    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.math.estimation;
    018
    019    import java.io.Serializable;
    020    import java.util.Arrays;
    021
    022    import org.apache.commons.math.exception.util.LocalizedFormats;
    023    import org.apache.commons.math.util.FastMath;
    024
    025
    026    /**
    027     * This class solves a least squares problem.
    028     *
    029     * <p>This implementation <em>should</em> work even for over-determined systems
    030     * (i.e. systems having more variables than equations). Over-determined systems
    031     * are solved by ignoring the variables which have the smallest impact according
    032     * to their jacobian column norm. Only the rank of the matrix and some loop bounds
    033     * are changed to implement this.</p>
    034     *
    035     * <p>The resolution engine is a simple translation of the MINPACK <a
    036     * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
    037     * changes. The changes include the over-determined resolution and the Q.R.
    038     * decomposition which has been rewritten following the algorithm described in the
    039     * P. Lascaux and R. Theodor book <i>Analyse num&eacute;rique matricielle
    040     * appliqu&eacute;e &agrave; l'art de l'ing&eacute;nieur</i>, Masson 1986.</p>
    041     * <p>The authors of the original fortran version are:
    042     * <ul>
    043     * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
    044     * <li>Burton S. Garbow</li>
    045     * <li>Kenneth E. Hillstrom</li>
    046     * <li>Jorge J. More</li>
    047     * </ul>
    048     * The redistribution policy for MINPACK is available <a
    049     * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
    050     * is reproduced below.</p>
    051     *
    052     * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
    053     * <tr><td>
    054     *    Minpack Copyright Notice (1999) University of Chicago.
    055     *    All rights reserved
    056     * </td></tr>
    057     * <tr><td>
    058     * Redistribution and use in source and binary forms, with or without
    059     * modification, are permitted provided that the following conditions
    060     * are met:
    061     * <ol>
    062     *  <li>Redistributions of source code must retain the above copyright
    063     *      notice, this list of conditions and the following disclaimer.</li>
    064     * <li>Redistributions in binary form must reproduce the above
    065     *     copyright notice, this list of conditions and the following
    066     *     disclaimer in the documentation and/or other materials provided
    067     *     with the distribution.</li>
    068     * <li>The end-user documentation included with the redistribution, if any,
    069     *     must include the following acknowledgment:
    070     *     <code>This product includes software developed by the University of
    071     *           Chicago, as Operator of Argonne National Laboratory.</code>
    072     *     Alternately, this acknowledgment may appear in the software itself,
    073     *     if and wherever such third-party acknowledgments normally appear.</li>
    074     * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
    075     *     WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
    076     *     UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
    077     *     THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
    078     *     IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
    079     *     OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
    080     *     OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
    081     *     OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
    082     *     USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
    083     *     THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
    084     *     DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
    085     *     UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
    086     *     BE CORRECTED.</strong></li>
    087     * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
    088     *     HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
    089     *     ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
    090     *     INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
    091     *     ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
    092     *     PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
    093     *     SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
    094     *     (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
    095     *     EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
    096     *     POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
    097     * <ol></td></tr>
    098     * </table>
    099
    100     * @version $Revision: 990655 $ $Date: 2010-08-29 23:49:40 +0200 (dim. 29 ao??t 2010) $
    101     * @since 1.2
    102     * @deprecated as of 2.0, everything in package org.apache.commons.math.estimation has
    103     * been deprecated and replaced by package org.apache.commons.math.optimization.general
    104     *
    105     */
    106    @Deprecated
    107    public class LevenbergMarquardtEstimator extends AbstractEstimator implements Serializable {
    108
    109        /** Serializable version identifier */
    110        private static final long serialVersionUID = -5705952631533171019L;
    111
    112        /** Number of solved variables. */
    113        private int solvedCols;
    114
    115        /** Diagonal elements of the R matrix in the Q.R. decomposition. */
    116        private double[] diagR;
    117
    118        /** Norms of the columns of the jacobian matrix. */
    119        private double[] jacNorm;
    120
    121        /** Coefficients of the Householder transforms vectors. */
    122        private double[] beta;
    123
    124        /** Columns permutation array. */
    125        private int[] permutation;
    126
    127        /** Rank of the jacobian matrix. */
    128        private int rank;
    129
    130        /** Levenberg-Marquardt parameter. */
    131        private double lmPar;
    132
    133        /** Parameters evolution direction associated with lmPar. */
    134        private double[] lmDir;
    135
    136        /** Positive input variable used in determining the initial step bound. */
    137        private double initialStepBoundFactor;
    138
    139        /** Desired relative error in the sum of squares. */
    140        private double costRelativeTolerance;
    141
    142        /**  Desired relative error in the approximate solution parameters. */
    143        private double parRelativeTolerance;
    144
    145        /** Desired max cosine on the orthogonality between the function vector
    146         * and the columns of the jacobian. */
    147        private double orthoTolerance;
    148
    149      /**
    150       * Build an estimator for least squares problems.
    151       * <p>The default values for the algorithm settings are:
    152       *   <ul>
    153       *    <li>{@link #setInitialStepBoundFactor initial step bound factor}: 100.0</li>
    154       *    <li>{@link #setMaxCostEval maximal cost evaluations}: 1000</li>
    155       *    <li>{@link #setCostRelativeTolerance cost relative tolerance}: 1.0e-10</li>
    156       *    <li>{@link #setParRelativeTolerance parameters relative tolerance}: 1.0e-10</li>
    157       *    <li>{@link #setOrthoTolerance orthogonality tolerance}: 1.0e-10</li>
    158       *   </ul>
    159       * </p>
    160       */
    161      public LevenbergMarquardtEstimator() {
    162
    163        // set up the superclass with a default  max cost evaluations setting
    164        setMaxCostEval(1000);
    165
    166        // default values for the tuning parameters
    167        setInitialStepBoundFactor(100.0);
    168        setCostRelativeTolerance(1.0e-10);
    169        setParRelativeTolerance(1.0e-10);
    170        setOrthoTolerance(1.0e-10);
    171
    172      }
    173
    174      /**
    175       * Set the positive input variable used in determining the initial step bound.
    176       * This bound is set to the product of initialStepBoundFactor and the euclidean norm of diag*x if nonzero,
    177       * or else to initialStepBoundFactor itself. In most cases factor should lie
    178       * in the interval (0.1, 100.0). 100.0 is a generally recommended value
    179       *
    180       * @param initialStepBoundFactor initial step bound factor
    181       * @see #estimate
    182       */
    183      public void setInitialStepBoundFactor(double initialStepBoundFactor) {
    184        this.initialStepBoundFactor = initialStepBoundFactor;
    185      }
    186
    187      /**
    188       * Set the desired relative error in the sum of squares.
    189       *
    190       * @param costRelativeTolerance desired relative error in the sum of squares
    191       * @see #estimate
    192       */
    193      public void setCostRelativeTolerance(double costRelativeTolerance) {
    194        this.costRelativeTolerance = costRelativeTolerance;
    195      }
    196
    197      /**
    198       * Set the desired relative error in the approximate solution parameters.
    199       *
    200       * @param parRelativeTolerance desired relative error
    201       * in the approximate solution parameters
    202       * @see #estimate
    203       */
    204      public void setParRelativeTolerance(double parRelativeTolerance) {
    205        this.parRelativeTolerance = parRelativeTolerance;
    206      }
    207
    208      /**
    209       * Set the desired max cosine on the orthogonality.
    210       *
    211       * @param orthoTolerance desired max cosine on the orthogonality
    212       * between the function vector and the columns of the jacobian
    213       * @see #estimate
    214       */
    215      public void setOrthoTolerance(double orthoTolerance) {
    216        this.orthoTolerance = orthoTolerance;
    217      }
    218
    219      /**
    220       * Solve an estimation problem using the Levenberg-Marquardt algorithm.
    221       * <p>The algorithm used is a modified Levenberg-Marquardt one, based
    222       * on the MINPACK <a href="http://www.netlib.org/minpack/lmder.f">lmder</a>
    223       * routine. The algorithm settings must have been set up before this method
    224       * is called with the {@link #setInitialStepBoundFactor},
    225       * {@link #setMaxCostEval}, {@link #setCostRelativeTolerance},
    226       * {@link #setParRelativeTolerance} and {@link #setOrthoTolerance} methods.
    227       * If these methods have not been called, the default values set up by the
    228       * {@link #LevenbergMarquardtEstimator() constructor} will be used.</p>
    229       * <p>The authors of the original fortran function are:</p>
    230       * <ul>
    231       *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
    232       *   <li>Burton  S. Garbow</li>
    233       *   <li>Kenneth E. Hillstrom</li>
    234       *   <li>Jorge   J. More</li>
    235       *   </ul>
    236       * <p>Luc Maisonobe did the Java translation.</p>
    237       *
    238       * @param problem estimation problem to solve
    239       * @exception EstimationException if convergence cannot be
    240       * reached with the specified algorithm settings or if there are more variables
    241       * than equations
    242       * @see #setInitialStepBoundFactor
    243       * @see #setCostRelativeTolerance
    244       * @see #setParRelativeTolerance
    245       * @see #setOrthoTolerance
    246       */
    247      @Override
    248      public void estimate(EstimationProblem problem)
    249        throws EstimationException {
    250
    251        initializeEstimate(problem);
    252
    253        // arrays shared with the other private methods
    254        solvedCols  = FastMath.min(rows, cols);
    255        diagR       = new double[cols];
    256        jacNorm     = new double[cols];
    257        beta        = new double[cols];
    258        permutation = new int[cols];
    259        lmDir       = new double[cols];
    260
    261        // local variables
    262        double   delta   = 0;
    263        double   xNorm = 0;
    264        double[] diag    = new double[cols];
    265        double[] oldX    = new double[cols];
    266        double[] oldRes  = new double[rows];
    267        double[] work1   = new double[cols];
    268        double[] work2   = new double[cols];
    269        double[] work3   = new double[cols];
    270
    271        // evaluate the function at the starting point and calculate its norm
    272        updateResidualsAndCost();
    273
    274        // outer loop
    275        lmPar = 0;
    276        boolean firstIteration = true;
    277        while (true) {
    278
    279          // compute the Q.R. decomposition of the jacobian matrix
    280          updateJacobian();
    281          qrDecomposition();
    282
    283          // compute Qt.res
    284          qTy(residuals);
    285
    286          // now we don't need Q anymore,
    287          // so let jacobian contain the R matrix with its diagonal elements
    288          for (int k = 0; k < solvedCols; ++k) {
    289            int pk = permutation[k];
    290            jacobian[k * cols + pk] = diagR[pk];
    291          }
    292
    293          if (firstIteration) {
    294
    295            // scale the variables according to the norms of the columns
    296            // of the initial jacobian
    297            xNorm = 0;
    298            for (int k = 0; k < cols; ++k) {
    299              double dk = jacNorm[k];
    300              if (dk == 0) {
    301                dk = 1.0;
    302              }
    303              double xk = dk * parameters[k].getEstimate();
    304              xNorm  += xk * xk;
    305              diag[k] = dk;
    306            }
    307            xNorm = FastMath.sqrt(xNorm);
    308
    309            // initialize the step bound delta
    310            delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
    311
    312          }
    313
    314          // check orthogonality between function vector and jacobian columns
    315          double maxCosine = 0;
    316          if (cost != 0) {
    317            for (int j = 0; j < solvedCols; ++j) {
    318              int    pj = permutation[j];
    319              double s  = jacNorm[pj];
    320              if (s != 0) {
    321                double sum = 0;
    322                int index = pj;
    323                for (int i = 0; i <= j; ++i) {
    324                  sum += jacobian[index] * residuals[i];
    325                  index += cols;
    326                }
    327                maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * cost));
    328              }
    329            }
    330          }
    331          if (maxCosine <= orthoTolerance) {
    332            return;
    333          }
    334
    335          // rescale if necessary
    336          for (int j = 0; j < cols; ++j) {
    337            diag[j] = FastMath.max(diag[j], jacNorm[j]);
    338          }
    339
    340          // inner loop
    341          for (double ratio = 0; ratio < 1.0e-4;) {
    342
    343            // save the state
    344            for (int j = 0; j < solvedCols; ++j) {
    345              int pj = permutation[j];
    346              oldX[pj] = parameters[pj].getEstimate();
    347            }
    348            double previousCost = cost;
    349            double[] tmpVec = residuals;
    350            residuals = oldRes;
    351            oldRes    = tmpVec;
    352
    353            // determine the Levenberg-Marquardt parameter
    354            determineLMParameter(oldRes, delta, diag, work1, work2, work3);
    355
    356            // compute the new point and the norm of the evolution direction
    357            double lmNorm = 0;
    358            for (int j = 0; j < solvedCols; ++j) {
    359              int pj = permutation[j];
    360              lmDir[pj] = -lmDir[pj];
    361              parameters[pj].setEstimate(oldX[pj] + lmDir[pj]);
    362              double s = diag[pj] * lmDir[pj];
    363              lmNorm  += s * s;
    364            }
    365            lmNorm = FastMath.sqrt(lmNorm);
    366
    367            // on the first iteration, adjust the initial step bound.
    368            if (firstIteration) {
    369              delta = FastMath.min(delta, lmNorm);
    370            }
    371
    372            // evaluate the function at x + p and calculate its norm
    373            updateResidualsAndCost();
    374
    375            // compute the scaled actual reduction
    376            double actRed = -1.0;
    377            if (0.1 * cost < previousCost) {
    378              double r = cost / previousCost;
    379              actRed = 1.0 - r * r;
    380            }
    381
    382            // compute the scaled predicted reduction
    383            // and the scaled directional derivative
    384            for (int j = 0; j < solvedCols; ++j) {
    385              int pj = permutation[j];
    386              double dirJ = lmDir[pj];
    387              work1[j] = 0;
    388              int index = pj;
    389              for (int i = 0; i <= j; ++i) {
    390                work1[i] += jacobian[index] * dirJ;
    391                index += cols;
    392              }
    393            }
    394            double coeff1 = 0;
    395            for (int j = 0; j < solvedCols; ++j) {
    396             coeff1 += work1[j] * work1[j];
    397            }
    398            double pc2 = previousCost * previousCost;
    399            coeff1 = coeff1 / pc2;
    400            double coeff2 = lmPar * lmNorm * lmNorm / pc2;
    401            double preRed = coeff1 + 2 * coeff2;
    402            double dirDer = -(coeff1 + coeff2);
    403
    404            // ratio of the actual to the predicted reduction
    405            ratio = (preRed == 0) ? 0 : (actRed / preRed);
    406
    407            // update the step bound
    408            if (ratio <= 0.25) {
    409              double tmp =
    410                (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
    411              if ((0.1 * cost >= previousCost) || (tmp < 0.1)) {
    412                tmp = 0.1;
    413              }
    414              delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
    415              lmPar /= tmp;
    416            } else if ((lmPar == 0) || (ratio >= 0.75)) {
    417              delta = 2 * lmNorm;
    418              lmPar *= 0.5;
    419            }
    420
    421            // test for successful iteration.
    422            if (ratio >= 1.0e-4) {
    423              // successful iteration, update the norm
    424              firstIteration = false;
    425              xNorm = 0;
    426              for (int k = 0; k < cols; ++k) {
    427                double xK = diag[k] * parameters[k].getEstimate();
    428                xNorm    += xK * xK;
    429              }
    430              xNorm = FastMath.sqrt(xNorm);
    431            } else {
    432              // failed iteration, reset the previous values
    433              cost = previousCost;
    434              for (int j = 0; j < solvedCols; ++j) {
    435                int pj = permutation[j];
    436                parameters[pj].setEstimate(oldX[pj]);
    437              }
    438              tmpVec    = residuals;
    439              residuals = oldRes;
    440              oldRes    = tmpVec;
    441            }
    442
    443            // tests for convergence.
    444            if (((FastMath.abs(actRed) <= costRelativeTolerance) &&
    445                 (preRed <= costRelativeTolerance) &&
    446                 (ratio <= 2.0)) ||
    447                 (delta <= parRelativeTolerance * xNorm)) {
    448              return;
    449            }
    450
    451            // tests for termination and stringent tolerances
    452            // (2.2204e-16 is the machine epsilon for IEEE754)
    453            if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
    454              throw new EstimationException("cost relative tolerance is too small ({0})," +
    455                                            " no further reduction in the" +
    456                                            " sum of squares is possible",
    457                                            costRelativeTolerance);
    458            } else if (delta <= 2.2204e-16 * xNorm) {
    459              throw new EstimationException("parameters relative tolerance is too small" +
    460                                            " ({0}), no further improvement in" +
    461                                            " the approximate solution is possible",
    462                                            parRelativeTolerance);
    463            } else if (maxCosine <= 2.2204e-16)  {
    464              throw new EstimationException("orthogonality tolerance is too small ({0})," +
    465                                            " solution is orthogonal to the jacobian",
    466                                            orthoTolerance);
    467            }
    468
    469          }
    470
    471        }
    472
    473      }
    474
    475      /**
    476       * Determine the Levenberg-Marquardt parameter.
    477       * <p>This implementation is a translation in Java of the MINPACK
    478       * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
    479       * routine.</p>
    480       * <p>This method sets the lmPar and lmDir attributes.</p>
    481       * <p>The authors of the original fortran function are:</p>
    482       * <ul>
    483       *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
    484       *   <li>Burton  S. Garbow</li>
    485       *   <li>Kenneth E. Hillstrom</li>
    486       *   <li>Jorge   J. More</li>
    487       * </ul>
    488       * <p>Luc Maisonobe did the Java translation.</p>
    489       *
    490       * @param qy array containing qTy
    491       * @param delta upper bound on the euclidean norm of diagR * lmDir
    492       * @param diag diagonal matrix
    493       * @param work1 work array
    494       * @param work2 work array
    495       * @param work3 work array
    496       */
    497      private void determineLMParameter(double[] qy, double delta, double[] diag,
    498                                        double[] work1, double[] work2, double[] work3) {
    499
    500        // compute and store in x the gauss-newton direction, if the
    501        // jacobian is rank-deficient, obtain a least squares solution
    502        for (int j = 0; j < rank; ++j) {
    503          lmDir[permutation[j]] = qy[j];
    504        }
    505        for (int j = rank; j < cols; ++j) {
    506          lmDir[permutation[j]] = 0;
    507        }
    508        for (int k = rank - 1; k >= 0; --k) {
    509          int pk = permutation[k];
    510          double ypk = lmDir[pk] / diagR[pk];
    511          int index = pk;
    512          for (int i = 0; i < k; ++i) {
    513            lmDir[permutation[i]] -= ypk * jacobian[index];
    514            index += cols;
    515          }
    516          lmDir[pk] = ypk;
    517        }
    518
    519        // evaluate the function at the origin, and test
    520        // for acceptance of the Gauss-Newton direction
    521        double dxNorm = 0;
    522        for (int j = 0; j < solvedCols; ++j) {
    523          int pj = permutation[j];
    524          double s = diag[pj] * lmDir[pj];
    525          work1[pj] = s;
    526          dxNorm += s * s;
    527        }
    528        dxNorm = FastMath.sqrt(dxNorm);
    529        double fp = dxNorm - delta;
    530        if (fp <= 0.1 * delta) {
    531          lmPar = 0;
    532          return;
    533        }
    534
    535        // if the jacobian is not rank deficient, the Newton step provides
    536        // a lower bound, parl, for the zero of the function,
    537        // otherwise set this bound to zero
    538        double sum2;
    539        double parl = 0;
    540        if (rank == solvedCols) {
    541          for (int j = 0; j < solvedCols; ++j) {
    542            int pj = permutation[j];
    543            work1[pj] *= diag[pj] / dxNorm;
    544          }
    545          sum2 = 0;
    546          for (int j = 0; j < solvedCols; ++j) {
    547            int pj = permutation[j];
    548            double sum = 0;
    549            int index = pj;
    550            for (int i = 0; i < j; ++i) {
    551              sum += jacobian[index] * work1[permutation[i]];
    552              index += cols;
    553            }
    554            double s = (work1[pj] - sum) / diagR[pj];
    555            work1[pj] = s;
    556            sum2 += s * s;
    557          }
    558          parl = fp / (delta * sum2);
    559        }
    560
    561        // calculate an upper bound, paru, for the zero of the function
    562        sum2 = 0;
    563        for (int j = 0; j < solvedCols; ++j) {
    564          int pj = permutation[j];
    565          double sum = 0;
    566          int index = pj;
    567          for (int i = 0; i <= j; ++i) {
    568            sum += jacobian[index] * qy[i];
    569            index += cols;
    570          }
    571          sum /= diag[pj];
    572          sum2 += sum * sum;
    573        }
    574        double gNorm = FastMath.sqrt(sum2);
    575        double paru = gNorm / delta;
    576        if (paru == 0) {
    577          // 2.2251e-308 is the smallest positive real for IEE754
    578          paru = 2.2251e-308 / FastMath.min(delta, 0.1);
    579        }
    580
    581        // if the input par lies outside of the interval (parl,paru),
    582        // set par to the closer endpoint
    583        lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
    584        if (lmPar == 0) {
    585          lmPar = gNorm / dxNorm;
    586        }
    587
    588        for (int countdown = 10; countdown >= 0; --countdown) {
    589
    590          // evaluate the function at the current value of lmPar
    591          if (lmPar == 0) {
    592            lmPar = FastMath.max(2.2251e-308, 0.001 * paru);
    593          }
    594          double sPar = FastMath.sqrt(lmPar);
    595          for (int j = 0; j < solvedCols; ++j) {
    596            int pj = permutation[j];
    597            work1[pj] = sPar * diag[pj];
    598          }
    599          determineLMDirection(qy, work1, work2, work3);
    600
    601          dxNorm = 0;
    602          for (int j = 0; j < solvedCols; ++j) {
    603            int pj = permutation[j];
    604            double s = diag[pj] * lmDir[pj];
    605            work3[pj] = s;
    606            dxNorm += s * s;
    607          }
    608          dxNorm = FastMath.sqrt(dxNorm);
    609          double previousFP = fp;
    610          fp = dxNorm - delta;
    611
    612          // if the function is small enough, accept the current value
    613          // of lmPar, also test for the exceptional cases where parl is zero
    614          if ((FastMath.abs(fp) <= 0.1 * delta) ||
    615              ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
    616            return;
    617          }
    618
    619          // compute the Newton correction
    620          for (int j = 0; j < solvedCols; ++j) {
    621           int pj = permutation[j];
    622            work1[pj] = work3[pj] * diag[pj] / dxNorm;
    623          }
    624          for (int j = 0; j < solvedCols; ++j) {
    625            int pj = permutation[j];
    626            work1[pj] /= work2[j];
    627            double tmp = work1[pj];
    628            for (int i = j + 1; i < solvedCols; ++i) {
    629              work1[permutation[i]] -= jacobian[i * cols + pj] * tmp;
    630            }
    631          }
    632          sum2 = 0;
    633          for (int j = 0; j < solvedCols; ++j) {
    634            double s = work1[permutation[j]];
    635            sum2 += s * s;
    636          }
    637          double correction = fp / (delta * sum2);
    638
    639          // depending on the sign of the function, update parl or paru.
    640          if (fp > 0) {
    641            parl = FastMath.max(parl, lmPar);
    642          } else if (fp < 0) {
    643            paru = FastMath.min(paru, lmPar);
    644          }
    645
    646          // compute an improved estimate for lmPar
    647          lmPar = FastMath.max(parl, lmPar + correction);
    648
    649        }
    650      }
    651
    652      /**
    653       * Solve a*x = b and d*x = 0 in the least squares sense.
    654       * <p>This implementation is a translation in Java of the MINPACK
    655       * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
    656       * routine.</p>
    657       * <p>This method sets the lmDir and lmDiag attributes.</p>
    658       * <p>The authors of the original fortran function are:</p>
    659       * <ul>
    660       *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
    661       *   <li>Burton  S. Garbow</li>
    662       *   <li>Kenneth E. Hillstrom</li>
    663       *   <li>Jorge   J. More</li>
    664       * </ul>
    665       * <p>Luc Maisonobe did the Java translation.</p>
    666       *
    667       * @param qy array containing qTy
    668       * @param diag diagonal matrix
    669       * @param lmDiag diagonal elements associated with lmDir
    670       * @param work work array
    671       */
    672      private void determineLMDirection(double[] qy, double[] diag,
    673                                        double[] lmDiag, double[] work) {
    674
    675        // copy R and Qty to preserve input and initialize s
    676        //  in particular, save the diagonal elements of R in lmDir
    677        for (int j = 0; j < solvedCols; ++j) {
    678          int pj = permutation[j];
    679          for (int i = j + 1; i < solvedCols; ++i) {
    680            jacobian[i * cols + pj] = jacobian[j * cols + permutation[i]];
    681          }
    682          lmDir[j] = diagR[pj];
    683          work[j]  = qy[j];
    684        }
    685
    686        // eliminate the diagonal matrix d using a Givens rotation
    687        for (int j = 0; j < solvedCols; ++j) {
    688
    689          // prepare the row of d to be eliminated, locating the
    690          // diagonal element using p from the Q.R. factorization
    691          int pj = permutation[j];
    692          double dpj = diag[pj];
    693          if (dpj != 0) {
    694            Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
    695          }
    696          lmDiag[j] = dpj;
    697
    698          //  the transformations to eliminate the row of d
    699          // modify only a single element of Qty
    700          // beyond the first n, which is initially zero.
    701          double qtbpj = 0;
    702          for (int k = j; k < solvedCols; ++k) {
    703            int pk = permutation[k];
    704
    705            // determine a Givens rotation which eliminates the
    706            // appropriate element in the current row of d
    707            if (lmDiag[k] != 0) {
    708
    709              final double sin;
    710              final double cos;
    711              double rkk = jacobian[k * cols + pk];
    712              if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
    713                final double cotan = rkk / lmDiag[k];
    714                sin   = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
    715                cos   = sin * cotan;
    716              } else {
    717                final double tan = lmDiag[k] / rkk;
    718                cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
    719                sin = cos * tan;
    720              }
    721
    722              // compute the modified diagonal element of R and
    723              // the modified element of (Qty,0)
    724              jacobian[k * cols + pk] = cos * rkk + sin * lmDiag[k];
    725              final double temp = cos * work[k] + sin * qtbpj;
    726              qtbpj = -sin * work[k] + cos * qtbpj;
    727              work[k] = temp;
    728
    729              // accumulate the tranformation in the row of s
    730              for (int i = k + 1; i < solvedCols; ++i) {
    731                double rik = jacobian[i * cols + pk];
    732                final double temp2 = cos * rik + sin * lmDiag[i];
    733                lmDiag[i] = -sin * rik + cos * lmDiag[i];
    734                jacobian[i * cols + pk] = temp2;
    735              }
    736
    737            }
    738          }
    739
    740          // store the diagonal element of s and restore
    741          // the corresponding diagonal element of R
    742          int index = j * cols + permutation[j];
    743          lmDiag[j]       = jacobian[index];
    744          jacobian[index] = lmDir[j];
    745
    746        }
    747
    748        // solve the triangular system for z, if the system is
    749        // singular, then obtain a least squares solution
    750        int nSing = solvedCols;
    751        for (int j = 0; j < solvedCols; ++j) {
    752          if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
    753            nSing = j;
    754          }
    755          if (nSing < solvedCols) {
    756            work[j] = 0;
    757          }
    758        }
    759        if (nSing > 0) {
    760          for (int j = nSing - 1; j >= 0; --j) {
    761            int pj = permutation[j];
    762            double sum = 0;
    763            for (int i = j + 1; i < nSing; ++i) {
    764              sum += jacobian[i * cols + pj] * work[i];
    765            }
    766            work[j] = (work[j] - sum) / lmDiag[j];
    767          }
    768        }
    769
    770        // permute the components of z back to components of lmDir
    771        for (int j = 0; j < lmDir.length; ++j) {
    772          lmDir[permutation[j]] = work[j];
    773        }
    774
    775      }
    776
    777      /**
    778       * Decompose a matrix A as A.P = Q.R using Householder transforms.
    779       * <p>As suggested in the P. Lascaux and R. Theodor book
    780       * <i>Analyse num&eacute;rique matricielle appliqu&eacute;e &agrave;
    781       * l'art de l'ing&eacute;nieur</i> (Masson, 1986), instead of representing
    782       * the Householder transforms with u<sub>k</sub> unit vectors such that:
    783       * <pre>
    784       * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
    785       * </pre>
    786       * we use <sub>k</sub> non-unit vectors such that:
    787       * <pre>
    788       * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
    789       * </pre>
    790       * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
    791       * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
    792       * them from the v<sub>k</sub> vectors would be costly.</p>
    793       * <p>This decomposition handles rank deficient cases since the tranformations
    794       * are performed in non-increasing columns norms order thanks to columns
    795       * pivoting. The diagonal elements of the R matrix are therefore also in
    796       * non-increasing absolute values order.</p>
    797       * @exception EstimationException if the decomposition cannot be performed
    798       */
    799      private void qrDecomposition() throws EstimationException {
    800
    801        // initializations
    802        for (int k = 0; k < cols; ++k) {
    803          permutation[k] = k;
    804          double norm2 = 0;
    805          for (int index = k; index < jacobian.length; index += cols) {
    806            double akk = jacobian[index];
    807            norm2 += akk * akk;
    808          }
    809          jacNorm[k] = FastMath.sqrt(norm2);
    810        }
    811
    812        // transform the matrix column after column
    813        for (int k = 0; k < cols; ++k) {
    814
    815          // select the column with the greatest norm on active components
    816          int nextColumn = -1;
    817          double ak2 = Double.NEGATIVE_INFINITY;
    818          for (int i = k; i < cols; ++i) {
    819            double norm2 = 0;
    820            int iDiag = k * cols + permutation[i];
    821            for (int index = iDiag; index < jacobian.length; index += cols) {
    822              double aki = jacobian[index];
    823              norm2 += aki * aki;
    824            }
    825            if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
    826                throw new EstimationException(
    827                        LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
    828                        rows, cols);
    829            }
    830            if (norm2 > ak2) {
    831              nextColumn = i;
    832              ak2        = norm2;
    833            }
    834          }
    835          if (ak2 == 0) {
    836            rank = k;
    837            return;
    838          }
    839          int pk                  = permutation[nextColumn];
    840          permutation[nextColumn] = permutation[k];
    841          permutation[k]          = pk;
    842
    843          // choose alpha such that Hk.u = alpha ek
    844          int    kDiag = k * cols + pk;
    845          double akk   = jacobian[kDiag];
    846          double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
    847          double betak = 1.0 / (ak2 - akk * alpha);
    848          beta[pk]     = betak;
    849
    850          // transform the current column
    851          diagR[pk]        = alpha;
    852          jacobian[kDiag] -= alpha;
    853
    854          // transform the remaining columns
    855          for (int dk = cols - 1 - k; dk > 0; --dk) {
    856            int dkp = permutation[k + dk] - pk;
    857            double gamma = 0;
    858            for (int index = kDiag; index < jacobian.length; index += cols) {
    859              gamma += jacobian[index] * jacobian[index + dkp];
    860            }
    861            gamma *= betak;
    862            for (int index = kDiag; index < jacobian.length; index += cols) {
    863              jacobian[index + dkp] -= gamma * jacobian[index];
    864            }
    865          }
    866
    867        }
    868
    869        rank = solvedCols;
    870
    871      }
    872
    873      /**
    874       * Compute the product Qt.y for some Q.R. decomposition.
    875       *
    876       * @param y vector to multiply (will be overwritten with the result)
    877       */
    878      private void qTy(double[] y) {
    879        for (int k = 0; k < cols; ++k) {
    880          int pk = permutation[k];
    881          int kDiag = k * cols + pk;
    882          double gamma = 0;
    883          int index = kDiag;
    884          for (int i = k; i < rows; ++i) {
    885            gamma += jacobian[index] * y[i];
    886            index += cols;
    887          }
    888          gamma *= beta[pk];
    889          index = kDiag;
    890          for (int i = k; i < rows; ++i) {
    891            y[i] -= gamma * jacobian[index];
    892            index += cols;
    893          }
    894        }
    895      }
    896
    897    }