001    /*
002     * Licensed to the Apache Software Foundation (ASF) under one or more
003     * contributor license agreements.  See the NOTICE file distributed with
004     * this work for additional information regarding copyright ownership.
005     * The ASF licenses this file to You under the Apache License, Version 2.0
006     * (the "License"); you may not use this file except in compliance with
007     * the License.  You may obtain a copy of the License at
008     *
009     *      http://www.apache.org/licenses/LICENSE-2.0
010     *
011     * Unless required by applicable law or agreed to in writing, software
012     * distributed under the License is distributed on an "AS IS" BASIS,
013     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014     * See the License for the specific language governing permissions and
015     * limitations under the License.
016     */
017    package org.apache.commons.math.estimation;
018    
019    import java.io.Serializable;
020    import java.util.Arrays;
021    
022    import org.apache.commons.math.exception.util.LocalizedFormats;
023    import org.apache.commons.math.util.FastMath;
024    
025    
026    /**
027     * This class solves a least squares problem.
028     *
029     * <p>This implementation <em>should</em> work even for over-determined systems
030     * (i.e. systems having more variables than equations). Over-determined systems
031     * are solved by ignoring the variables which have the smallest impact according
032     * to their jacobian column norm. Only the rank of the matrix and some loop bounds
033     * are changed to implement this.</p>
034     *
035     * <p>The resolution engine is a simple translation of the MINPACK <a
036     * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
037     * changes. The changes include the over-determined resolution and the Q.R.
038     * decomposition which has been rewritten following the algorithm described in the
039     * P. Lascaux and R. Theodor book <i>Analyse num&eacute;rique matricielle
040     * appliqu&eacute;e &agrave; l'art de l'ing&eacute;nieur</i>, Masson 1986.</p>
041     * <p>The authors of the original fortran version are:
042     * <ul>
043     * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
044     * <li>Burton S. Garbow</li>
045     * <li>Kenneth E. Hillstrom</li>
046     * <li>Jorge J. More</li>
047     * </ul>
048     * The redistribution policy for MINPACK is available <a
049     * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
050     * is reproduced below.</p>
051     *
052     * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
053     * <tr><td>
054     *    Minpack Copyright Notice (1999) University of Chicago.
055     *    All rights reserved
056     * </td></tr>
057     * <tr><td>
058     * Redistribution and use in source and binary forms, with or without
059     * modification, are permitted provided that the following conditions
060     * are met:
061     * <ol>
062     *  <li>Redistributions of source code must retain the above copyright
063     *      notice, this list of conditions and the following disclaimer.</li>
064     * <li>Redistributions in binary form must reproduce the above
065     *     copyright notice, this list of conditions and the following
066     *     disclaimer in the documentation and/or other materials provided
067     *     with the distribution.</li>
068     * <li>The end-user documentation included with the redistribution, if any,
069     *     must include the following acknowledgment:
070     *     <code>This product includes software developed by the University of
071     *           Chicago, as Operator of Argonne National Laboratory.</code>
072     *     Alternately, this acknowledgment may appear in the software itself,
073     *     if and wherever such third-party acknowledgments normally appear.</li>
074     * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
075     *     WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
076     *     UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
077     *     THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
078     *     IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
079     *     OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
080     *     OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
081     *     OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
082     *     USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
083     *     THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
084     *     DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
085     *     UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
086     *     BE CORRECTED.</strong></li>
087     * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
088     *     HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
089     *     ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
090     *     INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
091     *     ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
092     *     PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
093     *     SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
094     *     (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
095     *     EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
096     *     POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
097     * <ol></td></tr>
098     * </table>
099    
100     * @version $Revision: 990655 $ $Date: 2010-08-29 23:49:40 +0200 (dim. 29 ao??t 2010) $
101     * @since 1.2
102     * @deprecated as of 2.0, everything in package org.apache.commons.math.estimation has
103     * been deprecated and replaced by package org.apache.commons.math.optimization.general
104     *
105     */
106    @Deprecated
107    public class LevenbergMarquardtEstimator extends AbstractEstimator implements Serializable {
108    
109        /** Serializable version identifier */
110        private static final long serialVersionUID = -5705952631533171019L;
111    
112        /** Number of solved variables. */
113        private int solvedCols;
114    
115        /** Diagonal elements of the R matrix in the Q.R. decomposition. */
116        private double[] diagR;
117    
118        /** Norms of the columns of the jacobian matrix. */
119        private double[] jacNorm;
120    
121        /** Coefficients of the Householder transforms vectors. */
122        private double[] beta;
123    
124        /** Columns permutation array. */
125        private int[] permutation;
126    
127        /** Rank of the jacobian matrix. */
128        private int rank;
129    
130        /** Levenberg-Marquardt parameter. */
131        private double lmPar;
132    
133        /** Parameters evolution direction associated with lmPar. */
134        private double[] lmDir;
135    
136        /** Positive input variable used in determining the initial step bound. */
137        private double initialStepBoundFactor;
138    
139        /** Desired relative error in the sum of squares. */
140        private double costRelativeTolerance;
141    
142        /**  Desired relative error in the approximate solution parameters. */
143        private double parRelativeTolerance;
144    
145        /** Desired max cosine on the orthogonality between the function vector
146         * and the columns of the jacobian. */
147        private double orthoTolerance;
148    
149      /**
150       * Build an estimator for least squares problems.
151       * <p>The default values for the algorithm settings are:
152       *   <ul>
153       *    <li>{@link #setInitialStepBoundFactor initial step bound factor}: 100.0</li>
154       *    <li>{@link #setMaxCostEval maximal cost evaluations}: 1000</li>
155       *    <li>{@link #setCostRelativeTolerance cost relative tolerance}: 1.0e-10</li>
156       *    <li>{@link #setParRelativeTolerance parameters relative tolerance}: 1.0e-10</li>
157       *    <li>{@link #setOrthoTolerance orthogonality tolerance}: 1.0e-10</li>
158       *   </ul>
159       * </p>
160       */
161      public LevenbergMarquardtEstimator() {
162    
163        // set up the superclass with a default  max cost evaluations setting
164        setMaxCostEval(1000);
165    
166        // default values for the tuning parameters
167        setInitialStepBoundFactor(100.0);
168        setCostRelativeTolerance(1.0e-10);
169        setParRelativeTolerance(1.0e-10);
170        setOrthoTolerance(1.0e-10);
171    
172      }
173    
174      /**
175       * Set the positive input variable used in determining the initial step bound.
176       * This bound is set to the product of initialStepBoundFactor and the euclidean norm of diag*x if nonzero,
177       * or else to initialStepBoundFactor itself. In most cases factor should lie
178       * in the interval (0.1, 100.0). 100.0 is a generally recommended value
179       *
180       * @param initialStepBoundFactor initial step bound factor
181       * @see #estimate
182       */
183      public void setInitialStepBoundFactor(double initialStepBoundFactor) {
184        this.initialStepBoundFactor = initialStepBoundFactor;
185      }
186    
187      /**
188       * Set the desired relative error in the sum of squares.
189       *
190       * @param costRelativeTolerance desired relative error in the sum of squares
191       * @see #estimate
192       */
193      public void setCostRelativeTolerance(double costRelativeTolerance) {
194        this.costRelativeTolerance = costRelativeTolerance;
195      }
196    
197      /**
198       * Set the desired relative error in the approximate solution parameters.
199       *
200       * @param parRelativeTolerance desired relative error
201       * in the approximate solution parameters
202       * @see #estimate
203       */
204      public void setParRelativeTolerance(double parRelativeTolerance) {
205        this.parRelativeTolerance = parRelativeTolerance;
206      }
207    
208      /**
209       * Set the desired max cosine on the orthogonality.
210       *
211       * @param orthoTolerance desired max cosine on the orthogonality
212       * between the function vector and the columns of the jacobian
213       * @see #estimate
214       */
215      public void setOrthoTolerance(double orthoTolerance) {
216        this.orthoTolerance = orthoTolerance;
217      }
218    
219      /**
220       * Solve an estimation problem using the Levenberg-Marquardt algorithm.
221       * <p>The algorithm used is a modified Levenberg-Marquardt one, based
222       * on the MINPACK <a href="http://www.netlib.org/minpack/lmder.f">lmder</a>
223       * routine. The algorithm settings must have been set up before this method
224       * is called with the {@link #setInitialStepBoundFactor},
225       * {@link #setMaxCostEval}, {@link #setCostRelativeTolerance},
226       * {@link #setParRelativeTolerance} and {@link #setOrthoTolerance} methods.
227       * If these methods have not been called, the default values set up by the
228       * {@link #LevenbergMarquardtEstimator() constructor} will be used.</p>
229       * <p>The authors of the original fortran function are:</p>
230       * <ul>
231       *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
232       *   <li>Burton  S. Garbow</li>
233       *   <li>Kenneth E. Hillstrom</li>
234       *   <li>Jorge   J. More</li>
235       *   </ul>
236       * <p>Luc Maisonobe did the Java translation.</p>
237       *
238       * @param problem estimation problem to solve
239       * @exception EstimationException if convergence cannot be
240       * reached with the specified algorithm settings or if there are more variables
241       * than equations
242       * @see #setInitialStepBoundFactor
243       * @see #setCostRelativeTolerance
244       * @see #setParRelativeTolerance
245       * @see #setOrthoTolerance
246       */
247      @Override
248      public void estimate(EstimationProblem problem)
249        throws EstimationException {
250    
251        initializeEstimate(problem);
252    
253        // arrays shared with the other private methods
254        solvedCols  = FastMath.min(rows, cols);
255        diagR       = new double[cols];
256        jacNorm     = new double[cols];
257        beta        = new double[cols];
258        permutation = new int[cols];
259        lmDir       = new double[cols];
260    
261        // local variables
262        double   delta   = 0;
263        double   xNorm = 0;
264        double[] diag    = new double[cols];
265        double[] oldX    = new double[cols];
266        double[] oldRes  = new double[rows];
267        double[] work1   = new double[cols];
268        double[] work2   = new double[cols];
269        double[] work3   = new double[cols];
270    
271        // evaluate the function at the starting point and calculate its norm
272        updateResidualsAndCost();
273    
274        // outer loop
275        lmPar = 0;
276        boolean firstIteration = true;
277        while (true) {
278    
279          // compute the Q.R. decomposition of the jacobian matrix
280          updateJacobian();
281          qrDecomposition();
282    
283          // compute Qt.res
284          qTy(residuals);
285    
286          // now we don't need Q anymore,
287          // so let jacobian contain the R matrix with its diagonal elements
288          for (int k = 0; k < solvedCols; ++k) {
289            int pk = permutation[k];
290            jacobian[k * cols + pk] = diagR[pk];
291          }
292    
293          if (firstIteration) {
294    
295            // scale the variables according to the norms of the columns
296            // of the initial jacobian
297            xNorm = 0;
298            for (int k = 0; k < cols; ++k) {
299              double dk = jacNorm[k];
300              if (dk == 0) {
301                dk = 1.0;
302              }
303              double xk = dk * parameters[k].getEstimate();
304              xNorm  += xk * xk;
305              diag[k] = dk;
306            }
307            xNorm = FastMath.sqrt(xNorm);
308    
309            // initialize the step bound delta
310            delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
311    
312          }
313    
314          // check orthogonality between function vector and jacobian columns
315          double maxCosine = 0;
316          if (cost != 0) {
317            for (int j = 0; j < solvedCols; ++j) {
318              int    pj = permutation[j];
319              double s  = jacNorm[pj];
320              if (s != 0) {
321                double sum = 0;
322                int index = pj;
323                for (int i = 0; i <= j; ++i) {
324                  sum += jacobian[index] * residuals[i];
325                  index += cols;
326                }
327                maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * cost));
328              }
329            }
330          }
331          if (maxCosine <= orthoTolerance) {
332            return;
333          }
334    
335          // rescale if necessary
336          for (int j = 0; j < cols; ++j) {
337            diag[j] = FastMath.max(diag[j], jacNorm[j]);
338          }
339    
340          // inner loop
341          for (double ratio = 0; ratio < 1.0e-4;) {
342    
343            // save the state
344            for (int j = 0; j < solvedCols; ++j) {
345              int pj = permutation[j];
346              oldX[pj] = parameters[pj].getEstimate();
347            }
348            double previousCost = cost;
349            double[] tmpVec = residuals;
350            residuals = oldRes;
351            oldRes    = tmpVec;
352    
353            // determine the Levenberg-Marquardt parameter
354            determineLMParameter(oldRes, delta, diag, work1, work2, work3);
355    
356            // compute the new point and the norm of the evolution direction
357            double lmNorm = 0;
358            for (int j = 0; j < solvedCols; ++j) {
359              int pj = permutation[j];
360              lmDir[pj] = -lmDir[pj];
361              parameters[pj].setEstimate(oldX[pj] + lmDir[pj]);
362              double s = diag[pj] * lmDir[pj];
363              lmNorm  += s * s;
364            }
365            lmNorm = FastMath.sqrt(lmNorm);
366    
367            // on the first iteration, adjust the initial step bound.
368            if (firstIteration) {
369              delta = FastMath.min(delta, lmNorm);
370            }
371    
372            // evaluate the function at x + p and calculate its norm
373            updateResidualsAndCost();
374    
375            // compute the scaled actual reduction
376            double actRed = -1.0;
377            if (0.1 * cost < previousCost) {
378              double r = cost / previousCost;
379              actRed = 1.0 - r * r;
380            }
381    
382            // compute the scaled predicted reduction
383            // and the scaled directional derivative
384            for (int j = 0; j < solvedCols; ++j) {
385              int pj = permutation[j];
386              double dirJ = lmDir[pj];
387              work1[j] = 0;
388              int index = pj;
389              for (int i = 0; i <= j; ++i) {
390                work1[i] += jacobian[index] * dirJ;
391                index += cols;
392              }
393            }
394            double coeff1 = 0;
395            for (int j = 0; j < solvedCols; ++j) {
396             coeff1 += work1[j] * work1[j];
397            }
398            double pc2 = previousCost * previousCost;
399            coeff1 = coeff1 / pc2;
400            double coeff2 = lmPar * lmNorm * lmNorm / pc2;
401            double preRed = coeff1 + 2 * coeff2;
402            double dirDer = -(coeff1 + coeff2);
403    
404            // ratio of the actual to the predicted reduction
405            ratio = (preRed == 0) ? 0 : (actRed / preRed);
406    
407            // update the step bound
408            if (ratio <= 0.25) {
409              double tmp =
410                (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
411              if ((0.1 * cost >= previousCost) || (tmp < 0.1)) {
412                tmp = 0.1;
413              }
414              delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
415              lmPar /= tmp;
416            } else if ((lmPar == 0) || (ratio >= 0.75)) {
417              delta = 2 * lmNorm;
418              lmPar *= 0.5;
419            }
420    
421            // test for successful iteration.
422            if (ratio >= 1.0e-4) {
423              // successful iteration, update the norm
424              firstIteration = false;
425              xNorm = 0;
426              for (int k = 0; k < cols; ++k) {
427                double xK = diag[k] * parameters[k].getEstimate();
428                xNorm    += xK * xK;
429              }
430              xNorm = FastMath.sqrt(xNorm);
431            } else {
432              // failed iteration, reset the previous values
433              cost = previousCost;
434              for (int j = 0; j < solvedCols; ++j) {
435                int pj = permutation[j];
436                parameters[pj].setEstimate(oldX[pj]);
437              }
438              tmpVec    = residuals;
439              residuals = oldRes;
440              oldRes    = tmpVec;
441            }
442    
443            // tests for convergence.
444            if (((FastMath.abs(actRed) <= costRelativeTolerance) &&
445                 (preRed <= costRelativeTolerance) &&
446                 (ratio <= 2.0)) ||
447                 (delta <= parRelativeTolerance * xNorm)) {
448              return;
449            }
450    
451            // tests for termination and stringent tolerances
452            // (2.2204e-16 is the machine epsilon for IEEE754)
453            if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
454              throw new EstimationException("cost relative tolerance is too small ({0})," +
455                                            " no further reduction in the" +
456                                            " sum of squares is possible",
457                                            costRelativeTolerance);
458            } else if (delta <= 2.2204e-16 * xNorm) {
459              throw new EstimationException("parameters relative tolerance is too small" +
460                                            " ({0}), no further improvement in" +
461                                            " the approximate solution is possible",
462                                            parRelativeTolerance);
463            } else if (maxCosine <= 2.2204e-16)  {
464              throw new EstimationException("orthogonality tolerance is too small ({0})," +
465                                            " solution is orthogonal to the jacobian",
466                                            orthoTolerance);
467            }
468    
469          }
470    
471        }
472    
473      }
474    
475      /**
476       * Determine the Levenberg-Marquardt parameter.
477       * <p>This implementation is a translation in Java of the MINPACK
478       * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
479       * routine.</p>
480       * <p>This method sets the lmPar and lmDir attributes.</p>
481       * <p>The authors of the original fortran function are:</p>
482       * <ul>
483       *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
484       *   <li>Burton  S. Garbow</li>
485       *   <li>Kenneth E. Hillstrom</li>
486       *   <li>Jorge   J. More</li>
487       * </ul>
488       * <p>Luc Maisonobe did the Java translation.</p>
489       *
490       * @param qy array containing qTy
491       * @param delta upper bound on the euclidean norm of diagR * lmDir
492       * @param diag diagonal matrix
493       * @param work1 work array
494       * @param work2 work array
495       * @param work3 work array
496       */
497      private void determineLMParameter(double[] qy, double delta, double[] diag,
498                                        double[] work1, double[] work2, double[] work3) {
499    
500        // compute and store in x the gauss-newton direction, if the
501        // jacobian is rank-deficient, obtain a least squares solution
502        for (int j = 0; j < rank; ++j) {
503          lmDir[permutation[j]] = qy[j];
504        }
505        for (int j = rank; j < cols; ++j) {
506          lmDir[permutation[j]] = 0;
507        }
508        for (int k = rank - 1; k >= 0; --k) {
509          int pk = permutation[k];
510          double ypk = lmDir[pk] / diagR[pk];
511          int index = pk;
512          for (int i = 0; i < k; ++i) {
513            lmDir[permutation[i]] -= ypk * jacobian[index];
514            index += cols;
515          }
516          lmDir[pk] = ypk;
517        }
518    
519        // evaluate the function at the origin, and test
520        // for acceptance of the Gauss-Newton direction
521        double dxNorm = 0;
522        for (int j = 0; j < solvedCols; ++j) {
523          int pj = permutation[j];
524          double s = diag[pj] * lmDir[pj];
525          work1[pj] = s;
526          dxNorm += s * s;
527        }
528        dxNorm = FastMath.sqrt(dxNorm);
529        double fp = dxNorm - delta;
530        if (fp <= 0.1 * delta) {
531          lmPar = 0;
532          return;
533        }
534    
535        // if the jacobian is not rank deficient, the Newton step provides
536        // a lower bound, parl, for the zero of the function,
537        // otherwise set this bound to zero
538        double sum2;
539        double parl = 0;
540        if (rank == solvedCols) {
541          for (int j = 0; j < solvedCols; ++j) {
542            int pj = permutation[j];
543            work1[pj] *= diag[pj] / dxNorm;
544          }
545          sum2 = 0;
546          for (int j = 0; j < solvedCols; ++j) {
547            int pj = permutation[j];
548            double sum = 0;
549            int index = pj;
550            for (int i = 0; i < j; ++i) {
551              sum += jacobian[index] * work1[permutation[i]];
552              index += cols;
553            }
554            double s = (work1[pj] - sum) / diagR[pj];
555            work1[pj] = s;
556            sum2 += s * s;
557          }
558          parl = fp / (delta * sum2);
559        }
560    
561        // calculate an upper bound, paru, for the zero of the function
562        sum2 = 0;
563        for (int j = 0; j < solvedCols; ++j) {
564          int pj = permutation[j];
565          double sum = 0;
566          int index = pj;
567          for (int i = 0; i <= j; ++i) {
568            sum += jacobian[index] * qy[i];
569            index += cols;
570          }
571          sum /= diag[pj];
572          sum2 += sum * sum;
573        }
574        double gNorm = FastMath.sqrt(sum2);
575        double paru = gNorm / delta;
576        if (paru == 0) {
577          // 2.2251e-308 is the smallest positive real for IEE754
578          paru = 2.2251e-308 / FastMath.min(delta, 0.1);
579        }
580    
581        // if the input par lies outside of the interval (parl,paru),
582        // set par to the closer endpoint
583        lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
584        if (lmPar == 0) {
585          lmPar = gNorm / dxNorm;
586        }
587    
588        for (int countdown = 10; countdown >= 0; --countdown) {
589    
590          // evaluate the function at the current value of lmPar
591          if (lmPar == 0) {
592            lmPar = FastMath.max(2.2251e-308, 0.001 * paru);
593          }
594          double sPar = FastMath.sqrt(lmPar);
595          for (int j = 0; j < solvedCols; ++j) {
596            int pj = permutation[j];
597            work1[pj] = sPar * diag[pj];
598          }
599          determineLMDirection(qy, work1, work2, work3);
600    
601          dxNorm = 0;
602          for (int j = 0; j < solvedCols; ++j) {
603            int pj = permutation[j];
604            double s = diag[pj] * lmDir[pj];
605            work3[pj] = s;
606            dxNorm += s * s;
607          }
608          dxNorm = FastMath.sqrt(dxNorm);
609          double previousFP = fp;
610          fp = dxNorm - delta;
611    
612          // if the function is small enough, accept the current value
613          // of lmPar, also test for the exceptional cases where parl is zero
614          if ((FastMath.abs(fp) <= 0.1 * delta) ||
615              ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
616            return;
617          }
618    
619          // compute the Newton correction
620          for (int j = 0; j < solvedCols; ++j) {
621           int pj = permutation[j];
622            work1[pj] = work3[pj] * diag[pj] / dxNorm;
623          }
624          for (int j = 0; j < solvedCols; ++j) {
625            int pj = permutation[j];
626            work1[pj] /= work2[j];
627            double tmp = work1[pj];
628            for (int i = j + 1; i < solvedCols; ++i) {
629              work1[permutation[i]] -= jacobian[i * cols + pj] * tmp;
630            }
631          }
632          sum2 = 0;
633          for (int j = 0; j < solvedCols; ++j) {
634            double s = work1[permutation[j]];
635            sum2 += s * s;
636          }
637          double correction = fp / (delta * sum2);
638    
639          // depending on the sign of the function, update parl or paru.
640          if (fp > 0) {
641            parl = FastMath.max(parl, lmPar);
642          } else if (fp < 0) {
643            paru = FastMath.min(paru, lmPar);
644          }
645    
646          // compute an improved estimate for lmPar
647          lmPar = FastMath.max(parl, lmPar + correction);
648    
649        }
650      }
651    
652      /**
653       * Solve a*x = b and d*x = 0 in the least squares sense.
654       * <p>This implementation is a translation in Java of the MINPACK
655       * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
656       * routine.</p>
657       * <p>This method sets the lmDir and lmDiag attributes.</p>
658       * <p>The authors of the original fortran function are:</p>
659       * <ul>
660       *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
661       *   <li>Burton  S. Garbow</li>
662       *   <li>Kenneth E. Hillstrom</li>
663       *   <li>Jorge   J. More</li>
664       * </ul>
665       * <p>Luc Maisonobe did the Java translation.</p>
666       *
667       * @param qy array containing qTy
668       * @param diag diagonal matrix
669       * @param lmDiag diagonal elements associated with lmDir
670       * @param work work array
671       */
672      private void determineLMDirection(double[] qy, double[] diag,
673                                        double[] lmDiag, double[] work) {
674    
675        // copy R and Qty to preserve input and initialize s
676        //  in particular, save the diagonal elements of R in lmDir
677        for (int j = 0; j < solvedCols; ++j) {
678          int pj = permutation[j];
679          for (int i = j + 1; i < solvedCols; ++i) {
680            jacobian[i * cols + pj] = jacobian[j * cols + permutation[i]];
681          }
682          lmDir[j] = diagR[pj];
683          work[j]  = qy[j];
684        }
685    
686        // eliminate the diagonal matrix d using a Givens rotation
687        for (int j = 0; j < solvedCols; ++j) {
688    
689          // prepare the row of d to be eliminated, locating the
690          // diagonal element using p from the Q.R. factorization
691          int pj = permutation[j];
692          double dpj = diag[pj];
693          if (dpj != 0) {
694            Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
695          }
696          lmDiag[j] = dpj;
697    
698          //  the transformations to eliminate the row of d
699          // modify only a single element of Qty
700          // beyond the first n, which is initially zero.
701          double qtbpj = 0;
702          for (int k = j; k < solvedCols; ++k) {
703            int pk = permutation[k];
704    
705            // determine a Givens rotation which eliminates the
706            // appropriate element in the current row of d
707            if (lmDiag[k] != 0) {
708    
709              final double sin;
710              final double cos;
711              double rkk = jacobian[k * cols + pk];
712              if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
713                final double cotan = rkk / lmDiag[k];
714                sin   = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
715                cos   = sin * cotan;
716              } else {
717                final double tan = lmDiag[k] / rkk;
718                cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
719                sin = cos * tan;
720              }
721    
722              // compute the modified diagonal element of R and
723              // the modified element of (Qty,0)
724              jacobian[k * cols + pk] = cos * rkk + sin * lmDiag[k];
725              final double temp = cos * work[k] + sin * qtbpj;
726              qtbpj = -sin * work[k] + cos * qtbpj;
727              work[k] = temp;
728    
729              // accumulate the tranformation in the row of s
730              for (int i = k + 1; i < solvedCols; ++i) {
731                double rik = jacobian[i * cols + pk];
732                final double temp2 = cos * rik + sin * lmDiag[i];
733                lmDiag[i] = -sin * rik + cos * lmDiag[i];
734                jacobian[i * cols + pk] = temp2;
735              }
736    
737            }
738          }
739    
740          // store the diagonal element of s and restore
741          // the corresponding diagonal element of R
742          int index = j * cols + permutation[j];
743          lmDiag[j]       = jacobian[index];
744          jacobian[index] = lmDir[j];
745    
746        }
747    
748        // solve the triangular system for z, if the system is
749        // singular, then obtain a least squares solution
750        int nSing = solvedCols;
751        for (int j = 0; j < solvedCols; ++j) {
752          if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
753            nSing = j;
754          }
755          if (nSing < solvedCols) {
756            work[j] = 0;
757          }
758        }
759        if (nSing > 0) {
760          for (int j = nSing - 1; j >= 0; --j) {
761            int pj = permutation[j];
762            double sum = 0;
763            for (int i = j + 1; i < nSing; ++i) {
764              sum += jacobian[i * cols + pj] * work[i];
765            }
766            work[j] = (work[j] - sum) / lmDiag[j];
767          }
768        }
769    
770        // permute the components of z back to components of lmDir
771        for (int j = 0; j < lmDir.length; ++j) {
772          lmDir[permutation[j]] = work[j];
773        }
774    
775      }
776    
777      /**
778       * Decompose a matrix A as A.P = Q.R using Householder transforms.
779       * <p>As suggested in the P. Lascaux and R. Theodor book
780       * <i>Analyse num&eacute;rique matricielle appliqu&eacute;e &agrave;
781       * l'art de l'ing&eacute;nieur</i> (Masson, 1986), instead of representing
782       * the Householder transforms with u<sub>k</sub> unit vectors such that:
783       * <pre>
784       * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
785       * </pre>
786       * we use <sub>k</sub> non-unit vectors such that:
787       * <pre>
788       * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
789       * </pre>
790       * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
791       * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
792       * them from the v<sub>k</sub> vectors would be costly.</p>
793       * <p>This decomposition handles rank deficient cases since the tranformations
794       * are performed in non-increasing columns norms order thanks to columns
795       * pivoting. The diagonal elements of the R matrix are therefore also in
796       * non-increasing absolute values order.</p>
797       * @exception EstimationException if the decomposition cannot be performed
798       */
799      private void qrDecomposition() throws EstimationException {
800    
801        // initializations
802        for (int k = 0; k < cols; ++k) {
803          permutation[k] = k;
804          double norm2 = 0;
805          for (int index = k; index < jacobian.length; index += cols) {
806            double akk = jacobian[index];
807            norm2 += akk * akk;
808          }
809          jacNorm[k] = FastMath.sqrt(norm2);
810        }
811    
812        // transform the matrix column after column
813        for (int k = 0; k < cols; ++k) {
814    
815          // select the column with the greatest norm on active components
816          int nextColumn = -1;
817          double ak2 = Double.NEGATIVE_INFINITY;
818          for (int i = k; i < cols; ++i) {
819            double norm2 = 0;
820            int iDiag = k * cols + permutation[i];
821            for (int index = iDiag; index < jacobian.length; index += cols) {
822              double aki = jacobian[index];
823              norm2 += aki * aki;
824            }
825            if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
826                throw new EstimationException(
827                        LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
828                        rows, cols);
829            }
830            if (norm2 > ak2) {
831              nextColumn = i;
832              ak2        = norm2;
833            }
834          }
835          if (ak2 == 0) {
836            rank = k;
837            return;
838          }
839          int pk                  = permutation[nextColumn];
840          permutation[nextColumn] = permutation[k];
841          permutation[k]          = pk;
842    
843          // choose alpha such that Hk.u = alpha ek
844          int    kDiag = k * cols + pk;
845          double akk   = jacobian[kDiag];
846          double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
847          double betak = 1.0 / (ak2 - akk * alpha);
848          beta[pk]     = betak;
849    
850          // transform the current column
851          diagR[pk]        = alpha;
852          jacobian[kDiag] -= alpha;
853    
854          // transform the remaining columns
855          for (int dk = cols - 1 - k; dk > 0; --dk) {
856            int dkp = permutation[k + dk] - pk;
857            double gamma = 0;
858            for (int index = kDiag; index < jacobian.length; index += cols) {
859              gamma += jacobian[index] * jacobian[index + dkp];
860            }
861            gamma *= betak;
862            for (int index = kDiag; index < jacobian.length; index += cols) {
863              jacobian[index + dkp] -= gamma * jacobian[index];
864            }
865          }
866    
867        }
868    
869        rank = solvedCols;
870    
871      }
872    
873      /**
874       * Compute the product Qt.y for some Q.R. decomposition.
875       *
876       * @param y vector to multiply (will be overwritten with the result)
877       */
878      private void qTy(double[] y) {
879        for (int k = 0; k < cols; ++k) {
880          int pk = permutation[k];
881          int kDiag = k * cols + pk;
882          double gamma = 0;
883          int index = kDiag;
884          for (int i = k; i < rows; ++i) {
885            gamma += jacobian[index] * y[i];
886            index += cols;
887          }
888          gamma *= beta[pk];
889          index = kDiag;
890          for (int i = k; i < rows; ++i) {
891            y[i] -= gamma * jacobian[index];
892            index += cols;
893          }
894        }
895      }
896    
897    }