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1   /*
2    * Licensed to the Apache Software Foundation (ASF) under one or more
3    * contributor license agreements.  See the NOTICE file distributed with
4    * this work for additional information regarding copyright ownership.
5    * The ASF licenses this file to You under the Apache License, Version 2.0
6    * (the "License"); you may not use this file except in compliance with
7    * the License.  You may obtain a copy of the License at
8    *
9    *      http://www.apache.org/licenses/LICENSE-2.0
10   *
11   * Unless required by applicable law or agreed to in writing, software
12   * distributed under the License is distributed on an "AS IS" BASIS,
13   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14   * See the License for the specific language governing permissions and
15   * limitations under the License.
16   */
17  package org.apache.commons.math3.optim.nonlinear.vector.jacobian;
18  
19  import java.util.Arrays;
20  import org.apache.commons.math3.exception.ConvergenceException;
21  import org.apache.commons.math3.exception.MathUnsupportedOperationException;
22  import org.apache.commons.math3.exception.util.LocalizedFormats;
23  import org.apache.commons.math3.optim.PointVectorValuePair;
24  import org.apache.commons.math3.optim.ConvergenceChecker;
25  import org.apache.commons.math3.linear.RealMatrix;
26  import org.apache.commons.math3.util.Precision;
27  import org.apache.commons.math3.util.FastMath;
28  
29  
30  /**
31   * This class solves a least-squares problem using the Levenberg-Marquardt
32   * algorithm.
33   * <br/>
34   * Constraints are not supported: the call to
35   * {@link #optimize(OptimizationData[]) optimize} will throw
36   * {@link MathUnsupportedOperationException} if bounds are passed to it.
37   *
38   * <p>This implementation <em>should</em> work even for over-determined systems
39   * (i.e. systems having more point than equations). Over-determined systems
40   * are solved by ignoring the point which have the smallest impact according
41   * to their jacobian column norm. Only the rank of the matrix and some loop bounds
42   * are changed to implement this.</p>
43   *
44   * <p>The resolution engine is a simple translation of the MINPACK <a
45   * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
46   * changes. The changes include the over-determined resolution, the use of
47   * inherited convergence checker and the Q.R. decomposition which has been
48   * rewritten following the algorithm described in the
49   * P. Lascaux and R. Theodor book <i>Analyse num&eacute;rique matricielle
50   * appliqu&eacute;e &agrave; l'art de l'ing&eacute;nieur</i>, Masson 1986.</p>
51   * <p>The authors of the original fortran version are:
52   * <ul>
53   * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
54   * <li>Burton S. Garbow</li>
55   * <li>Kenneth E. Hillstrom</li>
56   * <li>Jorge J. More</li>
57   * </ul>
58   * The redistribution policy for MINPACK is available <a
59   * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
60   * is reproduced below.</p>
61   *
62   * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
63   * <tr><td>
64   *    Minpack Copyright Notice (1999) University of Chicago.
65   *    All rights reserved
66   * </td></tr>
67   * <tr><td>
68   * Redistribution and use in source and binary forms, with or without
69   * modification, are permitted provided that the following conditions
70   * are met:
71   * <ol>
72   *  <li>Redistributions of source code must retain the above copyright
73   *      notice, this list of conditions and the following disclaimer.</li>
74   * <li>Redistributions in binary form must reproduce the above
75   *     copyright notice, this list of conditions and the following
76   *     disclaimer in the documentation and/or other materials provided
77   *     with the distribution.</li>
78   * <li>The end-user documentation included with the redistribution, if any,
79   *     must include the following acknowledgment:
80   *     <code>This product includes software developed by the University of
81   *           Chicago, as Operator of Argonne National Laboratory.</code>
82   *     Alternately, this acknowledgment may appear in the software itself,
83   *     if and wherever such third-party acknowledgments normally appear.</li>
84   * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
85   *     WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
86   *     UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
87   *     THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
88   *     IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
89   *     OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
90   *     OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
91   *     OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
92   *     USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
93   *     THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
94   *     DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
95   *     UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
96   *     BE CORRECTED.</strong></li>
97   * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
98   *     HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
99   *     ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
100  *     INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
101  *     ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
102  *     PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
103  *     SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
104  *     (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
105  *     EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
106  *     POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
107  * <ol></td></tr>
108  * </table>
109  *
110  * @since 2.0
111  * @deprecated All classes and interfaces in this package are deprecated.
112  * The optimizers that were provided here were moved to the
113  * {@link org.apache.commons.math3.fitting.leastsquares} package
114  * (cf. MATH-1008).
115  */
116 @Deprecated
117 public class LevenbergMarquardtOptimizer
118     extends AbstractLeastSquaresOptimizer {
119     /** Twice the "epsilon machine". */
120     private static final double TWO_EPS = 2 * Precision.EPSILON;
121     /** Number of solved point. */
122     private int solvedCols;
123     /** Diagonal elements of the R matrix in the Q.R. decomposition. */
124     private double[] diagR;
125     /** Norms of the columns of the jacobian matrix. */
126     private double[] jacNorm;
127     /** Coefficients of the Householder transforms vectors. */
128     private double[] beta;
129     /** Columns permutation array. */
130     private int[] permutation;
131     /** Rank of the jacobian matrix. */
132     private int rank;
133     /** Levenberg-Marquardt parameter. */
134     private double lmPar;
135     /** Parameters evolution direction associated with lmPar. */
136     private double[] lmDir;
137     /** Positive input variable used in determining the initial step bound. */
138     private final double initialStepBoundFactor;
139     /** Desired relative error in the sum of squares. */
140     private final double costRelativeTolerance;
141     /**  Desired relative error in the approximate solution parameters. */
142     private final double parRelativeTolerance;
143     /** Desired max cosine on the orthogonality between the function vector
144      * and the columns of the jacobian. */
145     private final double orthoTolerance;
146     /** Threshold for QR ranking. */
147     private final double qrRankingThreshold;
148     /** Weighted residuals. */
149     private double[] weightedResidual;
150     /** Weighted Jacobian. */
151     private double[][] weightedJacobian;
152 
153     /**
154      * Build an optimizer for least squares problems with default values
155      * for all the tuning parameters (see the {@link
156      * #LevenbergMarquardtOptimizer(double,double,double,double,double)
157      * other contructor}.
158      * The default values for the algorithm settings are:
159      * <ul>
160      *  <li>Initial step bound factor: 100</li>
161      *  <li>Cost relative tolerance: 1e-10</li>
162      *  <li>Parameters relative tolerance: 1e-10</li>
163      *  <li>Orthogonality tolerance: 1e-10</li>
164      *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
165      * </ul>
166      */
167     public LevenbergMarquardtOptimizer() {
168         this(100, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
169     }
170 
171     /**
172      * Constructor that allows the specification of a custom convergence
173      * checker.
174      * Note that all the usual convergence checks will be <em>disabled</em>.
175      * The default values for the algorithm settings are:
176      * <ul>
177      *  <li>Initial step bound factor: 100</li>
178      *  <li>Cost relative tolerance: 1e-10</li>
179      *  <li>Parameters relative tolerance: 1e-10</li>
180      *  <li>Orthogonality tolerance: 1e-10</li>
181      *  <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
182      * </ul>
183      *
184      * @param checker Convergence checker.
185      */
186     public LevenbergMarquardtOptimizer(ConvergenceChecker<PointVectorValuePair> checker) {
187         this(100, checker, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
188     }
189 
190     /**
191      * Constructor that allows the specification of a custom convergence
192      * checker, in addition to the standard ones.
193      *
194      * @param initialStepBoundFactor Positive input variable used in
195      * determining the initial step bound. This bound is set to the
196      * product of initialStepBoundFactor and the euclidean norm of
197      * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
198      * itself. In most cases factor should lie in the interval
199      * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
200      * @param checker Convergence checker.
201      * @param costRelativeTolerance Desired relative error in the sum of
202      * squares.
203      * @param parRelativeTolerance Desired relative error in the approximate
204      * solution parameters.
205      * @param orthoTolerance Desired max cosine on the orthogonality between
206      * the function vector and the columns of the Jacobian.
207      * @param threshold Desired threshold for QR ranking. If the squared norm
208      * of a column vector is smaller or equal to this threshold during QR
209      * decomposition, it is considered to be a zero vector and hence the rank
210      * of the matrix is reduced.
211      */
212     public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
213                                        ConvergenceChecker<PointVectorValuePair> checker,
214                                        double costRelativeTolerance,
215                                        double parRelativeTolerance,
216                                        double orthoTolerance,
217                                        double threshold) {
218         super(checker);
219         this.initialStepBoundFactor = initialStepBoundFactor;
220         this.costRelativeTolerance = costRelativeTolerance;
221         this.parRelativeTolerance = parRelativeTolerance;
222         this.orthoTolerance = orthoTolerance;
223         this.qrRankingThreshold = threshold;
224     }
225 
226     /**
227      * Build an optimizer for least squares problems with default values
228      * for some of the tuning parameters (see the {@link
229      * #LevenbergMarquardtOptimizer(double,double,double,double,double)
230      * other contructor}.
231      * The default values for the algorithm settings are:
232      * <ul>
233      *  <li>Initial step bound factor}: 100</li>
234      *  <li>QR ranking threshold}: {@link Precision#SAFE_MIN}</li>
235      * </ul>
236      *
237      * @param costRelativeTolerance Desired relative error in the sum of
238      * squares.
239      * @param parRelativeTolerance Desired relative error in the approximate
240      * solution parameters.
241      * @param orthoTolerance Desired max cosine on the orthogonality between
242      * the function vector and the columns of the Jacobian.
243      */
244     public LevenbergMarquardtOptimizer(double costRelativeTolerance,
245                                        double parRelativeTolerance,
246                                        double orthoTolerance) {
247         this(100,
248              costRelativeTolerance, parRelativeTolerance, orthoTolerance,
249              Precision.SAFE_MIN);
250     }
251 
252     /**
253      * The arguments control the behaviour of the default convergence checking
254      * procedure.
255      * Additional criteria can defined through the setting of a {@link
256      * ConvergenceChecker}.
257      *
258      * @param initialStepBoundFactor Positive input variable used in
259      * determining the initial step bound. This bound is set to the
260      * product of initialStepBoundFactor and the euclidean norm of
261      * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
262      * itself. In most cases factor should lie in the interval
263      * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
264      * @param costRelativeTolerance Desired relative error in the sum of
265      * squares.
266      * @param parRelativeTolerance Desired relative error in the approximate
267      * solution parameters.
268      * @param orthoTolerance Desired max cosine on the orthogonality between
269      * the function vector and the columns of the Jacobian.
270      * @param threshold Desired threshold for QR ranking. If the squared norm
271      * of a column vector is smaller or equal to this threshold during QR
272      * decomposition, it is considered to be a zero vector and hence the rank
273      * of the matrix is reduced.
274      */
275     public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
276                                        double costRelativeTolerance,
277                                        double parRelativeTolerance,
278                                        double orthoTolerance,
279                                        double threshold) {
280         super(null); // No custom convergence criterion.
281         this.initialStepBoundFactor = initialStepBoundFactor;
282         this.costRelativeTolerance = costRelativeTolerance;
283         this.parRelativeTolerance = parRelativeTolerance;
284         this.orthoTolerance = orthoTolerance;
285         this.qrRankingThreshold = threshold;
286     }
287 
288     /** {@inheritDoc} */
289     @Override
290     protected PointVectorValuePair doOptimize() {
291         checkParameters();
292 
293         final int nR = getTarget().length; // Number of observed data.
294         final double[] currentPoint = getStartPoint();
295         final int nC = currentPoint.length; // Number of parameters.
296 
297         // arrays shared with the other private methods
298         solvedCols  = FastMath.min(nR, nC);
299         diagR       = new double[nC];
300         jacNorm     = new double[nC];
301         beta        = new double[nC];
302         permutation = new int[nC];
303         lmDir       = new double[nC];
304 
305         // local point
306         double   delta   = 0;
307         double   xNorm   = 0;
308         double[] diag    = new double[nC];
309         double[] oldX    = new double[nC];
310         double[] oldRes  = new double[nR];
311         double[] oldObj  = new double[nR];
312         double[] qtf     = new double[nR];
313         double[] work1   = new double[nC];
314         double[] work2   = new double[nC];
315         double[] work3   = new double[nC];
316 
317         final RealMatrix weightMatrixSqrt = getWeightSquareRoot();
318 
319         // Evaluate the function at the starting point and calculate its norm.
320         double[] currentObjective = computeObjectiveValue(currentPoint);
321         double[] currentResiduals = computeResiduals(currentObjective);
322         PointVectorValuePair current = new PointVectorValuePair(currentPoint, currentObjective);
323         double currentCost = computeCost(currentResiduals);
324 
325         // Outer loop.
326         lmPar = 0;
327         boolean firstIteration = true;
328         final ConvergenceChecker<PointVectorValuePair> checker = getConvergenceChecker();
329         while (true) {
330             incrementIterationCount();
331 
332             final PointVectorValuePair previous = current;
333 
334             // QR decomposition of the jacobian matrix
335             qrDecomposition(computeWeightedJacobian(currentPoint));
336 
337             weightedResidual = weightMatrixSqrt.operate(currentResiduals);
338             for (int i = 0; i < nR; i++) {
339                 qtf[i] = weightedResidual[i];
340             }
341 
342             // compute Qt.res
343             qTy(qtf);
344 
345             // now we don't need Q anymore,
346             // so let jacobian contain the R matrix with its diagonal elements
347             for (int k = 0; k < solvedCols; ++k) {
348                 int pk = permutation[k];
349                 weightedJacobian[k][pk] = diagR[pk];
350             }
351 
352             if (firstIteration) {
353                 // scale the point according to the norms of the columns
354                 // of the initial jacobian
355                 xNorm = 0;
356                 for (int k = 0; k < nC; ++k) {
357                     double dk = jacNorm[k];
358                     if (dk == 0) {
359                         dk = 1.0;
360                     }
361                     double xk = dk * currentPoint[k];
362                     xNorm  += xk * xk;
363                     diag[k] = dk;
364                 }
365                 xNorm = FastMath.sqrt(xNorm);
366 
367                 // initialize the step bound delta
368                 delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
369             }
370 
371             // check orthogonality between function vector and jacobian columns
372             double maxCosine = 0;
373             if (currentCost != 0) {
374                 for (int j = 0; j < solvedCols; ++j) {
375                     int    pj = permutation[j];
376                     double s  = jacNorm[pj];
377                     if (s != 0) {
378                         double sum = 0;
379                         for (int i = 0; i <= j; ++i) {
380                             sum += weightedJacobian[i][pj] * qtf[i];
381                         }
382                         maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * currentCost));
383                     }
384                 }
385             }
386             if (maxCosine <= orthoTolerance) {
387                 // Convergence has been reached.
388                 setCost(currentCost);
389                 return current;
390             }
391 
392             // rescale if necessary
393             for (int j = 0; j < nC; ++j) {
394                 diag[j] = FastMath.max(diag[j], jacNorm[j]);
395             }
396 
397             // Inner loop.
398             for (double ratio = 0; ratio < 1.0e-4;) {
399 
400                 // save the state
401                 for (int j = 0; j < solvedCols; ++j) {
402                     int pj = permutation[j];
403                     oldX[pj] = currentPoint[pj];
404                 }
405                 final double previousCost = currentCost;
406                 double[] tmpVec = weightedResidual;
407                 weightedResidual = oldRes;
408                 oldRes    = tmpVec;
409                 tmpVec    = currentObjective;
410                 currentObjective = oldObj;
411                 oldObj    = tmpVec;
412 
413                 // determine the Levenberg-Marquardt parameter
414                 determineLMParameter(qtf, delta, diag, work1, work2, work3);
415 
416                 // compute the new point and the norm of the evolution direction
417                 double lmNorm = 0;
418                 for (int j = 0; j < solvedCols; ++j) {
419                     int pj = permutation[j];
420                     lmDir[pj] = -lmDir[pj];
421                     currentPoint[pj] = oldX[pj] + lmDir[pj];
422                     double s = diag[pj] * lmDir[pj];
423                     lmNorm  += s * s;
424                 }
425                 lmNorm = FastMath.sqrt(lmNorm);
426                 // on the first iteration, adjust the initial step bound.
427                 if (firstIteration) {
428                     delta = FastMath.min(delta, lmNorm);
429                 }
430 
431                 // Evaluate the function at x + p and calculate its norm.
432                 currentObjective = computeObjectiveValue(currentPoint);
433                 currentResiduals = computeResiduals(currentObjective);
434                 current = new PointVectorValuePair(currentPoint, currentObjective);
435                 currentCost = computeCost(currentResiduals);
436 
437                 // compute the scaled actual reduction
438                 double actRed = -1.0;
439                 if (0.1 * currentCost < previousCost) {
440                     double r = currentCost / previousCost;
441                     actRed = 1.0 - r * r;
442                 }
443 
444                 // compute the scaled predicted reduction
445                 // and the scaled directional derivative
446                 for (int j = 0; j < solvedCols; ++j) {
447                     int pj = permutation[j];
448                     double dirJ = lmDir[pj];
449                     work1[j] = 0;
450                     for (int i = 0; i <= j; ++i) {
451                         work1[i] += weightedJacobian[i][pj] * dirJ;
452                     }
453                 }
454                 double coeff1 = 0;
455                 for (int j = 0; j < solvedCols; ++j) {
456                     coeff1 += work1[j] * work1[j];
457                 }
458                 double pc2 = previousCost * previousCost;
459                 coeff1 /= pc2;
460                 double coeff2 = lmPar * lmNorm * lmNorm / pc2;
461                 double preRed = coeff1 + 2 * coeff2;
462                 double dirDer = -(coeff1 + coeff2);
463 
464                 // ratio of the actual to the predicted reduction
465                 ratio = (preRed == 0) ? 0 : (actRed / preRed);
466 
467                 // update the step bound
468                 if (ratio <= 0.25) {
469                     double tmp =
470                         (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
471                         if ((0.1 * currentCost >= previousCost) || (tmp < 0.1)) {
472                             tmp = 0.1;
473                         }
474                         delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
475                         lmPar /= tmp;
476                 } else if ((lmPar == 0) || (ratio >= 0.75)) {
477                     delta = 2 * lmNorm;
478                     lmPar *= 0.5;
479                 }
480 
481                 // test for successful iteration.
482                 if (ratio >= 1.0e-4) {
483                     // successful iteration, update the norm
484                     firstIteration = false;
485                     xNorm = 0;
486                     for (int k = 0; k < nC; ++k) {
487                         double xK = diag[k] * currentPoint[k];
488                         xNorm += xK * xK;
489                     }
490                     xNorm = FastMath.sqrt(xNorm);
491 
492                     // tests for convergence.
493                     if (checker != null && checker.converged(getIterations(), previous, current)) {
494                         setCost(currentCost);
495                         return current;
496                     }
497                 } else {
498                     // failed iteration, reset the previous values
499                     currentCost = previousCost;
500                     for (int j = 0; j < solvedCols; ++j) {
501                         int pj = permutation[j];
502                         currentPoint[pj] = oldX[pj];
503                     }
504                     tmpVec    = weightedResidual;
505                     weightedResidual = oldRes;
506                     oldRes    = tmpVec;
507                     tmpVec    = currentObjective;
508                     currentObjective = oldObj;
509                     oldObj    = tmpVec;
510                     // Reset "current" to previous values.
511                     current = new PointVectorValuePair(currentPoint, currentObjective);
512                 }
513 
514                 // Default convergence criteria.
515                 if ((FastMath.abs(actRed) <= costRelativeTolerance &&
516                      preRed <= costRelativeTolerance &&
517                      ratio <= 2.0) ||
518                     delta <= parRelativeTolerance * xNorm) {
519                     setCost(currentCost);
520                     return current;
521                 }
522 
523                 // tests for termination and stringent tolerances
524                 if (FastMath.abs(actRed) <= TWO_EPS &&
525                     preRed <= TWO_EPS &&
526                     ratio <= 2.0) {
527                     throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
528                                                    costRelativeTolerance);
529                 } else if (delta <= TWO_EPS * xNorm) {
530                     throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
531                                                    parRelativeTolerance);
532                 } else if (maxCosine <= TWO_EPS) {
533                     throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
534                                                    orthoTolerance);
535                 }
536             }
537         }
538     }
539 
540     /**
541      * Determine the Levenberg-Marquardt parameter.
542      * <p>This implementation is a translation in Java of the MINPACK
543      * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
544      * routine.</p>
545      * <p>This method sets the lmPar and lmDir attributes.</p>
546      * <p>The authors of the original fortran function are:</p>
547      * <ul>
548      *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
549      *   <li>Burton  S. Garbow</li>
550      *   <li>Kenneth E. Hillstrom</li>
551      *   <li>Jorge   J. More</li>
552      * </ul>
553      * <p>Luc Maisonobe did the Java translation.</p>
554      *
555      * @param qy array containing qTy
556      * @param delta upper bound on the euclidean norm of diagR * lmDir
557      * @param diag diagonal matrix
558      * @param work1 work array
559      * @param work2 work array
560      * @param work3 work array
561      */
562     private void determineLMParameter(double[] qy, double delta, double[] diag,
563                                       double[] work1, double[] work2, double[] work3) {
564         final int nC = weightedJacobian[0].length;
565 
566         // compute and store in x the gauss-newton direction, if the
567         // jacobian is rank-deficient, obtain a least squares solution
568         for (int j = 0; j < rank; ++j) {
569             lmDir[permutation[j]] = qy[j];
570         }
571         for (int j = rank; j < nC; ++j) {
572             lmDir[permutation[j]] = 0;
573         }
574         for (int k = rank - 1; k >= 0; --k) {
575             int pk = permutation[k];
576             double ypk = lmDir[pk] / diagR[pk];
577             for (int i = 0; i < k; ++i) {
578                 lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk];
579             }
580             lmDir[pk] = ypk;
581         }
582 
583         // evaluate the function at the origin, and test
584         // for acceptance of the Gauss-Newton direction
585         double dxNorm = 0;
586         for (int j = 0; j < solvedCols; ++j) {
587             int pj = permutation[j];
588             double s = diag[pj] * lmDir[pj];
589             work1[pj] = s;
590             dxNorm += s * s;
591         }
592         dxNorm = FastMath.sqrt(dxNorm);
593         double fp = dxNorm - delta;
594         if (fp <= 0.1 * delta) {
595             lmPar = 0;
596             return;
597         }
598 
599         // if the jacobian is not rank deficient, the Newton step provides
600         // a lower bound, parl, for the zero of the function,
601         // otherwise set this bound to zero
602         double sum2;
603         double parl = 0;
604         if (rank == solvedCols) {
605             for (int j = 0; j < solvedCols; ++j) {
606                 int pj = permutation[j];
607                 work1[pj] *= diag[pj] / dxNorm;
608             }
609             sum2 = 0;
610             for (int j = 0; j < solvedCols; ++j) {
611                 int pj = permutation[j];
612                 double sum = 0;
613                 for (int i = 0; i < j; ++i) {
614                     sum += weightedJacobian[i][pj] * work1[permutation[i]];
615                 }
616                 double s = (work1[pj] - sum) / diagR[pj];
617                 work1[pj] = s;
618                 sum2 += s * s;
619             }
620             parl = fp / (delta * sum2);
621         }
622 
623         // calculate an upper bound, paru, for the zero of the function
624         sum2 = 0;
625         for (int j = 0; j < solvedCols; ++j) {
626             int pj = permutation[j];
627             double sum = 0;
628             for (int i = 0; i <= j; ++i) {
629                 sum += weightedJacobian[i][pj] * qy[i];
630             }
631             sum /= diag[pj];
632             sum2 += sum * sum;
633         }
634         double gNorm = FastMath.sqrt(sum2);
635         double paru = gNorm / delta;
636         if (paru == 0) {
637             paru = Precision.SAFE_MIN / FastMath.min(delta, 0.1);
638         }
639 
640         // if the input par lies outside of the interval (parl,paru),
641         // set par to the closer endpoint
642         lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
643         if (lmPar == 0) {
644             lmPar = gNorm / dxNorm;
645         }
646 
647         for (int countdown = 10; countdown >= 0; --countdown) {
648 
649             // evaluate the function at the current value of lmPar
650             if (lmPar == 0) {
651                 lmPar = FastMath.max(Precision.SAFE_MIN, 0.001 * paru);
652             }
653             double sPar = FastMath.sqrt(lmPar);
654             for (int j = 0; j < solvedCols; ++j) {
655                 int pj = permutation[j];
656                 work1[pj] = sPar * diag[pj];
657             }
658             determineLMDirection(qy, work1, work2, work3);
659 
660             dxNorm = 0;
661             for (int j = 0; j < solvedCols; ++j) {
662                 int pj = permutation[j];
663                 double s = diag[pj] * lmDir[pj];
664                 work3[pj] = s;
665                 dxNorm += s * s;
666             }
667             dxNorm = FastMath.sqrt(dxNorm);
668             double previousFP = fp;
669             fp = dxNorm - delta;
670 
671             // if the function is small enough, accept the current value
672             // of lmPar, also test for the exceptional cases where parl is zero
673             if ((FastMath.abs(fp) <= 0.1 * delta) ||
674                     ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
675                 return;
676             }
677 
678             // compute the Newton correction
679             for (int j = 0; j < solvedCols; ++j) {
680                 int pj = permutation[j];
681                 work1[pj] = work3[pj] * diag[pj] / dxNorm;
682             }
683             for (int j = 0; j < solvedCols; ++j) {
684                 int pj = permutation[j];
685                 work1[pj] /= work2[j];
686                 double tmp = work1[pj];
687                 for (int i = j + 1; i < solvedCols; ++i) {
688                     work1[permutation[i]] -= weightedJacobian[i][pj] * tmp;
689                 }
690             }
691             sum2 = 0;
692             for (int j = 0; j < solvedCols; ++j) {
693                 double s = work1[permutation[j]];
694                 sum2 += s * s;
695             }
696             double correction = fp / (delta * sum2);
697 
698             // depending on the sign of the function, update parl or paru.
699             if (fp > 0) {
700                 parl = FastMath.max(parl, lmPar);
701             } else if (fp < 0) {
702                 paru = FastMath.min(paru, lmPar);
703             }
704 
705             // compute an improved estimate for lmPar
706             lmPar = FastMath.max(parl, lmPar + correction);
707 
708         }
709     }
710 
711     /**
712      * Solve a*x = b and d*x = 0 in the least squares sense.
713      * <p>This implementation is a translation in Java of the MINPACK
714      * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
715      * routine.</p>
716      * <p>This method sets the lmDir and lmDiag attributes.</p>
717      * <p>The authors of the original fortran function are:</p>
718      * <ul>
719      *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
720      *   <li>Burton  S. Garbow</li>
721      *   <li>Kenneth E. Hillstrom</li>
722      *   <li>Jorge   J. More</li>
723      * </ul>
724      * <p>Luc Maisonobe did the Java translation.</p>
725      *
726      * @param qy array containing qTy
727      * @param diag diagonal matrix
728      * @param lmDiag diagonal elements associated with lmDir
729      * @param work work array
730      */
731     private void determineLMDirection(double[] qy, double[] diag,
732                                       double[] lmDiag, double[] work) {
733 
734         // copy R and Qty to preserve input and initialize s
735         //  in particular, save the diagonal elements of R in lmDir
736         for (int j = 0; j < solvedCols; ++j) {
737             int pj = permutation[j];
738             for (int i = j + 1; i < solvedCols; ++i) {
739                 weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]];
740             }
741             lmDir[j] = diagR[pj];
742             work[j]  = qy[j];
743         }
744 
745         // eliminate the diagonal matrix d using a Givens rotation
746         for (int j = 0; j < solvedCols; ++j) {
747 
748             // prepare the row of d to be eliminated, locating the
749             // diagonal element using p from the Q.R. factorization
750             int pj = permutation[j];
751             double dpj = diag[pj];
752             if (dpj != 0) {
753                 Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
754             }
755             lmDiag[j] = dpj;
756 
757             //  the transformations to eliminate the row of d
758             // modify only a single element of Qty
759             // beyond the first n, which is initially zero.
760             double qtbpj = 0;
761             for (int k = j; k < solvedCols; ++k) {
762                 int pk = permutation[k];
763 
764                 // determine a Givens rotation which eliminates the
765                 // appropriate element in the current row of d
766                 if (lmDiag[k] != 0) {
767 
768                     final double sin;
769                     final double cos;
770                     double rkk = weightedJacobian[k][pk];
771                     if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
772                         final double cotan = rkk / lmDiag[k];
773                         sin   = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
774                         cos   = sin * cotan;
775                     } else {
776                         final double tan = lmDiag[k] / rkk;
777                         cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
778                         sin = cos * tan;
779                     }
780 
781                     // compute the modified diagonal element of R and
782                     // the modified element of (Qty,0)
783                     weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
784                     final double temp = cos * work[k] + sin * qtbpj;
785                     qtbpj = -sin * work[k] + cos * qtbpj;
786                     work[k] = temp;
787 
788                     // accumulate the tranformation in the row of s
789                     for (int i = k + 1; i < solvedCols; ++i) {
790                         double rik = weightedJacobian[i][pk];
791                         final double temp2 = cos * rik + sin * lmDiag[i];
792                         lmDiag[i] = -sin * rik + cos * lmDiag[i];
793                         weightedJacobian[i][pk] = temp2;
794                     }
795                 }
796             }
797 
798             // store the diagonal element of s and restore
799             // the corresponding diagonal element of R
800             lmDiag[j] = weightedJacobian[j][permutation[j]];
801             weightedJacobian[j][permutation[j]] = lmDir[j];
802         }
803 
804         // solve the triangular system for z, if the system is
805         // singular, then obtain a least squares solution
806         int nSing = solvedCols;
807         for (int j = 0; j < solvedCols; ++j) {
808             if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
809                 nSing = j;
810             }
811             if (nSing < solvedCols) {
812                 work[j] = 0;
813             }
814         }
815         if (nSing > 0) {
816             for (int j = nSing - 1; j >= 0; --j) {
817                 int pj = permutation[j];
818                 double sum = 0;
819                 for (int i = j + 1; i < nSing; ++i) {
820                     sum += weightedJacobian[i][pj] * work[i];
821                 }
822                 work[j] = (work[j] - sum) / lmDiag[j];
823             }
824         }
825 
826         // permute the components of z back to components of lmDir
827         for (int j = 0; j < lmDir.length; ++j) {
828             lmDir[permutation[j]] = work[j];
829         }
830     }
831 
832     /**
833      * Decompose a matrix A as A.P = Q.R using Householder transforms.
834      * <p>As suggested in the P. Lascaux and R. Theodor book
835      * <i>Analyse num&eacute;rique matricielle appliqu&eacute;e &agrave;
836      * l'art de l'ing&eacute;nieur</i> (Masson, 1986), instead of representing
837      * the Householder transforms with u<sub>k</sub> unit vectors such that:
838      * <pre>
839      * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
840      * </pre>
841      * we use <sub>k</sub> non-unit vectors such that:
842      * <pre>
843      * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
844      * </pre>
845      * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
846      * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
847      * them from the v<sub>k</sub> vectors would be costly.</p>
848      * <p>This decomposition handles rank deficient cases since the tranformations
849      * are performed in non-increasing columns norms order thanks to columns
850      * pivoting. The diagonal elements of the R matrix are therefore also in
851      * non-increasing absolute values order.</p>
852      *
853      * @param jacobian Weighted Jacobian matrix at the current point.
854      * @exception ConvergenceException if the decomposition cannot be performed
855      */
856     private void qrDecomposition(RealMatrix jacobian) throws ConvergenceException {
857         // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
858         // hence the multiplication by -1.
859         weightedJacobian = jacobian.scalarMultiply(-1).getData();
860 
861         final int nR = weightedJacobian.length;
862         final int nC = weightedJacobian[0].length;
863 
864         // initializations
865         for (int k = 0; k < nC; ++k) {
866             permutation[k] = k;
867             double norm2 = 0;
868             for (int i = 0; i < nR; ++i) {
869                 double akk = weightedJacobian[i][k];
870                 norm2 += akk * akk;
871             }
872             jacNorm[k] = FastMath.sqrt(norm2);
873         }
874 
875         // transform the matrix column after column
876         for (int k = 0; k < nC; ++k) {
877 
878             // select the column with the greatest norm on active components
879             int nextColumn = -1;
880             double ak2 = Double.NEGATIVE_INFINITY;
881             for (int i = k; i < nC; ++i) {
882                 double norm2 = 0;
883                 for (int j = k; j < nR; ++j) {
884                     double aki = weightedJacobian[j][permutation[i]];
885                     norm2 += aki * aki;
886                 }
887                 if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
888                     throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
889                                                    nR, nC);
890                 }
891                 if (norm2 > ak2) {
892                     nextColumn = i;
893                     ak2        = norm2;
894                 }
895             }
896             if (ak2 <= qrRankingThreshold) {
897                 rank = k;
898                 return;
899             }
900             int pk                  = permutation[nextColumn];
901             permutation[nextColumn] = permutation[k];
902             permutation[k]          = pk;
903 
904             // choose alpha such that Hk.u = alpha ek
905             double akk   = weightedJacobian[k][pk];
906             double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
907             double betak = 1.0 / (ak2 - akk * alpha);
908             beta[pk]     = betak;
909 
910             // transform the current column
911             diagR[pk]        = alpha;
912             weightedJacobian[k][pk] -= alpha;
913 
914             // transform the remaining columns
915             for (int dk = nC - 1 - k; dk > 0; --dk) {
916                 double gamma = 0;
917                 for (int j = k; j < nR; ++j) {
918                     gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]];
919                 }
920                 gamma *= betak;
921                 for (int j = k; j < nR; ++j) {
922                     weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk];
923                 }
924             }
925         }
926         rank = solvedCols;
927     }
928 
929     /**
930      * Compute the product Qt.y for some Q.R. decomposition.
931      *
932      * @param y vector to multiply (will be overwritten with the result)
933      */
934     private void qTy(double[] y) {
935         final int nR = weightedJacobian.length;
936         final int nC = weightedJacobian[0].length;
937 
938         for (int k = 0; k < nC; ++k) {
939             int pk = permutation[k];
940             double gamma = 0;
941             for (int i = k; i < nR; ++i) {
942                 gamma += weightedJacobian[i][pk] * y[i];
943             }
944             gamma *= beta[pk];
945             for (int i = k; i < nR; ++i) {
946                 y[i] -= gamma * weightedJacobian[i][pk];
947             }
948         }
949     }
950 
951     /**
952      * @throws MathUnsupportedOperationException if bounds were passed to the
953      * {@link #optimize(OptimizationData[]) optimize} method.
954      */
955     private void checkParameters() {
956         if (getLowerBound() != null ||
957             getUpperBound() != null) {
958             throw new MathUnsupportedOperationException(LocalizedFormats.CONSTRAINT);
959         }
960     }
961 }