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.optim.nonlinear.vector.jacobian;
018    
019    import java.util.Arrays;
020    import org.apache.commons.math3.exception.ConvergenceException;
021    import org.apache.commons.math3.exception.util.LocalizedFormats;
022    import org.apache.commons.math3.optim.PointVectorValuePair;
023    import org.apache.commons.math3.optim.ConvergenceChecker;
024    import org.apache.commons.math3.linear.RealMatrix;
025    import org.apache.commons.math3.util.Precision;
026    import org.apache.commons.math3.util.FastMath;
027    
028    
029    /**
030     * This class solves a least-squares problem using the Levenberg-Marquardt algorithm.
031     *
032     * <p>This implementation <em>should</em> work even for over-determined systems
033     * (i.e. systems having more point than equations). Over-determined systems
034     * are solved by ignoring the point which have the smallest impact according
035     * to their jacobian column norm. Only the rank of the matrix and some loop bounds
036     * are changed to implement this.</p>
037     *
038     * <p>The resolution engine is a simple translation of the MINPACK <a
039     * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
040     * changes. The changes include the over-determined resolution, the use of
041     * inherited convergence checker and the Q.R. decomposition which has been
042     * rewritten following the algorithm described in the
043     * P. Lascaux and R. Theodor book <i>Analyse num&eacute;rique matricielle
044     * appliqu&eacute;e &agrave; l'art de l'ing&eacute;nieur</i>, Masson 1986.</p>
045     * <p>The authors of the original fortran version are:
046     * <ul>
047     * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
048     * <li>Burton S. Garbow</li>
049     * <li>Kenneth E. Hillstrom</li>
050     * <li>Jorge J. More</li>
051     * </ul>
052     * The redistribution policy for MINPACK is available <a
053     * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
054     * is reproduced below.</p>
055     *
056     * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
057     * <tr><td>
058     *    Minpack Copyright Notice (1999) University of Chicago.
059     *    All rights reserved
060     * </td></tr>
061     * <tr><td>
062     * Redistribution and use in source and binary forms, with or without
063     * modification, are permitted provided that the following conditions
064     * are met:
065     * <ol>
066     *  <li>Redistributions of source code must retain the above copyright
067     *      notice, this list of conditions and the following disclaimer.</li>
068     * <li>Redistributions in binary form must reproduce the above
069     *     copyright notice, this list of conditions and the following
070     *     disclaimer in the documentation and/or other materials provided
071     *     with the distribution.</li>
072     * <li>The end-user documentation included with the redistribution, if any,
073     *     must include the following acknowledgment:
074     *     <code>This product includes software developed by the University of
075     *           Chicago, as Operator of Argonne National Laboratory.</code>
076     *     Alternately, this acknowledgment may appear in the software itself,
077     *     if and wherever such third-party acknowledgments normally appear.</li>
078     * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
079     *     WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
080     *     UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
081     *     THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
082     *     IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
083     *     OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
084     *     OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
085     *     OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
086     *     USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
087     *     THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
088     *     DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
089     *     UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
090     *     BE CORRECTED.</strong></li>
091     * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
092     *     HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
093     *     ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
094     *     INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
095     *     ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
096     *     PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
097     *     SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
098     *     (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
099     *     EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
100     *     POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
101     * <ol></td></tr>
102     * </table>
103     *
104     * @version $Id: LevenbergMarquardtOptimizer.java 1416643 2012-12-03 19:37:14Z tn $
105     * @since 2.0
106     */
107    public class LevenbergMarquardtOptimizer
108        extends AbstractLeastSquaresOptimizer {
109        /** Number of solved point. */
110        private int solvedCols;
111        /** Diagonal elements of the R matrix in the Q.R. decomposition. */
112        private double[] diagR;
113        /** Norms of the columns of the jacobian matrix. */
114        private double[] jacNorm;
115        /** Coefficients of the Householder transforms vectors. */
116        private double[] beta;
117        /** Columns permutation array. */
118        private int[] permutation;
119        /** Rank of the jacobian matrix. */
120        private int rank;
121        /** Levenberg-Marquardt parameter. */
122        private double lmPar;
123        /** Parameters evolution direction associated with lmPar. */
124        private double[] lmDir;
125        /** Positive input variable used in determining the initial step bound. */
126        private final double initialStepBoundFactor;
127        /** Desired relative error in the sum of squares. */
128        private final double costRelativeTolerance;
129        /**  Desired relative error in the approximate solution parameters. */
130        private final double parRelativeTolerance;
131        /** Desired max cosine on the orthogonality between the function vector
132         * and the columns of the jacobian. */
133        private final double orthoTolerance;
134        /** Threshold for QR ranking. */
135        private final double qrRankingThreshold;
136        /** Weighted residuals. */
137        private double[] weightedResidual;
138        /** Weighted Jacobian. */
139        private double[][] weightedJacobian;
140    
141        /**
142         * Build an optimizer for least squares problems with default values
143         * for all the tuning parameters (see the {@link
144         * #LevenbergMarquardtOptimizer(double,double,double,double,double)
145         * other contructor}.
146         * The default values for the algorithm settings are:
147         * <ul>
148         *  <li>Initial step bound factor: 100</li>
149         *  <li>Cost relative tolerance: 1e-10</li>
150         *  <li>Parameters relative tolerance: 1e-10</li>
151         *  <li>Orthogonality tolerance: 1e-10</li>
152         *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
153         * </ul>
154         */
155        public LevenbergMarquardtOptimizer() {
156            this(100, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
157        }
158    
159        /**
160         * Constructor that allows the specification of a custom convergence
161         * checker.
162         * Note that all the usual convergence checks will be <em>disabled</em>.
163         * The default values for the algorithm settings are:
164         * <ul>
165         *  <li>Initial step bound factor: 100</li>
166         *  <li>Cost relative tolerance: 1e-10</li>
167         *  <li>Parameters relative tolerance: 1e-10</li>
168         *  <li>Orthogonality tolerance: 1e-10</li>
169         *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
170         * </ul>
171         *
172         * @param checker Convergence checker.
173         */
174        public LevenbergMarquardtOptimizer(ConvergenceChecker<PointVectorValuePair> checker) {
175            this(100, checker, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
176        }
177    
178        /**
179         * Constructor that allows the specification of a custom convergence
180         * checker, in addition to the standard ones.
181         *
182         * @param initialStepBoundFactor Positive input variable used in
183         * determining the initial step bound. This bound is set to the
184         * product of initialStepBoundFactor and the euclidean norm of
185         * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
186         * itself. In most cases factor should lie in the interval
187         * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
188         * @param checker Convergence checker.
189         * @param costRelativeTolerance Desired relative error in the sum of
190         * squares.
191         * @param parRelativeTolerance Desired relative error in the approximate
192         * solution parameters.
193         * @param orthoTolerance Desired max cosine on the orthogonality between
194         * the function vector and the columns of the Jacobian.
195         * @param threshold Desired threshold for QR ranking. If the squared norm
196         * of a column vector is smaller or equal to this threshold during QR
197         * decomposition, it is considered to be a zero vector and hence the rank
198         * of the matrix is reduced.
199         */
200        public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
201                                           ConvergenceChecker<PointVectorValuePair> checker,
202                                           double costRelativeTolerance,
203                                           double parRelativeTolerance,
204                                           double orthoTolerance,
205                                           double threshold) {
206            super(checker);
207            this.initialStepBoundFactor = initialStepBoundFactor;
208            this.costRelativeTolerance = costRelativeTolerance;
209            this.parRelativeTolerance = parRelativeTolerance;
210            this.orthoTolerance = orthoTolerance;
211            this.qrRankingThreshold = threshold;
212        }
213    
214        /**
215         * Build an optimizer for least squares problems with default values
216         * for some of the tuning parameters (see the {@link
217         * #LevenbergMarquardtOptimizer(double,double,double,double,double)
218         * other contructor}.
219         * The default values for the algorithm settings are:
220         * <ul>
221         *  <li>Initial step bound factor}: 100</li>
222         *  <li>QR ranking threshold}: {@link Precision#SAFE_MIN}</li>
223         * </ul>
224         *
225         * @param costRelativeTolerance Desired relative error in the sum of
226         * squares.
227         * @param parRelativeTolerance Desired relative error in the approximate
228         * solution parameters.
229         * @param orthoTolerance Desired max cosine on the orthogonality between
230         * the function vector and the columns of the Jacobian.
231         */
232        public LevenbergMarquardtOptimizer(double costRelativeTolerance,
233                                           double parRelativeTolerance,
234                                           double orthoTolerance) {
235            this(100,
236                 costRelativeTolerance, parRelativeTolerance, orthoTolerance,
237                 Precision.SAFE_MIN);
238        }
239    
240        /**
241         * The arguments control the behaviour of the default convergence checking
242         * procedure.
243         * Additional criteria can defined through the setting of a {@link
244         * ConvergenceChecker}.
245         *
246         * @param initialStepBoundFactor Positive input variable used in
247         * determining the initial step bound. This bound is set to the
248         * product of initialStepBoundFactor and the euclidean norm of
249         * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
250         * itself. In most cases factor should lie in the interval
251         * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
252         * @param costRelativeTolerance Desired relative error in the sum of
253         * squares.
254         * @param parRelativeTolerance Desired relative error in the approximate
255         * solution parameters.
256         * @param orthoTolerance Desired max cosine on the orthogonality between
257         * the function vector and the columns of the Jacobian.
258         * @param threshold Desired threshold for QR ranking. If the squared norm
259         * of a column vector is smaller or equal to this threshold during QR
260         * decomposition, it is considered to be a zero vector and hence the rank
261         * of the matrix is reduced.
262         */
263        public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
264                                           double costRelativeTolerance,
265                                           double parRelativeTolerance,
266                                           double orthoTolerance,
267                                           double threshold) {
268            super(null); // No custom convergence criterion.
269            this.initialStepBoundFactor = initialStepBoundFactor;
270            this.costRelativeTolerance = costRelativeTolerance;
271            this.parRelativeTolerance = parRelativeTolerance;
272            this.orthoTolerance = orthoTolerance;
273            this.qrRankingThreshold = threshold;
274        }
275    
276        /** {@inheritDoc} */
277        @Override
278        protected PointVectorValuePair doOptimize() {
279            final int nR = getTarget().length; // Number of observed data.
280            final double[] currentPoint = getStartPoint();
281            final int nC = currentPoint.length; // Number of parameters.
282    
283            // arrays shared with the other private methods
284            solvedCols  = FastMath.min(nR, nC);
285            diagR       = new double[nC];
286            jacNorm     = new double[nC];
287            beta        = new double[nC];
288            permutation = new int[nC];
289            lmDir       = new double[nC];
290    
291            // local point
292            double   delta   = 0;
293            double   xNorm   = 0;
294            double[] diag    = new double[nC];
295            double[] oldX    = new double[nC];
296            double[] oldRes  = new double[nR];
297            double[] oldObj  = new double[nR];
298            double[] qtf     = new double[nR];
299            double[] work1   = new double[nC];
300            double[] work2   = new double[nC];
301            double[] work3   = new double[nC];
302    
303            final RealMatrix weightMatrixSqrt = getWeightSquareRoot();
304    
305            // Evaluate the function at the starting point and calculate its norm.
306            double[] currentObjective = computeObjectiveValue(currentPoint);
307            double[] currentResiduals = computeResiduals(currentObjective);
308            PointVectorValuePair current = new PointVectorValuePair(currentPoint, currentObjective);
309            double currentCost = computeCost(currentResiduals);
310    
311            // Outer loop.
312            lmPar = 0;
313            boolean firstIteration = true;
314            int iter = 0;
315            final ConvergenceChecker<PointVectorValuePair> checker = getConvergenceChecker();
316            while (true) {
317                ++iter;
318                final PointVectorValuePair previous = current;
319    
320                // QR decomposition of the jacobian matrix
321                qrDecomposition(computeWeightedJacobian(currentPoint));
322    
323                weightedResidual = weightMatrixSqrt.operate(currentResiduals);
324                for (int i = 0; i < nR; i++) {
325                    qtf[i] = weightedResidual[i];
326                }
327    
328                // compute Qt.res
329                qTy(qtf);
330    
331                // now we don't need Q anymore,
332                // so let jacobian contain the R matrix with its diagonal elements
333                for (int k = 0; k < solvedCols; ++k) {
334                    int pk = permutation[k];
335                    weightedJacobian[k][pk] = diagR[pk];
336                }
337    
338                if (firstIteration) {
339                    // scale the point according to the norms of the columns
340                    // of the initial jacobian
341                    xNorm = 0;
342                    for (int k = 0; k < nC; ++k) {
343                        double dk = jacNorm[k];
344                        if (dk == 0) {
345                            dk = 1.0;
346                        }
347                        double xk = dk * currentPoint[k];
348                        xNorm  += xk * xk;
349                        diag[k] = dk;
350                    }
351                    xNorm = FastMath.sqrt(xNorm);
352    
353                    // initialize the step bound delta
354                    delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
355                }
356    
357                // check orthogonality between function vector and jacobian columns
358                double maxCosine = 0;
359                if (currentCost != 0) {
360                    for (int j = 0; j < solvedCols; ++j) {
361                        int    pj = permutation[j];
362                        double s  = jacNorm[pj];
363                        if (s != 0) {
364                            double sum = 0;
365                            for (int i = 0; i <= j; ++i) {
366                                sum += weightedJacobian[i][pj] * qtf[i];
367                            }
368                            maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * currentCost));
369                        }
370                    }
371                }
372                if (maxCosine <= orthoTolerance) {
373                    // Convergence has been reached.
374                    setCost(currentCost);
375                    return current;
376                }
377    
378                // rescale if necessary
379                for (int j = 0; j < nC; ++j) {
380                    diag[j] = FastMath.max(diag[j], jacNorm[j]);
381                }
382    
383                // Inner loop.
384                for (double ratio = 0; ratio < 1.0e-4;) {
385    
386                    // save the state
387                    for (int j = 0; j < solvedCols; ++j) {
388                        int pj = permutation[j];
389                        oldX[pj] = currentPoint[pj];
390                    }
391                    final double previousCost = currentCost;
392                    double[] tmpVec = weightedResidual;
393                    weightedResidual = oldRes;
394                    oldRes    = tmpVec;
395                    tmpVec    = currentObjective;
396                    currentObjective = oldObj;
397                    oldObj    = tmpVec;
398    
399                    // determine the Levenberg-Marquardt parameter
400                    determineLMParameter(qtf, delta, diag, work1, work2, work3);
401    
402                    // compute the new point and the norm of the evolution direction
403                    double lmNorm = 0;
404                    for (int j = 0; j < solvedCols; ++j) {
405                        int pj = permutation[j];
406                        lmDir[pj] = -lmDir[pj];
407                        currentPoint[pj] = oldX[pj] + lmDir[pj];
408                        double s = diag[pj] * lmDir[pj];
409                        lmNorm  += s * s;
410                    }
411                    lmNorm = FastMath.sqrt(lmNorm);
412                    // on the first iteration, adjust the initial step bound.
413                    if (firstIteration) {
414                        delta = FastMath.min(delta, lmNorm);
415                    }
416    
417                    // Evaluate the function at x + p and calculate its norm.
418                    currentObjective = computeObjectiveValue(currentPoint);
419                    currentResiduals = computeResiduals(currentObjective);
420                    current = new PointVectorValuePair(currentPoint, currentObjective);
421                    currentCost = computeCost(currentResiduals);
422    
423                    // compute the scaled actual reduction
424                    double actRed = -1.0;
425                    if (0.1 * currentCost < previousCost) {
426                        double r = currentCost / previousCost;
427                        actRed = 1.0 - r * r;
428                    }
429    
430                    // compute the scaled predicted reduction
431                    // and the scaled directional derivative
432                    for (int j = 0; j < solvedCols; ++j) {
433                        int pj = permutation[j];
434                        double dirJ = lmDir[pj];
435                        work1[j] = 0;
436                        for (int i = 0; i <= j; ++i) {
437                            work1[i] += weightedJacobian[i][pj] * dirJ;
438                        }
439                    }
440                    double coeff1 = 0;
441                    for (int j = 0; j < solvedCols; ++j) {
442                        coeff1 += work1[j] * work1[j];
443                    }
444                    double pc2 = previousCost * previousCost;
445                    coeff1 = coeff1 / pc2;
446                    double coeff2 = lmPar * lmNorm * lmNorm / pc2;
447                    double preRed = coeff1 + 2 * coeff2;
448                    double dirDer = -(coeff1 + coeff2);
449    
450                    // ratio of the actual to the predicted reduction
451                    ratio = (preRed == 0) ? 0 : (actRed / preRed);
452    
453                    // update the step bound
454                    if (ratio <= 0.25) {
455                        double tmp =
456                            (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
457                            if ((0.1 * currentCost >= previousCost) || (tmp < 0.1)) {
458                                tmp = 0.1;
459                            }
460                            delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
461                            lmPar /= tmp;
462                    } else if ((lmPar == 0) || (ratio >= 0.75)) {
463                        delta = 2 * lmNorm;
464                        lmPar *= 0.5;
465                    }
466    
467                    // test for successful iteration.
468                    if (ratio >= 1.0e-4) {
469                        // successful iteration, update the norm
470                        firstIteration = false;
471                        xNorm = 0;
472                        for (int k = 0; k < nC; ++k) {
473                            double xK = diag[k] * currentPoint[k];
474                            xNorm += xK * xK;
475                        }
476                        xNorm = FastMath.sqrt(xNorm);
477    
478                        // tests for convergence.
479                        if (checker != null) {
480                            // we use the vectorial convergence checker
481                            if (checker.converged(iter, previous, current)) {
482                                setCost(currentCost);
483                                return current;
484                            }
485                        }
486                    } else {
487                        // failed iteration, reset the previous values
488                        currentCost = previousCost;
489                        for (int j = 0; j < solvedCols; ++j) {
490                            int pj = permutation[j];
491                            currentPoint[pj] = oldX[pj];
492                        }
493                        tmpVec    = weightedResidual;
494                        weightedResidual = oldRes;
495                        oldRes    = tmpVec;
496                        tmpVec    = currentObjective;
497                        currentObjective = oldObj;
498                        oldObj    = tmpVec;
499                        // Reset "current" to previous values.
500                        current = new PointVectorValuePair(currentPoint, currentObjective);
501                    }
502    
503                    // Default convergence criteria.
504                    if ((FastMath.abs(actRed) <= costRelativeTolerance &&
505                         preRed <= costRelativeTolerance &&
506                         ratio <= 2.0) ||
507                        delta <= parRelativeTolerance * xNorm) {
508                        setCost(currentCost);
509                        return current;
510                    }
511    
512                    // tests for termination and stringent tolerances
513                    // (2.2204e-16 is the machine epsilon for IEEE754)
514                    if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
515                        throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
516                                                       costRelativeTolerance);
517                    } else if (delta <= 2.2204e-16 * xNorm) {
518                        throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
519                                                       parRelativeTolerance);
520                    } else if (maxCosine <= 2.2204e-16)  {
521                        throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
522                                                       orthoTolerance);
523                    }
524                }
525            }
526        }
527    
528        /**
529         * Determine the Levenberg-Marquardt parameter.
530         * <p>This implementation is a translation in Java of the MINPACK
531         * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
532         * routine.</p>
533         * <p>This method sets the lmPar and lmDir attributes.</p>
534         * <p>The authors of the original fortran function are:</p>
535         * <ul>
536         *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
537         *   <li>Burton  S. Garbow</li>
538         *   <li>Kenneth E. Hillstrom</li>
539         *   <li>Jorge   J. More</li>
540         * </ul>
541         * <p>Luc Maisonobe did the Java translation.</p>
542         *
543         * @param qy array containing qTy
544         * @param delta upper bound on the euclidean norm of diagR * lmDir
545         * @param diag diagonal matrix
546         * @param work1 work array
547         * @param work2 work array
548         * @param work3 work array
549         */
550        private void determineLMParameter(double[] qy, double delta, double[] diag,
551                                          double[] work1, double[] work2, double[] work3) {
552            final int nC = weightedJacobian[0].length;
553    
554            // compute and store in x the gauss-newton direction, if the
555            // jacobian is rank-deficient, obtain a least squares solution
556            for (int j = 0; j < rank; ++j) {
557                lmDir[permutation[j]] = qy[j];
558            }
559            for (int j = rank; j < nC; ++j) {
560                lmDir[permutation[j]] = 0;
561            }
562            for (int k = rank - 1; k >= 0; --k) {
563                int pk = permutation[k];
564                double ypk = lmDir[pk] / diagR[pk];
565                for (int i = 0; i < k; ++i) {
566                    lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk];
567                }
568                lmDir[pk] = ypk;
569            }
570    
571            // evaluate the function at the origin, and test
572            // for acceptance of the Gauss-Newton direction
573            double dxNorm = 0;
574            for (int j = 0; j < solvedCols; ++j) {
575                int pj = permutation[j];
576                double s = diag[pj] * lmDir[pj];
577                work1[pj] = s;
578                dxNorm += s * s;
579            }
580            dxNorm = FastMath.sqrt(dxNorm);
581            double fp = dxNorm - delta;
582            if (fp <= 0.1 * delta) {
583                lmPar = 0;
584                return;
585            }
586    
587            // if the jacobian is not rank deficient, the Newton step provides
588            // a lower bound, parl, for the zero of the function,
589            // otherwise set this bound to zero
590            double sum2;
591            double parl = 0;
592            if (rank == solvedCols) {
593                for (int j = 0; j < solvedCols; ++j) {
594                    int pj = permutation[j];
595                    work1[pj] *= diag[pj] / dxNorm;
596                }
597                sum2 = 0;
598                for (int j = 0; j < solvedCols; ++j) {
599                    int pj = permutation[j];
600                    double sum = 0;
601                    for (int i = 0; i < j; ++i) {
602                        sum += weightedJacobian[i][pj] * work1[permutation[i]];
603                    }
604                    double s = (work1[pj] - sum) / diagR[pj];
605                    work1[pj] = s;
606                    sum2 += s * s;
607                }
608                parl = fp / (delta * sum2);
609            }
610    
611            // calculate an upper bound, paru, for the zero of the function
612            sum2 = 0;
613            for (int j = 0; j < solvedCols; ++j) {
614                int pj = permutation[j];
615                double sum = 0;
616                for (int i = 0; i <= j; ++i) {
617                    sum += weightedJacobian[i][pj] * qy[i];
618                }
619                sum /= diag[pj];
620                sum2 += sum * sum;
621            }
622            double gNorm = FastMath.sqrt(sum2);
623            double paru = gNorm / delta;
624            if (paru == 0) {
625                // 2.2251e-308 is the smallest positive real for IEE754
626                paru = 2.2251e-308 / FastMath.min(delta, 0.1);
627            }
628    
629            // if the input par lies outside of the interval (parl,paru),
630            // set par to the closer endpoint
631            lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
632            if (lmPar == 0) {
633                lmPar = gNorm / dxNorm;
634            }
635    
636            for (int countdown = 10; countdown >= 0; --countdown) {
637    
638                // evaluate the function at the current value of lmPar
639                if (lmPar == 0) {
640                    lmPar = FastMath.max(2.2251e-308, 0.001 * paru);
641                }
642                double sPar = FastMath.sqrt(lmPar);
643                for (int j = 0; j < solvedCols; ++j) {
644                    int pj = permutation[j];
645                    work1[pj] = sPar * diag[pj];
646                }
647                determineLMDirection(qy, work1, work2, work3);
648    
649                dxNorm = 0;
650                for (int j = 0; j < solvedCols; ++j) {
651                    int pj = permutation[j];
652                    double s = diag[pj] * lmDir[pj];
653                    work3[pj] = s;
654                    dxNorm += s * s;
655                }
656                dxNorm = FastMath.sqrt(dxNorm);
657                double previousFP = fp;
658                fp = dxNorm - delta;
659    
660                // if the function is small enough, accept the current value
661                // of lmPar, also test for the exceptional cases where parl is zero
662                if ((FastMath.abs(fp) <= 0.1 * delta) ||
663                        ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
664                    return;
665                }
666    
667                // compute the Newton correction
668                for (int j = 0; j < solvedCols; ++j) {
669                    int pj = permutation[j];
670                    work1[pj] = work3[pj] * diag[pj] / dxNorm;
671                }
672                for (int j = 0; j < solvedCols; ++j) {
673                    int pj = permutation[j];
674                    work1[pj] /= work2[j];
675                    double tmp = work1[pj];
676                    for (int i = j + 1; i < solvedCols; ++i) {
677                        work1[permutation[i]] -= weightedJacobian[i][pj] * tmp;
678                    }
679                }
680                sum2 = 0;
681                for (int j = 0; j < solvedCols; ++j) {
682                    double s = work1[permutation[j]];
683                    sum2 += s * s;
684                }
685                double correction = fp / (delta * sum2);
686    
687                // depending on the sign of the function, update parl or paru.
688                if (fp > 0) {
689                    parl = FastMath.max(parl, lmPar);
690                } else if (fp < 0) {
691                    paru = FastMath.min(paru, lmPar);
692                }
693    
694                // compute an improved estimate for lmPar
695                lmPar = FastMath.max(parl, lmPar + correction);
696    
697            }
698        }
699    
700        /**
701         * Solve a*x = b and d*x = 0 in the least squares sense.
702         * <p>This implementation is a translation in Java of the MINPACK
703         * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
704         * routine.</p>
705         * <p>This method sets the lmDir and lmDiag attributes.</p>
706         * <p>The authors of the original fortran function are:</p>
707         * <ul>
708         *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
709         *   <li>Burton  S. Garbow</li>
710         *   <li>Kenneth E. Hillstrom</li>
711         *   <li>Jorge   J. More</li>
712         * </ul>
713         * <p>Luc Maisonobe did the Java translation.</p>
714         *
715         * @param qy array containing qTy
716         * @param diag diagonal matrix
717         * @param lmDiag diagonal elements associated with lmDir
718         * @param work work array
719         */
720        private void determineLMDirection(double[] qy, double[] diag,
721                                          double[] lmDiag, double[] work) {
722    
723            // copy R and Qty to preserve input and initialize s
724            //  in particular, save the diagonal elements of R in lmDir
725            for (int j = 0; j < solvedCols; ++j) {
726                int pj = permutation[j];
727                for (int i = j + 1; i < solvedCols; ++i) {
728                    weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]];
729                }
730                lmDir[j] = diagR[pj];
731                work[j]  = qy[j];
732            }
733    
734            // eliminate the diagonal matrix d using a Givens rotation
735            for (int j = 0; j < solvedCols; ++j) {
736    
737                // prepare the row of d to be eliminated, locating the
738                // diagonal element using p from the Q.R. factorization
739                int pj = permutation[j];
740                double dpj = diag[pj];
741                if (dpj != 0) {
742                    Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
743                }
744                lmDiag[j] = dpj;
745    
746                //  the transformations to eliminate the row of d
747                // modify only a single element of Qty
748                // beyond the first n, which is initially zero.
749                double qtbpj = 0;
750                for (int k = j; k < solvedCols; ++k) {
751                    int pk = permutation[k];
752    
753                    // determine a Givens rotation which eliminates the
754                    // appropriate element in the current row of d
755                    if (lmDiag[k] != 0) {
756    
757                        final double sin;
758                        final double cos;
759                        double rkk = weightedJacobian[k][pk];
760                        if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
761                            final double cotan = rkk / lmDiag[k];
762                            sin   = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
763                            cos   = sin * cotan;
764                        } else {
765                            final double tan = lmDiag[k] / rkk;
766                            cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
767                            sin = cos * tan;
768                        }
769    
770                        // compute the modified diagonal element of R and
771                        // the modified element of (Qty,0)
772                        weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
773                        final double temp = cos * work[k] + sin * qtbpj;
774                        qtbpj = -sin * work[k] + cos * qtbpj;
775                        work[k] = temp;
776    
777                        // accumulate the tranformation in the row of s
778                        for (int i = k + 1; i < solvedCols; ++i) {
779                            double rik = weightedJacobian[i][pk];
780                            final double temp2 = cos * rik + sin * lmDiag[i];
781                            lmDiag[i] = -sin * rik + cos * lmDiag[i];
782                            weightedJacobian[i][pk] = temp2;
783                        }
784                    }
785                }
786    
787                // store the diagonal element of s and restore
788                // the corresponding diagonal element of R
789                lmDiag[j] = weightedJacobian[j][permutation[j]];
790                weightedJacobian[j][permutation[j]] = lmDir[j];
791            }
792    
793            // solve the triangular system for z, if the system is
794            // singular, then obtain a least squares solution
795            int nSing = solvedCols;
796            for (int j = 0; j < solvedCols; ++j) {
797                if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
798                    nSing = j;
799                }
800                if (nSing < solvedCols) {
801                    work[j] = 0;
802                }
803            }
804            if (nSing > 0) {
805                for (int j = nSing - 1; j >= 0; --j) {
806                    int pj = permutation[j];
807                    double sum = 0;
808                    for (int i = j + 1; i < nSing; ++i) {
809                        sum += weightedJacobian[i][pj] * work[i];
810                    }
811                    work[j] = (work[j] - sum) / lmDiag[j];
812                }
813            }
814    
815            // permute the components of z back to components of lmDir
816            for (int j = 0; j < lmDir.length; ++j) {
817                lmDir[permutation[j]] = work[j];
818            }
819        }
820    
821        /**
822         * Decompose a matrix A as A.P = Q.R using Householder transforms.
823         * <p>As suggested in the P. Lascaux and R. Theodor book
824         * <i>Analyse num&eacute;rique matricielle appliqu&eacute;e &agrave;
825         * l'art de l'ing&eacute;nieur</i> (Masson, 1986), instead of representing
826         * the Householder transforms with u<sub>k</sub> unit vectors such that:
827         * <pre>
828         * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
829         * </pre>
830         * we use <sub>k</sub> non-unit vectors such that:
831         * <pre>
832         * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
833         * </pre>
834         * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
835         * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
836         * them from the v<sub>k</sub> vectors would be costly.</p>
837         * <p>This decomposition handles rank deficient cases since the tranformations
838         * are performed in non-increasing columns norms order thanks to columns
839         * pivoting. The diagonal elements of the R matrix are therefore also in
840         * non-increasing absolute values order.</p>
841         *
842         * @param jacobian Weighted Jacobian matrix at the current point.
843         * @exception ConvergenceException if the decomposition cannot be performed
844         */
845        private void qrDecomposition(RealMatrix jacobian) throws ConvergenceException {
846            // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
847            // hence the multiplication by -1.
848            weightedJacobian = jacobian.scalarMultiply(-1).getData();
849    
850            final int nR = weightedJacobian.length;
851            final int nC = weightedJacobian[0].length;
852    
853            // initializations
854            for (int k = 0; k < nC; ++k) {
855                permutation[k] = k;
856                double norm2 = 0;
857                for (int i = 0; i < nR; ++i) {
858                    double akk = weightedJacobian[i][k];
859                    norm2 += akk * akk;
860                }
861                jacNorm[k] = FastMath.sqrt(norm2);
862            }
863    
864            // transform the matrix column after column
865            for (int k = 0; k < nC; ++k) {
866    
867                // select the column with the greatest norm on active components
868                int nextColumn = -1;
869                double ak2 = Double.NEGATIVE_INFINITY;
870                for (int i = k; i < nC; ++i) {
871                    double norm2 = 0;
872                    for (int j = k; j < nR; ++j) {
873                        double aki = weightedJacobian[j][permutation[i]];
874                        norm2 += aki * aki;
875                    }
876                    if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
877                        throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
878                                                       nR, nC);
879                    }
880                    if (norm2 > ak2) {
881                        nextColumn = i;
882                        ak2        = norm2;
883                    }
884                }
885                if (ak2 <= qrRankingThreshold) {
886                    rank = k;
887                    return;
888                }
889                int pk                  = permutation[nextColumn];
890                permutation[nextColumn] = permutation[k];
891                permutation[k]          = pk;
892    
893                // choose alpha such that Hk.u = alpha ek
894                double akk   = weightedJacobian[k][pk];
895                double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
896                double betak = 1.0 / (ak2 - akk * alpha);
897                beta[pk]     = betak;
898    
899                // transform the current column
900                diagR[pk]        = alpha;
901                weightedJacobian[k][pk] -= alpha;
902    
903                // transform the remaining columns
904                for (int dk = nC - 1 - k; dk > 0; --dk) {
905                    double gamma = 0;
906                    for (int j = k; j < nR; ++j) {
907                        gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]];
908                    }
909                    gamma *= betak;
910                    for (int j = k; j < nR; ++j) {
911                        weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk];
912                    }
913                }
914            }
915            rank = solvedCols;
916        }
917    
918        /**
919         * Compute the product Qt.y for some Q.R. decomposition.
920         *
921         * @param y vector to multiply (will be overwritten with the result)
922         */
923        private void qTy(double[] y) {
924            final int nR = weightedJacobian.length;
925            final int nC = weightedJacobian[0].length;
926    
927            for (int k = 0; k < nC; ++k) {
928                int pk = permutation[k];
929                double gamma = 0;
930                for (int i = k; i < nR; ++i) {
931                    gamma += weightedJacobian[i][pk] * y[i];
932                }
933                gamma *= beta[pk];
934                for (int i = k; i < nR; ++i) {
935                    y[i] -= gamma * weightedJacobian[i][pk];
936                }
937            }
938        }
939    }