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    
023    /**
024     * This class solves a least squares problem.
025     *
026     * <p>This implementation <em>should</em> work even for over-determined systems
027     * (i.e. systems having more variables than equations). Over-determined systems
028     * are solved by ignoring the variables which have the smallest impact according
029     * to their jacobian column norm. Only the rank of the matrix and some loop bounds
030     * are changed to implement this.</p>
031     *
032     * <p>The resolution engine is a simple translation of the MINPACK <a
033     * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
034     * changes. The changes include the over-determined resolution and the Q.R.
035     * decomposition which has been rewritten following the algorithm described in the
036     * P. Lascaux and R. Theodor book <i>Analyse num&eacute;rique matricielle
037     * appliqu&eacute;e &agrave; l'art de l'ing&eacute;nieur</i>, Masson 1986.</p>
038     * <p>The authors of the original fortran version are:
039     * <ul>
040     * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
041     * <li>Burton S. Garbow</li>
042     * <li>Kenneth E. Hillstrom</li>
043     * <li>Jorge J. More</li>
044     * </ul>
045     * The redistribution policy for MINPACK is available <a
046     * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
047     * is reproduced below.</p>
048     *
049     * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
050     * <tr><td>
051     *    Minpack Copyright Notice (1999) University of Chicago.
052     *    All rights reserved
053     * </td></tr>
054     * <tr><td>
055     * Redistribution and use in source and binary forms, with or without
056     * modification, are permitted provided that the following conditions
057     * are met:
058     * <ol>
059     *  <li>Redistributions of source code must retain the above copyright
060     *      notice, this list of conditions and the following disclaimer.</li>
061     * <li>Redistributions in binary form must reproduce the above
062     *     copyright notice, this list of conditions and the following
063     *     disclaimer in the documentation and/or other materials provided
064     *     with the distribution.</li>
065     * <li>The end-user documentation included with the redistribution, if any,
066     *     must include the following acknowledgment:
067     *     <code>This product includes software developed by the University of
068     *           Chicago, as Operator of Argonne National Laboratory.</code>
069     *     Alternately, this acknowledgment may appear in the software itself,
070     *     if and wherever such third-party acknowledgments normally appear.</li>
071     * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
072     *     WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
073     *     UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
074     *     THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
075     *     IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
076     *     OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
077     *     OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
078     *     OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
079     *     USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
080     *     THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
081     *     DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
082     *     UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
083     *     BE CORRECTED.</strong></li>
084     * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
085     *     HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
086     *     ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
087     *     INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
088     *     ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
089     *     PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
090     *     SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
091     *     (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
092     *     EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
093     *     POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
094     * <ol></td></tr>
095     * </table>
096    
097     * @version $Revision: 825919 $ $Date: 2009-10-16 10:51:55 -0400 (Fri, 16 Oct 2009) $
098     * @since 1.2
099     * @deprecated as of 2.0, everything in package org.apache.commons.math.estimation has
100     * been deprecated and replaced by package org.apache.commons.math.optimization.general
101     *
102     */
103    @Deprecated
104    public class LevenbergMarquardtEstimator extends AbstractEstimator implements Serializable {
105    
106        /** Serializable version identifier */
107        private static final long serialVersionUID = -5705952631533171019L;
108    
109        /** Number of solved variables. */
110        private int solvedCols;
111    
112        /** Diagonal elements of the R matrix in the Q.R. decomposition. */
113        private double[] diagR;
114    
115        /** Norms of the columns of the jacobian matrix. */
116        private double[] jacNorm;
117    
118        /** Coefficients of the Householder transforms vectors. */
119        private double[] beta;
120    
121        /** Columns permutation array. */
122        private int[] permutation;
123    
124        /** Rank of the jacobian matrix. */
125        private int rank;
126    
127        /** Levenberg-Marquardt parameter. */
128        private double lmPar;
129    
130        /** Parameters evolution direction associated with lmPar. */
131        private double[] lmDir;
132    
133        /** Positive input variable used in determining the initial step bound. */
134        private double initialStepBoundFactor;
135    
136        /** Desired relative error in the sum of squares. */
137        private double costRelativeTolerance;
138    
139        /**  Desired relative error in the approximate solution parameters. */
140        private double parRelativeTolerance;
141    
142        /** Desired max cosine on the orthogonality between the function vector
143         * and the columns of the jacobian. */
144        private double orthoTolerance;
145    
146      /**
147       * Build an estimator for least squares problems.
148       * <p>The default values for the algorithm settings are:
149       *   <ul>
150       *    <li>{@link #setInitialStepBoundFactor initial step bound factor}: 100.0</li>
151       *    <li>{@link #setMaxCostEval maximal cost evaluations}: 1000</li>
152       *    <li>{@link #setCostRelativeTolerance cost relative tolerance}: 1.0e-10</li>
153       *    <li>{@link #setParRelativeTolerance parameters relative tolerance}: 1.0e-10</li>
154       *    <li>{@link #setOrthoTolerance orthogonality tolerance}: 1.0e-10</li>
155       *   </ul>
156       * </p>
157       */
158      public LevenbergMarquardtEstimator() {
159    
160        // set up the superclass with a default  max cost evaluations setting
161        setMaxCostEval(1000);
162    
163        // default values for the tuning parameters
164        setInitialStepBoundFactor(100.0);
165        setCostRelativeTolerance(1.0e-10);
166        setParRelativeTolerance(1.0e-10);
167        setOrthoTolerance(1.0e-10);
168    
169      }
170    
171      /**
172       * Set the positive input variable used in determining the initial step bound.
173       * This bound is set to the product of initialStepBoundFactor and the euclidean norm of diag*x if nonzero,
174       * or else to initialStepBoundFactor itself. In most cases factor should lie
175       * in the interval (0.1, 100.0). 100.0 is a generally recommended value
176       *
177       * @param initialStepBoundFactor initial step bound factor
178       * @see #estimate
179       */
180      public void setInitialStepBoundFactor(double initialStepBoundFactor) {
181        this.initialStepBoundFactor = initialStepBoundFactor;
182      }
183    
184      /**
185       * Set the desired relative error in the sum of squares.
186       *
187       * @param costRelativeTolerance desired relative error in the sum of squares
188       * @see #estimate
189       */
190      public void setCostRelativeTolerance(double costRelativeTolerance) {
191        this.costRelativeTolerance = costRelativeTolerance;
192      }
193    
194      /**
195       * Set the desired relative error in the approximate solution parameters.
196       *
197       * @param parRelativeTolerance desired relative error
198       * in the approximate solution parameters
199       * @see #estimate
200       */
201      public void setParRelativeTolerance(double parRelativeTolerance) {
202        this.parRelativeTolerance = parRelativeTolerance;
203      }
204    
205      /**
206       * Set the desired max cosine on the orthogonality.
207       *
208       * @param orthoTolerance desired max cosine on the orthogonality
209       * between the function vector and the columns of the jacobian
210       * @see #estimate
211       */
212      public void setOrthoTolerance(double orthoTolerance) {
213        this.orthoTolerance = orthoTolerance;
214      }
215    
216      /**
217       * Solve an estimation problem using the Levenberg-Marquardt algorithm.
218       * <p>The algorithm used is a modified Levenberg-Marquardt one, based
219       * on the MINPACK <a href="http://www.netlib.org/minpack/lmder.f">lmder</a>
220       * routine. The algorithm settings must have been set up before this method
221       * is called with the {@link #setInitialStepBoundFactor},
222       * {@link #setMaxCostEval}, {@link #setCostRelativeTolerance},
223       * {@link #setParRelativeTolerance} and {@link #setOrthoTolerance} methods.
224       * If these methods have not been called, the default values set up by the
225       * {@link #LevenbergMarquardtEstimator() constructor} will be used.</p>
226       * <p>The authors of the original fortran function are:</p>
227       * <ul>
228       *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
229       *   <li>Burton  S. Garbow</li>
230       *   <li>Kenneth E. Hillstrom</li>
231       *   <li>Jorge   J. More</li>
232       *   </ul>
233       * <p>Luc Maisonobe did the Java translation.</p>
234       *
235       * @param problem estimation problem to solve
236       * @exception EstimationException if convergence cannot be
237       * reached with the specified algorithm settings or if there are more variables
238       * than equations
239       * @see #setInitialStepBoundFactor
240       * @see #setCostRelativeTolerance
241       * @see #setParRelativeTolerance
242       * @see #setOrthoTolerance
243       */
244      @Override
245      public void estimate(EstimationProblem problem)
246        throws EstimationException {
247    
248        initializeEstimate(problem);
249    
250        // arrays shared with the other private methods
251        solvedCols  = Math.min(rows, cols);
252        diagR       = new double[cols];
253        jacNorm     = new double[cols];
254        beta        = new double[cols];
255        permutation = new int[cols];
256        lmDir       = new double[cols];
257    
258        // local variables
259        double   delta   = 0;
260        double   xNorm = 0;
261        double[] diag    = new double[cols];
262        double[] oldX    = new double[cols];
263        double[] oldRes  = new double[rows];
264        double[] work1   = new double[cols];
265        double[] work2   = new double[cols];
266        double[] work3   = new double[cols];
267    
268        // evaluate the function at the starting point and calculate its norm
269        updateResidualsAndCost();
270    
271        // outer loop
272        lmPar = 0;
273        boolean firstIteration = true;
274        while (true) {
275    
276          // compute the Q.R. decomposition of the jacobian matrix
277          updateJacobian();
278          qrDecomposition();
279    
280          // compute Qt.res
281          qTy(residuals);
282    
283          // now we don't need Q anymore,
284          // so let jacobian contain the R matrix with its diagonal elements
285          for (int k = 0; k < solvedCols; ++k) {
286            int pk = permutation[k];
287            jacobian[k * cols + pk] = diagR[pk];
288          }
289    
290          if (firstIteration) {
291    
292            // scale the variables according to the norms of the columns
293            // of the initial jacobian
294            xNorm = 0;
295            for (int k = 0; k < cols; ++k) {
296              double dk = jacNorm[k];
297              if (dk == 0) {
298                dk = 1.0;
299              }
300              double xk = dk * parameters[k].getEstimate();
301              xNorm  += xk * xk;
302              diag[k] = dk;
303            }
304            xNorm = Math.sqrt(xNorm);
305    
306            // initialize the step bound delta
307            delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
308    
309          }
310    
311          // check orthogonality between function vector and jacobian columns
312          double maxCosine = 0;
313          if (cost != 0) {
314            for (int j = 0; j < solvedCols; ++j) {
315              int    pj = permutation[j];
316              double s  = jacNorm[pj];
317              if (s != 0) {
318                double sum = 0;
319                int index = pj;
320                for (int i = 0; i <= j; ++i) {
321                  sum += jacobian[index] * residuals[i];
322                  index += cols;
323                }
324                maxCosine = Math.max(maxCosine, Math.abs(sum) / (s * cost));
325              }
326            }
327          }
328          if (maxCosine <= orthoTolerance) {
329            return;
330          }
331    
332          // rescale if necessary
333          for (int j = 0; j < cols; ++j) {
334            diag[j] = Math.max(diag[j], jacNorm[j]);
335          }
336    
337          // inner loop
338          for (double ratio = 0; ratio < 1.0e-4;) {
339    
340            // save the state
341            for (int j = 0; j < solvedCols; ++j) {
342              int pj = permutation[j];
343              oldX[pj] = parameters[pj].getEstimate();
344            }
345            double previousCost = cost;
346            double[] tmpVec = residuals;
347            residuals = oldRes;
348            oldRes    = tmpVec;
349    
350            // determine the Levenberg-Marquardt parameter
351            determineLMParameter(oldRes, delta, diag, work1, work2, work3);
352    
353            // compute the new point and the norm of the evolution direction
354            double lmNorm = 0;
355            for (int j = 0; j < solvedCols; ++j) {
356              int pj = permutation[j];
357              lmDir[pj] = -lmDir[pj];
358              parameters[pj].setEstimate(oldX[pj] + lmDir[pj]);
359              double s = diag[pj] * lmDir[pj];
360              lmNorm  += s * s;
361            }
362            lmNorm = Math.sqrt(lmNorm);
363    
364            // on the first iteration, adjust the initial step bound.
365            if (firstIteration) {
366              delta = Math.min(delta, lmNorm);
367            }
368    
369            // evaluate the function at x + p and calculate its norm
370            updateResidualsAndCost();
371    
372            // compute the scaled actual reduction
373            double actRed = -1.0;
374            if (0.1 * cost < previousCost) {
375              double r = cost / previousCost;
376              actRed = 1.0 - r * r;
377            }
378    
379            // compute the scaled predicted reduction
380            // and the scaled directional derivative
381            for (int j = 0; j < solvedCols; ++j) {
382              int pj = permutation[j];
383              double dirJ = lmDir[pj];
384              work1[j] = 0;
385              int index = pj;
386              for (int i = 0; i <= j; ++i) {
387                work1[i] += jacobian[index] * dirJ;
388                index += cols;
389              }
390            }
391            double coeff1 = 0;
392            for (int j = 0; j < solvedCols; ++j) {
393             coeff1 += work1[j] * work1[j];
394            }
395            double pc2 = previousCost * previousCost;
396            coeff1 = coeff1 / pc2;
397            double coeff2 = lmPar * lmNorm * lmNorm / pc2;
398            double preRed = coeff1 + 2 * coeff2;
399            double dirDer = -(coeff1 + coeff2);
400    
401            // ratio of the actual to the predicted reduction
402            ratio = (preRed == 0) ? 0 : (actRed / preRed);
403    
404            // update the step bound
405            if (ratio <= 0.25) {
406              double tmp =
407                (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
408              if ((0.1 * cost >= previousCost) || (tmp < 0.1)) {
409                tmp = 0.1;
410              }
411              delta = tmp * Math.min(delta, 10.0 * lmNorm);
412              lmPar /= tmp;
413            } else if ((lmPar == 0) || (ratio >= 0.75)) {
414              delta = 2 * lmNorm;
415              lmPar *= 0.5;
416            }
417    
418            // test for successful iteration.
419            if (ratio >= 1.0e-4) {
420              // successful iteration, update the norm
421              firstIteration = false;
422              xNorm = 0;
423              for (int k = 0; k < cols; ++k) {
424                double xK = diag[k] * parameters[k].getEstimate();
425                xNorm    += xK * xK;
426              }
427              xNorm = Math.sqrt(xNorm);
428            } else {
429              // failed iteration, reset the previous values
430              cost = previousCost;
431              for (int j = 0; j < solvedCols; ++j) {
432                int pj = permutation[j];
433                parameters[pj].setEstimate(oldX[pj]);
434              }
435              tmpVec    = residuals;
436              residuals = oldRes;
437              oldRes    = tmpVec;
438            }
439    
440            // tests for convergence.
441            if (((Math.abs(actRed) <= costRelativeTolerance) &&
442                 (preRed <= costRelativeTolerance) &&
443                 (ratio <= 2.0)) ||
444                 (delta <= parRelativeTolerance * xNorm)) {
445              return;
446            }
447    
448            // tests for termination and stringent tolerances
449            // (2.2204e-16 is the machine epsilon for IEEE754)
450            if ((Math.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
451              throw new EstimationException("cost relative tolerance is too small ({0})," +
452                                            " no further reduction in the" +
453                                            " sum of squares is possible",
454                                            costRelativeTolerance);
455            } else if (delta <= 2.2204e-16 * xNorm) {
456              throw new EstimationException("parameters relative tolerance is too small" +
457                                            " ({0}), no further improvement in" +
458                                            " the approximate solution is possible",
459                                            parRelativeTolerance);
460            } else if (maxCosine <= 2.2204e-16)  {
461              throw new EstimationException("orthogonality tolerance is too small ({0})," +
462                                            " solution is orthogonal to the jacobian",
463                                            orthoTolerance);
464            }
465    
466          }
467    
468        }
469    
470      }
471    
472      /**
473       * Determine the Levenberg-Marquardt parameter.
474       * <p>This implementation is a translation in Java of the MINPACK
475       * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
476       * routine.</p>
477       * <p>This method sets the lmPar and lmDir attributes.</p>
478       * <p>The authors of the original fortran function are:</p>
479       * <ul>
480       *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
481       *   <li>Burton  S. Garbow</li>
482       *   <li>Kenneth E. Hillstrom</li>
483       *   <li>Jorge   J. More</li>
484       * </ul>
485       * <p>Luc Maisonobe did the Java translation.</p>
486       *
487       * @param qy array containing qTy
488       * @param delta upper bound on the euclidean norm of diagR * lmDir
489       * @param diag diagonal matrix
490       * @param work1 work array
491       * @param work2 work array
492       * @param work3 work array
493       */
494      private void determineLMParameter(double[] qy, double delta, double[] diag,
495                                        double[] work1, double[] work2, double[] work3) {
496    
497        // compute and store in x the gauss-newton direction, if the
498        // jacobian is rank-deficient, obtain a least squares solution
499        for (int j = 0; j < rank; ++j) {
500          lmDir[permutation[j]] = qy[j];
501        }
502        for (int j = rank; j < cols; ++j) {
503          lmDir[permutation[j]] = 0;
504        }
505        for (int k = rank - 1; k >= 0; --k) {
506          int pk = permutation[k];
507          double ypk = lmDir[pk] / diagR[pk];
508          int index = pk;
509          for (int i = 0; i < k; ++i) {
510            lmDir[permutation[i]] -= ypk * jacobian[index];
511            index += cols;
512          }
513          lmDir[pk] = ypk;
514        }
515    
516        // evaluate the function at the origin, and test
517        // for acceptance of the Gauss-Newton direction
518        double dxNorm = 0;
519        for (int j = 0; j < solvedCols; ++j) {
520          int pj = permutation[j];
521          double s = diag[pj] * lmDir[pj];
522          work1[pj] = s;
523          dxNorm += s * s;
524        }
525        dxNorm = Math.sqrt(dxNorm);
526        double fp = dxNorm - delta;
527        if (fp <= 0.1 * delta) {
528          lmPar = 0;
529          return;
530        }
531    
532        // if the jacobian is not rank deficient, the Newton step provides
533        // a lower bound, parl, for the zero of the function,
534        // otherwise set this bound to zero
535        double sum2;
536        double parl = 0;
537        if (rank == solvedCols) {
538          for (int j = 0; j < solvedCols; ++j) {
539            int pj = permutation[j];
540            work1[pj] *= diag[pj] / dxNorm;
541          }
542          sum2 = 0;
543          for (int j = 0; j < solvedCols; ++j) {
544            int pj = permutation[j];
545            double sum = 0;
546            int index = pj;
547            for (int i = 0; i < j; ++i) {
548              sum += jacobian[index] * work1[permutation[i]];
549              index += cols;
550            }
551            double s = (work1[pj] - sum) / diagR[pj];
552            work1[pj] = s;
553            sum2 += s * s;
554          }
555          parl = fp / (delta * sum2);
556        }
557    
558        // calculate an upper bound, paru, for the zero of the function
559        sum2 = 0;
560        for (int j = 0; j < solvedCols; ++j) {
561          int pj = permutation[j];
562          double sum = 0;
563          int index = pj;
564          for (int i = 0; i <= j; ++i) {
565            sum += jacobian[index] * qy[i];
566            index += cols;
567          }
568          sum /= diag[pj];
569          sum2 += sum * sum;
570        }
571        double gNorm = Math.sqrt(sum2);
572        double paru = gNorm / delta;
573        if (paru == 0) {
574          // 2.2251e-308 is the smallest positive real for IEE754
575          paru = 2.2251e-308 / Math.min(delta, 0.1);
576        }
577    
578        // if the input par lies outside of the interval (parl,paru),
579        // set par to the closer endpoint
580        lmPar = Math.min(paru, Math.max(lmPar, parl));
581        if (lmPar == 0) {
582          lmPar = gNorm / dxNorm;
583        }
584    
585        for (int countdown = 10; countdown >= 0; --countdown) {
586    
587          // evaluate the function at the current value of lmPar
588          if (lmPar == 0) {
589            lmPar = Math.max(2.2251e-308, 0.001 * paru);
590          }
591          double sPar = Math.sqrt(lmPar);
592          for (int j = 0; j < solvedCols; ++j) {
593            int pj = permutation[j];
594            work1[pj] = sPar * diag[pj];
595          }
596          determineLMDirection(qy, work1, work2, work3);
597    
598          dxNorm = 0;
599          for (int j = 0; j < solvedCols; ++j) {
600            int pj = permutation[j];
601            double s = diag[pj] * lmDir[pj];
602            work3[pj] = s;
603            dxNorm += s * s;
604          }
605          dxNorm = Math.sqrt(dxNorm);
606          double previousFP = fp;
607          fp = dxNorm - delta;
608    
609          // if the function is small enough, accept the current value
610          // of lmPar, also test for the exceptional cases where parl is zero
611          if ((Math.abs(fp) <= 0.1 * delta) ||
612              ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
613            return;
614          }
615    
616          // compute the Newton correction
617          for (int j = 0; j < solvedCols; ++j) {
618           int pj = permutation[j];
619            work1[pj] = work3[pj] * diag[pj] / dxNorm;
620          }
621          for (int j = 0; j < solvedCols; ++j) {
622            int pj = permutation[j];
623            work1[pj] /= work2[j];
624            double tmp = work1[pj];
625            for (int i = j + 1; i < solvedCols; ++i) {
626              work1[permutation[i]] -= jacobian[i * cols + pj] * tmp;
627            }
628          }
629          sum2 = 0;
630          for (int j = 0; j < solvedCols; ++j) {
631            double s = work1[permutation[j]];
632            sum2 += s * s;
633          }
634          double correction = fp / (delta * sum2);
635    
636          // depending on the sign of the function, update parl or paru.
637          if (fp > 0) {
638            parl = Math.max(parl, lmPar);
639          } else if (fp < 0) {
640            paru = Math.min(paru, lmPar);
641          }
642    
643          // compute an improved estimate for lmPar
644          lmPar = Math.max(parl, lmPar + correction);
645    
646        }
647      }
648    
649      /**
650       * Solve a*x = b and d*x = 0 in the least squares sense.
651       * <p>This implementation is a translation in Java of the MINPACK
652       * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
653       * routine.</p>
654       * <p>This method sets the lmDir and lmDiag attributes.</p>
655       * <p>The authors of the original fortran function are:</p>
656       * <ul>
657       *   <li>Argonne National Laboratory. MINPACK project. March 1980</li>
658       *   <li>Burton  S. Garbow</li>
659       *   <li>Kenneth E. Hillstrom</li>
660       *   <li>Jorge   J. More</li>
661       * </ul>
662       * <p>Luc Maisonobe did the Java translation.</p>
663       *
664       * @param qy array containing qTy
665       * @param diag diagonal matrix
666       * @param lmDiag diagonal elements associated with lmDir
667       * @param work work array
668       */
669      private void determineLMDirection(double[] qy, double[] diag,
670                                        double[] lmDiag, double[] work) {
671    
672        // copy R and Qty to preserve input and initialize s
673        //  in particular, save the diagonal elements of R in lmDir
674        for (int j = 0; j < solvedCols; ++j) {
675          int pj = permutation[j];
676          for (int i = j + 1; i < solvedCols; ++i) {
677            jacobian[i * cols + pj] = jacobian[j * cols + permutation[i]];
678          }
679          lmDir[j] = diagR[pj];
680          work[j]  = qy[j];
681        }
682    
683        // eliminate the diagonal matrix d using a Givens rotation
684        for (int j = 0; j < solvedCols; ++j) {
685    
686          // prepare the row of d to be eliminated, locating the
687          // diagonal element using p from the Q.R. factorization
688          int pj = permutation[j];
689          double dpj = diag[pj];
690          if (dpj != 0) {
691            Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
692          }
693          lmDiag[j] = dpj;
694    
695          //  the transformations to eliminate the row of d
696          // modify only a single element of Qty
697          // beyond the first n, which is initially zero.
698          double qtbpj = 0;
699          for (int k = j; k < solvedCols; ++k) {
700            int pk = permutation[k];
701    
702            // determine a Givens rotation which eliminates the
703            // appropriate element in the current row of d
704            if (lmDiag[k] != 0) {
705    
706              final double sin;
707              final double cos;
708              double rkk = jacobian[k * cols + pk];
709              if (Math.abs(rkk) < Math.abs(lmDiag[k])) {
710                final double cotan = rkk / lmDiag[k];
711                sin   = 1.0 / Math.sqrt(1.0 + cotan * cotan);
712                cos   = sin * cotan;
713              } else {
714                final double tan = lmDiag[k] / rkk;
715                cos = 1.0 / Math.sqrt(1.0 + tan * tan);
716                sin = cos * tan;
717              }
718    
719              // compute the modified diagonal element of R and
720              // the modified element of (Qty,0)
721              jacobian[k * cols + pk] = cos * rkk + sin * lmDiag[k];
722              final double temp = cos * work[k] + sin * qtbpj;
723              qtbpj = -sin * work[k] + cos * qtbpj;
724              work[k] = temp;
725    
726              // accumulate the tranformation in the row of s
727              for (int i = k + 1; i < solvedCols; ++i) {
728                double rik = jacobian[i * cols + pk];
729                final double temp2 = cos * rik + sin * lmDiag[i];
730                lmDiag[i] = -sin * rik + cos * lmDiag[i];
731                jacobian[i * cols + pk] = temp2;
732              }
733    
734            }
735          }
736    
737          // store the diagonal element of s and restore
738          // the corresponding diagonal element of R
739          int index = j * cols + permutation[j];
740          lmDiag[j]       = jacobian[index];
741          jacobian[index] = lmDir[j];
742    
743        }
744    
745        // solve the triangular system for z, if the system is
746        // singular, then obtain a least squares solution
747        int nSing = solvedCols;
748        for (int j = 0; j < solvedCols; ++j) {
749          if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
750            nSing = j;
751          }
752          if (nSing < solvedCols) {
753            work[j] = 0;
754          }
755        }
756        if (nSing > 0) {
757          for (int j = nSing - 1; j >= 0; --j) {
758            int pj = permutation[j];
759            double sum = 0;
760            for (int i = j + 1; i < nSing; ++i) {
761              sum += jacobian[i * cols + pj] * work[i];
762            }
763            work[j] = (work[j] - sum) / lmDiag[j];
764          }
765        }
766    
767        // permute the components of z back to components of lmDir
768        for (int j = 0; j < lmDir.length; ++j) {
769          lmDir[permutation[j]] = work[j];
770        }
771    
772      }
773    
774      /**
775       * Decompose a matrix A as A.P = Q.R using Householder transforms.
776       * <p>As suggested in the P. Lascaux and R. Theodor book
777       * <i>Analyse num&eacute;rique matricielle appliqu&eacute;e &agrave;
778       * l'art de l'ing&eacute;nieur</i> (Masson, 1986), instead of representing
779       * the Householder transforms with u<sub>k</sub> unit vectors such that:
780       * <pre>
781       * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
782       * </pre>
783       * we use <sub>k</sub> non-unit vectors such that:
784       * <pre>
785       * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
786       * </pre>
787       * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
788       * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
789       * them from the v<sub>k</sub> vectors would be costly.</p>
790       * <p>This decomposition handles rank deficient cases since the tranformations
791       * are performed in non-increasing columns norms order thanks to columns
792       * pivoting. The diagonal elements of the R matrix are therefore also in
793       * non-increasing absolute values order.</p>
794       * @exception EstimationException if the decomposition cannot be performed
795       */
796      private void qrDecomposition() throws EstimationException {
797    
798        // initializations
799        for (int k = 0; k < cols; ++k) {
800          permutation[k] = k;
801          double norm2 = 0;
802          for (int index = k; index < jacobian.length; index += cols) {
803            double akk = jacobian[index];
804            norm2 += akk * akk;
805          }
806          jacNorm[k] = Math.sqrt(norm2);
807        }
808    
809        // transform the matrix column after column
810        for (int k = 0; k < cols; ++k) {
811    
812          // select the column with the greatest norm on active components
813          int nextColumn = -1;
814          double ak2 = Double.NEGATIVE_INFINITY;
815          for (int i = k; i < cols; ++i) {
816            double norm2 = 0;
817            int iDiag = k * cols + permutation[i];
818            for (int index = iDiag; index < jacobian.length; index += cols) {
819              double aki = jacobian[index];
820              norm2 += aki * aki;
821            }
822            if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
823                throw new EstimationException(
824                        "unable to perform Q.R decomposition on the {0}x{1} jacobian matrix",
825                        rows, cols);
826            }
827            if (norm2 > ak2) {
828              nextColumn = i;
829              ak2        = norm2;
830            }
831          }
832          if (ak2 == 0) {
833            rank = k;
834            return;
835          }
836          int pk                  = permutation[nextColumn];
837          permutation[nextColumn] = permutation[k];
838          permutation[k]          = pk;
839    
840          // choose alpha such that Hk.u = alpha ek
841          int    kDiag = k * cols + pk;
842          double akk   = jacobian[kDiag];
843          double alpha = (akk > 0) ? -Math.sqrt(ak2) : Math.sqrt(ak2);
844          double betak = 1.0 / (ak2 - akk * alpha);
845          beta[pk]     = betak;
846    
847          // transform the current column
848          diagR[pk]        = alpha;
849          jacobian[kDiag] -= alpha;
850    
851          // transform the remaining columns
852          for (int dk = cols - 1 - k; dk > 0; --dk) {
853            int dkp = permutation[k + dk] - pk;
854            double gamma = 0;
855            for (int index = kDiag; index < jacobian.length; index += cols) {
856              gamma += jacobian[index] * jacobian[index + dkp];
857            }
858            gamma *= betak;
859            for (int index = kDiag; index < jacobian.length; index += cols) {
860              jacobian[index + dkp] -= gamma * jacobian[index];
861            }
862          }
863    
864        }
865    
866        rank = solvedCols;
867    
868      }
869    
870      /**
871       * Compute the product Qt.y for some Q.R. decomposition.
872       *
873       * @param y vector to multiply (will be overwritten with the result)
874       */
875      private void qTy(double[] y) {
876        for (int k = 0; k < cols; ++k) {
877          int pk = permutation[k];
878          int kDiag = k * cols + pk;
879          double gamma = 0;
880          int index = kDiag;
881          for (int i = k; i < rows; ++i) {
882            gamma += jacobian[index] * y[i];
883            index += cols;
884          }
885          gamma *= beta[pk];
886          index = kDiag;
887          for (int i = k; i < rows; ++i) {
888            y[i] -= gamma * jacobian[index];
889            index += cols;
890          }
891        }
892      }
893    
894    }