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.optimization.general;
018
019 import java.util.Arrays;
020
021 import org.apache.commons.math.FunctionEvaluationException;
022 import org.apache.commons.math.optimization.OptimizationException;
023 import org.apache.commons.math.optimization.VectorialPointValuePair;
024
025
026 /**
027 * This class solves a least squares problem using the Levenberg-Marquardt algorithm.
028 *
029 * <p>This implementation <em>should</em> work even for over-determined systems
030 * (i.e. systems having more point than equations). Over-determined systems
031 * are solved by ignoring the point 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 numérique matricielle
040 * appliquée à l'art de l'ingénieur</i>, Masson 1986. The
041 * redistribution policy for MINPACK is available <a
042 * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
043 * is reproduced below.</p>
044 *
045 * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
046 * <tr><td>
047 * Minpack Copyright Notice (1999) University of Chicago.
048 * All rights reserved
049 * </td></tr>
050 * <tr><td>
051 * Redistribution and use in source and binary forms, with or without
052 * modification, are permitted provided that the following conditions
053 * are met:
054 * <ol>
055 * <li>Redistributions of source code must retain the above copyright
056 * notice, this list of conditions and the following disclaimer.</li>
057 * <li>Redistributions in binary form must reproduce the above
058 * copyright notice, this list of conditions and the following
059 * disclaimer in the documentation and/or other materials provided
060 * with the distribution.</li>
061 * <li>The end-user documentation included with the redistribution, if any,
062 * must include the following acknowledgment:
063 * <code>This product includes software developed by the University of
064 * Chicago, as Operator of Argonne National Laboratory.</code>
065 * Alternately, this acknowledgment may appear in the software itself,
066 * if and wherever such third-party acknowledgments normally appear.</li>
067 * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
068 * WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
069 * UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
070 * THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
071 * IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
072 * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
073 * OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
074 * OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
075 * USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
076 * THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
077 * DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
078 * UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
079 * BE CORRECTED.</strong></li>
080 * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
081 * HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
082 * ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
083 * INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
084 * ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
085 * PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
086 * SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
087 * (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
088 * EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
089 * POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
090 * <ol></td></tr>
091 * </table>
092
093 * @author Argonne National Laboratory. MINPACK project. March 1980 (original fortran)
094 * @author Burton S. Garbow (original fortran)
095 * @author Kenneth E. Hillstrom (original fortran)
096 * @author Jorge J. More (original fortran)
097
098 * @version $Revision: 795978 $ $Date: 2009-07-20 15:57:08 -0400 (Mon, 20 Jul 2009) $
099 * @since 2.0
100 *
101 */
102 public class LevenbergMarquardtOptimizer extends AbstractLeastSquaresOptimizer {
103
104 /** Number of solved point. */
105 private int solvedCols;
106
107 /** Diagonal elements of the R matrix in the Q.R. decomposition. */
108 private double[] diagR;
109
110 /** Norms of the columns of the jacobian matrix. */
111 private double[] jacNorm;
112
113 /** Coefficients of the Householder transforms vectors. */
114 private double[] beta;
115
116 /** Columns permutation array. */
117 private int[] permutation;
118
119 /** Rank of the jacobian matrix. */
120 private int rank;
121
122 /** Levenberg-Marquardt parameter. */
123 private double lmPar;
124
125 /** Parameters evolution direction associated with lmPar. */
126 private double[] lmDir;
127
128 /** Positive input variable used in determining the initial step bound. */
129 private double initialStepBoundFactor;
130
131 /** Desired relative error in the sum of squares. */
132 private double costRelativeTolerance;
133
134 /** Desired relative error in the approximate solution parameters. */
135 private double parRelativeTolerance;
136
137 /** Desired max cosine on the orthogonality between the function vector
138 * and the columns of the jacobian. */
139 private double orthoTolerance;
140
141 /**
142 * Build an optimizer for least squares problems.
143 * <p>The default values for the algorithm settings are:
144 * <ul>
145 * <li>{@link #setInitialStepBoundFactor initial step bound factor}: 100.0</li>
146 * <li>{@link #setMaxIterations maximal iterations}: 1000</li>
147 * <li>{@link #setCostRelativeTolerance cost relative tolerance}: 1.0e-10</li>
148 * <li>{@link #setParRelativeTolerance parameters relative tolerance}: 1.0e-10</li>
149 * <li>{@link #setOrthoTolerance orthogonality tolerance}: 1.0e-10</li>
150 * </ul>
151 * </p>
152 */
153 public LevenbergMarquardtOptimizer() {
154
155 // set up the superclass with a default max cost evaluations setting
156 setMaxIterations(1000);
157
158 // default values for the tuning parameters
159 setInitialStepBoundFactor(100.0);
160 setCostRelativeTolerance(1.0e-10);
161 setParRelativeTolerance(1.0e-10);
162 setOrthoTolerance(1.0e-10);
163
164 }
165
166 /**
167 * Set the positive input variable used in determining the initial step bound.
168 * This bound is set to the product of initialStepBoundFactor and the euclidean
169 * norm of diag*x if nonzero, or else to initialStepBoundFactor itself. In most
170 * cases factor should lie in the interval (0.1, 100.0). 100.0 is a generally
171 * recommended value.
172 *
173 * @param initialStepBoundFactor initial step bound factor
174 */
175 public void setInitialStepBoundFactor(double initialStepBoundFactor) {
176 this.initialStepBoundFactor = initialStepBoundFactor;
177 }
178
179 /**
180 * Set the desired relative error in the sum of squares.
181 *
182 * @param costRelativeTolerance desired relative error in the sum of squares
183 */
184 public void setCostRelativeTolerance(double costRelativeTolerance) {
185 this.costRelativeTolerance = costRelativeTolerance;
186 }
187
188 /**
189 * Set the desired relative error in the approximate solution parameters.
190 *
191 * @param parRelativeTolerance desired relative error
192 * in the approximate solution parameters
193 */
194 public void setParRelativeTolerance(double parRelativeTolerance) {
195 this.parRelativeTolerance = parRelativeTolerance;
196 }
197
198 /**
199 * Set the desired max cosine on the orthogonality.
200 *
201 * @param orthoTolerance desired max cosine on the orthogonality
202 * between the function vector and the columns of the jacobian
203 */
204 public void setOrthoTolerance(double orthoTolerance) {
205 this.orthoTolerance = orthoTolerance;
206 }
207
208 /** {@inheritDoc} */
209 @Override
210 protected VectorialPointValuePair doOptimize()
211 throws FunctionEvaluationException, OptimizationException, IllegalArgumentException {
212
213 // arrays shared with the other private methods
214 solvedCols = Math.min(rows, cols);
215 diagR = new double[cols];
216 jacNorm = new double[cols];
217 beta = new double[cols];
218 permutation = new int[cols];
219 lmDir = new double[cols];
220
221 // local point
222 double delta = 0, xNorm = 0;
223 double[] diag = new double[cols];
224 double[] oldX = new double[cols];
225 double[] oldRes = new double[rows];
226 double[] work1 = new double[cols];
227 double[] work2 = new double[cols];
228 double[] work3 = new double[cols];
229
230 // evaluate the function at the starting point and calculate its norm
231 updateResidualsAndCost();
232
233 // outer loop
234 lmPar = 0;
235 boolean firstIteration = true;
236 while (true) {
237
238 incrementIterationsCounter();
239
240 // compute the Q.R. decomposition of the jacobian matrix
241 updateJacobian();
242 qrDecomposition();
243
244 // compute Qt.res
245 qTy(residuals);
246
247 // now we don't need Q anymore,
248 // so let jacobian contain the R matrix with its diagonal elements
249 for (int k = 0; k < solvedCols; ++k) {
250 int pk = permutation[k];
251 jacobian[k][pk] = diagR[pk];
252 }
253
254 if (firstIteration) {
255
256 // scale the point according to the norms of the columns
257 // of the initial jacobian
258 xNorm = 0;
259 for (int k = 0; k < cols; ++k) {
260 double dk = jacNorm[k];
261 if (dk == 0) {
262 dk = 1.0;
263 }
264 double xk = dk * point[k];
265 xNorm += xk * xk;
266 diag[k] = dk;
267 }
268 xNorm = Math.sqrt(xNorm);
269
270 // initialize the step bound delta
271 delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
272
273 }
274
275 // check orthogonality between function vector and jacobian columns
276 double maxCosine = 0;
277 if (cost != 0) {
278 for (int j = 0; j < solvedCols; ++j) {
279 int pj = permutation[j];
280 double s = jacNorm[pj];
281 if (s != 0) {
282 double sum = 0;
283 for (int i = 0; i <= j; ++i) {
284 sum += jacobian[i][pj] * residuals[i];
285 }
286 maxCosine = Math.max(maxCosine, Math.abs(sum) / (s * cost));
287 }
288 }
289 }
290 if (maxCosine <= orthoTolerance) {
291 // convergence has been reached
292 return new VectorialPointValuePair(point, objective);
293 }
294
295 // rescale if necessary
296 for (int j = 0; j < cols; ++j) {
297 diag[j] = Math.max(diag[j], jacNorm[j]);
298 }
299
300 // inner loop
301 for (double ratio = 0; ratio < 1.0e-4;) {
302
303 // save the state
304 for (int j = 0; j < solvedCols; ++j) {
305 int pj = permutation[j];
306 oldX[pj] = point[pj];
307 }
308 double previousCost = cost;
309 double[] tmpVec = residuals;
310 residuals = oldRes;
311 oldRes = tmpVec;
312
313 // determine the Levenberg-Marquardt parameter
314 determineLMParameter(oldRes, delta, diag, work1, work2, work3);
315
316 // compute the new point and the norm of the evolution direction
317 double lmNorm = 0;
318 for (int j = 0; j < solvedCols; ++j) {
319 int pj = permutation[j];
320 lmDir[pj] = -lmDir[pj];
321 point[pj] = oldX[pj] + lmDir[pj];
322 double s = diag[pj] * lmDir[pj];
323 lmNorm += s * s;
324 }
325 lmNorm = Math.sqrt(lmNorm);
326
327 // on the first iteration, adjust the initial step bound.
328 if (firstIteration) {
329 delta = Math.min(delta, lmNorm);
330 }
331
332 // evaluate the function at x + p and calculate its norm
333 updateResidualsAndCost();
334
335 // compute the scaled actual reduction
336 double actRed = -1.0;
337 if (0.1 * cost < previousCost) {
338 double r = cost / previousCost;
339 actRed = 1.0 - r * r;
340 }
341
342 // compute the scaled predicted reduction
343 // and the scaled directional derivative
344 for (int j = 0; j < solvedCols; ++j) {
345 int pj = permutation[j];
346 double dirJ = lmDir[pj];
347 work1[j] = 0;
348 for (int i = 0; i <= j; ++i) {
349 work1[i] += jacobian[i][pj] * dirJ;
350 }
351 }
352 double coeff1 = 0;
353 for (int j = 0; j < solvedCols; ++j) {
354 coeff1 += work1[j] * work1[j];
355 }
356 double pc2 = previousCost * previousCost;
357 coeff1 = coeff1 / pc2;
358 double coeff2 = lmPar * lmNorm * lmNorm / pc2;
359 double preRed = coeff1 + 2 * coeff2;
360 double dirDer = -(coeff1 + coeff2);
361
362 // ratio of the actual to the predicted reduction
363 ratio = (preRed == 0) ? 0 : (actRed / preRed);
364
365 // update the step bound
366 if (ratio <= 0.25) {
367 double tmp =
368 (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
369 if ((0.1 * cost >= previousCost) || (tmp < 0.1)) {
370 tmp = 0.1;
371 }
372 delta = tmp * Math.min(delta, 10.0 * lmNorm);
373 lmPar /= tmp;
374 } else if ((lmPar == 0) || (ratio >= 0.75)) {
375 delta = 2 * lmNorm;
376 lmPar *= 0.5;
377 }
378
379 // test for successful iteration.
380 if (ratio >= 1.0e-4) {
381 // successful iteration, update the norm
382 firstIteration = false;
383 xNorm = 0;
384 for (int k = 0; k < cols; ++k) {
385 double xK = diag[k] * point[k];
386 xNorm += xK * xK;
387 }
388 xNorm = Math.sqrt(xNorm);
389 } else {
390 // failed iteration, reset the previous values
391 cost = previousCost;
392 for (int j = 0; j < solvedCols; ++j) {
393 int pj = permutation[j];
394 point[pj] = oldX[pj];
395 }
396 tmpVec = residuals;
397 residuals = oldRes;
398 oldRes = tmpVec;
399 }
400
401 // tests for convergence.
402 if (((Math.abs(actRed) <= costRelativeTolerance) &&
403 (preRed <= costRelativeTolerance) &&
404 (ratio <= 2.0)) ||
405 (delta <= parRelativeTolerance * xNorm)) {
406 return new VectorialPointValuePair(point, objective);
407 }
408
409 // tests for termination and stringent tolerances
410 // (2.2204e-16 is the machine epsilon for IEEE754)
411 if ((Math.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
412 throw new OptimizationException("cost relative tolerance is too small ({0})," +
413 " no further reduction in the" +
414 " sum of squares is possible",
415 costRelativeTolerance);
416 } else if (delta <= 2.2204e-16 * xNorm) {
417 throw new OptimizationException("parameters relative tolerance is too small" +
418 " ({0}), no further improvement in" +
419 " the approximate solution is possible",
420 parRelativeTolerance);
421 } else if (maxCosine <= 2.2204e-16) {
422 throw new OptimizationException("orthogonality tolerance is too small ({0})," +
423 " solution is orthogonal to the jacobian",
424 orthoTolerance);
425 }
426
427 }
428
429 }
430
431 }
432
433 /**
434 * Determine the Levenberg-Marquardt parameter.
435 * <p>This implementation is a translation in Java of the MINPACK
436 * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
437 * routine.</p>
438 * <p>This method sets the lmPar and lmDir attributes.</p>
439 * <p>The authors of the original fortran function are:</p>
440 * <ul>
441 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
442 * <li>Burton S. Garbow</li>
443 * <li>Kenneth E. Hillstrom</li>
444 * <li>Jorge J. More</li>
445 * </ul>
446 * <p>Luc Maisonobe did the Java translation.</p>
447 *
448 * @param qy array containing qTy
449 * @param delta upper bound on the euclidean norm of diagR * lmDir
450 * @param diag diagonal matrix
451 * @param work1 work array
452 * @param work2 work array
453 * @param work3 work array
454 */
455 private void determineLMParameter(double[] qy, double delta, double[] diag,
456 double[] work1, double[] work2, double[] work3) {
457
458 // compute and store in x the gauss-newton direction, if the
459 // jacobian is rank-deficient, obtain a least squares solution
460 for (int j = 0; j < rank; ++j) {
461 lmDir[permutation[j]] = qy[j];
462 }
463 for (int j = rank; j < cols; ++j) {
464 lmDir[permutation[j]] = 0;
465 }
466 for (int k = rank - 1; k >= 0; --k) {
467 int pk = permutation[k];
468 double ypk = lmDir[pk] / diagR[pk];
469 for (int i = 0; i < k; ++i) {
470 lmDir[permutation[i]] -= ypk * jacobian[i][pk];
471 }
472 lmDir[pk] = ypk;
473 }
474
475 // evaluate the function at the origin, and test
476 // for acceptance of the Gauss-Newton direction
477 double dxNorm = 0;
478 for (int j = 0; j < solvedCols; ++j) {
479 int pj = permutation[j];
480 double s = diag[pj] * lmDir[pj];
481 work1[pj] = s;
482 dxNorm += s * s;
483 }
484 dxNorm = Math.sqrt(dxNorm);
485 double fp = dxNorm - delta;
486 if (fp <= 0.1 * delta) {
487 lmPar = 0;
488 return;
489 }
490
491 // if the jacobian is not rank deficient, the Newton step provides
492 // a lower bound, parl, for the zero of the function,
493 // otherwise set this bound to zero
494 double sum2, parl = 0;
495 if (rank == solvedCols) {
496 for (int j = 0; j < solvedCols; ++j) {
497 int pj = permutation[j];
498 work1[pj] *= diag[pj] / dxNorm;
499 }
500 sum2 = 0;
501 for (int j = 0; j < solvedCols; ++j) {
502 int pj = permutation[j];
503 double sum = 0;
504 for (int i = 0; i < j; ++i) {
505 sum += jacobian[i][pj] * work1[permutation[i]];
506 }
507 double s = (work1[pj] - sum) / diagR[pj];
508 work1[pj] = s;
509 sum2 += s * s;
510 }
511 parl = fp / (delta * sum2);
512 }
513
514 // calculate an upper bound, paru, for the zero of the function
515 sum2 = 0;
516 for (int j = 0; j < solvedCols; ++j) {
517 int pj = permutation[j];
518 double sum = 0;
519 for (int i = 0; i <= j; ++i) {
520 sum += jacobian[i][pj] * qy[i];
521 }
522 sum /= diag[pj];
523 sum2 += sum * sum;
524 }
525 double gNorm = Math.sqrt(sum2);
526 double paru = gNorm / delta;
527 if (paru == 0) {
528 // 2.2251e-308 is the smallest positive real for IEE754
529 paru = 2.2251e-308 / Math.min(delta, 0.1);
530 }
531
532 // if the input par lies outside of the interval (parl,paru),
533 // set par to the closer endpoint
534 lmPar = Math.min(paru, Math.max(lmPar, parl));
535 if (lmPar == 0) {
536 lmPar = gNorm / dxNorm;
537 }
538
539 for (int countdown = 10; countdown >= 0; --countdown) {
540
541 // evaluate the function at the current value of lmPar
542 if (lmPar == 0) {
543 lmPar = Math.max(2.2251e-308, 0.001 * paru);
544 }
545 double sPar = Math.sqrt(lmPar);
546 for (int j = 0; j < solvedCols; ++j) {
547 int pj = permutation[j];
548 work1[pj] = sPar * diag[pj];
549 }
550 determineLMDirection(qy, work1, work2, work3);
551
552 dxNorm = 0;
553 for (int j = 0; j < solvedCols; ++j) {
554 int pj = permutation[j];
555 double s = diag[pj] * lmDir[pj];
556 work3[pj] = s;
557 dxNorm += s * s;
558 }
559 dxNorm = Math.sqrt(dxNorm);
560 double previousFP = fp;
561 fp = dxNorm - delta;
562
563 // if the function is small enough, accept the current value
564 // of lmPar, also test for the exceptional cases where parl is zero
565 if ((Math.abs(fp) <= 0.1 * delta) ||
566 ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
567 return;
568 }
569
570 // compute the Newton correction
571 for (int j = 0; j < solvedCols; ++j) {
572 int pj = permutation[j];
573 work1[pj] = work3[pj] * diag[pj] / dxNorm;
574 }
575 for (int j = 0; j < solvedCols; ++j) {
576 int pj = permutation[j];
577 work1[pj] /= work2[j];
578 double tmp = work1[pj];
579 for (int i = j + 1; i < solvedCols; ++i) {
580 work1[permutation[i]] -= jacobian[i][pj] * tmp;
581 }
582 }
583 sum2 = 0;
584 for (int j = 0; j < solvedCols; ++j) {
585 double s = work1[permutation[j]];
586 sum2 += s * s;
587 }
588 double correction = fp / (delta * sum2);
589
590 // depending on the sign of the function, update parl or paru.
591 if (fp > 0) {
592 parl = Math.max(parl, lmPar);
593 } else if (fp < 0) {
594 paru = Math.min(paru, lmPar);
595 }
596
597 // compute an improved estimate for lmPar
598 lmPar = Math.max(parl, lmPar + correction);
599
600 }
601 }
602
603 /**
604 * Solve a*x = b and d*x = 0 in the least squares sense.
605 * <p>This implementation is a translation in Java of the MINPACK
606 * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
607 * routine.</p>
608 * <p>This method sets the lmDir and lmDiag attributes.</p>
609 * <p>The authors of the original fortran function are:</p>
610 * <ul>
611 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
612 * <li>Burton S. Garbow</li>
613 * <li>Kenneth E. Hillstrom</li>
614 * <li>Jorge J. More</li>
615 * </ul>
616 * <p>Luc Maisonobe did the Java translation.</p>
617 *
618 * @param qy array containing qTy
619 * @param diag diagonal matrix
620 * @param lmDiag diagonal elements associated with lmDir
621 * @param work work array
622 */
623 private void determineLMDirection(double[] qy, double[] diag,
624 double[] lmDiag, double[] work) {
625
626 // copy R and Qty to preserve input and initialize s
627 // in particular, save the diagonal elements of R in lmDir
628 for (int j = 0; j < solvedCols; ++j) {
629 int pj = permutation[j];
630 for (int i = j + 1; i < solvedCols; ++i) {
631 jacobian[i][pj] = jacobian[j][permutation[i]];
632 }
633 lmDir[j] = diagR[pj];
634 work[j] = qy[j];
635 }
636
637 // eliminate the diagonal matrix d using a Givens rotation
638 for (int j = 0; j < solvedCols; ++j) {
639
640 // prepare the row of d to be eliminated, locating the
641 // diagonal element using p from the Q.R. factorization
642 int pj = permutation[j];
643 double dpj = diag[pj];
644 if (dpj != 0) {
645 Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
646 }
647 lmDiag[j] = dpj;
648
649 // the transformations to eliminate the row of d
650 // modify only a single element of Qty
651 // beyond the first n, which is initially zero.
652 double qtbpj = 0;
653 for (int k = j; k < solvedCols; ++k) {
654 int pk = permutation[k];
655
656 // determine a Givens rotation which eliminates the
657 // appropriate element in the current row of d
658 if (lmDiag[k] != 0) {
659
660 double sin, cos;
661 double rkk = jacobian[k][pk];
662 if (Math.abs(rkk) < Math.abs(lmDiag[k])) {
663 double cotan = rkk / lmDiag[k];
664 sin = 1.0 / Math.sqrt(1.0 + cotan * cotan);
665 cos = sin * cotan;
666 } else {
667 double tan = lmDiag[k] / rkk;
668 cos = 1.0 / Math.sqrt(1.0 + tan * tan);
669 sin = cos * tan;
670 }
671
672 // compute the modified diagonal element of R and
673 // the modified element of (Qty,0)
674 jacobian[k][pk] = cos * rkk + sin * lmDiag[k];
675 double temp = cos * work[k] + sin * qtbpj;
676 qtbpj = -sin * work[k] + cos * qtbpj;
677 work[k] = temp;
678
679 // accumulate the tranformation in the row of s
680 for (int i = k + 1; i < solvedCols; ++i) {
681 double rik = jacobian[i][pk];
682 temp = cos * rik + sin * lmDiag[i];
683 lmDiag[i] = -sin * rik + cos * lmDiag[i];
684 jacobian[i][pk] = temp;
685 }
686
687 }
688 }
689
690 // store the diagonal element of s and restore
691 // the corresponding diagonal element of R
692 lmDiag[j] = jacobian[j][permutation[j]];
693 jacobian[j][permutation[j]] = lmDir[j];
694
695 }
696
697 // solve the triangular system for z, if the system is
698 // singular, then obtain a least squares solution
699 int nSing = solvedCols;
700 for (int j = 0; j < solvedCols; ++j) {
701 if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
702 nSing = j;
703 }
704 if (nSing < solvedCols) {
705 work[j] = 0;
706 }
707 }
708 if (nSing > 0) {
709 for (int j = nSing - 1; j >= 0; --j) {
710 int pj = permutation[j];
711 double sum = 0;
712 for (int i = j + 1; i < nSing; ++i) {
713 sum += jacobian[i][pj] * work[i];
714 }
715 work[j] = (work[j] - sum) / lmDiag[j];
716 }
717 }
718
719 // permute the components of z back to components of lmDir
720 for (int j = 0; j < lmDir.length; ++j) {
721 lmDir[permutation[j]] = work[j];
722 }
723
724 }
725
726 /**
727 * Decompose a matrix A as A.P = Q.R using Householder transforms.
728 * <p>As suggested in the P. Lascaux and R. Theodor book
729 * <i>Analyse numérique matricielle appliquée à
730 * l'art de l'ingénieur</i> (Masson, 1986), instead of representing
731 * the Householder transforms with u<sub>k</sub> unit vectors such that:
732 * <pre>
733 * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
734 * </pre>
735 * we use <sub>k</sub> non-unit vectors such that:
736 * <pre>
737 * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
738 * </pre>
739 * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
740 * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
741 * them from the v<sub>k</sub> vectors would be costly.</p>
742 * <p>This decomposition handles rank deficient cases since the tranformations
743 * are performed in non-increasing columns norms order thanks to columns
744 * pivoting. The diagonal elements of the R matrix are therefore also in
745 * non-increasing absolute values order.</p>
746 * @exception OptimizationException if the decomposition cannot be performed
747 */
748 private void qrDecomposition() throws OptimizationException {
749
750 // initializations
751 for (int k = 0; k < cols; ++k) {
752 permutation[k] = k;
753 double norm2 = 0;
754 for (int i = 0; i < jacobian.length; ++i) {
755 double akk = jacobian[i][k];
756 norm2 += akk * akk;
757 }
758 jacNorm[k] = Math.sqrt(norm2);
759 }
760
761 // transform the matrix column after column
762 for (int k = 0; k < cols; ++k) {
763
764 // select the column with the greatest norm on active components
765 int nextColumn = -1;
766 double ak2 = Double.NEGATIVE_INFINITY;
767 for (int i = k; i < cols; ++i) {
768 double norm2 = 0;
769 for (int j = k; j < jacobian.length; ++j) {
770 double aki = jacobian[j][permutation[i]];
771 norm2 += aki * aki;
772 }
773 if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
774 throw new OptimizationException(
775 "unable to perform Q.R decomposition on the {0}x{1} jacobian matrix",
776 rows, cols);
777 }
778 if (norm2 > ak2) {
779 nextColumn = i;
780 ak2 = norm2;
781 }
782 }
783 if (ak2 == 0) {
784 rank = k;
785 return;
786 }
787 int pk = permutation[nextColumn];
788 permutation[nextColumn] = permutation[k];
789 permutation[k] = pk;
790
791 // choose alpha such that Hk.u = alpha ek
792 double akk = jacobian[k][pk];
793 double alpha = (akk > 0) ? -Math.sqrt(ak2) : Math.sqrt(ak2);
794 double betak = 1.0 / (ak2 - akk * alpha);
795 beta[pk] = betak;
796
797 // transform the current column
798 diagR[pk] = alpha;
799 jacobian[k][pk] -= alpha;
800
801 // transform the remaining columns
802 for (int dk = cols - 1 - k; dk > 0; --dk) {
803 double gamma = 0;
804 for (int j = k; j < jacobian.length; ++j) {
805 gamma += jacobian[j][pk] * jacobian[j][permutation[k + dk]];
806 }
807 gamma *= betak;
808 for (int j = k; j < jacobian.length; ++j) {
809 jacobian[j][permutation[k + dk]] -= gamma * jacobian[j][pk];
810 }
811 }
812
813 }
814
815 rank = solvedCols;
816
817 }
818
819 /**
820 * Compute the product Qt.y for some Q.R. decomposition.
821 *
822 * @param y vector to multiply (will be overwritten with the result)
823 */
824 private void qTy(double[] y) {
825 for (int k = 0; k < cols; ++k) {
826 int pk = permutation[k];
827 double gamma = 0;
828 for (int i = k; i < rows; ++i) {
829 gamma += jacobian[i][pk] * y[i];
830 }
831 gamma *= beta[pk];
832 for (int i = k; i < rows; ++i) {
833 y[i] -= gamma * jacobian[i][pk];
834 }
835 }
836 }
837
838 }