    001    /*
    002     * Licensed to the Apache Software Foundation (ASF) under one or more
    003     * contributor license agreements.  See the NOTICE file distributed with
    004     * this work for additional information regarding copyright ownership.
    005     * The ASF licenses this file to You under the Apache License, Version 2.0
    006     * (the "License"); you may not use this file except in compliance with
    007     * the License.  You may obtain a copy of the License at
    008     *
    009     *      http://www.apache.org/licenses/LICENSE-2.0
    010     *
    011     * Unless required by applicable law or agreed to in writing, software
    012     * distributed under the License is distributed on an "AS IS" BASIS,
    013     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
    014     * See the License for the specific language governing permissions and
    015     * limitations under the License.
    016     */
    017    package org.apache.commons.math3.optimization.general;
    018
    019    import java.util.Arrays;
    020
    021    import org.apache.commons.math3.exception.ConvergenceException;
    022    import org.apache.commons.math3.exception.util.LocalizedFormats;
    023    import org.apache.commons.math3.optimization.PointVectorValuePair;
    024    import org.apache.commons.math3.optimization.ConvergenceChecker;
    025    import org.apache.commons.math3.linear.RealMatrix;
    026    import org.apache.commons.math3.util.Precision;
    027    import org.apache.commons.math3.util.FastMath;
    028
    029
    030    /**
    031     * This class solves a least squares problem using the Levenberg-Marquardt algorithm.
    032     *
    033     * <p>This implementation <em>should</em> work even for over-determined systems
    034     * (i.e. systems having more point than equations). Over-determined systems
    035     * are solved by ignoring the point which have the smallest impact according
    036     * to their jacobian column norm. Only the rank of the matrix and some loop bounds
    037     * are changed to implement this.</p>
    038     *
    039     * <p>The resolution engine is a simple translation of the MINPACK <a
    040     * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
    041     * changes. The changes include the over-determined resolution, the use of
    042     * inherited convergence checker and the Q.R. decomposition which has been
    043     * rewritten following the algorithm described in the
    044     * P. Lascaux and R. Theodor book <i>Analyse num&eacute;rique matricielle
    045     * appliqu&eacute;e &agrave; l'art de l'ing&eacute;nieur</i>, Masson 1986.</p>
    046     * <p>The authors of the original fortran version are:
    047     * <ul>
    048     * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
    049     * <li>Burton S. Garbow</li>
    050     * <li>Kenneth E. Hillstrom</li>
    051     * <li>Jorge J. More</li>
    052     * </ul>
    053     * The redistribution policy for MINPACK is available <a
    054     * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
    055     * is reproduced below.</p>
    056     *
    057     * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
    058     * <tr><td>
    059     *    Minpack Copyright Notice (1999) University of Chicago.
    060     *    All rights reserved
    061     * </td></tr>
    062     * <tr><td>
    063     * Redistribution and use in source and binary forms, with or without
    064     * modification, are permitted provided that the following conditions
    065     * are met:
    066     * <ol>
    067     *  <li>Redistributions of source code must retain the above copyright
    068     *      notice, this list of conditions and the following disclaimer.</li>
    069     * <li>Redistributions in binary form must reproduce the above
    070     *     copyright notice, this list of conditions and the following
    071     *     disclaimer in the documentation and/or other materials provided
    072     *     with the distribution.</li>
    073     * <li>The end-user documentation included with the redistribution, if any,
    074     *     must include the following acknowledgment:
    075     *     <code>This product includes software developed by the University of
    076     *           Chicago, as Operator of Argonne National Laboratory.</code>
    077     *     Alternately, this acknowledgment may appear in the software itself,
    078     *     if and wherever such third-party acknowledgments normally appear.</li>
    079     * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
    080     *     WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
    081     *     UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
    082     *     THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
    083     *     IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
    084     *     OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
    085     *     OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
    086     *     OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
    087     *     USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
    088     *     THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
    089     *     DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
    090     *     UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
    091     *     BE CORRECTED.</strong></li>
    092     * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
    093     *     HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
    094     *     ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
    095     *     INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
    096     *     ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
    097     *     PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
    098     *     SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
    099     *     (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     * @version $Id: LevenbergMarquardtOptimizer.java 1423555 2012-12-18 18:02:52Z erans $
    105     * @deprecated As of 3.1 (to be removed in 4.0).
    106     * @since 2.0
    107     *
    108     */
    109    @Deprecated
    110    public class LevenbergMarquardtOptimizer extends AbstractLeastSquaresOptimizer {
    111        /** Number of solved point. */
    112        private int solvedCols;
    113        /** Diagonal elements of the R matrix in the Q.R. decomposition. */
    114        private double[] diagR;
    115        /** Norms of the columns of the jacobian matrix. */
    116        private double[] jacNorm;
    117        /** Coefficients of the Householder transforms vectors. */
    118        private double[] beta;
    119        /** Columns permutation array. */
    120        private int[] permutation;
    121        /** Rank of the jacobian matrix. */
    122        private int rank;
    123        /** Levenberg-Marquardt parameter. */
    124        private double lmPar;
    125        /** Parameters evolution direction associated with lmPar. */
    126        private double[] lmDir;
    127        /** Positive input variable used in determining the initial step bound. */
    128        private final double initialStepBoundFactor;
    129        /** Desired relative error in the sum of squares. */
    130        private final double costRelativeTolerance;
    131        /**  Desired relative error in the approximate solution parameters. */
    132        private final double parRelativeTolerance;
    133        /** Desired max cosine on the orthogonality between the function vector
    134         * and the columns of the jacobian. */
    135        private final double orthoTolerance;
    136        /** Threshold for QR ranking. */
    137        private final double qrRankingThreshold;
    138        /** Weighted residuals. */
    139        private double[] weightedResidual;
    140        /** Weighted Jacobian. */
    141        private double[][] weightedJacobian;
    142
    143        /**
    144         * Build an optimizer for least squares problems with default values
    145         * for all the tuning parameters (see the {@link
    146         * #LevenbergMarquardtOptimizer(double,double,double,double,double)
    147         * other contructor}.
    148         * The default values for the algorithm settings are:
    149         * <ul>
    150         *  <li>Initial step bound factor: 100</li>
    151         *  <li>Cost relative tolerance: 1e-10</li>
    152         *  <li>Parameters relative tolerance: 1e-10</li>
    153         *  <li>Orthogonality tolerance: 1e-10</li>
    154         *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
    155         * </ul>
    156         */
    157        public LevenbergMarquardtOptimizer() {
    158            this(100, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
    159        }
    160
    161        /**
    162         * Constructor that allows the specification of a custom convergence
    163         * checker.
    164         * Note that all the usual convergence checks will be <em>disabled</em>.
    165         * The default values for the algorithm settings are:
    166         * <ul>
    167         *  <li>Initial step bound factor: 100</li>
    168         *  <li>Cost relative tolerance: 1e-10</li>
    169         *  <li>Parameters relative tolerance: 1e-10</li>
    170         *  <li>Orthogonality tolerance: 1e-10</li>
    171         *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
    172         * </ul>
    173         *
    174         * @param checker Convergence checker.
    175         */
    176        public LevenbergMarquardtOptimizer(ConvergenceChecker<PointVectorValuePair> checker) {
    177            this(100, checker, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
    178        }
    179
    180        /**
    181         * Constructor that allows the specification of a custom convergence
    182         * checker, in addition to the standard ones.
    183         *
    184         * @param initialStepBoundFactor Positive input variable used in
    185         * determining the initial step bound. This bound is set to the
    186         * product of initialStepBoundFactor and the euclidean norm of
    187         * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
    188         * itself. In most cases factor should lie in the interval
    189         * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
    190         * @param checker Convergence checker.
    191         * @param costRelativeTolerance Desired relative error in the sum of
    192         * squares.
    193         * @param parRelativeTolerance Desired relative error in the approximate
    194         * solution parameters.
    195         * @param orthoTolerance Desired max cosine on the orthogonality between
    196         * the function vector and the columns of the Jacobian.
    197         * @param threshold Desired threshold for QR ranking. If the squared norm
    198         * of a column vector is smaller or equal to this threshold during QR
    199         * decomposition, it is considered to be a zero vector and hence the rank
    200         * of the matrix is reduced.
    201         */
    202        public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
    203                                           ConvergenceChecker<PointVectorValuePair> checker,
    204                                           double costRelativeTolerance,
    205                                           double parRelativeTolerance,
    206                                           double orthoTolerance,
    207                                           double threshold) {
    208            super(checker);
    209            this.initialStepBoundFactor = initialStepBoundFactor;
    210            this.costRelativeTolerance = costRelativeTolerance;
    211            this.parRelativeTolerance = parRelativeTolerance;
    212            this.orthoTolerance = orthoTolerance;
    213            this.qrRankingThreshold = threshold;
    214        }
    215
    216        /**
    217         * Build an optimizer for least squares problems with default values
    218         * for some of the tuning parameters (see the {@link
    219         * #LevenbergMarquardtOptimizer(double,double,double,double,double)
    220         * other contructor}.
    221         * The default values for the algorithm settings are:
    222         * <ul>
    223         *  <li>Initial step bound factor}: 100</li>
    224         *  <li>QR ranking threshold}: {@link Precision#SAFE_MIN}</li>
    225         * </ul>
    226         *
    227         * @param costRelativeTolerance Desired relative error in the sum of
    228         * squares.
    229         * @param parRelativeTolerance Desired relative error in the approximate
    230         * solution parameters.
    231         * @param orthoTolerance Desired max cosine on the orthogonality between
    232         * the function vector and the columns of the Jacobian.
    233         */
    234        public LevenbergMarquardtOptimizer(double costRelativeTolerance,
    235                                           double parRelativeTolerance,
    236                                           double orthoTolerance) {
    237            this(100,
    238                 costRelativeTolerance, parRelativeTolerance, orthoTolerance,
    239                 Precision.SAFE_MIN);
    240        }
    241
    242        /**
    243         * The arguments control the behaviour of the default convergence checking
    244         * procedure.
    245         * Additional criteria can defined through the setting of a {@link
    246         * ConvergenceChecker}.
    247         *
    248         * @param initialStepBoundFactor Positive input variable used in
    249         * determining the initial step bound. This bound is set to the
    250         * product of initialStepBoundFactor and the euclidean norm of
    251         * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
    252         * itself. In most cases factor should lie in the interval
    253         * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
    254         * @param costRelativeTolerance Desired relative error in the sum of
    255         * squares.
    256         * @param parRelativeTolerance Desired relative error in the approximate
    257         * solution parameters.
    258         * @param orthoTolerance Desired max cosine on the orthogonality between
    259         * the function vector and the columns of the Jacobian.
    260         * @param threshold Desired threshold for QR ranking. If the squared norm
    261         * of a column vector is smaller or equal to this threshold during QR
    262         * decomposition, it is considered to be a zero vector and hence the rank
    263         * of the matrix is reduced.
    264         */
    265        public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
    266                                           double costRelativeTolerance,
    267                                           double parRelativeTolerance,
    268                                           double orthoTolerance,
    269                                           double threshold) {
    270            super(null); // No custom convergence criterion.
    271            this.initialStepBoundFactor = initialStepBoundFactor;
    272            this.costRelativeTolerance = costRelativeTolerance;
    273            this.parRelativeTolerance = parRelativeTolerance;
    274            this.orthoTolerance = orthoTolerance;
    275            this.qrRankingThreshold = threshold;
    276        }
    277
    278        /** {@inheritDoc} */
    279        @Override
    280        protected PointVectorValuePair doOptimize() {
    281            final int nR = getTarget().length; // Number of observed data.
    282            final double[] currentPoint = getStartPoint();
    283            final int nC = currentPoint.length; // Number of parameters.
    284
    285            // arrays shared with the other private methods
    286            solvedCols  = FastMath.min(nR, nC);
    287            diagR       = new double[nC];
    288            jacNorm     = new double[nC];
    289            beta        = new double[nC];
    290            permutation = new int[nC];
    291            lmDir       = new double[nC];
    292
    293            // local point
    294            double   delta   = 0;
    295            double   xNorm   = 0;
    296            double[] diag    = new double[nC];
    297            double[] oldX    = new double[nC];
    298            double[] oldRes  = new double[nR];
    299            double[] oldObj  = new double[nR];
    300            double[] qtf     = new double[nR];
    301            double[] work1   = new double[nC];
    302            double[] work2   = new double[nC];
    303            double[] work3   = new double[nC];
    304
    305            final RealMatrix weightMatrixSqrt = getWeightSquareRoot();
    306
    307            // Evaluate the function at the starting point and calculate its norm.
    308            double[] currentObjective = computeObjectiveValue(currentPoint);
    309            double[] currentResiduals = computeResiduals(currentObjective);
    310            PointVectorValuePair current = new PointVectorValuePair(currentPoint, currentObjective);
    311            double currentCost = computeCost(currentResiduals);
    312
    313            // Outer loop.
    314            lmPar = 0;
    315            boolean firstIteration = true;
    316            int iter = 0;
    317            final ConvergenceChecker<PointVectorValuePair> checker = getConvergenceChecker();
    318            while (true) {
    319                ++iter;
    320                final PointVectorValuePair previous = current;
    321
    322                // QR decomposition of the jacobian matrix
    323                qrDecomposition(computeWeightedJacobian(currentPoint));
    324
    325                weightedResidual = weightMatrixSqrt.operate(currentResiduals);
    326                for (int i = 0; i < nR; i++) {
    327                    qtf[i] = weightedResidual[i];
    328                }
    329
    330                // compute Qt.res
    331                qTy(qtf);
    332
    333                // now we don't need Q anymore,
    334                // so let jacobian contain the R matrix with its diagonal elements
    335                for (int k = 0; k < solvedCols; ++k) {
    336                    int pk = permutation[k];
    337                    weightedJacobian[k][pk] = diagR[pk];
    338                }
    339
    340                if (firstIteration) {
    341                    // scale the point according to the norms of the columns
    342                    // of the initial jacobian
    343                    xNorm = 0;
    344                    for (int k = 0; k < nC; ++k) {
    345                        double dk = jacNorm[k];
    346                        if (dk == 0) {
    347                            dk = 1.0;
    348                        }
    349                        double xk = dk * currentPoint[k];
    350                        xNorm  += xk * xk;
    351                        diag[k] = dk;
    352                    }
    353                    xNorm = FastMath.sqrt(xNorm);
    354
    355                    // initialize the step bound delta
    356                    delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
    357                }
    358
    359                // check orthogonality between function vector and jacobian columns
    360                double maxCosine = 0;
    361                if (currentCost != 0) {
    362                    for (int j = 0; j < solvedCols; ++j) {
    363                        int    pj = permutation[j];
    364                        double s  = jacNorm[pj];
    365                        if (s != 0) {
    366                            double sum = 0;
    367                            for (int i = 0; i <= j; ++i) {
    368                                sum += weightedJacobian[i][pj] * qtf[i];
    369                            }
    370                            maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * currentCost));
    371                        }
    372                    }
    373                }
    374                if (maxCosine <= orthoTolerance) {
    375                    // Convergence has been reached.
    376                    setCost(currentCost);
    377                    // Update (deprecated) "point" field.
    378                    point = current.getPoint();
    379                    return current;
    380                }
    381
    382                // rescale if necessary
    383                for (int j = 0; j < nC; ++j) {
    384                    diag[j] = FastMath.max(diag[j], jacNorm[j]);
    385                }
    386
    387                // Inner loop.
    388                for (double ratio = 0; ratio < 1.0e-4;) {
    389
    390                    // save the state
    391                    for (int j = 0; j < solvedCols; ++j) {
    392                        int pj = permutation[j];
    393                        oldX[pj] = currentPoint[pj];
    394                    }
    395                    final double previousCost = currentCost;
    396                    double[] tmpVec = weightedResidual;
    397                    weightedResidual = oldRes;
    398                    oldRes    = tmpVec;
    399                    tmpVec    = currentObjective;
    400                    currentObjective = oldObj;
    401                    oldObj    = tmpVec;
    402
    403                    // determine the Levenberg-Marquardt parameter
    404                    determineLMParameter(qtf, delta, diag, work1, work2, work3);
    405
    406                    // compute the new point and the norm of the evolution direction
    407                    double lmNorm = 0;
    408                    for (int j = 0; j < solvedCols; ++j) {
    409                        int pj = permutation[j];
    410                        lmDir[pj] = -lmDir[pj];
    411                        currentPoint[pj] = oldX[pj] + lmDir[pj];
    412                        double s = diag[pj] * lmDir[pj];
    413                        lmNorm  += s * s;
    414                    }
    415                    lmNorm = FastMath.sqrt(lmNorm);
    416                    // on the first iteration, adjust the initial step bound.
    417                    if (firstIteration) {
    418                        delta = FastMath.min(delta, lmNorm);
    419                    }
    420
    421                    // Evaluate the function at x + p and calculate its norm.
    422                    currentObjective = computeObjectiveValue(currentPoint);
    423                    currentResiduals = computeResiduals(currentObjective);
    424                    current = new PointVectorValuePair(currentPoint, currentObjective);
    425                    currentCost = computeCost(currentResiduals);
    426
    427                    // compute the scaled actual reduction
    428                    double actRed = -1.0;
    429                    if (0.1 * currentCost < previousCost) {
    430                        double r = currentCost / previousCost;
    431                        actRed = 1.0 - r * r;
    432                    }
    433
    434                    // compute the scaled predicted reduction
    435                    // and the scaled directional derivative
    436                    for (int j = 0; j < solvedCols; ++j) {
    437                        int pj = permutation[j];
    438                        double dirJ = lmDir[pj];
    439                        work1[j] = 0;
    440                        for (int i = 0; i <= j; ++i) {
    441                            work1[i] += weightedJacobian[i][pj] * dirJ;
    442                        }
    443                    }
    444                    double coeff1 = 0;
    445                    for (int j = 0; j < solvedCols; ++j) {
    446                        coeff1 += work1[j] * work1[j];
    447                    }
    448                    double pc2 = previousCost * previousCost;
    449                    coeff1 = coeff1 / pc2;
    450                    double coeff2 = lmPar * lmNorm * lmNorm / pc2;
    451                    double preRed = coeff1 + 2 * coeff2;
    452                    double dirDer = -(coeff1 + coeff2);
    453
    454                    // ratio of the actual to the predicted reduction
    455                    ratio = (preRed == 0) ? 0 : (actRed / preRed);
    456
    457                    // update the step bound
    458                    if (ratio <= 0.25) {
    459                        double tmp =
    460                            (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
    461                            if ((0.1 * currentCost >= previousCost) || (tmp < 0.1)) {
    462                                tmp = 0.1;
    463                            }
    464                            delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
    465                            lmPar /= tmp;
    466                    } else if ((lmPar == 0) || (ratio >= 0.75)) {
    467                        delta = 2 * lmNorm;
    468                        lmPar *= 0.5;
    469                    }
    470
    471                    // test for successful iteration.
    472                    if (ratio >= 1.0e-4) {
    473                        // successful iteration, update the norm
    474                        firstIteration = false;
    475                        xNorm = 0;
    476                        for (int k = 0; k < nC; ++k) {
    477                            double xK = diag[k] * currentPoint[k];
    478                            xNorm += xK * xK;
    479                        }
    480                        xNorm = FastMath.sqrt(xNorm);
    481
    482                        // tests for convergence.
    483                        if (checker != null) {
    484                            // we use the vectorial convergence checker
    485                            if (checker.converged(iter, previous, current)) {
    486                                setCost(currentCost);
    487                                // Update (deprecated) "point" field.
    488                                point = current.getPoint();
    489                                return current;
    490                            }
    491                        }
    492                    } else {
    493                        // failed iteration, reset the previous values
    494                        currentCost = previousCost;
    495                        for (int j = 0; j < solvedCols; ++j) {
    496                            int pj = permutation[j];
    497                            currentPoint[pj] = oldX[pj];
    498                        }
    499                        tmpVec    = weightedResidual;
    500                        weightedResidual = oldRes;
    501                        oldRes    = tmpVec;
    502                        tmpVec    = currentObjective;
    503                        currentObjective = oldObj;
    504                        oldObj    = tmpVec;
    505                        // Reset "current" to previous values.
    506                        current = new PointVectorValuePair(currentPoint, currentObjective);
    507                    }
    508
    509                    // Default convergence criteria.
    510                    if ((FastMath.abs(actRed) <= costRelativeTolerance &&
    511                         preRed <= costRelativeTolerance &&
    512                         ratio <= 2.0) ||
    513                        delta <= parRelativeTolerance * xNorm) {
    514                        setCost(currentCost);
    515                        // Update (deprecated) "point" field.
    516                        point = current.getPoint();
    517                        return current;
    518                    }
    519
    520                    // tests for termination and stringent tolerances
    521                    // (2.2204e-16 is the machine epsilon for IEEE754)
    522                    if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
    523                        throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
    524                                                       costRelativeTolerance);
    525                    } else if (delta <= 2.2204e-16 * xNorm) {
    526                        throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
    527                                                       parRelativeTolerance);
    528                    } else if (maxCosine <= 2.2204e-16)  {
    529                        throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
    530                                                       orthoTolerance);
    531                    }
    532                }
    533            }
    534        }
    535
    536        /**
    537         * Determine the Levenberg-Marquardt parameter.
    538         * <p>This implementation is a translation in Java of the MINPACK
    539         * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
    540         * routine.</p>
    541         * <p>This method sets the lmPar and lmDir attributes.</p>
    542         * <p>The authors of the original fortran function are:</p>
    543         * <ul>
    544         *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
    545         *   <li>Burton  S. Garbow</li>
    546         *   <li>Kenneth E. Hillstrom</li>
    547         *   <li>Jorge   J. More</li>
    548         * </ul>
    549         * <p>Luc Maisonobe did the Java translation.</p>
    550         *
    551         * @param qy array containing qTy
    552         * @param delta upper bound on the euclidean norm of diagR * lmDir
    553         * @param diag diagonal matrix
    554         * @param work1 work array
    555         * @param work2 work array
    556         * @param work3 work array
    557         */
    558        private void determineLMParameter(double[] qy, double delta, double[] diag,
    559                                          double[] work1, double[] work2, double[] work3) {
    560            final int nC = weightedJacobian[0].length;
    561
    562            // compute and store in x the gauss-newton direction, if the
    563            // jacobian is rank-deficient, obtain a least squares solution
    564            for (int j = 0; j < rank; ++j) {
    565                lmDir[permutation[j]] = qy[j];
    566            }
    567            for (int j = rank; j < nC; ++j) {
    568                lmDir[permutation[j]] = 0;
    569            }
    570            for (int k = rank - 1; k >= 0; --k) {
    571                int pk = permutation[k];
    572                double ypk = lmDir[pk] / diagR[pk];
    573                for (int i = 0; i < k; ++i) {
    574                    lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk];
    575                }
    576                lmDir[pk] = ypk;
    577            }
    578
    579            // evaluate the function at the origin, and test
    580            // for acceptance of the Gauss-Newton direction
    581            double dxNorm = 0;
    582            for (int j = 0; j < solvedCols; ++j) {
    583                int pj = permutation[j];
    584                double s = diag[pj] * lmDir[pj];
    585                work1[pj] = s;
    586                dxNorm += s * s;
    587            }
    588            dxNorm = FastMath.sqrt(dxNorm);
    589            double fp = dxNorm - delta;
    590            if (fp <= 0.1 * delta) {
    591                lmPar = 0;
    592                return;
    593            }
    594
    595            // if the jacobian is not rank deficient, the Newton step provides
    596            // a lower bound, parl, for the zero of the function,
    597            // otherwise set this bound to zero
    598            double sum2;
    599            double parl = 0;
    600            if (rank == solvedCols) {
    601                for (int j = 0; j < solvedCols; ++j) {
    602                    int pj = permutation[j];
    603                    work1[pj] *= diag[pj] / dxNorm;
    604                }
    605                sum2 = 0;
    606                for (int j = 0; j < solvedCols; ++j) {
    607                    int pj = permutation[j];
    608                    double sum = 0;
    609                    for (int i = 0; i < j; ++i) {
    610                        sum += weightedJacobian[i][pj] * work1[permutation[i]];
    611                    }
    612                    double s = (work1[pj] - sum) / diagR[pj];
    613                    work1[pj] = s;
    614                    sum2 += s * s;
    615                }
    616                parl = fp / (delta * sum2);
    617            }
    618
    619            // calculate an upper bound, paru, for the zero of the function
    620            sum2 = 0;
    621            for (int j = 0; j < solvedCols; ++j) {
    622                int pj = permutation[j];
    623                double sum = 0;
    624                for (int i = 0; i <= j; ++i) {
    625                    sum += weightedJacobian[i][pj] * qy[i];
    626                }
    627                sum /= diag[pj];
    628                sum2 += sum * sum;
    629            }
    630            double gNorm = FastMath.sqrt(sum2);
    631            double paru = gNorm / delta;
    632            if (paru == 0) {
    633                // 2.2251e-308 is the smallest positive real for IEE754
    634                paru = 2.2251e-308 / FastMath.min(delta, 0.1);
    635            }
    636
    637            // if the input par lies outside of the interval (parl,paru),
    638            // set par to the closer endpoint
    639            lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
    640            if (lmPar == 0) {
    641                lmPar = gNorm / dxNorm;
    642            }
    643
    644            for (int countdown = 10; countdown >= 0; --countdown) {
    645
    646                // evaluate the function at the current value of lmPar
    647                if (lmPar == 0) {
    648                    lmPar = FastMath.max(2.2251e-308, 0.001 * paru);
    649                }
    650                double sPar = FastMath.sqrt(lmPar);
    651                for (int j = 0; j < solvedCols; ++j) {
    652                    int pj = permutation[j];
    653                    work1[pj] = sPar * diag[pj];
    654                }
    655                determineLMDirection(qy, work1, work2, work3);
    656
    657                dxNorm = 0;
    658                for (int j = 0; j < solvedCols; ++j) {
    659                    int pj = permutation[j];
    660                    double s = diag[pj] * lmDir[pj];
    661                    work3[pj] = s;
    662                    dxNorm += s * s;
    663                }
    664                dxNorm = FastMath.sqrt(dxNorm);
    665                double previousFP = fp;
    666                fp = dxNorm - delta;
    667
    668                // if the function is small enough, accept the current value
    669                // of lmPar, also test for the exceptional cases where parl is zero
    670                if ((FastMath.abs(fp) <= 0.1 * delta) ||
    671                        ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
    672                    return;
    673                }
    674
    675                // compute the Newton correction
    676                for (int j = 0; j < solvedCols; ++j) {
    677                    int pj = permutation[j];
    678                    work1[pj] = work3[pj] * diag[pj] / dxNorm;
    679                }
    680                for (int j = 0; j < solvedCols; ++j) {
    681                    int pj = permutation[j];
    682                    work1[pj] /= work2[j];
    683                    double tmp = work1[pj];
    684                    for (int i = j + 1; i < solvedCols; ++i) {
    685                        work1[permutation[i]] -= weightedJacobian[i][pj] * tmp;
    686                    }
    687                }
    688                sum2 = 0;
    689                for (int j = 0; j < solvedCols; ++j) {
    690                    double s = work1[permutation[j]];
    691                    sum2 += s * s;
    692                }
    693                double correction = fp / (delta * sum2);
    694
    695                // depending on the sign of the function, update parl or paru.
    696                if (fp > 0) {
    697                    parl = FastMath.max(parl, lmPar);
    698                } else if (fp < 0) {
    699                    paru = FastMath.min(paru, lmPar);
    700                }
    701
    702                // compute an improved estimate for lmPar
    703                lmPar = FastMath.max(parl, lmPar + correction);
    704
    705            }
    706        }
    707
    708        /**
    709         * Solve a*x = b and d*x = 0 in the least squares sense.
    710         * <p>This implementation is a translation in Java of the MINPACK
    711         * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
    712         * routine.</p>
    713         * <p>This method sets the lmDir and lmDiag attributes.</p>
    714         * <p>The authors of the original fortran function are:</p>
    715         * <ul>
    716         *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
    717         *   <li>Burton  S. Garbow</li>
    718         *   <li>Kenneth E. Hillstrom</li>
    719         *   <li>Jorge   J. More</li>
    720         * </ul>
    721         * <p>Luc Maisonobe did the Java translation.</p>
    722         *
    723         * @param qy array containing qTy
    724         * @param diag diagonal matrix
    725         * @param lmDiag diagonal elements associated with lmDir
    726         * @param work work array
    727         */
    728        private void determineLMDirection(double[] qy, double[] diag,
    729                                          double[] lmDiag, double[] work) {
    730
    731            // copy R and Qty to preserve input and initialize s
    732            //  in particular, save the diagonal elements of R in lmDir
    733            for (int j = 0; j < solvedCols; ++j) {
    734                int pj = permutation[j];
    735                for (int i = j + 1; i < solvedCols; ++i) {
    736                    weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]];
    737                }
    738                lmDir[j] = diagR[pj];
    739                work[j]  = qy[j];
    740            }
    741
    742            // eliminate the diagonal matrix d using a Givens rotation
    743            for (int j = 0; j < solvedCols; ++j) {
    744
    745                // prepare the row of d to be eliminated, locating the
    746                // diagonal element using p from the Q.R. factorization
    747                int pj = permutation[j];
    748                double dpj = diag[pj];
    749                if (dpj != 0) {
    750                    Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
    751                }
    752                lmDiag[j] = dpj;
    753
    754                //  the transformations to eliminate the row of d
    755                // modify only a single element of Qty
    756                // beyond the first n, which is initially zero.
    757                double qtbpj = 0;
    758                for (int k = j; k < solvedCols; ++k) {
    759                    int pk = permutation[k];
    760
    761                    // determine a Givens rotation which eliminates the
    762                    // appropriate element in the current row of d
    763                    if (lmDiag[k] != 0) {
    764
    765                        final double sin;
    766                        final double cos;
    767                        double rkk = weightedJacobian[k][pk];
    768                        if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
    769                            final double cotan = rkk / lmDiag[k];
    770                            sin   = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
    771                            cos   = sin * cotan;
    772                        } else {
    773                            final double tan = lmDiag[k] / rkk;
    774                            cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
    775                            sin = cos * tan;
    776                        }
    777
    778                        // compute the modified diagonal element of R and
    779                        // the modified element of (Qty,0)
    780                        weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
    781                        final double temp = cos * work[k] + sin * qtbpj;
    782                        qtbpj = -sin * work[k] + cos * qtbpj;
    783                        work[k] = temp;
    784
    785                        // accumulate the tranformation in the row of s
    786                        for (int i = k + 1; i < solvedCols; ++i) {
    787                            double rik = weightedJacobian[i][pk];
    788                            final double temp2 = cos * rik + sin * lmDiag[i];
    789                            lmDiag[i] = -sin * rik + cos * lmDiag[i];
    790                            weightedJacobian[i][pk] = temp2;
    791                        }
    792                    }
    793                }
    794
    795                // store the diagonal element of s and restore
    796                // the corresponding diagonal element of R
    797                lmDiag[j] = weightedJacobian[j][permutation[j]];
    798                weightedJacobian[j][permutation[j]] = lmDir[j];
    799            }
    800
    801            // solve the triangular system for z, if the system is
    802            // singular, then obtain a least squares solution
    803            int nSing = solvedCols;
    804            for (int j = 0; j < solvedCols; ++j) {
    805                if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
    806                    nSing = j;
    807                }
    808                if (nSing < solvedCols) {
    809                    work[j] = 0;
    810                }
    811            }
    812            if (nSing > 0) {
    813                for (int j = nSing - 1; j >= 0; --j) {
    814                    int pj = permutation[j];
    815                    double sum = 0;
    816                    for (int i = j + 1; i < nSing; ++i) {
    817                        sum += weightedJacobian[i][pj] * work[i];
    818                    }
    819                    work[j] = (work[j] - sum) / lmDiag[j];
    820                }
    821            }
    822
    823            // permute the components of z back to components of lmDir
    824            for (int j = 0; j < lmDir.length; ++j) {
    825                lmDir[permutation[j]] = work[j];
    826            }
    827        }
    828
    829        /**
    830         * Decompose a matrix A as A.P = Q.R using Householder transforms.
    831         * <p>As suggested in the P. Lascaux and R. Theodor book
    832         * <i>Analyse num&eacute;rique matricielle appliqu&eacute;e &agrave;
    833         * l'art de l'ing&eacute;nieur</i> (Masson, 1986), instead of representing
    834         * the Householder transforms with u<sub>k</sub> unit vectors such that:
    835         * <pre>
    836         * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
    837         * </pre>
    838         * we use <sub>k</sub> non-unit vectors such that:
    839         * <pre>
    840         * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
    841         * </pre>
    842         * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
    843         * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
    844         * them from the v<sub>k</sub> vectors would be costly.</p>
    845         * <p>This decomposition handles rank deficient cases since the tranformations
    846         * are performed in non-increasing columns norms order thanks to columns
    847         * pivoting. The diagonal elements of the R matrix are therefore also in
    848         * non-increasing absolute values order.</p>
    849         *
    850         * @param jacobian Weighted Jacobian matrix at the current point.
    851         * @exception ConvergenceException if the decomposition cannot be performed
    852         */
    853        private void qrDecomposition(RealMatrix jacobian) throws ConvergenceException {
    854            // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
    855            // hence the multiplication by -1.
    856            weightedJacobian = jacobian.scalarMultiply(-1).getData();
    857
    858            final int nR = weightedJacobian.length;
    859            final int nC = weightedJacobian[0].length;
    860
    861            // initializations
    862            for (int k = 0; k < nC; ++k) {
    863                permutation[k] = k;
    864                double norm2 = 0;
    865                for (int i = 0; i < nR; ++i) {
    866                    double akk = weightedJacobian[i][k];
    867                    norm2 += akk * akk;
    868                }
    869                jacNorm[k] = FastMath.sqrt(norm2);
    870            }
    871
    872            // transform the matrix column after column
    873            for (int k = 0; k < nC; ++k) {
    874
    875                // select the column with the greatest norm on active components
    876                int nextColumn = -1;
    877                double ak2 = Double.NEGATIVE_INFINITY;
    878                for (int i = k; i < nC; ++i) {
    879                    double norm2 = 0;
    880                    for (int j = k; j < nR; ++j) {
    881                        double aki = weightedJacobian[j][permutation[i]];
    882                        norm2 += aki * aki;
    883                    }
    884                    if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
    885                        throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
    886                                                       nR, nC);
    887                    }
    888                    if (norm2 > ak2) {
    889                        nextColumn = i;
    890                        ak2        = norm2;
    891                    }
    892                }
    893                if (ak2 <= qrRankingThreshold) {
    894                    rank = k;
    895                    return;
    896                }
    897                int pk                  = permutation[nextColumn];
    898                permutation[nextColumn] = permutation[k];
    899                permutation[k]          = pk;
    900
    901                // choose alpha such that Hk.u = alpha ek
    902                double akk   = weightedJacobian[k][pk];
    903                double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
    904                double betak = 1.0 / (ak2 - akk * alpha);
    905                beta[pk]     = betak;
    906
    907                // transform the current column
    908                diagR[pk]        = alpha;
    909                weightedJacobian[k][pk] -= alpha;
    910
    911                // transform the remaining columns
    912                for (int dk = nC - 1 - k; dk > 0; --dk) {
    913                    double gamma = 0;
    914                    for (int j = k; j < nR; ++j) {
    915                        gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]];
    916                    }
    917                    gamma *= betak;
    918                    for (int j = k; j < nR; ++j) {
    919                        weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk];
    920                    }
    921                }
    922            }
    923            rank = solvedCols;
    924        }
    925
    926        /**
    927         * Compute the product Qt.y for some Q.R. decomposition.
    928         *
    929         * @param y vector to multiply (will be overwritten with the result)
    930         */
    931        private void qTy(double[] y) {
    932            final int nR = weightedJacobian.length;
    933            final int nC = weightedJacobian[0].length;
    934
    935            for (int k = 0; k < nC; ++k) {
    936                int pk = permutation[k];
    937                double gamma = 0;
    938                for (int i = k; i < nR; ++i) {
    939                    gamma += weightedJacobian[i][pk] * y[i];
    940                }
    941                gamma *= beta[pk];
    942                for (int i = k; i < nR; ++i) {
    943                    y[i] -= gamma * weightedJacobian[i][pk];
    944                }
    945            }
    946        }
    947    }