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 */
017package org.apache.commons.math3.optimization.general;
018
019import java.util.Arrays;
020
021import org.apache.commons.math3.exception.ConvergenceException;
022import org.apache.commons.math3.exception.util.LocalizedFormats;
023import org.apache.commons.math3.optimization.PointVectorValuePair;
024import org.apache.commons.math3.optimization.ConvergenceChecker;
025import org.apache.commons.math3.linear.RealMatrix;
026import org.apache.commons.math3.util.Precision;
027import 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 1591835 2014-05-02 09:04:01Z tn $
105 * @deprecated As of 3.1 (to be removed in 4.0).
106 * @since 2.0
107 *
108 */
109@Deprecated
110public 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 /= 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 && checker.converged(iter, previous, current)) {
484                        setCost(currentCost);
485                        // Update (deprecated) "point" field.
486                        point = current.getPoint();
487                        return current;
488                    }
489                } else {
490                    // failed iteration, reset the previous values
491                    currentCost = previousCost;
492                    for (int j = 0; j < solvedCols; ++j) {
493                        int pj = permutation[j];
494                        currentPoint[pj] = oldX[pj];
495                    }
496                    tmpVec    = weightedResidual;
497                    weightedResidual = oldRes;
498                    oldRes    = tmpVec;
499                    tmpVec    = currentObjective;
500                    currentObjective = oldObj;
501                    oldObj    = tmpVec;
502                    // Reset "current" to previous values.
503                    current = new PointVectorValuePair(currentPoint, currentObjective);
504                }
505
506                // Default convergence criteria.
507                if ((FastMath.abs(actRed) <= costRelativeTolerance &&
508                     preRed <= costRelativeTolerance &&
509                     ratio <= 2.0) ||
510                    delta <= parRelativeTolerance * xNorm) {
511                    setCost(currentCost);
512                    // Update (deprecated) "point" field.
513                    point = current.getPoint();
514                    return current;
515                }
516
517                // tests for termination and stringent tolerances
518                // (2.2204e-16 is the machine epsilon for IEEE754)
519                if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
520                    throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
521                                                   costRelativeTolerance);
522                } else if (delta <= 2.2204e-16 * xNorm) {
523                    throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
524                                                   parRelativeTolerance);
525                } else if (maxCosine <= 2.2204e-16)  {
526                    throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
527                                                   orthoTolerance);
528                }
529            }
530        }
531    }
532
533    /**
534     * Determine the Levenberg-Marquardt parameter.
535     * <p>This implementation is a translation in Java of the MINPACK
536     * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
537     * routine.</p>
538     * <p>This method sets the lmPar and lmDir attributes.</p>
539     * <p>The authors of the original fortran function are:</p>
540     * <ul>
541     *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
542     *   <li>Burton  S. Garbow</li>
543     *   <li>Kenneth E. Hillstrom</li>
544     *   <li>Jorge   J. More</li>
545     * </ul>
546     * <p>Luc Maisonobe did the Java translation.</p>
547     *
548     * @param qy array containing qTy
549     * @param delta upper bound on the euclidean norm of diagR * lmDir
550     * @param diag diagonal matrix
551     * @param work1 work array
552     * @param work2 work array
553     * @param work3 work array
554     */
555    private void determineLMParameter(double[] qy, double delta, double[] diag,
556                                      double[] work1, double[] work2, double[] work3) {
557        final int nC = weightedJacobian[0].length;
558
559        // compute and store in x the gauss-newton direction, if the
560        // jacobian is rank-deficient, obtain a least squares solution
561        for (int j = 0; j < rank; ++j) {
562            lmDir[permutation[j]] = qy[j];
563        }
564        for (int j = rank; j < nC; ++j) {
565            lmDir[permutation[j]] = 0;
566        }
567        for (int k = rank - 1; k >= 0; --k) {
568            int pk = permutation[k];
569            double ypk = lmDir[pk] / diagR[pk];
570            for (int i = 0; i < k; ++i) {
571                lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk];
572            }
573            lmDir[pk] = ypk;
574        }
575
576        // evaluate the function at the origin, and test
577        // for acceptance of the Gauss-Newton direction
578        double dxNorm = 0;
579        for (int j = 0; j < solvedCols; ++j) {
580            int pj = permutation[j];
581            double s = diag[pj] * lmDir[pj];
582            work1[pj] = s;
583            dxNorm += s * s;
584        }
585        dxNorm = FastMath.sqrt(dxNorm);
586        double fp = dxNorm - delta;
587        if (fp <= 0.1 * delta) {
588            lmPar = 0;
589            return;
590        }
591
592        // if the jacobian is not rank deficient, the Newton step provides
593        // a lower bound, parl, for the zero of the function,
594        // otherwise set this bound to zero
595        double sum2;
596        double parl = 0;
597        if (rank == solvedCols) {
598            for (int j = 0; j < solvedCols; ++j) {
599                int pj = permutation[j];
600                work1[pj] *= diag[pj] / dxNorm;
601            }
602            sum2 = 0;
603            for (int j = 0; j < solvedCols; ++j) {
604                int pj = permutation[j];
605                double sum = 0;
606                for (int i = 0; i < j; ++i) {
607                    sum += weightedJacobian[i][pj] * work1[permutation[i]];
608                }
609                double s = (work1[pj] - sum) / diagR[pj];
610                work1[pj] = s;
611                sum2 += s * s;
612            }
613            parl = fp / (delta * sum2);
614        }
615
616        // calculate an upper bound, paru, for the zero of the function
617        sum2 = 0;
618        for (int j = 0; j < solvedCols; ++j) {
619            int pj = permutation[j];
620            double sum = 0;
621            for (int i = 0; i <= j; ++i) {
622                sum += weightedJacobian[i][pj] * qy[i];
623            }
624            sum /= diag[pj];
625            sum2 += sum * sum;
626        }
627        double gNorm = FastMath.sqrt(sum2);
628        double paru = gNorm / delta;
629        if (paru == 0) {
630            // 2.2251e-308 is the smallest positive real for IEE754
631            paru = 2.2251e-308 / FastMath.min(delta, 0.1);
632        }
633
634        // if the input par lies outside of the interval (parl,paru),
635        // set par to the closer endpoint
636        lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
637        if (lmPar == 0) {
638            lmPar = gNorm / dxNorm;
639        }
640
641        for (int countdown = 10; countdown >= 0; --countdown) {
642
643            // evaluate the function at the current value of lmPar
644            if (lmPar == 0) {
645                lmPar = FastMath.max(2.2251e-308, 0.001 * paru);
646            }
647            double sPar = FastMath.sqrt(lmPar);
648            for (int j = 0; j < solvedCols; ++j) {
649                int pj = permutation[j];
650                work1[pj] = sPar * diag[pj];
651            }
652            determineLMDirection(qy, work1, work2, work3);
653
654            dxNorm = 0;
655            for (int j = 0; j < solvedCols; ++j) {
656                int pj = permutation[j];
657                double s = diag[pj] * lmDir[pj];
658                work3[pj] = s;
659                dxNorm += s * s;
660            }
661            dxNorm = FastMath.sqrt(dxNorm);
662            double previousFP = fp;
663            fp = dxNorm - delta;
664
665            // if the function is small enough, accept the current value
666            // of lmPar, also test for the exceptional cases where parl is zero
667            if ((FastMath.abs(fp) <= 0.1 * delta) ||
668                    ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
669                return;
670            }
671
672            // compute the Newton correction
673            for (int j = 0; j < solvedCols; ++j) {
674                int pj = permutation[j];
675                work1[pj] = work3[pj] * diag[pj] / dxNorm;
676            }
677            for (int j = 0; j < solvedCols; ++j) {
678                int pj = permutation[j];
679                work1[pj] /= work2[j];
680                double tmp = work1[pj];
681                for (int i = j + 1; i < solvedCols; ++i) {
682                    work1[permutation[i]] -= weightedJacobian[i][pj] * tmp;
683                }
684            }
685            sum2 = 0;
686            for (int j = 0; j < solvedCols; ++j) {
687                double s = work1[permutation[j]];
688                sum2 += s * s;
689            }
690            double correction = fp / (delta * sum2);
691
692            // depending on the sign of the function, update parl or paru.
693            if (fp > 0) {
694                parl = FastMath.max(parl, lmPar);
695            } else if (fp < 0) {
696                paru = FastMath.min(paru, lmPar);
697            }
698
699            // compute an improved estimate for lmPar
700            lmPar = FastMath.max(parl, lmPar + correction);
701
702        }
703    }
704
705    /**
706     * Solve a*x = b and d*x = 0 in the least squares sense.
707     * <p>This implementation is a translation in Java of the MINPACK
708     * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
709     * routine.</p>
710     * <p>This method sets the lmDir and lmDiag attributes.</p>
711     * <p>The authors of the original fortran function are:</p>
712     * <ul>
713     *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
714     *   <li>Burton  S. Garbow</li>
715     *   <li>Kenneth E. Hillstrom</li>
716     *   <li>Jorge   J. More</li>
717     * </ul>
718     * <p>Luc Maisonobe did the Java translation.</p>
719     *
720     * @param qy array containing qTy
721     * @param diag diagonal matrix
722     * @param lmDiag diagonal elements associated with lmDir
723     * @param work work array
724     */
725    private void determineLMDirection(double[] qy, double[] diag,
726                                      double[] lmDiag, double[] work) {
727
728        // copy R and Qty to preserve input and initialize s
729        //  in particular, save the diagonal elements of R in lmDir
730        for (int j = 0; j < solvedCols; ++j) {
731            int pj = permutation[j];
732            for (int i = j + 1; i < solvedCols; ++i) {
733                weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]];
734            }
735            lmDir[j] = diagR[pj];
736            work[j]  = qy[j];
737        }
738
739        // eliminate the diagonal matrix d using a Givens rotation
740        for (int j = 0; j < solvedCols; ++j) {
741
742            // prepare the row of d to be eliminated, locating the
743            // diagonal element using p from the Q.R. factorization
744            int pj = permutation[j];
745            double dpj = diag[pj];
746            if (dpj != 0) {
747                Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
748            }
749            lmDiag[j] = dpj;
750
751            //  the transformations to eliminate the row of d
752            // modify only a single element of Qty
753            // beyond the first n, which is initially zero.
754            double qtbpj = 0;
755            for (int k = j; k < solvedCols; ++k) {
756                int pk = permutation[k];
757
758                // determine a Givens rotation which eliminates the
759                // appropriate element in the current row of d
760                if (lmDiag[k] != 0) {
761
762                    final double sin;
763                    final double cos;
764                    double rkk = weightedJacobian[k][pk];
765                    if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
766                        final double cotan = rkk / lmDiag[k];
767                        sin   = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
768                        cos   = sin * cotan;
769                    } else {
770                        final double tan = lmDiag[k] / rkk;
771                        cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
772                        sin = cos * tan;
773                    }
774
775                    // compute the modified diagonal element of R and
776                    // the modified element of (Qty,0)
777                    weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
778                    final double temp = cos * work[k] + sin * qtbpj;
779                    qtbpj = -sin * work[k] + cos * qtbpj;
780                    work[k] = temp;
781
782                    // accumulate the tranformation in the row of s
783                    for (int i = k + 1; i < solvedCols; ++i) {
784                        double rik = weightedJacobian[i][pk];
785                        final double temp2 = cos * rik + sin * lmDiag[i];
786                        lmDiag[i] = -sin * rik + cos * lmDiag[i];
787                        weightedJacobian[i][pk] = temp2;
788                    }
789                }
790            }
791
792            // store the diagonal element of s and restore
793            // the corresponding diagonal element of R
794            lmDiag[j] = weightedJacobian[j][permutation[j]];
795            weightedJacobian[j][permutation[j]] = lmDir[j];
796        }
797
798        // solve the triangular system for z, if the system is
799        // singular, then obtain a least squares solution
800        int nSing = solvedCols;
801        for (int j = 0; j < solvedCols; ++j) {
802            if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
803                nSing = j;
804            }
805            if (nSing < solvedCols) {
806                work[j] = 0;
807            }
808        }
809        if (nSing > 0) {
810            for (int j = nSing - 1; j >= 0; --j) {
811                int pj = permutation[j];
812                double sum = 0;
813                for (int i = j + 1; i < nSing; ++i) {
814                    sum += weightedJacobian[i][pj] * work[i];
815                }
816                work[j] = (work[j] - sum) / lmDiag[j];
817            }
818        }
819
820        // permute the components of z back to components of lmDir
821        for (int j = 0; j < lmDir.length; ++j) {
822            lmDir[permutation[j]] = work[j];
823        }
824    }
825
826    /**
827     * Decompose a matrix A as A.P = Q.R using Householder transforms.
828     * <p>As suggested in the P. Lascaux and R. Theodor book
829     * <i>Analyse num&eacute;rique matricielle appliqu&eacute;e &agrave;
830     * l'art de l'ing&eacute;nieur</i> (Masson, 1986), instead of representing
831     * the Householder transforms with u<sub>k</sub> unit vectors such that:
832     * <pre>
833     * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
834     * </pre>
835     * we use <sub>k</sub> non-unit vectors such that:
836     * <pre>
837     * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
838     * </pre>
839     * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
840     * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
841     * them from the v<sub>k</sub> vectors would be costly.</p>
842     * <p>This decomposition handles rank deficient cases since the tranformations
843     * are performed in non-increasing columns norms order thanks to columns
844     * pivoting. The diagonal elements of the R matrix are therefore also in
845     * non-increasing absolute values order.</p>
846     *
847     * @param jacobian Weighted Jacobian matrix at the current point.
848     * @exception ConvergenceException if the decomposition cannot be performed
849     */
850    private void qrDecomposition(RealMatrix jacobian) throws ConvergenceException {
851        // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
852        // hence the multiplication by -1.
853        weightedJacobian = jacobian.scalarMultiply(-1).getData();
854
855        final int nR = weightedJacobian.length;
856        final int nC = weightedJacobian[0].length;
857
858        // initializations
859        for (int k = 0; k < nC; ++k) {
860            permutation[k] = k;
861            double norm2 = 0;
862            for (int i = 0; i < nR; ++i) {
863                double akk = weightedJacobian[i][k];
864                norm2 += akk * akk;
865            }
866            jacNorm[k] = FastMath.sqrt(norm2);
867        }
868
869        // transform the matrix column after column
870        for (int k = 0; k < nC; ++k) {
871
872            // select the column with the greatest norm on active components
873            int nextColumn = -1;
874            double ak2 = Double.NEGATIVE_INFINITY;
875            for (int i = k; i < nC; ++i) {
876                double norm2 = 0;
877                for (int j = k; j < nR; ++j) {
878                    double aki = weightedJacobian[j][permutation[i]];
879                    norm2 += aki * aki;
880                }
881                if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
882                    throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
883                                                   nR, nC);
884                }
885                if (norm2 > ak2) {
886                    nextColumn = i;
887                    ak2        = norm2;
888                }
889            }
890            if (ak2 <= qrRankingThreshold) {
891                rank = k;
892                return;
893            }
894            int pk                  = permutation[nextColumn];
895            permutation[nextColumn] = permutation[k];
896            permutation[k]          = pk;
897
898            // choose alpha such that Hk.u = alpha ek
899            double akk   = weightedJacobian[k][pk];
900            double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
901            double betak = 1.0 / (ak2 - akk * alpha);
902            beta[pk]     = betak;
903
904            // transform the current column
905            diagR[pk]        = alpha;
906            weightedJacobian[k][pk] -= alpha;
907
908            // transform the remaining columns
909            for (int dk = nC - 1 - k; dk > 0; --dk) {
910                double gamma = 0;
911                for (int j = k; j < nR; ++j) {
912                    gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]];
913                }
914                gamma *= betak;
915                for (int j = k; j < nR; ++j) {
916                    weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk];
917                }
918            }
919        }
920        rank = solvedCols;
921    }
922
923    /**
924     * Compute the product Qt.y for some Q.R. decomposition.
925     *
926     * @param y vector to multiply (will be overwritten with the result)
927     */
928    private void qTy(double[] y) {
929        final int nR = weightedJacobian.length;
930        final int nC = weightedJacobian[0].length;
931
932        for (int k = 0; k < nC; ++k) {
933            int pk = permutation[k];
934            double gamma = 0;
935            for (int i = k; i < nR; ++i) {
936                gamma += weightedJacobian[i][pk] * y[i];
937            }
938            gamma *= beta[pk];
939            for (int i = k; i < nR; ++i) {
940                y[i] -= gamma * weightedJacobian[i][pk];
941            }
942        }
943    }
944}