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.math4.legacy.fitting.leastsquares;
018
019import java.util.Arrays;
020
021import org.apache.commons.math4.legacy.exception.ConvergenceException;
022import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
023import org.apache.commons.math4.legacy.fitting.leastsquares.LeastSquaresProblem.Evaluation;
024import org.apache.commons.math4.legacy.linear.ArrayRealVector;
025import org.apache.commons.math4.legacy.linear.RealMatrix;
026import org.apache.commons.math4.legacy.optim.ConvergenceChecker;
027import org.apache.commons.math4.core.jdkmath.JdkMath;
028import org.apache.commons.math4.legacy.core.IntegerSequence;
029import org.apache.commons.numbers.core.Precision;
030
031
032/**
033 * This class solves a least-squares problem using the Levenberg-Marquardt
034 * algorithm.
035 *
036 * <p>This implementation <em>should</em> work even for over-determined systems
037 * (i.e. systems having more point than equations). Over-determined systems
038 * are solved by ignoring the point which have the smallest impact according
039 * to their jacobian column norm. Only the rank of the matrix and some loop bounds
040 * are changed to implement this.</p>
041 *
042 * <p>The resolution engine is a simple translation of the MINPACK <a
043 * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
044 * changes. The changes include the over-determined resolution, the use of
045 * inherited convergence checker and the Q.R. decomposition which has been
046 * rewritten following the algorithm described in the
047 * P. Lascaux and R. Theodor book <i>Analyse num&eacute;rique matricielle
048 * appliqu&eacute;e &agrave; l'art de l'ing&eacute;nieur</i>, Masson 1986.</p>
049 * <p>The authors of the original fortran version are:
050 * <ul>
051 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
052 * <li>Burton S. Garbow</li>
053 * <li>Kenneth E. Hillstrom</li>
054 * <li>Jorge J. More</li>
055 * </ul>
056 * The redistribution policy for MINPACK is available <a
057 * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
058 * is reproduced below.
059 *
060 * <table style="text-align: center; background-color: #E0E0E0" border="">
061 * <caption>MINPACK redistribution policy</caption>
062 * <tr><td>
063 *    Minpack Copyright Notice (1999) University of Chicago.
064 *    All rights reserved
065 * </td></tr>
066 * <tr><td>
067 * Redistribution and use in source and binary forms, with or without
068 * modification, are permitted provided that the following conditions
069 * are met:
070 * <ol>
071 *  <li>Redistributions of source code must retain the above copyright
072 *      notice, this list of conditions and the following disclaimer.</li>
073 * <li>Redistributions in binary form must reproduce the above
074 *     copyright notice, this list of conditions and the following
075 *     disclaimer in the documentation and/or other materials provided
076 *     with the distribution.</li>
077 * <li>The end-user documentation included with the redistribution, if any,
078 *     must include the following acknowledgment:
079 *     <code>This product includes software developed by the University of
080 *           Chicago, as Operator of Argonne National Laboratory.</code>
081 *     Alternately, this acknowledgment may appear in the software itself,
082 *     if and wherever such third-party acknowledgments normally appear.</li>
083 * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
084 *     WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
085 *     UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
086 *     THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
087 *     IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
088 *     OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
089 *     OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
090 *     OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
091 *     USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
092 *     THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
093 *     DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
094 *     UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
095 *     BE CORRECTED.</strong></li>
096 * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
097 *     HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
098 *     ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
099 *     INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
100 *     ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
101 *     PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
102 *     SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
103 *     (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
104 *     EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
105 *     POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
106 * </ol></td></tr>
107 * </table>
108 *
109 * @since 3.3
110 */
111public class LevenbergMarquardtOptimizer implements LeastSquaresOptimizer {
112
113    /** Twice the "epsilon machine". */
114    private static final double TWO_EPS = 2 * Precision.EPSILON;
115
116    /* configuration parameters */
117    /** Positive input variable used in determining the initial step bound. */
118    private final double initialStepBoundFactor;
119    /** Desired relative error in the sum of squares. */
120    private final double costRelativeTolerance;
121    /**  Desired relative error in the approximate solution parameters. */
122    private final double parRelativeTolerance;
123    /** Desired max cosine on the orthogonality between the function vector
124     * and the columns of the jacobian. */
125    private final double orthoTolerance;
126    /** Threshold for QR ranking. */
127    private final double qrRankingThreshold;
128
129    /** Default constructor.
130     * <p>
131     * The default values for the algorithm settings are:
132     * <ul>
133     *  <li>Initial step bound factor: 100</li>
134     *  <li>Cost relative tolerance: 1e-10</li>
135     *  <li>Parameters relative tolerance: 1e-10</li>
136     *  <li>Orthogonality tolerance: 1e-10</li>
137     *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
138     * </ul>
139     **/
140    public LevenbergMarquardtOptimizer() {
141        this(100, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
142    }
143
144    /**
145     * Construct an instance with all parameters specified.
146     *
147     * @param initialStepBoundFactor initial step bound factor
148     * @param costRelativeTolerance  cost relative tolerance
149     * @param parRelativeTolerance   parameters relative tolerance
150     * @param orthoTolerance         orthogonality tolerance
151     * @param qrRankingThreshold     threshold in the QR decomposition. Columns with a 2
152     *                               norm less than this threshold are considered to be
153     *                               all 0s.
154     */
155    public LevenbergMarquardtOptimizer(
156            final double initialStepBoundFactor,
157            final double costRelativeTolerance,
158            final double parRelativeTolerance,
159            final double orthoTolerance,
160            final double qrRankingThreshold) {
161        this.initialStepBoundFactor = initialStepBoundFactor;
162        this.costRelativeTolerance = costRelativeTolerance;
163        this.parRelativeTolerance = parRelativeTolerance;
164        this.orthoTolerance = orthoTolerance;
165        this.qrRankingThreshold = qrRankingThreshold;
166    }
167
168    /**
169     * @param newInitialStepBoundFactor Positive input variable used in
170     * determining the initial step bound. This bound is set to the
171     * product of initialStepBoundFactor and the euclidean norm of
172     * {@code diag * x} if non-zero, or else to {@code newInitialStepBoundFactor}
173     * itself. In most cases factor should lie in the interval
174     * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
175     * of the matrix is reduced.
176     * @return a new instance.
177     */
178    public LevenbergMarquardtOptimizer withInitialStepBoundFactor(double newInitialStepBoundFactor) {
179        return new LevenbergMarquardtOptimizer(
180                newInitialStepBoundFactor,
181                costRelativeTolerance,
182                parRelativeTolerance,
183                orthoTolerance,
184                qrRankingThreshold);
185    }
186
187    /**
188     * @param newCostRelativeTolerance Desired relative error in the sum of squares.
189     * @return a new instance.
190     */
191    public LevenbergMarquardtOptimizer withCostRelativeTolerance(double newCostRelativeTolerance) {
192        return new LevenbergMarquardtOptimizer(
193                initialStepBoundFactor,
194                newCostRelativeTolerance,
195                parRelativeTolerance,
196                orthoTolerance,
197                qrRankingThreshold);
198    }
199
200    /**
201     * @param newParRelativeTolerance Desired relative error in the approximate solution
202     * parameters.
203     * @return a new instance.
204     */
205    public LevenbergMarquardtOptimizer withParameterRelativeTolerance(double newParRelativeTolerance) {
206        return new LevenbergMarquardtOptimizer(
207                initialStepBoundFactor,
208                costRelativeTolerance,
209                newParRelativeTolerance,
210                orthoTolerance,
211                qrRankingThreshold);
212    }
213
214    /**
215     * Modifies the given parameter.
216     *
217     * @param newOrthoTolerance Desired max cosine on the orthogonality between
218     * the function vector and the columns of the Jacobian.
219     * @return a new instance.
220     */
221    public LevenbergMarquardtOptimizer withOrthoTolerance(double newOrthoTolerance) {
222        return new LevenbergMarquardtOptimizer(
223                initialStepBoundFactor,
224                costRelativeTolerance,
225                parRelativeTolerance,
226                newOrthoTolerance,
227                qrRankingThreshold);
228    }
229
230    /**
231     * @param newQRRankingThreshold Desired threshold for QR ranking.
232     * If the squared norm of a column vector is smaller or equal to this
233     * threshold during QR decomposition, it is considered to be a zero vector
234     * and hence the rank of the matrix is reduced.
235     * @return a new instance.
236     */
237    public LevenbergMarquardtOptimizer withRankingThreshold(double newQRRankingThreshold) {
238        return new LevenbergMarquardtOptimizer(
239                initialStepBoundFactor,
240                costRelativeTolerance,
241                parRelativeTolerance,
242                orthoTolerance,
243                newQRRankingThreshold);
244    }
245
246    /**
247     * Gets the value of a tuning parameter.
248     * @see #withInitialStepBoundFactor(double)
249     *
250     * @return the parameter's value.
251     */
252    public double getInitialStepBoundFactor() {
253        return initialStepBoundFactor;
254    }
255
256    /**
257     * Gets the value of a tuning parameter.
258     * @see #withCostRelativeTolerance(double)
259     *
260     * @return the parameter's value.
261     */
262    public double getCostRelativeTolerance() {
263        return costRelativeTolerance;
264    }
265
266    /**
267     * Gets the value of a tuning parameter.
268     * @see #withParameterRelativeTolerance(double)
269     *
270     * @return the parameter's value.
271     */
272    public double getParameterRelativeTolerance() {
273        return parRelativeTolerance;
274    }
275
276    /**
277     * Gets the value of a tuning parameter.
278     * @see #withOrthoTolerance(double)
279     *
280     * @return the parameter's value.
281     */
282    public double getOrthoTolerance() {
283        return orthoTolerance;
284    }
285
286    /**
287     * Gets the value of a tuning parameter.
288     * @see #withRankingThreshold(double)
289     *
290     * @return the parameter's value.
291     */
292    public double getRankingThreshold() {
293        return qrRankingThreshold;
294    }
295
296    /** {@inheritDoc} */
297    @Override
298    public Optimum optimize(final LeastSquaresProblem problem) {
299        // Pull in relevant data from the problem as locals.
300        final int nR = problem.getObservationSize(); // Number of observed data.
301        final int nC = problem.getParameterSize(); // Number of parameters.
302        // Counters.
303        final IntegerSequence.Incrementor iterationCounter = problem.getIterationCounter();
304        final IntegerSequence.Incrementor evaluationCounter = problem.getEvaluationCounter();
305        // Convergence criterion.
306        final ConvergenceChecker<Evaluation> checker = problem.getConvergenceChecker();
307
308        // arrays shared with the other private methods
309        final int solvedCols  = JdkMath.min(nR, nC);
310        /* Parameters evolution direction associated with lmPar. */
311        double[] lmDir = new double[nC];
312        /* Levenberg-Marquardt parameter. */
313        double lmPar = 0;
314
315        // local point
316        double   delta   = 0;
317        double   xNorm   = 0;
318        double[] diag    = new double[nC];
319        double[] oldX    = new double[nC];
320        double[] oldRes  = new double[nR];
321        double[] qtf     = new double[nR];
322        double[] work1   = new double[nC];
323        double[] work2   = new double[nC];
324        double[] work3   = new double[nC];
325
326
327        // Evaluate the function at the starting point and calculate its norm.
328        evaluationCounter.increment();
329        //value will be reassigned in the loop
330        Evaluation current = problem.evaluate(problem.getStart());
331        double[] currentResiduals = current.getResiduals().toArray();
332        double currentCost = current.getCost();
333        double[] currentPoint = current.getPoint().toArray();
334
335        // Outer loop.
336        boolean firstIteration = true;
337        while (true) {
338            iterationCounter.increment();
339
340            final Evaluation previous = current;
341
342            // QR decomposition of the jacobian matrix
343            final InternalData internalData
344                    = qrDecomposition(current.getJacobian(), solvedCols);
345            final double[][] weightedJacobian = internalData.weightedJacobian;
346            final int[] permutation = internalData.permutation;
347            final double[] diagR = internalData.diagR;
348            final double[] jacNorm = internalData.jacNorm;
349
350            //residuals already have weights applied
351            double[] weightedResidual = currentResiduals;
352            System.arraycopy(weightedResidual, 0, qtf, 0, nR);
353
354            // compute Qt.res
355            qTy(qtf, internalData);
356
357            // now we don't need Q anymore,
358            // so let jacobian contain the R matrix with its diagonal elements
359            for (int k = 0; k < solvedCols; ++k) {
360                int pk = permutation[k];
361                weightedJacobian[k][pk] = diagR[pk];
362            }
363
364            if (firstIteration) {
365                // scale the point according to the norms of the columns
366                // of the initial jacobian
367                xNorm = 0;
368                for (int k = 0; k < nC; ++k) {
369                    double dk = jacNorm[k];
370                    if (dk == 0) {
371                        dk = 1.0;
372                    }
373                    double xk = dk * currentPoint[k];
374                    xNorm  += xk * xk;
375                    diag[k] = dk;
376                }
377                xNorm = JdkMath.sqrt(xNorm);
378
379                // initialize the step bound delta
380                delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
381            }
382
383            // check orthogonality between function vector and jacobian columns
384            double maxCosine = 0;
385            if (currentCost != 0) {
386                for (int j = 0; j < solvedCols; ++j) {
387                    int    pj = permutation[j];
388                    double s  = jacNorm[pj];
389                    if (s != 0) {
390                        double sum = 0;
391                        for (int i = 0; i <= j; ++i) {
392                            sum += weightedJacobian[i][pj] * qtf[i];
393                        }
394                        maxCosine = JdkMath.max(maxCosine, JdkMath.abs(sum) / (s * currentCost));
395                    }
396                }
397            }
398            if (maxCosine <= orthoTolerance) {
399                // Convergence has been reached.
400                return new OptimumImpl(
401                        current,
402                        evaluationCounter.getCount(),
403                        iterationCounter.getCount());
404            }
405
406            // rescale if necessary
407            for (int j = 0; j < nC; ++j) {
408                diag[j] = JdkMath.max(diag[j], jacNorm[j]);
409            }
410
411            // Inner loop.
412            for (double ratio = 0; ratio < 1.0e-4;) {
413
414                // save the state
415                for (int j = 0; j < solvedCols; ++j) {
416                    int pj = permutation[j];
417                    oldX[pj] = currentPoint[pj];
418                }
419                final double previousCost = currentCost;
420                double[] tmpVec = weightedResidual;
421                weightedResidual = oldRes;
422                oldRes    = tmpVec;
423
424                // determine the Levenberg-Marquardt parameter
425                lmPar = determineLMParameter(qtf, delta, diag,
426                                     internalData, solvedCols,
427                                     work1, work2, work3, lmDir, lmPar);
428
429                // compute the new point and the norm of the evolution direction
430                double lmNorm = 0;
431                for (int j = 0; j < solvedCols; ++j) {
432                    int pj = permutation[j];
433                    lmDir[pj] = -lmDir[pj];
434                    currentPoint[pj] = oldX[pj] + lmDir[pj];
435                    double s = diag[pj] * lmDir[pj];
436                    lmNorm  += s * s;
437                }
438                lmNorm = JdkMath.sqrt(lmNorm);
439                // on the first iteration, adjust the initial step bound.
440                if (firstIteration) {
441                    delta = JdkMath.min(delta, lmNorm);
442                }
443
444                // Evaluate the function at x + p and calculate its norm.
445                evaluationCounter.increment();
446                current = problem.evaluate(new ArrayRealVector(currentPoint));
447                currentResiduals = current.getResiduals().toArray();
448                currentCost = current.getCost();
449                currentPoint = current.getPoint().toArray();
450
451                // compute the scaled actual reduction
452                double actRed = -1.0;
453                if (0.1 * currentCost < previousCost) {
454                    double r = currentCost / previousCost;
455                    actRed = 1.0 - r * r;
456                }
457
458                // compute the scaled predicted reduction
459                // and the scaled directional derivative
460                for (int j = 0; j < solvedCols; ++j) {
461                    int pj = permutation[j];
462                    double dirJ = lmDir[pj];
463                    work1[j] = 0;
464                    for (int i = 0; i <= j; ++i) {
465                        work1[i] += weightedJacobian[i][pj] * dirJ;
466                    }
467                }
468                double coeff1 = 0;
469                for (int j = 0; j < solvedCols; ++j) {
470                    coeff1 += work1[j] * work1[j];
471                }
472                double pc2 = previousCost * previousCost;
473                coeff1 /= pc2;
474                double coeff2 = lmPar * lmNorm * lmNorm / pc2;
475                double preRed = coeff1 + 2 * coeff2;
476                double dirDer = -(coeff1 + coeff2);
477
478                // ratio of the actual to the predicted reduction
479                ratio = (preRed == 0) ? 0 : (actRed / preRed);
480
481                // update the step bound
482                if (ratio <= 0.25) {
483                    double tmp =
484                        (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
485                        if (0.1 * currentCost >= previousCost || tmp < 0.1) {
486                            tmp = 0.1;
487                        }
488                        delta = tmp * JdkMath.min(delta, 10.0 * lmNorm);
489                        lmPar /= tmp;
490                } else if (lmPar == 0 || ratio >= 0.75) {
491                    delta = 2 * lmNorm;
492                    lmPar *= 0.5;
493                }
494
495                // test for successful iteration.
496                if (ratio >= 1.0e-4) {
497                    // successful iteration, update the norm
498                    firstIteration = false;
499                    xNorm = 0;
500                    for (int k = 0; k < nC; ++k) {
501                        double xK = diag[k] * currentPoint[k];
502                        xNorm += xK * xK;
503                    }
504                    xNorm = JdkMath.sqrt(xNorm);
505
506                    // tests for convergence.
507                    if (checker != null && checker.converged(iterationCounter.getCount(), previous, current)) {
508                        return new OptimumImpl(current, evaluationCounter.getCount(), iterationCounter.getCount());
509                    }
510                } else {
511                    // failed iteration, reset the previous values
512                    currentCost = previousCost;
513                    for (int j = 0; j < solvedCols; ++j) {
514                        int pj = permutation[j];
515                        currentPoint[pj] = oldX[pj];
516                    }
517                    tmpVec    = weightedResidual;
518                    weightedResidual = oldRes;
519                    oldRes    = tmpVec;
520                    // Reset "current" to previous values.
521                    current = previous;
522                }
523
524                // Default convergence criteria.
525                if ((JdkMath.abs(actRed) <= costRelativeTolerance &&
526                     preRed <= costRelativeTolerance &&
527                     ratio <= 2.0) ||
528                    delta <= parRelativeTolerance * xNorm) {
529                    return new OptimumImpl(current, evaluationCounter.getCount(), iterationCounter.getCount());
530                }
531
532                // tests for termination and stringent tolerances
533                if (JdkMath.abs(actRed) <= TWO_EPS &&
534                    preRed <= TWO_EPS &&
535                    ratio <= 2.0) {
536                    throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
537                                                   costRelativeTolerance);
538                } else if (delta <= TWO_EPS * xNorm) {
539                    throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
540                                                   parRelativeTolerance);
541                } else if (maxCosine <= TWO_EPS) {
542                    throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
543                                                   orthoTolerance);
544                }
545            }
546        }
547    }
548
549    /**
550     * Holds internal data.
551     * This structure was created so that all optimizer fields can be "final".
552     * Code should be further refactored in order to not pass around arguments
553     * that will modified in-place (cf. "work" arrays).
554     */
555    private static class InternalData {
556        /** Weighted Jacobian. */
557        private final double[][] weightedJacobian;
558        /** Columns permutation array. */
559        private final int[] permutation;
560        /** Rank of the Jacobian matrix. */
561        private final int rank;
562        /** Diagonal elements of the R matrix in the QR decomposition. */
563        private final double[] diagR;
564        /** Norms of the columns of the jacobian matrix. */
565        private final double[] jacNorm;
566        /** Coefficients of the Householder transforms vectors. */
567        private final double[] beta;
568
569        /**
570         * @param weightedJacobian Weighted Jacobian.
571         * @param permutation Columns permutation array.
572         * @param rank Rank of the Jacobian matrix.
573         * @param diagR Diagonal elements of the R matrix in the QR decomposition.
574         * @param jacNorm Norms of the columns of the jacobian matrix.
575         * @param beta Coefficients of the Householder transforms vectors.
576         */
577        InternalData(double[][] weightedJacobian,
578                     int[] permutation,
579                     int rank,
580                     double[] diagR,
581                     double[] jacNorm,
582                     double[] beta) {
583            this.weightedJacobian = weightedJacobian;
584            this.permutation = permutation;
585            this.rank = rank;
586            this.diagR = diagR;
587            this.jacNorm = jacNorm;
588            this.beta = beta;
589        }
590    }
591
592    /**
593     * Determines the Levenberg-Marquardt parameter.
594     *
595     * <p>This implementation is a translation in Java of the MINPACK
596     * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
597     * routine.</p>
598     * <p>This method sets the lmPar and lmDir attributes.</p>
599     * <p>The authors of the original fortran function are:</p>
600     * <ul>
601     *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
602     *   <li>Burton  S. Garbow</li>
603     *   <li>Kenneth E. Hillstrom</li>
604     *   <li>Jorge   J. More</li>
605     * </ul>
606     * <p>Luc Maisonobe did the Java translation.</p>
607     *
608     * @param qy Array containing qTy.
609     * @param delta Upper bound on the euclidean norm of diagR * lmDir.
610     * @param diag Diagonal matrix.
611     * @param internalData Data (modified in-place in this method).
612     * @param solvedCols Number of solved point.
613     * @param work1 work array
614     * @param work2 work array
615     * @param work3 work array
616     * @param lmDir the "returned" LM direction will be stored in this array.
617     * @param lmPar the value of the LM parameter from the previous iteration.
618     * @return the new LM parameter
619     */
620    private double determineLMParameter(double[] qy, double delta, double[] diag,
621                                      InternalData internalData, int solvedCols,
622                                      double[] work1, double[] work2, double[] work3,
623                                      double[] lmDir, double lmPar) {
624        final double[][] weightedJacobian = internalData.weightedJacobian;
625        final int[] permutation = internalData.permutation;
626        final int rank = internalData.rank;
627        final double[] diagR = internalData.diagR;
628
629        final int nC = weightedJacobian[0].length;
630
631        // compute and store in x the gauss-newton direction, if the
632        // jacobian is rank-deficient, obtain a least squares solution
633        for (int j = 0; j < rank; ++j) {
634            lmDir[permutation[j]] = qy[j];
635        }
636        for (int j = rank; j < nC; ++j) {
637            lmDir[permutation[j]] = 0;
638        }
639        for (int k = rank - 1; k >= 0; --k) {
640            int pk = permutation[k];
641            double ypk = lmDir[pk] / diagR[pk];
642            for (int i = 0; i < k; ++i) {
643                lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk];
644            }
645            lmDir[pk] = ypk;
646        }
647
648        // evaluate the function at the origin, and test
649        // for acceptance of the Gauss-Newton direction
650        double dxNorm = 0;
651        for (int j = 0; j < solvedCols; ++j) {
652            int pj = permutation[j];
653            double s = diag[pj] * lmDir[pj];
654            work1[pj] = s;
655            dxNorm += s * s;
656        }
657        dxNorm = JdkMath.sqrt(dxNorm);
658        double fp = dxNorm - delta;
659        if (fp <= 0.1 * delta) {
660            lmPar = 0;
661            return lmPar;
662        }
663
664        // if the jacobian is not rank deficient, the Newton step provides
665        // a lower bound, parl, for the zero of the function,
666        // otherwise set this bound to zero
667        double sum2;
668        double parl = 0;
669        if (rank == solvedCols) {
670            for (int j = 0; j < solvedCols; ++j) {
671                int pj = permutation[j];
672                work1[pj] *= diag[pj] / dxNorm;
673            }
674            sum2 = 0;
675            for (int j = 0; j < solvedCols; ++j) {
676                int pj = permutation[j];
677                double sum = 0;
678                for (int i = 0; i < j; ++i) {
679                    sum += weightedJacobian[i][pj] * work1[permutation[i]];
680                }
681                double s = (work1[pj] - sum) / diagR[pj];
682                work1[pj] = s;
683                sum2 += s * s;
684            }
685            parl = fp / (delta * sum2);
686        }
687
688        // calculate an upper bound, paru, for the zero of the function
689        sum2 = 0;
690        for (int j = 0; j < solvedCols; ++j) {
691            int pj = permutation[j];
692            double sum = 0;
693            for (int i = 0; i <= j; ++i) {
694                sum += weightedJacobian[i][pj] * qy[i];
695            }
696            sum /= diag[pj];
697            sum2 += sum * sum;
698        }
699        double gNorm = JdkMath.sqrt(sum2);
700        double paru = gNorm / delta;
701        if (paru == 0) {
702            paru = Precision.SAFE_MIN / JdkMath.min(delta, 0.1);
703        }
704
705        // if the input par lies outside of the interval (parl,paru),
706        // set par to the closer endpoint
707        lmPar = JdkMath.min(paru, JdkMath.max(lmPar, parl));
708        if (lmPar == 0) {
709            lmPar = gNorm / dxNorm;
710        }
711
712        for (int countdown = 10; countdown >= 0; --countdown) {
713
714            // evaluate the function at the current value of lmPar
715            if (lmPar == 0) {
716                lmPar = JdkMath.max(Precision.SAFE_MIN, 0.001 * paru);
717            }
718            double sPar = JdkMath.sqrt(lmPar);
719            for (int j = 0; j < solvedCols; ++j) {
720                int pj = permutation[j];
721                work1[pj] = sPar * diag[pj];
722            }
723            determineLMDirection(qy, work1, work2, internalData, solvedCols, work3, lmDir);
724
725            dxNorm = 0;
726            for (int j = 0; j < solvedCols; ++j) {
727                int pj = permutation[j];
728                double s = diag[pj] * lmDir[pj];
729                work3[pj] = s;
730                dxNorm += s * s;
731            }
732            dxNorm = JdkMath.sqrt(dxNorm);
733            double previousFP = fp;
734            fp = dxNorm - delta;
735
736            // if the function is small enough, accept the current value
737            // of lmPar, also test for the exceptional cases where parl is zero
738            if (JdkMath.abs(fp) <= 0.1 * delta ||
739                (parl == 0 &&
740                 fp <= previousFP &&
741                 previousFP < 0)) {
742                return lmPar;
743            }
744
745            // compute the Newton correction
746            for (int j = 0; j < solvedCols; ++j) {
747                int pj = permutation[j];
748                work1[pj] = work3[pj] * diag[pj] / dxNorm;
749            }
750            for (int j = 0; j < solvedCols; ++j) {
751                int pj = permutation[j];
752                work1[pj] /= work2[j];
753                double tmp = work1[pj];
754                for (int i = j + 1; i < solvedCols; ++i) {
755                    work1[permutation[i]] -= weightedJacobian[i][pj] * tmp;
756                }
757            }
758            sum2 = 0;
759            for (int j = 0; j < solvedCols; ++j) {
760                double s = work1[permutation[j]];
761                sum2 += s * s;
762            }
763            double correction = fp / (delta * sum2);
764
765            // depending on the sign of the function, update parl or paru.
766            if (fp > 0) {
767                parl = JdkMath.max(parl, lmPar);
768            } else if (fp < 0) {
769                paru = JdkMath.min(paru, lmPar);
770            }
771
772            // compute an improved estimate for lmPar
773            lmPar = JdkMath.max(parl, lmPar + correction);
774        }
775
776        return lmPar;
777    }
778
779    /**
780     * Solve a*x = b and d*x = 0 in the least squares sense.
781     * <p>This implementation is a translation in Java of the MINPACK
782     * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
783     * routine.</p>
784     * <p>This method sets the lmDir and lmDiag attributes.</p>
785     * <p>The authors of the original fortran function are:</p>
786     * <ul>
787     *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
788     *   <li>Burton  S. Garbow</li>
789     *   <li>Kenneth E. Hillstrom</li>
790     *   <li>Jorge   J. More</li>
791     * </ul>
792     * <p>Luc Maisonobe did the Java translation.</p>
793     *
794     * @param qy array containing qTy
795     * @param diag diagonal matrix
796     * @param lmDiag diagonal elements associated with lmDir
797     * @param internalData Data (modified in-place in this method).
798     * @param solvedCols Number of sloved point.
799     * @param work work array
800     * @param lmDir the "returned" LM direction is stored in this array
801     */
802    private void determineLMDirection(double[] qy, double[] diag,
803                                      double[] lmDiag,
804                                      InternalData internalData,
805                                      int solvedCols,
806                                      double[] work,
807                                      double[] lmDir) {
808        final int[] permutation = internalData.permutation;
809        final double[][] weightedJacobian = internalData.weightedJacobian;
810        final double[] diagR = internalData.diagR;
811
812        // copy R and Qty to preserve input and initialize s
813        //  in particular, save the diagonal elements of R in lmDir
814        for (int j = 0; j < solvedCols; ++j) {
815            int pj = permutation[j];
816            for (int i = j + 1; i < solvedCols; ++i) {
817                weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]];
818            }
819            lmDir[j] = diagR[pj];
820            work[j]  = qy[j];
821        }
822
823        // eliminate the diagonal matrix d using a Givens rotation
824        for (int j = 0; j < solvedCols; ++j) {
825
826            // prepare the row of d to be eliminated, locating the
827            // diagonal element using p from the Q.R. factorization
828            int pj = permutation[j];
829            double dpj = diag[pj];
830            if (dpj != 0) {
831                Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
832            }
833            lmDiag[j] = dpj;
834
835            //  the transformations to eliminate the row of d
836            // modify only a single element of Qty
837            // beyond the first n, which is initially zero.
838            double qtbpj = 0;
839            for (int k = j; k < solvedCols; ++k) {
840                int pk = permutation[k];
841
842                // determine a Givens rotation which eliminates the
843                // appropriate element in the current row of d
844                if (lmDiag[k] != 0) {
845
846                    final double sin;
847                    final double cos;
848                    double rkk = weightedJacobian[k][pk];
849                    if (JdkMath.abs(rkk) < JdkMath.abs(lmDiag[k])) {
850                        final double cotan = rkk / lmDiag[k];
851                        sin   = 1.0 / JdkMath.sqrt(1.0 + cotan * cotan);
852                        cos   = sin * cotan;
853                    } else {
854                        final double tan = lmDiag[k] / rkk;
855                        cos = 1.0 / JdkMath.sqrt(1.0 + tan * tan);
856                        sin = cos * tan;
857                    }
858
859                    // compute the modified diagonal element of R and
860                    // the modified element of (Qty,0)
861                    weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
862                    final double temp = cos * work[k] + sin * qtbpj;
863                    qtbpj = -sin * work[k] + cos * qtbpj;
864                    work[k] = temp;
865
866                    // accumulate the tranformation in the row of s
867                    for (int i = k + 1; i < solvedCols; ++i) {
868                        double rik = weightedJacobian[i][pk];
869                        final double temp2 = cos * rik + sin * lmDiag[i];
870                        lmDiag[i] = -sin * rik + cos * lmDiag[i];
871                        weightedJacobian[i][pk] = temp2;
872                    }
873                }
874            }
875
876            // store the diagonal element of s and restore
877            // the corresponding diagonal element of R
878            lmDiag[j] = weightedJacobian[j][permutation[j]];
879            weightedJacobian[j][permutation[j]] = lmDir[j];
880        }
881
882        // solve the triangular system for z, if the system is
883        // singular, then obtain a least squares solution
884        int nSing = solvedCols;
885        for (int j = 0; j < solvedCols; ++j) {
886            if (lmDiag[j] == 0 && nSing == solvedCols) {
887                nSing = j;
888            }
889            if (nSing < solvedCols) {
890                work[j] = 0;
891            }
892        }
893        if (nSing > 0) {
894            for (int j = nSing - 1; j >= 0; --j) {
895                int pj = permutation[j];
896                double sum = 0;
897                for (int i = j + 1; i < nSing; ++i) {
898                    sum += weightedJacobian[i][pj] * work[i];
899                }
900                work[j] = (work[j] - sum) / lmDiag[j];
901            }
902        }
903
904        // permute the components of z back to components of lmDir
905        for (int j = 0; j < lmDir.length; ++j) {
906            lmDir[permutation[j]] = work[j];
907        }
908    }
909
910    /**
911     * Decompose a matrix A as A.P = Q.R using Householder transforms.
912     * <p>As suggested in the P. Lascaux and R. Theodor book
913     * <i>Analyse num&eacute;rique matricielle appliqu&eacute;e &agrave;
914     * l'art de l'ing&eacute;nieur</i> (Masson, 1986), instead of representing
915     * the Householder transforms with u<sub>k</sub> unit vectors such that:
916     * <pre>
917     * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
918     * </pre>
919     * we use <sub>k</sub> non-unit vectors such that:
920     * <pre>
921     * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
922     * </pre>
923     * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
924     * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
925     * them from the v<sub>k</sub> vectors would be costly.</p>
926     * <p>This decomposition handles rank deficient cases since the tranformations
927     * are performed in non-increasing columns norms order thanks to columns
928     * pivoting. The diagonal elements of the R matrix are therefore also in
929     * non-increasing absolute values order.</p>
930     *
931     * @param jacobian Weighted Jacobian matrix at the current point.
932     * @param solvedCols Number of solved point.
933     * @return data used in other methods of this class.
934     * @throws ConvergenceException if the decomposition cannot be performed.
935     */
936    private InternalData qrDecomposition(RealMatrix jacobian,
937                                         int solvedCols) throws ConvergenceException {
938        // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
939        // hence the multiplication by -1.
940        final double[][] weightedJacobian = jacobian.scalarMultiply(-1).getData();
941
942        final int nR = weightedJacobian.length;
943        final int nC = weightedJacobian[0].length;
944
945        final int[] permutation = new int[nC];
946        final double[] diagR = new double[nC];
947        final double[] jacNorm = new double[nC];
948        final double[] beta = new double[nC];
949
950        // initializations
951        for (int k = 0; k < nC; ++k) {
952            permutation[k] = k;
953            double norm2 = 0;
954            for (int i = 0; i < nR; ++i) {
955                double akk = weightedJacobian[i][k];
956                norm2 += akk * akk;
957            }
958            jacNorm[k] = JdkMath.sqrt(norm2);
959        }
960
961        // transform the matrix column after column
962        for (int k = 0; k < nC; ++k) {
963
964            // select the column with the greatest norm on active components
965            int nextColumn = -1;
966            double ak2 = Double.NEGATIVE_INFINITY;
967            for (int i = k; i < nC; ++i) {
968                double norm2 = 0;
969                for (int j = k; j < nR; ++j) {
970                    double aki = weightedJacobian[j][permutation[i]];
971                    norm2 += aki * aki;
972                }
973                if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
974                    throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
975                                                   nR, nC);
976                }
977                if (norm2 > ak2) {
978                    nextColumn = i;
979                    ak2        = norm2;
980                }
981            }
982            if (ak2 <= qrRankingThreshold) {
983                return new InternalData(weightedJacobian, permutation, k, diagR, jacNorm, beta);
984            }
985            int pk = permutation[nextColumn];
986            permutation[nextColumn] = permutation[k];
987            permutation[k] = pk;
988
989            // choose alpha such that Hk.u = alpha ek
990            double akk = weightedJacobian[k][pk];
991            double alpha = (akk > 0) ? -JdkMath.sqrt(ak2) : JdkMath.sqrt(ak2);
992            double betak = 1.0 / (ak2 - akk * alpha);
993            beta[pk] = betak;
994
995            // transform the current column
996            diagR[pk] = alpha;
997            weightedJacobian[k][pk] -= alpha;
998
999            // transform the remaining columns
1000            for (int dk = nC - 1 - k; dk > 0; --dk) {
1001                double gamma = 0;
1002                for (int j = k; j < nR; ++j) {
1003                    gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]];
1004                }
1005                gamma *= betak;
1006                for (int j = k; j < nR; ++j) {
1007                    weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk];
1008                }
1009            }
1010        }
1011
1012        return new InternalData(weightedJacobian, permutation, solvedCols, diagR, jacNorm, beta);
1013    }
1014
1015    /**
1016     * Compute the product Qt.y for some Q.R. decomposition.
1017     *
1018     * @param y vector to multiply (will be overwritten with the result)
1019     * @param internalData Data.
1020     */
1021    private void qTy(double[] y,
1022                     InternalData internalData) {
1023        final double[][] weightedJacobian = internalData.weightedJacobian;
1024        final int[] permutation = internalData.permutation;
1025        final double[] beta = internalData.beta;
1026
1027        final int nR = weightedJacobian.length;
1028        final int nC = weightedJacobian[0].length;
1029
1030        for (int k = 0; k < nC; ++k) {
1031            int pk = permutation[k];
1032            double gamma = 0;
1033            for (int i = k; i < nR; ++i) {
1034                gamma += weightedJacobian[i][pk] * y[i];
1035            }
1036            gamma *= beta[pk];
1037            for (int i = k; i < nR; ++i) {
1038                y[i] -= gamma * weightedJacobian[i][pk];
1039            }
1040        }
1041    }
1042}