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.optim.nonlinear.vector.jacobian;

  19. import java.util.Arrays;
  20. import org.apache.commons.math3.exception.ConvergenceException;
  21. import org.apache.commons.math3.exception.MathUnsupportedOperationException;
  22. import org.apache.commons.math3.exception.util.LocalizedFormats;
  23. import org.apache.commons.math3.optim.PointVectorValuePair;
  24. import org.apache.commons.math3.optim.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
  32.  * algorithm.
  33.  * <br/>
  34.  * Constraints are not supported: the call to
  35.  * {@link #optimize(OptimizationData[]) optimize} will throw
  36.  * {@link MathUnsupportedOperationException} if bounds are passed to it.
  37.  *
  38.  * <p>This implementation <em>should</em> work even for over-determined systems
  39.  * (i.e. systems having more point than equations). Over-determined systems
  40.  * are solved by ignoring the point which have the smallest impact according
  41.  * to their jacobian column norm. Only the rank of the matrix and some loop bounds
  42.  * are changed to implement this.</p>
  43.  *
  44.  * <p>The resolution engine is a simple translation of the MINPACK <a
  45.  * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
  46.  * changes. The changes include the over-determined resolution, the use of
  47.  * inherited convergence checker and the Q.R. decomposition which has been
  48.  * rewritten following the algorithm described in the
  49.  * P. Lascaux and R. Theodor book <i>Analyse num&eacute;rique matricielle
  50.  * appliqu&eacute;e &agrave; l'art de l'ing&eacute;nieur</i>, Masson 1986.</p>
  51.  * <p>The authors of the original fortran version are:
  52.  * <ul>
  53.  * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
  54.  * <li>Burton S. Garbow</li>
  55.  * <li>Kenneth E. Hillstrom</li>
  56.  * <li>Jorge J. More</li>
  57.  * </ul>
  58.  * The redistribution policy for MINPACK is available <a
  59.  * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
  60.  * is reproduced below.</p>
  61.  *
  62.  * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
  63.  * <tr><td>
  64.  *    Minpack Copyright Notice (1999) University of Chicago.
  65.  *    All rights reserved
  66.  * </td></tr>
  67.  * <tr><td>
  68.  * Redistribution and use in source and binary forms, with or without
  69.  * modification, are permitted provided that the following conditions
  70.  * are met:
  71.  * <ol>
  72.  *  <li>Redistributions of source code must retain the above copyright
  73.  *      notice, this list of conditions and the following disclaimer.</li>
  74.  * <li>Redistributions in binary form must reproduce the above
  75.  *     copyright notice, this list of conditions and the following
  76.  *     disclaimer in the documentation and/or other materials provided
  77.  *     with the distribution.</li>
  78.  * <li>The end-user documentation included with the redistribution, if any,
  79.  *     must include the following acknowledgment:
  80.  *     <code>This product includes software developed by the University of
  81.  *           Chicago, as Operator of Argonne National Laboratory.</code>
  82.  *     Alternately, this acknowledgment may appear in the software itself,
  83.  *     if and wherever such third-party acknowledgments normally appear.</li>
  84.  * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
  85.  *     WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
  86.  *     UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
  87.  *     THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
  88.  *     IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
  89.  *     OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
  90.  *     OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
  91.  *     OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
  92.  *     USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
  93.  *     THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
  94.  *     DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
  95.  *     UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
  96.  *     BE CORRECTED.</strong></li>
  97.  * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
  98.  *     HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
  99.  *     ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
  100.  *     INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
  101.  *     ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
  102.  *     PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
  103.  *     SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
  104.  *     (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
  105.  *     EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
  106.  *     POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
  107.  * <ol></td></tr>
  108.  * </table>
  109.  *
  110.  * @since 2.0
  111.  * @deprecated All classes and interfaces in this package are deprecated.
  112.  * The optimizers that were provided here were moved to the
  113.  * {@link org.apache.commons.math3.fitting.leastsquares} package
  114.  * (cf. MATH-1008).
  115.  */
  116. @Deprecated
  117. public class LevenbergMarquardtOptimizer
  118.     extends AbstractLeastSquaresOptimizer {
  119.     /** Twice the "epsilon machine". */
  120.     private static final double TWO_EPS = 2 * Precision.EPSILON;
  121.     /** Number of solved point. */
  122.     private int solvedCols;
  123.     /** Diagonal elements of the R matrix in the Q.R. decomposition. */
  124.     private double[] diagR;
  125.     /** Norms of the columns of the jacobian matrix. */
  126.     private double[] jacNorm;
  127.     /** Coefficients of the Householder transforms vectors. */
  128.     private double[] beta;
  129.     /** Columns permutation array. */
  130.     private int[] permutation;
  131.     /** Rank of the jacobian matrix. */
  132.     private int rank;
  133.     /** Levenberg-Marquardt parameter. */
  134.     private double lmPar;
  135.     /** Parameters evolution direction associated with lmPar. */
  136.     private double[] lmDir;
  137.     /** Positive input variable used in determining the initial step bound. */
  138.     private final double initialStepBoundFactor;
  139.     /** Desired relative error in the sum of squares. */
  140.     private final double costRelativeTolerance;
  141.     /**  Desired relative error in the approximate solution parameters. */
  142.     private final double parRelativeTolerance;
  143.     /** Desired max cosine on the orthogonality between the function vector
  144.      * and the columns of the jacobian. */
  145.     private final double orthoTolerance;
  146.     /** Threshold for QR ranking. */
  147.     private final double qrRankingThreshold;
  148.     /** Weighted residuals. */
  149.     private double[] weightedResidual;
  150.     /** Weighted Jacobian. */
  151.     private double[][] weightedJacobian;

  153.     /**
  154.      * Build an optimizer for least squares problems with default values
  155.      * for all the tuning parameters (see the {@link
  156.      * #LevenbergMarquardtOptimizer(double,double,double,double,double)
  157.      * other contructor}.
  158.      * The default values for the algorithm settings are:
  159.      * <ul>
  160.      *  <li>Initial step bound factor: 100</li>
  161.      *  <li>Cost relative tolerance: 1e-10</li>
  162.      *  <li>Parameters relative tolerance: 1e-10</li>
  163.      *  <li>Orthogonality tolerance: 1e-10</li>
  164.      *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
  165.      * </ul>
  166.      */
  167.     public LevenbergMarquardtOptimizer() {
  168.         this(100, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
  169.     }

  171.     /**
  172.      * Constructor that allows the specification of a custom convergence
  173.      * checker.
  174.      * Note that all the usual convergence checks will be <em>disabled</em>.
  175.      * The default values for the algorithm settings are:
  176.      * <ul>
  177.      *  <li>Initial step bound factor: 100</li>
  178.      *  <li>Cost relative tolerance: 1e-10</li>
  179.      *  <li>Parameters relative tolerance: 1e-10</li>
  180.      *  <li>Orthogonality tolerance: 1e-10</li>
  181.      *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
  182.      * </ul>
  183.      *
  184.      * @param checker Convergence checker.
  185.      */
  186.     public LevenbergMarquardtOptimizer(ConvergenceChecker<PointVectorValuePair> checker) {
  187.         this(100, checker, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
  188.     }

  190.     /**
  191.      * Constructor that allows the specification of a custom convergence
  192.      * checker, in addition to the standard ones.
  193.      *
  194.      * @param initialStepBoundFactor Positive input variable used in
  195.      * determining the initial step bound. This bound is set to the
  196.      * product of initialStepBoundFactor and the euclidean norm of
  197.      * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
  198.      * itself. In most cases factor should lie in the interval
  199.      * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
  200.      * @param checker Convergence checker.
  201.      * @param costRelativeTolerance Desired relative error in the sum of
  202.      * squares.
  203.      * @param parRelativeTolerance Desired relative error in the approximate
  204.      * solution parameters.
  205.      * @param orthoTolerance Desired max cosine on the orthogonality between
  206.      * the function vector and the columns of the Jacobian.
  207.      * @param threshold Desired threshold for QR ranking. If the squared norm
  208.      * of a column vector is smaller or equal to this threshold during QR
  209.      * decomposition, it is considered to be a zero vector and hence the rank
  210.      * of the matrix is reduced.
  211.      */
  212.     public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
  213.                                        ConvergenceChecker<PointVectorValuePair> checker,
  214.                                        double costRelativeTolerance,
  215.                                        double parRelativeTolerance,
  216.                                        double orthoTolerance,
  217.                                        double threshold) {
  218.         super(checker);
  219.         this.initialStepBoundFactor = initialStepBoundFactor;
  220.         this.costRelativeTolerance = costRelativeTolerance;
  221.         this.parRelativeTolerance = parRelativeTolerance;
  222.         this.orthoTolerance = orthoTolerance;
  223.         this.qrRankingThreshold = threshold;
  224.     }

  226.     /**
  227.      * Build an optimizer for least squares problems with default values
  228.      * for some of the tuning parameters (see the {@link
  229.      * #LevenbergMarquardtOptimizer(double,double,double,double,double)
  230.      * other contructor}.
  231.      * The default values for the algorithm settings are:
  232.      * <ul>
  233.      *  <li>Initial step bound factor}: 100</li>
  234.      *  <li>QR ranking threshold}: {@link Precision#SAFE_MIN}</li>
  235.      * </ul>
  236.      *
  237.      * @param costRelativeTolerance Desired relative error in the sum of
  238.      * squares.
  239.      * @param parRelativeTolerance Desired relative error in the approximate
  240.      * solution parameters.
  241.      * @param orthoTolerance Desired max cosine on the orthogonality between
  242.      * the function vector and the columns of the Jacobian.
  243.      */
  244.     public LevenbergMarquardtOptimizer(double costRelativeTolerance,
  245.                                        double parRelativeTolerance,
  246.                                        double orthoTolerance) {
  247.         this(100,
  248.              costRelativeTolerance, parRelativeTolerance, orthoTolerance,
  249.              Precision.SAFE_MIN);
  250.     }

  252.     /**
  253.      * The arguments control the behaviour of the default convergence checking
  254.      * procedure.
  255.      * Additional criteria can defined through the setting of a {@link
  256.      * ConvergenceChecker}.
  257.      *
  258.      * @param initialStepBoundFactor Positive input variable used in
  259.      * determining the initial step bound. This bound is set to the
  260.      * product of initialStepBoundFactor and the euclidean norm of
  261.      * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
  262.      * itself. In most cases factor should lie in the interval
  263.      * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
  264.      * @param costRelativeTolerance Desired relative error in the sum of
  265.      * squares.
  266.      * @param parRelativeTolerance Desired relative error in the approximate
  267.      * solution parameters.
  268.      * @param orthoTolerance Desired max cosine on the orthogonality between
  269.      * the function vector and the columns of the Jacobian.
  270.      * @param threshold Desired threshold for QR ranking. If the squared norm
  271.      * of a column vector is smaller or equal to this threshold during QR
  272.      * decomposition, it is considered to be a zero vector and hence the rank
  273.      * of the matrix is reduced.
  274.      */
  275.     public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
  276.                                        double costRelativeTolerance,
  277.                                        double parRelativeTolerance,
  278.                                        double orthoTolerance,
  279.                                        double threshold) {
  280.         super(null); // No custom convergence criterion.
  281.         this.initialStepBoundFactor = initialStepBoundFactor;
  282.         this.costRelativeTolerance = costRelativeTolerance;
  283.         this.parRelativeTolerance = parRelativeTolerance;
  284.         this.orthoTolerance = orthoTolerance;
  285.         this.qrRankingThreshold = threshold;
  286.     }

  288.     /** {@inheritDoc} */
  289.     @Override
  290.     protected PointVectorValuePair doOptimize() {
  291.         checkParameters();

  293.         final int nR = getTarget().length; // Number of observed data.
  294.         final double[] currentPoint = getStartPoint();
  295.         final int nC = currentPoint.length; // Number of parameters.

  297.         // arrays shared with the other private methods
  298.         solvedCols  = FastMath.min(nR, nC);
  299.         diagR       = new double[nC];
  300.         jacNorm     = new double[nC];
  301.         beta        = new double[nC];
  302.         permutation = new int[nC];
  303.         lmDir       = new double[nC];

  305.         // local point
  306.         double   delta   = 0;
  307.         double   xNorm   = 0;
  308.         double[] diag    = new double[nC];
  309.         double[] oldX    = new double[nC];
  310.         double[] oldRes  = new double[nR];
  311.         double[] oldObj  = new double[nR];
  312.         double[] qtf     = new double[nR];
  313.         double[] work1   = new double[nC];
  314.         double[] work2   = new double[nC];
  315.         double[] work3   = new double[nC];

  317.         final RealMatrix weightMatrixSqrt = getWeightSquareRoot();

  319.         // Evaluate the function at the starting point and calculate its norm.
  320.         double[] currentObjective = computeObjectiveValue(currentPoint);
  321.         double[] currentResiduals = computeResiduals(currentObjective);
  322.         PointVectorValuePair current = new PointVectorValuePair(currentPoint, currentObjective);
  323.         double currentCost = computeCost(currentResiduals);

  325.         // Outer loop.
  326.         lmPar = 0;
  327.         boolean firstIteration = true;
  328.         final ConvergenceChecker<PointVectorValuePair> checker = getConvergenceChecker();
  329.         while (true) {
  330.             incrementIterationCount();

  332.             final PointVectorValuePair previous = current;

  334.             // QR decomposition of the jacobian matrix
  335.             qrDecomposition(computeWeightedJacobian(currentPoint));

  337.             weightedResidual = weightMatrixSqrt.operate(currentResiduals);
  338.             for (int i = 0; i < nR; i++) {
  339.                 qtf[i] = weightedResidual[i];
  340.             }

  342.             // compute Qt.res
  343.             qTy(qtf);

  345.             // now we don't need Q anymore,
  346.             // so let jacobian contain the R matrix with its diagonal elements
  347.             for (int k = 0; k < solvedCols; ++k) {
  348.                 int pk = permutation[k];
  349.                 weightedJacobian[k][pk] = diagR[pk];
  350.             }

  352.             if (firstIteration) {
  353.                 // scale the point according to the norms of the columns
  354.                 // of the initial jacobian
  355.                 xNorm = 0;
  356.                 for (int k = 0; k < nC; ++k) {
  357.                     double dk = jacNorm[k];
  358.                     if (dk == 0) {
  359.                         dk = 1.0;
  360.                     }
  361.                     double xk = dk * currentPoint[k];
  362.                     xNorm  += xk * xk;
  363.                     diag[k] = dk;
  364.                 }
  365.                 xNorm = FastMath.sqrt(xNorm);

  367.                 // initialize the step bound delta
  368.                 delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
  369.             }

  371.             // check orthogonality between function vector and jacobian columns
  372.             double maxCosine = 0;
  373.             if (currentCost != 0) {
  374.                 for (int j = 0; j < solvedCols; ++j) {
  375.                     int    pj = permutation[j];
  376.                     double s  = jacNorm[pj];
  377.                     if (s != 0) {
  378.                         double sum = 0;
  379.                         for (int i = 0; i <= j; ++i) {
  380.                             sum += weightedJacobian[i][pj] * qtf[i];
  381.                         }
  382.                         maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * currentCost));
  383.                     }
  384.                 }
  385.             }
  386.             if (maxCosine <= orthoTolerance) {
  387.                 // Convergence has been reached.
  388.                 setCost(currentCost);
  389.                 return current;
  390.             }

  392.             // rescale if necessary
  393.             for (int j = 0; j < nC; ++j) {
  394.                 diag[j] = FastMath.max(diag[j], jacNorm[j]);
  395.             }

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

  400.                 // save the state
  401.                 for (int j = 0; j < solvedCols; ++j) {
  402.                     int pj = permutation[j];
  403.                     oldX[pj] = currentPoint[pj];
  404.                 }
  405.                 final double previousCost = currentCost;
  406.                 double[] tmpVec = weightedResidual;
  407.                 weightedResidual = oldRes;
  408.                 oldRes    = tmpVec;
  409.                 tmpVec    = currentObjective;
  410.                 currentObjective = oldObj;
  411.                 oldObj    = tmpVec;

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

  416.                 // compute the new point and the norm of the evolution direction
  417.                 double lmNorm = 0;
  418.                 for (int j = 0; j < solvedCols; ++j) {
  419.                     int pj = permutation[j];
  420.                     lmDir[pj] = -lmDir[pj];
  421.                     currentPoint[pj] = oldX[pj] + lmDir[pj];
  422.                     double s = diag[pj] * lmDir[pj];
  423.                     lmNorm  += s * s;
  424.                 }
  425.                 lmNorm = FastMath.sqrt(lmNorm);
  426.                 // on the first iteration, adjust the initial step bound.
  427.                 if (firstIteration) {
  428.                     delta = FastMath.min(delta, lmNorm);
  429.                 }

  431.                 // Evaluate the function at x + p and calculate its norm.
  432.                 currentObjective = computeObjectiveValue(currentPoint);
  433.                 currentResiduals = computeResiduals(currentObjective);
  434.                 current = new PointVectorValuePair(currentPoint, currentObjective);
  435.                 currentCost = computeCost(currentResiduals);

  437.                 // compute the scaled actual reduction
  438.                 double actRed = -1.0;
  439.                 if (0.1 * currentCost < previousCost) {
  440.                     double r = currentCost / previousCost;
  441.                     actRed = 1.0 - r * r;
  442.                 }

  444.                 // compute the scaled predicted reduction
  445.                 // and the scaled directional derivative
  446.                 for (int j = 0; j < solvedCols; ++j) {
  447.                     int pj = permutation[j];
  448.                     double dirJ = lmDir[pj];
  449.                     work1[j] = 0;
  450.                     for (int i = 0; i <= j; ++i) {
  451.                         work1[i] += weightedJacobian[i][pj] * dirJ;
  452.                     }
  453.                 }
  454.                 double coeff1 = 0;
  455.                 for (int j = 0; j < solvedCols; ++j) {
  456.                     coeff1 += work1[j] * work1[j];
  457.                 }
  458.                 double pc2 = previousCost * previousCost;
  459.                 coeff1 /= pc2;
  460.                 double coeff2 = lmPar * lmNorm * lmNorm / pc2;
  461.                 double preRed = coeff1 + 2 * coeff2;
  462.                 double dirDer = -(coeff1 + coeff2);

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

  467.                 // update the step bound
  468.                 if (ratio <= 0.25) {
  469.                     double tmp =
  470.                         (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
  471.                         if ((0.1 * currentCost >= previousCost) || (tmp < 0.1)) {
  472.                             tmp = 0.1;
  473.                         }
  474.                         delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
  475.                         lmPar /= tmp;
  476.                 } else if ((lmPar == 0) || (ratio >= 0.75)) {
  477.                     delta = 2 * lmNorm;
  478.                     lmPar *= 0.5;
  479.                 }

  481.                 // test for successful iteration.
  482.                 if (ratio >= 1.0e-4) {
  483.                     // successful iteration, update the norm
  484.                     firstIteration = false;
  485.                     xNorm = 0;
  486.                     for (int k = 0; k < nC; ++k) {
  487.                         double xK = diag[k] * currentPoint[k];
  488.                         xNorm += xK * xK;
  489.                     }
  490.                     xNorm = FastMath.sqrt(xNorm);

  492.                     // tests for convergence.
  493.                     if (checker != null && checker.converged(getIterations(), previous, current)) {
  494.                         setCost(currentCost);
  495.                         return current;
  496.                     }
  497.                 } else {
  498.                     // failed iteration, reset the previous values
  499.                     currentCost = previousCost;
  500.                     for (int j = 0; j < solvedCols; ++j) {
  501.                         int pj = permutation[j];
  502.                         currentPoint[pj] = oldX[pj];
  503.                     }
  504.                     tmpVec    = weightedResidual;
  505.                     weightedResidual = oldRes;
  506.                     oldRes    = tmpVec;
  507.                     tmpVec    = currentObjective;
  508.                     currentObjective = oldObj;
  509.                     oldObj    = tmpVec;
  510.                     // Reset "current" to previous values.
  511.                     current = new PointVectorValuePair(currentPoint, currentObjective);
  512.                 }

  514.                 // Default convergence criteria.
  515.                 if ((FastMath.abs(actRed) <= costRelativeTolerance &&
  516.                      preRed <= costRelativeTolerance &&
  517.                      ratio <= 2.0) ||
  518.                     delta <= parRelativeTolerance * xNorm) {
  519.                     setCost(currentCost);
  520.                     return current;
  521.                 }

  523.                 // tests for termination and stringent tolerances
  524.                 if (FastMath.abs(actRed) <= TWO_EPS &&
  525.                     preRed <= TWO_EPS &&
  526.                     ratio <= 2.0) {
  527.                     throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
  528.                                                    costRelativeTolerance);
  529.                 } else if (delta <= TWO_EPS * xNorm) {
  530.                     throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
  531.                                                    parRelativeTolerance);
  532.                 } else if (maxCosine <= TWO_EPS) {
  533.                     throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
  534.                                                    orthoTolerance);
  535.                 }
  536.             }
  537.         }
  538.     }

  540.     /**
  541.      * Determine the Levenberg-Marquardt parameter.
  542.      * <p>This implementation is a translation in Java of the MINPACK
  543.      * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
  544.      * routine.</p>
  545.      * <p>This method sets the lmPar and lmDir attributes.</p>
  546.      * <p>The authors of the original fortran function are:</p>
  547.      * <ul>
  548.      *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
  549.      *   <li>Burton  S. Garbow</li>
  550.      *   <li>Kenneth E. Hillstrom</li>
  551.      *   <li>Jorge   J. More</li>
  552.      * </ul>
  553.      * <p>Luc Maisonobe did the Java translation.</p>
  554.      *
  555.      * @param qy array containing qTy
  556.      * @param delta upper bound on the euclidean norm of diagR * lmDir
  557.      * @param diag diagonal matrix
  558.      * @param work1 work array
  559.      * @param work2 work array
  560.      * @param work3 work array
  561.      */
  562.     private void determineLMParameter(double[] qy, double delta, double[] diag,
  563.                                       double[] work1, double[] work2, double[] work3) {
  564.         final int nC = weightedJacobian[0].length;

  566.         // compute and store in x the gauss-newton direction, if the
  567.         // jacobian is rank-deficient, obtain a least squares solution
  568.         for (int j = 0; j < rank; ++j) {
  569.             lmDir[permutation[j]] = qy[j];
  570.         }
  571.         for (int j = rank; j < nC; ++j) {
  572.             lmDir[permutation[j]] = 0;
  573.         }
  574.         for (int k = rank - 1; k >= 0; --k) {
  575.             int pk = permutation[k];
  576.             double ypk = lmDir[pk] / diagR[pk];
  577.             for (int i = 0; i < k; ++i) {
  578.                 lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk];
  579.             }
  580.             lmDir[pk] = ypk;
  581.         }

  583.         // evaluate the function at the origin, and test
  584.         // for acceptance of the Gauss-Newton direction
  585.         double dxNorm = 0;
  586.         for (int j = 0; j < solvedCols; ++j) {
  587.             int pj = permutation[j];
  588.             double s = diag[pj] * lmDir[pj];
  589.             work1[pj] = s;
  590.             dxNorm += s * s;
  591.         }
  592.         dxNorm = FastMath.sqrt(dxNorm);
  593.         double fp = dxNorm - delta;
  594.         if (fp <= 0.1 * delta) {
  595.             lmPar = 0;
  596.             return;
  597.         }

  599.         // if the jacobian is not rank deficient, the Newton step provides
  600.         // a lower bound, parl, for the zero of the function,
  601.         // otherwise set this bound to zero
  602.         double sum2;
  603.         double parl = 0;
  604.         if (rank == solvedCols) {
  605.             for (int j = 0; j < solvedCols; ++j) {
  606.                 int pj = permutation[j];
  607.                 work1[pj] *= diag[pj] / dxNorm;
  608.             }
  609.             sum2 = 0;
  610.             for (int j = 0; j < solvedCols; ++j) {
  611.                 int pj = permutation[j];
  612.                 double sum = 0;
  613.                 for (int i = 0; i < j; ++i) {
  614.                     sum += weightedJacobian[i][pj] * work1[permutation[i]];
  615.                 }
  616.                 double s = (work1[pj] - sum) / diagR[pj];
  617.                 work1[pj] = s;
  618.                 sum2 += s * s;
  619.             }
  620.             parl = fp / (delta * sum2);
  621.         }

  623.         // calculate an upper bound, paru, for the zero of the function
  624.         sum2 = 0;
  625.         for (int j = 0; j < solvedCols; ++j) {
  626.             int pj = permutation[j];
  627.             double sum = 0;
  628.             for (int i = 0; i <= j; ++i) {
  629.                 sum += weightedJacobian[i][pj] * qy[i];
  630.             }
  631.             sum /= diag[pj];
  632.             sum2 += sum * sum;
  633.         }
  634.         double gNorm = FastMath.sqrt(sum2);
  635.         double paru = gNorm / delta;
  636.         if (paru == 0) {
  637.             paru = Precision.SAFE_MIN / FastMath.min(delta, 0.1);
  638.         }

  640.         // if the input par lies outside of the interval (parl,paru),
  641.         // set par to the closer endpoint
  642.         lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
  643.         if (lmPar == 0) {
  644.             lmPar = gNorm / dxNorm;
  645.         }

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

  649.             // evaluate the function at the current value of lmPar
  650.             if (lmPar == 0) {
  651.                 lmPar = FastMath.max(Precision.SAFE_MIN, 0.001 * paru);
  652.             }
  653.             double sPar = FastMath.sqrt(lmPar);
  654.             for (int j = 0; j < solvedCols; ++j) {
  655.                 int pj = permutation[j];
  656.                 work1[pj] = sPar * diag[pj];
  657.             }
  658.             determineLMDirection(qy, work1, work2, work3);

  660.             dxNorm = 0;
  661.             for (int j = 0; j < solvedCols; ++j) {
  662.                 int pj = permutation[j];
  663.                 double s = diag[pj] * lmDir[pj];
  664.                 work3[pj] = s;
  665.                 dxNorm += s * s;
  666.             }
  667.             dxNorm = FastMath.sqrt(dxNorm);
  668.             double previousFP = fp;
  669.             fp = dxNorm - delta;

  671.             // if the function is small enough, accept the current value
  672.             // of lmPar, also test for the exceptional cases where parl is zero
  673.             if ((FastMath.abs(fp) <= 0.1 * delta) ||
  674.                     ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
  675.                 return;
  676.             }

  678.             // compute the Newton correction
  679.             for (int j = 0; j < solvedCols; ++j) {
  680.                 int pj = permutation[j];
  681.                 work1[pj] = work3[pj] * diag[pj] / dxNorm;
  682.             }
  683.             for (int j = 0; j < solvedCols; ++j) {
  684.                 int pj = permutation[j];
  685.                 work1[pj] /= work2[j];
  686.                 double tmp = work1[pj];
  687.                 for (int i = j + 1; i < solvedCols; ++i) {
  688.                     work1[permutation[i]] -= weightedJacobian[i][pj] * tmp;
  689.                 }
  690.             }
  691.             sum2 = 0;
  692.             for (int j = 0; j < solvedCols; ++j) {
  693.                 double s = work1[permutation[j]];
  694.                 sum2 += s * s;
  695.             }
  696.             double correction = fp / (delta * sum2);

  698.             // depending on the sign of the function, update parl or paru.
  699.             if (fp > 0) {
  700.                 parl = FastMath.max(parl, lmPar);
  701.             } else if (fp < 0) {
  702.                 paru = FastMath.min(paru, lmPar);
  703.             }

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

  708.         }
  709.     }

  711.     /**
  712.      * Solve a*x = b and d*x = 0 in the least squares sense.
  713.      * <p>This implementation is a translation in Java of the MINPACK
  714.      * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
  715.      * routine.</p>
  716.      * <p>This method sets the lmDir and lmDiag attributes.</p>
  717.      * <p>The authors of the original fortran function are:</p>
  718.      * <ul>
  719.      *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
  720.      *   <li>Burton  S. Garbow</li>
  721.      *   <li>Kenneth E. Hillstrom</li>
  722.      *   <li>Jorge   J. More</li>
  723.      * </ul>
  724.      * <p>Luc Maisonobe did the Java translation.</p>
  725.      *
  726.      * @param qy array containing qTy
  727.      * @param diag diagonal matrix
  728.      * @param lmDiag diagonal elements associated with lmDir
  729.      * @param work work array
  730.      */
  731.     private void determineLMDirection(double[] qy, double[] diag,
  732.                                       double[] lmDiag, double[] work) {

  734.         // copy R and Qty to preserve input and initialize s
  735.         //  in particular, save the diagonal elements of R in lmDir
  736.         for (int j = 0; j < solvedCols; ++j) {
  737.             int pj = permutation[j];
  738.             for (int i = j + 1; i < solvedCols; ++i) {
  739.                 weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]];
  740.             }
  741.             lmDir[j] = diagR[pj];
  742.             work[j]  = qy[j];
  743.         }

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

  748.             // prepare the row of d to be eliminated, locating the
  749.             // diagonal element using p from the Q.R. factorization
  750.             int pj = permutation[j];
  751.             double dpj = diag[pj];
  752.             if (dpj != 0) {
  753.                 Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
  754.             }
  755.             lmDiag[j] = dpj;

  757.             //  the transformations to eliminate the row of d
  758.             // modify only a single element of Qty
  759.             // beyond the first n, which is initially zero.
  760.             double qtbpj = 0;
  761.             for (int k = j; k < solvedCols; ++k) {
  762.                 int pk = permutation[k];

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

  768.                     final double sin;
  769.                     final double cos;
  770.                     double rkk = weightedJacobian[k][pk];
  771.                     if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
  772.                         final double cotan = rkk / lmDiag[k];
  773.                         sin   = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
  774.                         cos   = sin * cotan;
  775.                     } else {
  776.                         final double tan = lmDiag[k] / rkk;
  777.                         cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
  778.                         sin = cos * tan;
  779.                     }

  781.                     // compute the modified diagonal element of R and
  782.                     // the modified element of (Qty,0)
  783.                     weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
  784.                     final double temp = cos * work[k] + sin * qtbpj;
  785.                     qtbpj = -sin * work[k] + cos * qtbpj;
  786.                     work[k] = temp;

  788.                     // accumulate the tranformation in the row of s
  789.                     for (int i = k + 1; i < solvedCols; ++i) {
  790.                         double rik = weightedJacobian[i][pk];
  791.                         final double temp2 = cos * rik + sin * lmDiag[i];
  792.                         lmDiag[i] = -sin * rik + cos * lmDiag[i];
  793.                         weightedJacobian[i][pk] = temp2;
  794.                     }
  795.                 }
  796.             }

  798.             // store the diagonal element of s and restore
  799.             // the corresponding diagonal element of R
  800.             lmDiag[j] = weightedJacobian[j][permutation[j]];
  801.             weightedJacobian[j][permutation[j]] = lmDir[j];
  802.         }

  804.         // solve the triangular system for z, if the system is
  805.         // singular, then obtain a least squares solution
  806.         int nSing = solvedCols;
  807.         for (int j = 0; j < solvedCols; ++j) {
  808.             if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
  809.                 nSing = j;
  810.             }
  811.             if (nSing < solvedCols) {
  812.                 work[j] = 0;
  813.             }
  814.         }
  815.         if (nSing > 0) {
  816.             for (int j = nSing - 1; j >= 0; --j) {
  817.                 int pj = permutation[j];
  818.                 double sum = 0;
  819.                 for (int i = j + 1; i < nSing; ++i) {
  820.                     sum += weightedJacobian[i][pj] * work[i];
  821.                 }
  822.                 work[j] = (work[j] - sum) / lmDiag[j];
  823.             }
  824.         }

  826.         // permute the components of z back to components of lmDir
  827.         for (int j = 0; j < lmDir.length; ++j) {
  828.             lmDir[permutation[j]] = work[j];
  829.         }
  830.     }

  832.     /**
  833.      * Decompose a matrix A as A.P = Q.R using Householder transforms.
  834.      * <p>As suggested in the P. Lascaux and R. Theodor book
  835.      * <i>Analyse num&eacute;rique matricielle appliqu&eacute;e &agrave;
  836.      * l'art de l'ing&eacute;nieur</i> (Masson, 1986), instead of representing
  837.      * the Householder transforms with u<sub>k</sub> unit vectors such that:
  838.      * <pre>
  839.      * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
  840.      * </pre>
  841.      * we use <sub>k</sub> non-unit vectors such that:
  842.      * <pre>
  843.      * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
  844.      * </pre>
  845.      * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
  846.      * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
  847.      * them from the v<sub>k</sub> vectors would be costly.</p>
  848.      * <p>This decomposition handles rank deficient cases since the tranformations
  849.      * are performed in non-increasing columns norms order thanks to columns
  850.      * pivoting. The diagonal elements of the R matrix are therefore also in
  851.      * non-increasing absolute values order.</p>
  852.      *
  853.      * @param jacobian Weighted Jacobian matrix at the current point.
  854.      * @exception ConvergenceException if the decomposition cannot be performed
  855.      */
  856.     private void qrDecomposition(RealMatrix jacobian) throws ConvergenceException {
  857.         // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
  858.         // hence the multiplication by -1.
  859.         weightedJacobian = jacobian.scalarMultiply(-1).getData();

  861.         final int nR = weightedJacobian.length;
  862.         final int nC = weightedJacobian[0].length;

  864.         // initializations
  865.         for (int k = 0; k < nC; ++k) {
  866.             permutation[k] = k;
  867.             double norm2 = 0;
  868.             for (int i = 0; i < nR; ++i) {
  869.                 double akk = weightedJacobian[i][k];
  870.                 norm2 += akk * akk;
  871.             }
  872.             jacNorm[k] = FastMath.sqrt(norm2);
  873.         }

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

  878.             // select the column with the greatest norm on active components
  879.             int nextColumn = -1;
  880.             double ak2 = Double.NEGATIVE_INFINITY;
  881.             for (int i = k; i < nC; ++i) {
  882.                 double norm2 = 0;
  883.                 for (int j = k; j < nR; ++j) {
  884.                     double aki = weightedJacobian[j][permutation[i]];
  885.                     norm2 += aki * aki;
  886.                 }
  887.                 if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
  888.                     throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
  889.                                                    nR, nC);
  890.                 }
  891.                 if (norm2 > ak2) {
  892.                     nextColumn = i;
  893.                     ak2        = norm2;
  894.                 }
  895.             }
  896.             if (ak2 <= qrRankingThreshold) {
  897.                 rank = k;
  898.                 return;
  899.             }
  900.             int pk                  = permutation[nextColumn];
  901.             permutation[nextColumn] = permutation[k];
  902.             permutation[k]          = pk;

  904.             // choose alpha such that Hk.u = alpha ek
  905.             double akk   = weightedJacobian[k][pk];
  906.             double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
  907.             double betak = 1.0 / (ak2 - akk * alpha);
  908.             beta[pk]     = betak;

  910.             // transform the current column
  911.             diagR[pk]        = alpha;
  912.             weightedJacobian[k][pk] -= alpha;

  914.             // transform the remaining columns
  915.             for (int dk = nC - 1 - k; dk > 0; --dk) {
  916.                 double gamma = 0;
  917.                 for (int j = k; j < nR; ++j) {
  918.                     gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]];
  919.                 }
  920.                 gamma *= betak;
  921.                 for (int j = k; j < nR; ++j) {
  922.                     weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk];
  923.                 }
  924.             }
  925.         }
  926.         rank = solvedCols;
  927.     }

  929.     /**
  930.      * Compute the product Qt.y for some Q.R. decomposition.
  931.      *
  932.      * @param y vector to multiply (will be overwritten with the result)
  933.      */
  934.     private void qTy(double[] y) {
  935.         final int nR = weightedJacobian.length;
  936.         final int nC = weightedJacobian[0].length;

  938.         for (int k = 0; k < nC; ++k) {
  939.             int pk = permutation[k];
  940.             double gamma = 0;
  941.             for (int i = k; i < nR; ++i) {
  942.                 gamma += weightedJacobian[i][pk] * y[i];
  943.             }
  944.             gamma *= beta[pk];
  945.             for (int i = k; i < nR; ++i) {
  946.                 y[i] -= gamma * weightedJacobian[i][pk];
  947.             }
  948.         }
  949.     }

  951.     /**
  952.      * @throws MathUnsupportedOperationException if bounds were passed to the
  953.      * {@link #optimize(OptimizationData[]) optimize} method.
  954.      */
  955.     private void checkParameters() {
  956.         if (getLowerBound() != null ||
  957.             getUpperBound() != null) {
  958.             throw new MathUnsupportedOperationException(LocalizedFormats.CONSTRAINT);
  959.         }
  960.     }
  961. }
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