of solved point. */
  111.     private int solvedCols;
  112.     /** Diagonal elements of the R matrix in the Q.R. decomposition. */
  113.     private double[] diagR;
  114.     /** Norms of the columns of the jacobian matrix. */
  115.     private double[] jacNorm;
  116.     /** Coefficients of the Householder transforms vectors. */
  117.     private double[] beta;
  118.     /** Columns permutation array. */
  119.     private int[] permutation;
  120.     /** Rank of the jacobian matrix. */
  121.     private int rank;
  122.     /** Levenberg-Marquardt parameter. */
  123.     private double lmPar;
  124.     /** Parameters evolution direction associated with lmPar. */
  125.     private double[] lmDir;
  126.     /** Positive input variable used in determining the initial step bound. */
  127.     private final double initialStepBoundFactor;
  128.     /** Desired relative error in the sum of squares. */
  129.     private final double costRelativeTolerance;
  130.     /**  Desired relative error in the approximate solution parameters. */
  131.     private final double parRelativeTolerance;
  132.     /** Desired max cosine on the orthogonality between the function vector
  133.      * and the columns of the jacobian. */
  134.     private final double orthoTolerance;
  135.     /** Threshold for QR ranking. */
  136.     private final double qrRankingThreshold;
  137.     /** Weighted residuals. */
  138.     private double[] weightedResidual;
  139.     /** Weighted Jacobian. */
  140.     private double[][] weightedJacobian;

  142.     /**
  143.      * Build an optimizer for least squares problems with default values
  144.      * for all the tuning parameters (see the {@link
  145.      * #LevenbergMarquardtOptimizer(double,double,double,double,double)
  146.      * other contructor}.
  147.      * The default values for the algorithm settings are:
  148.      * <ul>
  149.      *  <li>Initial step bound factor: 100</li>
  150.      *  <li>Cost relative tolerance: 1e-10</li>
  151.      *  <li>Parameters relative tolerance: 1e-10</li>
  152.      *  <li>Orthogonality tolerance: 1e-10</li>
  153.      *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
  154.      * </ul>
  155.      */
  156.     public LevenbergMarquardtOptimizer() {
  157.         this(100, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
  158.     }

  160.     /**
  161.      * Constructor that allows the specification of a custom convergence
  162.      * checker.
  163.      * Note that all the usual convergence checks will be <em>disabled</em>.
  164.      * The default values for the algorithm settings are:
  165.      * <ul>
  166.      *  <li>Initial step bound factor: 100</li>
  167.      *  <li>Cost relative tolerance: 1e-10</li>
  168.      *  <li>Parameters relative tolerance: 1e-10</li>
  169.      *  <li>Orthogonality tolerance: 1e-10</li>
  170.      *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
  171.      * </ul>
  172.      *
  173.      * @param checker Convergence checker.
  174.      */
  175.     public LevenbergMarquardtOptimizer(ConvergenceChecker<PointVectorValuePair> checker) {
  176.         this(100, checker, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
  177.     }

  179.     /**
  180.      * Constructor that allows the specification of a custom convergence
  181.      * checker, in addition to the standard ones.
  182.      *
  183.      * @param initialStepBoundFactor Positive input variable used in
  184.      * determining the initial step bound. This bound is set to the
  185.      * product of initialStepBoundFactor and the euclidean norm of
  186.      * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
  187.      * itself. In most cases factor should lie in the interval
  188.      * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
  189.      * @param checker Convergence checker.
  190.      * @param costRelativeTolerance Desired relative error in the sum of
  191.      * squares.
  192.      * @param parRelativeTolerance Desired relative error in the approximate
  193.      * solution parameters.
  194.      * @param orthoTolerance Desired max cosine on the orthogonality between
  195.      * the function vector and the columns of the Jacobian.
  196.      * @param threshold Desired threshold for QR ranking. If the squared norm
  197.      * of a column vector is smaller or equal to this threshold during QR
  198.      * decomposition, it is considered to be a zero vector and hence the rank
  199.      * of the matrix is reduced.
  200.      */
  201.     public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
  202.                                        ConvergenceChecker<PointVectorValuePair> checker,
  203.                                        double costRelativeTolerance,
  204.                                        double parRelativeTolerance,
  205.                                        double orthoTolerance,
  206.                                        double threshold) {
  207.         super(checker);
  208.         this.initialStepBoundFactor = initialStepBoundFactor;
  209.         this.costRelativeTolerance = costRelativeTolerance;
  210.         this.parRelativeTolerance = parRelativeTolerance;
  211.         this.orthoTolerance = orthoTolerance;
  212.         this.qrRankingThreshold = threshold;
  213.     }

  215.     /**
  216.      * Build an optimizer for least squares problems with default values
  217.      * for some of the tuning parameters (see the {@link
  218.      * #LevenbergMarquardtOptimizer(double,double,double,double,double)
  219.      * other contructor}.
  220.      * The default values for the algorithm settings are:
  221.      * <ul>
  222.      *  <li>Initial step bound factor}: 100</li>
  223.      *  <li>QR ranking threshold}: {@link Precision#SAFE_MIN}</li>
  224.      * </ul>
  225.      *
  226.      * @param costRelativeTolerance Desired relative error in the sum of
  227.      * squares.
  228.      * @param parRelativeTolerance Desired relative error in the approximate
  229.      * solution parameters.
  230.      * @param orthoTolerance Desired max cosine on the orthogonality between
  231.      * the function vector and the columns of the Jacobian.
  232.      */
  233.     public LevenbergMarquardtOptimizer(double costRelativeTolerance,
  234.                                        double parRelativeTolerance,
  235.                                        double orthoTolerance) {
  236.         this(100,
  237.              costRelativeTolerance, parRelativeTolerance, orthoTolerance,
  238.              Precision.SAFE_MIN);
  239.     }

  241.     /**
  242.      * The arguments control the behaviour of the default convergence checking
  243.      * procedure.
  244.      * Additional criteria can defined through the setting of a {@link
  245.      * ConvergenceChecker}.
  246.      *
  247.      * @param initialStepBoundFactor Positive input variable used in
  248.      * determining the initial step bound. This bound is set to the
  249.      * product of initialStepBoundFactor and the euclidean norm of
  250.      * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
  251.      * itself. In most cases factor should lie in the interval
  252.      * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
  253.      * @param costRelativeTolerance Desired relative error in the sum of
  254.      * squares.
  255.      * @param parRelativeTolerance Desired relative error in the approximate
  256.      * solution parameters.
  257.      * @param orthoTolerance Desired max cosine on the orthogonality between
  258.      * the function vector and the columns of the Jacobian.
  259.      * @param threshold Desired threshold for QR ranking. If the squared norm
  260.      * of a column vector is smaller or equal to this threshold during QR
  261.      * decomposition, it is considered to be a zero vector and hence the rank
  262.      * of the matrix is reduced.
  263.      */
  264.     public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
  265.                                        double costRelativeTolerance,
  266.                                        double parRelativeTolerance,
  267.                                        double orthoTolerance,
  268.                                        double threshold) {
  269.         super(null); // No custom convergence criterion.
  270.         this.initialStepBoundFactor = initialStepBoundFactor;
  271.         this.costRelativeTolerance = costRelativeTolerance;
  272.         this.parRelativeTolerance = parRelativeTolerance;
  273.         this.orthoTolerance = orthoTolerance;
  274.         this.qrRankingThreshold = threshold;
  275.     }

  277.     /** {@inheritDoc} */
  278.     @Override
  279.     protected PointVectorValuePair doOptimize() {
  280.         final int nR = getTarget().length; // Number of observed data.
  281.         final double[] currentPoint = getStartPoint();
  282.         final int nC = currentPoint.length; // Number of parameters.

  284.         // arrays shared with the other private methods
  285.         solvedCols  = FastMath.min(nR, nC);
  286.         diagR       = new double[nC];
  287.         jacNorm     = new double[nC];
  288.         beta        = new double[nC];
  289.         permutation = new int[nC];
  290.         lmDir       = new double[nC];

  292.         // local point
  293.         double   delta   = 0;
  294.         double   xNorm   = 0;
  295.         double[] diag    = new double[nC];
  296.         double[] oldX    = new double[nC];
  297.         double[] oldRes  = new double[nR];
  298.         double[] oldObj  = new double[nR];
  299.         double[] qtf     = new double[nR];
  300.         double[] work1   = new double[nC];
  301.         double[] work2   = new double[nC];
  302.         double[] work3   = new double[nC];

  304.         final RealMatrix weightMatrixSqrt = getWeightSquareRoot();

  306.         // Evaluate the function at the starting point and calculate its norm.
  307.         double[] currentObjective = computeObjectiveValue(currentPoint);
  308.         double[] currentResiduals = computeResiduals(currentObjective);
  309.         PointVectorValuePair current = new PointVectorValuePair(currentPoint, currentObjective);
  310.         double currentCost = computeCost(currentResiduals);

  312.         // Outer loop.
  313.         lmPar = 0;
  314.         boolean firstIteration = true;
  315.         int iter = 0;
  316.         final ConvergenceChecker<PointVectorValuePair> checker = getConvergenceChecker();
  317.         while (true) {
  318.             ++iter;
  319.             final PointVectorValuePair previous = current;

  321.             // QR decomposition of the jacobian matrix
  322.             qrDecomposition(computeWeightedJacobian(currentPoint));

  324.             weightedResidual = weightMatrixSqrt.operate(currentResiduals);
  325.             for (int i = 0; i < nR; i++) {
  326.                 qtf[i] = weightedResidual[i];
  327.             }

  329.             // compute Qt.res
  330.             qTy(qtf);

  332.             // now we don't need Q anymore,
  333.             // so let jacobian contain the R matrix with its diagonal elements
  334.             for (int k = 0; k < solvedCols; ++k) {
  335.                 int pk = permutation[k];
  336.                 weightedJacobian[k][pk] = diagR[pk];
  337.             }

  339.             if (firstIteration) {
  340.                 // scale the point according to the norms of the columns
  341.                 // of the initial jacobian
  342.                 xNorm = 0;
  343.                 for (int k = 0; k < nC; ++k) {
  344.                     double dk = jacNorm[k];
  345.                     if (dk == 0) {
  346.                         dk = 1.0;
  347.                     }
  348.                     double xk = dk * currentPoint[k];
  349.                     xNorm  += xk * xk;
  350.                     diag[k] = dk;
  351.                 }
  352.                 xNorm = FastMath.sqrt(xNorm);

  354.                 // initialize the step bound delta
  355.                 delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
  356.             }

  358.             // check orthogonality between function vector and jacobian columns
  359.             double maxCosine = 0;
  360.             if (currentCost != 0) {
  361.                 for (int j = 0; j < solvedCols; ++j) {
  362.                     int    pj = permutation[j];
  363.                     double s  = jacNorm[pj];
  364.                     if (s != 0) {
  365.                         double sum = 0;
  366.                         for (int i = 0; i <= j; ++i) {
  367.                             sum += weightedJacobian[i][pj] * qtf[i];
  368.                         }
  369.                         maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * currentCost));
  370.                     }
  371.                 }
  372.             }
  373.             if (maxCosine <= orthoTolerance) {
  374.                 // Convergence has been reached.
  375.                 setCost(currentCost);
  376.                 // Update (deprecated) "point" field.
  377.                 point = current.getPoint();
  378.                 return current;
  379.             }

  381.             // rescale if necessary
  382.             for (int j = 0; j < nC; ++j) {
  383.                 diag[j] = FastMath.max(diag[j], jacNorm[j]);
  384.             }

  386.             // Inner loop.
  387.             for (double ratio = 0; ratio < 1.0e-4;) {

  389.                 // save the state
  390.                 for (int j = 0; j < solvedCols; ++j) {
  391.                     int pj = permutation[j];
  392.                     oldX[pj] = currentPoint[pj];
  393.                 }
  394.                 final double previousCost = currentCost;
  395.                 double[] tmpVec = weightedResidual;
  396.                 weightedResidual = oldRes;
  397.                 oldRes    = tmpVec;
  398.                 tmpVec    = currentObjective;
  399.                 currentObjective = oldObj;
  400.                 oldObj    = tmpVec;

  402.                 // determine the Levenberg-Marquardt parameter
  403.                 determineLMParameter(qtf, delta, diag, work1, work2, work3);

  405.                 // compute the new point and the norm of the evolution direction
  406.                 double lmNorm = 0;
  407.                 for (int j = 0; j < solvedCols; ++j) {
  408.                     int pj = permutation[j];
  409.                     lmDir[pj] = -lmDir[pj];
  410.                     currentPoint[pj] = oldX[pj] + lmDir[pj];
  411.                     double s = diag[pj] * lmDir[pj];
  412.                     lmNorm  += s * s;
  413.                 }
  414.                 lmNorm = FastMath.sqrt(lmNorm);
  415.                 // on the first iteration, adjust the initial step bound.
  416.                 if (firstIteration) {
  417.                     delta = FastMath.min(delta, lmNorm);
  418.                 }

  420.                 // Evaluate the function at x + p and calculate its norm.
  421.                 currentObjective = computeObjectiveValue(currentPoint);
  422.                 currentResiduals = computeResiduals(currentObjective);
  423.                 current = new PointVectorValuePair(currentPoint, currentObjective);
  424.                 currentCost = computeCost(currentResiduals);

  426.                 // compute the scaled actual reduction
  427.                 double actRed = -1.0;
  428.                 if (0.1 * currentCost < previousCost) {
  429.                     double r = currentCost / previousCost;
  430.                     actRed = 1.0 - r * r;
  431.                 }

  433.                 // compute the scaled predicted reduction
  434.                 // and the scaled directional derivative
  435.                 for (int j = 0; j < solvedCols; ++j) {
  436.                     int pj = permutation[j];
  437.                     double dirJ = lmDir[pj];
  438.                     work1[j] = 0;
  439.                     for (int i = 0; i <= j; ++i) {
  440.                         work1[i] += weightedJacobian[i][pj] * dirJ;
  441.                     }
  442.                 }
  443.                 double coeff1 = 0;
  444.                 for (int j = 0; j < solvedCols; ++j) {
  445.                     coeff1 += work1[j] * work1[j];
  446.                 }
  447.                 double pc2 = previousCost * previousCost;
  448.                 coeff1 /= pc2;
  449.                 double coeff2 = lmPar * lmNorm * lmNorm / pc2;
  450.                 double preRed = coeff1 + 2 * coeff2;
  451.                 double dirDer = -(coeff1 + coeff2);

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

  456.                 // update the step bound
  457.                 if (ratio <= 0.25) {
  458.                     double tmp =
  459.                         (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
  460.                         if ((0.1 * currentCost >= previousCost) || (tmp < 0.1)) {
  461.                             tmp = 0.1;
  462.                         }
  463.                         delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
  464.                         lmPar /= tmp;
  465.                 } else if ((lmPar == 0) || (ratio >= 0.75)) {
  466.                     delta = 2 * lmNorm;
  467.                     lmPar *= 0.5;
  468.                 }

  470.                 // test for successful iteration.
  471.                 if (ratio >= 1.0e-4) {
  472.                     // successful iteration, update the norm
  473.                     firstIteration = false;
  474.                     xNorm = 0;
  475.                     for (int k = 0; k < nC; ++k) {
  476.                         double xK = diag[k] * currentPoint[k];
  477.                         xNorm += xK * xK;
  478.                     }
  479.                     xNorm = FastMath.sqrt(xNorm);

  481.                     // tests for convergence.
  482.                     if (checker != null && checker.converged(iter, previous, current)) {
  483.                         setCost(currentCost);
  484.                         // Update (deprecated) "point" field.
  485.                         point = current.getPoint();
  486.                         return current;
  487.                     }
  488.                 } else {
  489.                     // failed iteration, reset the previous values
  490.                     currentCost = previousCost;
  491.                     for (int j = 0; j < solvedCols; ++j) {
  492.                         int pj = permutation[j];
  493.                         currentPoint[pj] = oldX[pj];
  494.                     }
  495.                     tmpVec    = weightedResidual;
  496.                     weightedResidual = oldRes;
  497.                     oldRes    = tmpVec;
  498.                     tmpVec    = currentObjective;
  499.                     currentObjective = oldObj;
  500.                     oldObj    = tmpVec;
  501.                     // Reset "current" to previous values.
  502.                     current = new PointVectorValuePair(currentPoint, currentObjective);
  503.                 }

  505.                 // Default convergence criteria.
  506.                 if ((FastMath.abs(actRed) <= costRelativeTolerance &&
  507.                      preRed <= costRelativeTolerance &&
  508.                      ratio <= 2.0) ||
  509.                     delta <= parRelativeTolerance * xNorm) {
  510.                     setCost(currentCost);
  511.                     // Update (deprecated) "point" field.
  512.                     point = current.getPoint();
  513.                     return current;
  514.                 }

  516.                 // tests for termination and stringent tolerances
  517.                 // (2.2204e-16 is the machine epsilon for IEEE754)
  518.                 if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
  519.                     throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
  520.                                                    costRelativeTolerance);
  521.                 } else if (delta <= 2.2204e-16 * xNorm) {
  522.                     throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
  523.                                                    parRelativeTolerance);
  524.                 } else if (maxCosine <= 2.2204e-16)  {
  525.                     throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
  526.                                                    orthoTolerance);
  527.                 }
  528.             }
  529.         }
  530.     }

  532.     /**
  533.      * Determine the Levenberg-Marquardt parameter.
  534.      * <p>This implementation is a translation in Java of the MINPACK
  535.      * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
  536.      * routine.</p>
  537.      * <p>This method sets the lmPar and lmDir attributes.</p>
  538.      * <p>The authors of the original fortran function are:</p>
  539.      * <ul>
  540.      *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
  541.      *   <li>Burton  S. Garbow</li>
  542.      *   <li>Kenneth E. Hillstrom</li>
  543.      *   <li>Jorge   J. More</li>
  544.      * </ul>
  545.      * <p>Luc Maisonobe did the Java translation.</p>
  546.      *
  547.      * @param qy array containing qTy
  548.      * @param delta upper bound on the euclidean norm of diagR * lmDir
  549.      * @param diag diagonal matrix
  550.      * @param work1 work array
  551.      * @param work2 work array
  552.      * @param work3 work array
  553.      */
  554.     private void determineLMParameter(double[] qy, double delta, double[] diag,
  555.                                       double[] work1, double[] work2, double[] work3) {
  556.         final int nC = weightedJacobian[0].length;

  558.         // compute and store in x the gauss-newton direction, if the
  559.         // jacobian is rank-deficient, obtain a least squares solution
  560.         for (int j = 0; j < rank; ++j) {
  561.             lmDir[permutation[j]] = qy[j];
  562.         }
  563.         for (int j = rank; j < nC; ++j) {
  564.             lmDir[permutation[j]] = 0;
  565.         }
  566.         for (int k = rank - 1; k >= 0; --k) {
  567.             int pk = permutation[k];
  568.             double ypk = lmDir[pk] / diagR[pk];
  569.             for (int i = 0; i < k; ++i) {
  570.                 lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk];
  571.             }
  572.             lmDir[pk] = ypk;
  573.         }

  575.         // evaluate the function at the origin, and test
  576.         // for acceptance of the Gauss-Newton direction
  577.         double dxNorm = 0;
  578.         for (int j = 0; j < solvedCols; ++j) {
  579.             int pj = permutation[j];
  580.             double s = diag[pj] * lmDir[pj];
  581.             work1[pj] = s;
  582.             dxNorm += s * s;
  583.         }
  584.         dxNorm = FastMath.sqrt(dxNorm);
  585.         double fp = dxNorm - delta;
  586.         if (fp <= 0.1 * delta) {
  587.             lmPar = 0;
  588.             return;
  589.         }

  591.         // if the jacobian is not rank deficient, the Newton step provides
  592.         // a lower bound, parl, for the zero of the function,
  593.         // otherwise set this bound to zero
  594.         double sum2;
  595.         double parl = 0;
  596.         if (rank == solvedCols) {
  597.             for (int j = 0; j < solvedCols; ++j) {
  598.                 int pj = permutation[j];
  599.                 work1[pj] *= diag[pj] / dxNorm;
  600.             }
  601.             sum2 = 0;
  602.             for (int j = 0; j < solvedCols; ++j) {
  603.                 int pj = permutation[j];
  604.                 double sum = 0;
  605.                 for (int i = 0; i < j; ++i) {
  606.                     sum += weightedJacobian[i][pj] * work1[permutation[i]];
  607.                 }
  608.                 double s = (work1[pj] - sum) / diagR[pj];
  609.                 work1[pj] = s;
  610.                 sum2 += s * s;
  611.             }
  612.             parl = fp / (delta * sum2);
  613.         }

  615.         // calculate an upper bound, paru, for the zero of the function
  616.         sum2 = 0;
  617.         for (int j = 0; j < solvedCols; ++j) {
  618.             int pj = permutation[j];
  619.             double sum = 0;
  620.             for (int i = 0; i <= j; ++i) {
  621.                 sum += weightedJacobian[i][pj] * qy[i];
  622.             }
  623.             sum /= diag[pj];
  624.             sum2 += sum * sum;
  625.         }
  626.         double gNorm = FastMath.sqrt(sum2);
  627.         double paru = gNorm / delta;
  628.         if (paru == 0) {
  629.             // 2.2251e-308 is the smallest positive real for IEE754
  630.             paru = 2.2251e-308 / FastMath.min(delta, 0.1);
  631.         }

  633.         // if the input par lies outside of the interval (parl,paru),
  634.         // set par to the closer endpoint
  635.         lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
  636.         if (lmPar == 0) {
  637.             lmPar = gNorm / dxNorm;
  638.         }

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

  642.             // evaluate the function at the current value of lmPar
  643.             if (lmPar == 0) {
  644.                 lmPar = FastMath.max(2.2251e-308, 0.001 * paru);
  645.             }
  646.             double sPar = FastMath.sqrt(lmPar);
  647.             for (int j = 0; j < solvedCols; ++j) {
  648.                 int pj = permutation[j];
  649.                 work1[pj] = sPar * diag[pj];
  650.             }
  651.             determineLMDirection(qy, work1, work2, work3);

  653.             dxNorm = 0;
  654.             for (int j = 0; j < solvedCols; ++j) {
  655.                 int pj = permutation[j];
  656.                 double s = diag[pj] * lmDir[pj];
  657.                 work3[pj] = s;
  658.                 dxNorm += s * s;
  659.             }
  660.             dxNorm = FastMath.sqrt(dxNorm);
  661.             double previousFP = fp;
  662.             fp = dxNorm - delta;

  664.             // if the function is small enough, accept the current value
  665.             // of lmPar, also test for the exceptional cases where parl is zero
  666.             if ((FastMath.abs(fp) <= 0.1 * delta) ||
  667.                     ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
  668.                 return;
  669.             }

  671.             // compute the Newton correction
  672.             for (int j = 0; j < solvedCols; ++j) {
  673.                 int pj = permutation[j];
  674.                 work1[pj] = work3[pj] * diag[pj] / dxNorm;
  675.             }
  676.             for (int j = 0; j < solvedCols; ++j) {
  677.                 int pj = permutation[j];
  678.                 work1[pj] /= work2[j];
  679.                 double tmp = work1[pj];
  680.                 for (int i = j + 1; i < solvedCols; ++i) {
  681.                     work1[permutation[i]] -= weightedJacobian[i][pj] * tmp;
  682.                 }
  683.             }
  684.             sum2 = 0;
  685.             for (int j = 0; j < solvedCols; ++j) {
  686.                 double s = work1[permutation[j]];
  687.                 sum2 += s * s;
  688.             }
  689.             double correction = fp / (delta * sum2);

  691.             // depending on the sign of the function, update parl or paru.
  692.             if (fp > 0) {
  693.                 parl = FastMath.max(parl, lmPar);
  694.             } else if (fp < 0) {
  695.                 paru = FastMath.min(paru, lmPar);
  696.             }

  698.             // compute an improved estimate for lmPar
  699.             lmPar = FastMath.max(parl, lmPar + correction);

  701.         }
  702.     }

  704.     /**
  705.      * Solve a*x = b and d*x = 0 in the least squares sense.
  706.      * <p>This implementation is a translation in Java of the MINPACK
  707.      * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
  708.      * routine.</p>
  709.      * <p>This method sets the lmDir and lmDiag attributes.</p>
  710.      * <p>The authors of the original fortran function are:</p>
  711.      * <ul>
  712.      *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
  713.      *   <li>Burton  S. Garbow</li>
  714.      *   <li>Kenneth E. Hillstrom</li>
  715.      *   <li>Jorge   J. More</li>
  716.      * </ul>
  717.      * <p>Luc Maisonobe did the Java translation.</p>
  718.      *
  719.      * @param qy array containing qTy
  720.      * @param diag diagonal matrix
  721.      * @param lmDiag diagonal elements associated with lmDir
  722.      * @param work work array
  723.      */
  724.     private void determineLMDirection(double[] qy, double[] diag,
  725.                                       double[] lmDiag, double[] work) {

  727.         // copy R and Qty to preserve input and initialize s
  728.         //  in particular, save the diagonal elements of R in lmDir
  729.         for (int j = 0; j < solvedCols; ++j) {
  730.             int pj = permutation[j];
  731.             for (int i = j + 1; i < solvedCols; ++i) {
  732.                 weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]];
  733.             }
  734.             lmDir[j] = diagR[pj];
  735.             work[j]  = qy[j];
  736.         }

  738.         // eliminate the diagonal matrix d using a Givens rotation
  739.         for (int j = 0; j < solvedCols; ++j) {

  741.             // prepare the row of d to be eliminated, locating the
  742.             // diagonal element using p from the Q.R. factorization
  743.             int pj = permutation[j];
  744.             double dpj = diag[pj];
  745.             if (dpj != 0) {
  746.                 Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
  747.             }
  748.             lmDiag[j] = dpj;

  750.             //  the transformations to eliminate the row of d
  751.             // modify only a single element of Qty
  752.             // beyond the first n, which is initially zero.
  753.             double qtbpj = 0;
  754.             for (int k = j; k < solvedCols; ++k) {
  755.                 int pk = permutation[k];

  757.                 // determine a Givens rotation which eliminates the
  758.                 // appropriate element in the current row of d
  759.                 if (lmDiag[k] != 0) {

  761.                     final double sin;
  762.                     final double cos;
  763.                     double rkk = weightedJacobian[k][pk];
  764.                     if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
  765.                         final double cotan = rkk / lmDiag[k];
  766.                         sin   = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
  767.                         cos   = sin * cotan;
  768.                     } else {
  769.                         final double tan = lmDiag[k] / rkk;
  770.                         cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
  771.                         sin = cos * tan;
  772.                     }

  774.                     // compute the modified diagonal element of R and
  775.                     // the modified element of (Qty,0)
  776.                     weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
  777.                     final double temp = cos * work[k] + sin * qtbpj;
  778.                     qtbpj = -sin * work[k] + cos * qtbpj;
  779.                     work[k] = temp;

  781.                     // accumulate the tranformation in the row of s
  782.                     for (int i = k + 1; i < solvedCols; ++i) {
  783.                         double rik = weightedJacobian[i][pk];
  784.                         final double temp2 = cos * rik + sin * lmDiag[i];
  785.                         lmDiag[i] = -sin * rik + cos * lmDiag[i];
  786.                         weightedJacobian[i][pk] = temp2;
  787.                     }
  788.                 }
  789.             }

  791.             // store the diagonal element of s and restore
  792.             // the corresponding diagonal element of R
  793.             lmDiag[j] = weightedJacobian[j][permutation[j]];
  794.             weightedJacobian[j][permutation[j]] = lmDir[j];
  795.         }

  797.         // solve the triangular system for z, if the system is
  798.         // singular, then obtain a least squares solution
  799.         int nSing = solvedCols;
  800.         for (int j = 0; j < solvedCols; ++j) {
  801.             if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
  802.                 nSing = j;
  803.             }
  804.             if (nSing < solvedCols) {
  805.                 work[j] = 0;
  806.             }
  807.         }
  808.         if (nSing > 0) {
  809.             for (int j = nSing - 1; j >= 0; --j) {
  810.                 int pj = permutation[j];
  811.                 double sum = 0;
  812.                 for (int i = j + 1; i < nSing; ++i) {
  813.                     sum += weightedJacobian[i][pj] * work[i];
  814.                 }
  815.                 work[j] = (work[j] - sum) / lmDiag[j];
  816.             }
  817.         }

  819.         // permute the components of z back to components of lmDir
  820.         for (int j = 0; j < lmDir.length; ++j) {
  821.             lmDir[permutation[j]] = work[j];
  822.         }
  823.     }

  825.     /**
  826.      * Decompose a matrix A as A.P = Q.R using Householder transforms.
  827.      * <p>As suggested in the P. Lascaux and R. Theodor book
  828.      * <i>Analyse num&eacute;rique matricielle appliqu&eacute;e &agrave;
  829.      * l'art de l'ing&eacute;nieur</i> (Masson, 1986), instead of representing
  830.      * the Householder transforms with u<sub>k</sub> unit vectors such that:
  831.      * <pre>
  832.      * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
  833.      * </pre>
  834.      * we use <sub>k</sub> non-unit vectors such that:
  835.      * <pre>
  836.      * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
  837.      * </pre>
  838.      * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
  839.      * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
  840.      * them from the v<sub>k</sub> vectors would be costly.</p>
  841.      * <p>This decomposition handles rank deficient cases since the tranformations
  842.      * are performed in non-increasing columns norms order thanks to columns
  843.      * pivoting. The diagonal elements of the R matrix are therefore also in
  844.      * non-increasing absolute values order.</p>
  845.      *
  846.      * @param jacobian Weighted Jacobian matrix at the current point.
  847.      * @exception ConvergenceException if the decomposition cannot be performed
  848.      */
  849.     private void qrDecomposition(RealMatrix jacobian) throws ConvergenceException {
  850.         // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
  851.         // hence the multiplication by -1.
  852.         weightedJacobian = jacobian.scalarMultiply(-1).getData();

  854.         final int nR = weightedJacobian.length;
  855.         final int nC = weightedJacobian[0].length;

  857.         // initializations
  858.         for (int k = 0; k < nC; ++k) {
  859.             permutation[k] = k;
  860.             double norm2 = 0;
  861.             for (int i = 0; i < nR; ++i) {
  862.                 double akk = weightedJacobian[i][k];
  863.                 norm2 += akk * akk;
  864.             }
  865.             jacNorm[k] = FastMath.sqrt(norm2);
  866.         }

  868.         // transform the matrix column after column
  869.         for (int k = 0; k < nC; ++k) {

  871.             // select the column with the greatest norm on active components
  872.             int nextColumn = -1;
  873.             double ak2 = Double.NEGATIVE_INFINITY;
  874.             for (int i = k; i < nC; ++i) {
  875.                 double norm2 = 0;
  876.                 for (int j = k; j < nR; ++j) {
  877.                     double aki = weightedJacobian[j][permutation[i]];
  878.                     norm2 += aki * aki;
  879.                 }
  880.                 if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
  881.                     throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
  882.                                                    nR, nC);
  883.                 }
  884.                 if (norm2 > ak2) {
  885.                     nextColumn = i;
  886.                     ak2        = norm2;
  887.                 }
  888.             }
  889.             if (ak2 <= qrRankingThreshold) {
  890.                 rank = k;
  891.                 return;
  892.             }
  893.             int pk                  = permutation[nextColumn];
  894.             permutation[nextColumn] = permutation[k];
  895.             permutation[k]          = pk;

  897.             // choose alpha such that Hk.u = alpha ek
  898.             double akk   = weightedJacobian[k][pk];
  899.             double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
  900.             double betak = 1.0 / (ak2 - akk * alpha);
  901.             beta[pk]     = betak;

  903.             // transform the current column
  904.             diagR[pk]        = alpha;
  905.             weightedJacobian[k][pk] -= alpha;

  907.             // transform the remaining columns
  908.             for (int dk = nC - 1 - k; dk > 0; --dk) {
  909.                 double gamma = 0;
  910.                 for (int j = k; j < nR; ++j) {
  911.                     gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]];
  912.                 }
  913.                 gamma *= betak;
  914.                 for (int j = k; j < nR; ++j) {
  915.                     weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk];
  916.                 }
  917.             }
  918.         }
  919.         rank = solvedCols;
  920.     }

  922.     /**
  923.      * Compute the product Qt.y for some Q.R. decomposition.
  924.      *
  925.      * @param y vector to multiply (will be overwritten with the result)
  926.      */
  927.     private void qTy(double[] y) {
  928.         final int nR = weightedJacobian.length;
  929.         final int nC = weightedJacobian[0].length;

  931.         for (int k = 0; k < nC; ++k) {
  932.             int pk = permutation[k];
  933.             double gamma = 0;
  934.             for (int i = k; i < nR; ++i) {
  935.                 gamma += weightedJacobian[i][pk] * y[i];
  936.             }
  937.             gamma *= beta[pk];
  938.             for (int i = k; i < nR; ++i) {
  939.                 y[i] -= gamma * weightedJacobian[i][pk];
  940.             }
  941.         }
  942.     }
  943. }
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