1 /*
2 * Licensed to the Apache Software Foundation (ASF) under one or more
3 * contributor license agreements. See the NOTICE file distributed with
4 * this work for additional information regarding copyright ownership.
5 * The ASF licenses this file to You under the Apache License, Version 2.0
6 * (the "License"); you may not use this file except in compliance with
7 * the License. You may obtain a copy of the License at
8 *
9 * http://www.apache.org/licenses/LICENSE-2.0
10 *
11 * Unless required by applicable law or agreed to in writing, software
12 * distributed under the License is distributed on an "AS IS" BASIS,
13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14 * See the License for the specific language governing permissions and
15 * limitations under the License.
16 */
17 package org.apache.commons.math.estimation;
18
19 import java.io.Serializable;
20 import java.util.Arrays;
21
22
23 /**
24 * This class solves a least squares problem.
25 *
26 * <p>This implementation <em>should</em> work even for over-determined systems
27 * (i.e. systems having more variables than equations). Over-determined systems
28 * are solved by ignoring the variables which have the smallest impact according
29 * to their jacobian column norm. Only the rank of the matrix and some loop bounds
30 * are changed to implement this. This feature has undergone only basic testing
31 * for now and should still be considered experimental.</p>
32 *
33 * <p>The resolution engine is a simple translation of the MINPACK <a
34 * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
35 * changes. The changes include the over-determined resolution and the Q.R.
36 * decomposition which has been rewritten following the algorithm described in the
37 * P. Lascaux and R. Theodor book <i>Analyse numérique matricielle
38 * appliquée à l'art de l'ingénieur</i>, Masson 1986. The
39 * redistribution policy for MINPACK is available <a
40 * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
41 * is reproduced below.</p>
42 *
43 * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
44 * <tr><td>
45 * Minpack Copyright Notice (1999) University of Chicago.
46 * All rights reserved
47 * </td></tr>
48 * <tr><td>
49 * Redistribution and use in source and binary forms, with or without
50 * modification, are permitted provided that the following conditions
51 * are met:
52 * <ol>
53 * <li>Redistributions of source code must retain the above copyright
54 * notice, this list of conditions and the following disclaimer.</li>
55 * <li>Redistributions in binary form must reproduce the above
56 * copyright notice, this list of conditions and the following
57 * disclaimer in the documentation and/or other materials provided
58 * with the distribution.</li>
59 * <li>The end-user documentation included with the redistribution, if any,
60 * must include the following acknowledgment:
61 * <code>This product includes software developed by the University of
62 * Chicago, as Operator of Argonne National Laboratory.</code>
63 * Alternately, this acknowledgment may appear in the software itself,
64 * if and wherever such third-party acknowledgments normally appear.</li>
65 * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
66 * WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
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78 * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
79 * HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
80 * ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
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83 * PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
84 * SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
85 * (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
86 * EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
87 * POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
88 * <ol></td></tr>
89 * </table>
90
91 * @author Argonne National Laboratory. MINPACK project. March 1980 (original fortran)
92 * @author Burton S. Garbow (original fortran)
93 * @author Kenneth E. Hillstrom (original fortran)
94 * @author Jorge J. More (original fortran)
95
96 * @version $Revision: 620312 $ $Date: 2008-02-10 12:28:59 -0700 (Sun, 10 Feb 2008) $
97 * @since 1.2
98 *
99 */
100 public class LevenbergMarquardtEstimator extends AbstractEstimator implements Serializable {
101
102 /**
103 * Build an estimator for least squares problems.
104 * <p>The default values for the algorithm settings are:
105 * <ul>
106 * <li>{@link #setInitialStepBoundFactor initial step bound factor}: 100.0</li>
107 * <li>{@link #setMaxCostEval maximal cost evaluations}: 1000</li>
108 * <li>{@link #setCostRelativeTolerance cost relative tolerance}: 1.0e-10</li>
109 * <li>{@link #setParRelativeTolerance parameters relative tolerance}: 1.0e-10</li>
110 * <li>{@link #setOrthoTolerance orthogonality tolerance}: 1.0e-10</li>
111 * </ul>
112 * </p>
113 */
114 public LevenbergMarquardtEstimator() {
115
116 // set up the superclass with a default max cost evaluations setting
117 setMaxCostEval(1000);
118
119 // default values for the tuning parameters
120 setInitialStepBoundFactor(100.0);
121 setCostRelativeTolerance(1.0e-10);
122 setParRelativeTolerance(1.0e-10);
123 setOrthoTolerance(1.0e-10);
124
125 }
126
127 /**
128 * Set the positive input variable used in determining the initial step bound.
129 * This bound is set to the product of initialStepBoundFactor and the euclidean norm of diag*x if nonzero,
130 * or else to initialStepBoundFactor itself. In most cases factor should lie
131 * in the interval (0.1, 100.0). 100.0 is a generally recommended value
132 *
133 * @param initialStepBoundFactor initial step bound factor
134 * @see #estimate
135 */
136 public void setInitialStepBoundFactor(double initialStepBoundFactor) {
137 this.initialStepBoundFactor = initialStepBoundFactor;
138 }
139
140 /**
141 * Set the desired relative error in the sum of squares.
142 *
143 * @param costRelativeTolerance desired relative error in the sum of squares
144 * @see #estimate
145 */
146 public void setCostRelativeTolerance(double costRelativeTolerance) {
147 this.costRelativeTolerance = costRelativeTolerance;
148 }
149
150 /**
151 * Set the desired relative error in the approximate solution parameters.
152 *
153 * @param parRelativeTolerance desired relative error
154 * in the approximate solution parameters
155 * @see #estimate
156 */
157 public void setParRelativeTolerance(double parRelativeTolerance) {
158 this.parRelativeTolerance = parRelativeTolerance;
159 }
160
161 /**
162 * Set the desired max cosine on the orthogonality.
163 *
164 * @param orthoTolerance desired max cosine on the orthogonality
165 * between the function vector and the columns of the jacobian
166 * @see #estimate
167 */
168 public void setOrthoTolerance(double orthoTolerance) {
169 this.orthoTolerance = orthoTolerance;
170 }
171
172 /**
173 * Solve an estimation problem using the Levenberg-Marquardt algorithm.
174 * <p>The algorithm used is a modified Levenberg-Marquardt one, based
175 * on the MINPACK <a href="http://www.netlib.org/minpack/lmder.f">lmder</a>
176 * routine. The algorithm settings must have been set up before this method
177 * is called with the {@link #setInitialStepBoundFactor},
178 * {@link #setMaxCostEval}, {@link #setCostRelativeTolerance},
179 * {@link #setParRelativeTolerance} and {@link #setOrthoTolerance} methods.
180 * If these methods have not been called, the default values set up by the
181 * {@link #LevenbergMarquardtEstimator() constructor} will be used.</p>
182 * <p>The authors of the original fortran function are:</p>
183 * <ul>
184 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
185 * <li>Burton S. Garbow</li>
186 * <li>Kenneth E. Hillstrom</li>
187 * <li>Jorge J. More</li>
188 * </ul>
189 * <p>Luc Maisonobe did the Java translation.</p>
190 *
191 * @param problem estimation problem to solve
192 * @exception EstimationException if convergence cannot be
193 * reached with the specified algorithm settings or if there are more variables
194 * than equations
195 * @see #setInitialStepBoundFactor
196 * @see #setCostRelativeTolerance
197 * @see #setParRelativeTolerance
198 * @see #setOrthoTolerance
199 */
200 public void estimate(EstimationProblem problem)
201 throws EstimationException {
202
203 initializeEstimate(problem);
204
205 // arrays shared with the other private methods
206 solvedCols = Math.min(rows, cols);
207 diagR = new double[cols];
208 jacNorm = new double[cols];
209 beta = new double[cols];
210 permutation = new int[cols];
211 lmDir = new double[cols];
212
213 // local variables
214 double delta = 0, xNorm = 0;
215 double[] diag = new double[cols];
216 double[] oldX = new double[cols];
217 double[] oldRes = new double[rows];
218 double[] work1 = new double[cols];
219 double[] work2 = new double[cols];
220 double[] work3 = new double[cols];
221
222 // evaluate the function at the starting point and calculate its norm
223 updateResidualsAndCost();
224
225 // outer loop
226 lmPar = 0;
227 boolean firstIteration = true;
228 while (true) {
229
230 // compute the Q.R. decomposition of the jacobian matrix
231 updateJacobian();
232 qrDecomposition();
233
234 // compute Qt.res
235 qTy(residuals);
236
237 // now we don't need Q anymore,
238 // so let jacobian contain the R matrix with its diagonal elements
239 for (int k = 0; k < solvedCols; ++k) {
240 int pk = permutation[k];
241 jacobian[k * cols + pk] = diagR[pk];
242 }
243
244 if (firstIteration) {
245
246 // scale the variables according to the norms of the columns
247 // of the initial jacobian
248 xNorm = 0;
249 for (int k = 0; k < cols; ++k) {
250 double dk = jacNorm[k];
251 if (dk == 0) {
252 dk = 1.0;
253 }
254 double xk = dk * parameters[k].getEstimate();
255 xNorm += xk * xk;
256 diag[k] = dk;
257 }
258 xNorm = Math.sqrt(xNorm);
259
260 // initialize the step bound delta
261 delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
262
263 }
264
265 // check orthogonality between function vector and jacobian columns
266 double maxCosine = 0;
267 if (cost != 0) {
268 for (int j = 0; j < solvedCols; ++j) {
269 int pj = permutation[j];
270 double s = jacNorm[pj];
271 if (s != 0) {
272 double sum = 0;
273 for (int i = 0, index = pj; i <= j; ++i, index += cols) {
274 sum += jacobian[index] * residuals[i];
275 }
276 maxCosine = Math.max(maxCosine, Math.abs(sum) / (s * cost));
277 }
278 }
279 }
280 if (maxCosine <= orthoTolerance) {
281 return;
282 }
283
284 // rescale if necessary
285 for (int j = 0; j < cols; ++j) {
286 diag[j] = Math.max(diag[j], jacNorm[j]);
287 }
288
289 // inner loop
290 for (double ratio = 0; ratio < 1.0e-4;) {
291
292 // save the state
293 for (int j = 0; j < solvedCols; ++j) {
294 int pj = permutation[j];
295 oldX[pj] = parameters[pj].getEstimate();
296 }
297 double previousCost = cost;
298 double[] tmpVec = residuals;
299 residuals = oldRes;
300 oldRes = tmpVec;
301
302 // determine the Levenberg-Marquardt parameter
303 determineLMParameter(oldRes, delta, diag, work1, work2, work3);
304
305 // compute the new point and the norm of the evolution direction
306 double lmNorm = 0;
307 for (int j = 0; j < solvedCols; ++j) {
308 int pj = permutation[j];
309 lmDir[pj] = -lmDir[pj];
310 parameters[pj].setEstimate(oldX[pj] + lmDir[pj]);
311 double s = diag[pj] * lmDir[pj];
312 lmNorm += s * s;
313 }
314 lmNorm = Math.sqrt(lmNorm);
315
316 // on the first iteration, adjust the initial step bound.
317 if (firstIteration) {
318 delta = Math.min(delta, lmNorm);
319 }
320
321 // evaluate the function at x + p and calculate its norm
322 updateResidualsAndCost();
323
324 // compute the scaled actual reduction
325 double actRed = -1.0;
326 if (0.1 * cost < previousCost) {
327 double r = cost / previousCost;
328 actRed = 1.0 - r * r;
329 }
330
331 // compute the scaled predicted reduction
332 // and the scaled directional derivative
333 for (int j = 0; j < solvedCols; ++j) {
334 int pj = permutation[j];
335 double dirJ = lmDir[pj];
336 work1[j] = 0;
337 for (int i = 0, index = pj; i <= j; ++i, index += cols) {
338 work1[i] += jacobian[index] * dirJ;
339 }
340 }
341 double coeff1 = 0;
342 for (int j = 0; j < solvedCols; ++j) {
343 coeff1 += work1[j] * work1[j];
344 }
345 double pc2 = previousCost * previousCost;
346 coeff1 = coeff1 / pc2;
347 double coeff2 = lmPar * lmNorm * lmNorm / pc2;
348 double preRed = coeff1 + 2 * coeff2;
349 double dirDer = -(coeff1 + coeff2);
350
351 // ratio of the actual to the predicted reduction
352 ratio = (preRed == 0) ? 0 : (actRed / preRed);
353
354 // update the step bound
355 if (ratio <= 0.25) {
356 double tmp =
357 (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
358 if ((0.1 * cost >= previousCost) || (tmp < 0.1)) {
359 tmp = 0.1;
360 }
361 delta = tmp * Math.min(delta, 10.0 * lmNorm);
362 lmPar /= tmp;
363 } else if ((lmPar == 0) || (ratio >= 0.75)) {
364 delta = 2 * lmNorm;
365 lmPar *= 0.5;
366 }
367
368 // test for successful iteration.
369 if (ratio >= 1.0e-4) {
370 // successful iteration, update the norm
371 firstIteration = false;
372 xNorm = 0;
373 for (int k = 0; k < cols; ++k) {
374 double xK = diag[k] * parameters[k].getEstimate();
375 xNorm += xK * xK;
376 }
377 xNorm = Math.sqrt(xNorm);
378 } else {
379 // failed iteration, reset the previous values
380 cost = previousCost;
381 for (int j = 0; j < solvedCols; ++j) {
382 int pj = permutation[j];
383 parameters[pj].setEstimate(oldX[pj]);
384 }
385 tmpVec = residuals;
386 residuals = oldRes;
387 oldRes = tmpVec;
388 }
389
390 // tests for convergence.
391 if (((Math.abs(actRed) <= costRelativeTolerance) &&
392 (preRed <= costRelativeTolerance) &&
393 (ratio <= 2.0)) ||
394 (delta <= parRelativeTolerance * xNorm)) {
395 return;
396 }
397
398 // tests for termination and stringent tolerances
399 // (2.2204e-16 is the machine epsilon for IEEE754)
400 if ((Math.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
401 throw new EstimationException("cost relative tolerance is too small ({0})," +
402 " no further reduction in the" +
403 " sum of squares is possible",
404 new Object[] { new Double(costRelativeTolerance) });
405 } else if (delta <= 2.2204e-16 * xNorm) {
406 throw new EstimationException("parameters relative tolerance is too small" +
407 " ({0}), no further improvement in" +
408 " the approximate solution is possible",
409 new Object[] { new Double(parRelativeTolerance) });
410 } else if (maxCosine <= 2.2204e-16) {
411 throw new EstimationException("orthogonality tolerance is too small ({0})," +
412 " solution is orthogonal to the jacobian",
413 new Object[] { new Double(orthoTolerance) });
414 }
415
416 }
417
418 }
419
420 }
421
422 /**
423 * Determine the Levenberg-Marquardt parameter.
424 * <p>This implementation is a translation in Java of the MINPACK
425 * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
426 * routine.</p>
427 * <p>This method sets the lmPar and lmDir attributes.</p>
428 * <p>The authors of the original fortran function are:</p>
429 * <ul>
430 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
431 * <li>Burton S. Garbow</li>
432 * <li>Kenneth E. Hillstrom</li>
433 * <li>Jorge J. More</li>
434 * </ul>
435 * <p>Luc Maisonobe did the Java translation.</p>
436 *
437 * @param qy array containing qTy
438 * @param delta upper bound on the euclidean norm of diagR * lmDir
439 * @param diag diagonal matrix
440 * @param work1 work array
441 * @param work2 work array
442 * @param work3 work array
443 */
444 private void determineLMParameter(double[] qy, double delta, double[] diag,
445 double[] work1, double[] work2, double[] work3) {
446
447 // compute and store in x the gauss-newton direction, if the
448 // jacobian is rank-deficient, obtain a least squares solution
449 for (int j = 0; j < rank; ++j) {
450 lmDir[permutation[j]] = qy[j];
451 }
452 for (int j = rank; j < cols; ++j) {
453 lmDir[permutation[j]] = 0;
454 }
455 for (int k = rank - 1; k >= 0; --k) {
456 int pk = permutation[k];
457 double ypk = lmDir[pk] / diagR[pk];
458 for (int i = 0, index = pk; i < k; ++i, index += cols) {
459 lmDir[permutation[i]] -= ypk * jacobian[index];
460 }
461 lmDir[pk] = ypk;
462 }
463
464 // evaluate the function at the origin, and test
465 // for acceptance of the Gauss-Newton direction
466 double dxNorm = 0;
467 for (int j = 0; j < solvedCols; ++j) {
468 int pj = permutation[j];
469 double s = diag[pj] * lmDir[pj];
470 work1[pj] = s;
471 dxNorm += s * s;
472 }
473 dxNorm = Math.sqrt(dxNorm);
474 double fp = dxNorm - delta;
475 if (fp <= 0.1 * delta) {
476 lmPar = 0;
477 return;
478 }
479
480 // if the jacobian is not rank deficient, the Newton step provides
481 // a lower bound, parl, for the zero of the function,
482 // otherwise set this bound to zero
483 double sum2, parl = 0;
484 if (rank == solvedCols) {
485 for (int j = 0; j < solvedCols; ++j) {
486 int pj = permutation[j];
487 work1[pj] *= diag[pj] / dxNorm;
488 }
489 sum2 = 0;
490 for (int j = 0; j < solvedCols; ++j) {
491 int pj = permutation[j];
492 double sum = 0;
493 for (int i = 0, index = pj; i < j; ++i, index += cols) {
494 sum += jacobian[index] * work1[permutation[i]];
495 }
496 double s = (work1[pj] - sum) / diagR[pj];
497 work1[pj] = s;
498 sum2 += s * s;
499 }
500 parl = fp / (delta * sum2);
501 }
502
503 // calculate an upper bound, paru, for the zero of the function
504 sum2 = 0;
505 for (int j = 0; j < solvedCols; ++j) {
506 int pj = permutation[j];
507 double sum = 0;
508 for (int i = 0, index = pj; i <= j; ++i, index += cols) {
509 sum += jacobian[index] * qy[i];
510 }
511 sum /= diag[pj];
512 sum2 += sum * sum;
513 }
514 double gNorm = Math.sqrt(sum2);
515 double paru = gNorm / delta;
516 if (paru == 0) {
517 // 2.2251e-308 is the smallest positive real for IEE754
518 paru = 2.2251e-308 / Math.min(delta, 0.1);
519 }
520
521 // if the input par lies outside of the interval (parl,paru),
522 // set par to the closer endpoint
523 lmPar = Math.min(paru, Math.max(lmPar, parl));
524 if (lmPar == 0) {
525 lmPar = gNorm / dxNorm;
526 }
527
528 for (int countdown = 10; countdown >= 0; --countdown) {
529
530 // evaluate the function at the current value of lmPar
531 if (lmPar == 0) {
532 lmPar = Math.max(2.2251e-308, 0.001 * paru);
533 }
534 double sPar = Math.sqrt(lmPar);
535 for (int j = 0; j < solvedCols; ++j) {
536 int pj = permutation[j];
537 work1[pj] = sPar * diag[pj];
538 }
539 determineLMDirection(qy, work1, work2, work3);
540
541 dxNorm = 0;
542 for (int j = 0; j < solvedCols; ++j) {
543 int pj = permutation[j];
544 double s = diag[pj] * lmDir[pj];
545 work3[pj] = s;
546 dxNorm += s * s;
547 }
548 dxNorm = Math.sqrt(dxNorm);
549 double previousFP = fp;
550 fp = dxNorm - delta;
551
552 // if the function is small enough, accept the current value
553 // of lmPar, also test for the exceptional cases where parl is zero
554 if ((Math.abs(fp) <= 0.1 * delta) ||
555 ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
556 return;
557 }
558
559 // compute the Newton correction
560 for (int j = 0; j < solvedCols; ++j) {
561 int pj = permutation[j];
562 work1[pj] = work3[pj] * diag[pj] / dxNorm;
563 }
564 for (int j = 0; j < solvedCols; ++j) {
565 int pj = permutation[j];
566 work1[pj] /= work2[j];
567 double tmp = work1[pj];
568 for (int i = j + 1; i < solvedCols; ++i) {
569 work1[permutation[i]] -= jacobian[i * cols + pj] * tmp;
570 }
571 }
572 sum2 = 0;
573 for (int j = 0; j < solvedCols; ++j) {
574 double s = work1[permutation[j]];
575 sum2 += s * s;
576 }
577 double correction = fp / (delta * sum2);
578
579 // depending on the sign of the function, update parl or paru.
580 if (fp > 0) {
581 parl = Math.max(parl, lmPar);
582 } else if (fp < 0) {
583 paru = Math.min(paru, lmPar);
584 }
585
586 // compute an improved estimate for lmPar
587 lmPar = Math.max(parl, lmPar + correction);
588
589 }
590 }
591
592 /**
593 * Solve a*x = b and d*x = 0 in the least squares sense.
594 * <p>This implementation is a translation in Java of the MINPACK
595 * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
596 * routine.</p>
597 * <p>This method sets the lmDir and lmDiag attributes.</p>
598 * <p>The authors of the original fortran function are:</p>
599 * <ul>
600 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
601 * <li>Burton S. Garbow</li>
602 * <li>Kenneth E. Hillstrom</li>
603 * <li>Jorge J. More</li>
604 * </ul>
605 * <p>Luc Maisonobe did the Java translation.</p>
606 *
607 * @param qy array containing qTy
608 * @param diag diagonal matrix
609 * @param lmDiag diagonal elements associated with lmDir
610 * @param work work array
611 */
612 private void determineLMDirection(double[] qy, double[] diag,
613 double[] lmDiag, double[] work) {
614
615 // copy R and Qty to preserve input and initialize s
616 // in particular, save the diagonal elements of R in lmDir
617 for (int j = 0; j < solvedCols; ++j) {
618 int pj = permutation[j];
619 for (int i = j + 1; i < solvedCols; ++i) {
620 jacobian[i * cols + pj] = jacobian[j * cols + permutation[i]];
621 }
622 lmDir[j] = diagR[pj];
623 work[j] = qy[j];
624 }
625
626 // eliminate the diagonal matrix d using a Givens rotation
627 for (int j = 0; j < solvedCols; ++j) {
628
629 // prepare the row of d to be eliminated, locating the
630 // diagonal element using p from the Q.R. factorization
631 int pj = permutation[j];
632 double dpj = diag[pj];
633 if (dpj != 0) {
634 Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
635 }
636 lmDiag[j] = dpj;
637
638 // the transformations to eliminate the row of d
639 // modify only a single element of Qty
640 // beyond the first n, which is initially zero.
641 double qtbpj = 0;
642 for (int k = j; k < solvedCols; ++k) {
643 int pk = permutation[k];
644
645 // determine a Givens rotation which eliminates the
646 // appropriate element in the current row of d
647 if (lmDiag[k] != 0) {
648
649 double sin, cos;
650 double rkk = jacobian[k * cols + pk];
651 if (Math.abs(rkk) < Math.abs(lmDiag[k])) {
652 double cotan = rkk / lmDiag[k];
653 sin = 1.0 / Math.sqrt(1.0 + cotan * cotan);
654 cos = sin * cotan;
655 } else {
656 double tan = lmDiag[k] / rkk;
657 cos = 1.0 / Math.sqrt(1.0 + tan * tan);
658 sin = cos * tan;
659 }
660
661 // compute the modified diagonal element of R and
662 // the modified element of (Qty,0)
663 jacobian[k * cols + pk] = cos * rkk + sin * lmDiag[k];
664 double temp = cos * work[k] + sin * qtbpj;
665 qtbpj = -sin * work[k] + cos * qtbpj;
666 work[k] = temp;
667
668 // accumulate the tranformation in the row of s
669 for (int i = k + 1; i < solvedCols; ++i) {
670 double rik = jacobian[i * cols + pk];
671 temp = cos * rik + sin * lmDiag[i];
672 lmDiag[i] = -sin * rik + cos * lmDiag[i];
673 jacobian[i * cols + pk] = temp;
674 }
675
676 }
677 }
678
679 // store the diagonal element of s and restore
680 // the corresponding diagonal element of R
681 int index = j * cols + permutation[j];
682 lmDiag[j] = jacobian[index];
683 jacobian[index] = lmDir[j];
684
685 }
686
687 // solve the triangular system for z, if the system is
688 // singular, then obtain a least squares solution
689 int nSing = solvedCols;
690 for (int j = 0; j < solvedCols; ++j) {
691 if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
692 nSing = j;
693 }
694 if (nSing < solvedCols) {
695 work[j] = 0;
696 }
697 }
698 if (nSing > 0) {
699 for (int j = nSing - 1; j >= 0; --j) {
700 int pj = permutation[j];
701 double sum = 0;
702 for (int i = j + 1; i < nSing; ++i) {
703 sum += jacobian[i * cols + pj] * work[i];
704 }
705 work[j] = (work[j] - sum) / lmDiag[j];
706 }
707 }
708
709 // permute the components of z back to components of lmDir
710 for (int j = 0; j < lmDir.length; ++j) {
711 lmDir[permutation[j]] = work[j];
712 }
713
714 }
715
716 /**
717 * Decompose a matrix A as A.P = Q.R using Householder transforms.
718 * <p>As suggested in the P. Lascaux and R. Theodor book
719 * <i>Analyse numérique matricielle appliquée à
720 * l'art de l'ingénieur</i> (Masson, 1986), instead of representing
721 * the Householder transforms with u<sub>k</sub> unit vectors such that:
722 * <pre>
723 * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
724 * </pre>
725 * we use <sub>k</sub> non-unit vectors such that:
726 * <pre>
727 * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
728 * </pre>
729 * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
730 * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
731 * them from the v<sub>k</sub> vectors would be costly.</p>
732 * <p>This decomposition handles rank deficient cases since the tranformations
733 * are performed in non-increasing columns norms order thanks to columns
734 * pivoting. The diagonal elements of the R matrix are therefore also in
735 * non-increasing absolute values order.</p>
736 */
737 private void qrDecomposition() {
738
739 // initializations
740 for (int k = 0; k < cols; ++k) {
741 permutation[k] = k;
742 double norm2 = 0;
743 for (int index = k; index < jacobian.length; index += cols) {
744 double akk = jacobian[index];
745 norm2 += akk * akk;
746 }
747 jacNorm[k] = Math.sqrt(norm2);
748 }
749
750 // transform the matrix column after column
751 for (int k = 0; k < cols; ++k) {
752
753 // select the column with the greatest norm on active components
754 int nextColumn = -1;
755 double ak2 = Double.NEGATIVE_INFINITY;
756 for (int i = k; i < cols; ++i) {
757 double norm2 = 0;
758 int iDiag = k * cols + permutation[i];
759 for (int index = iDiag; index < jacobian.length; index += cols) {
760 double aki = jacobian[index];
761 norm2 += aki * aki;
762 }
763 if (norm2 > ak2) {
764 nextColumn = i;
765 ak2 = norm2;
766 }
767 }
768 if (ak2 == 0) {
769 rank = k;
770 return;
771 }
772 int pk = permutation[nextColumn];
773 permutation[nextColumn] = permutation[k];
774 permutation[k] = pk;
775
776 // choose alpha such that Hk.u = alpha ek
777 int kDiag = k * cols + pk;
778 double akk = jacobian[kDiag];
779 double alpha = (akk > 0) ? -Math.sqrt(ak2) : Math.sqrt(ak2);
780 double betak = 1.0 / (ak2 - akk * alpha);
781 beta[pk] = betak;
782
783 // transform the current column
784 diagR[pk] = alpha;
785 jacobian[kDiag] -= alpha;
786
787 // transform the remaining columns
788 for (int dk = cols - 1 - k; dk > 0; --dk) {
789 int dkp = permutation[k + dk] - pk;
790 double gamma = 0;
791 for (int index = kDiag; index < jacobian.length; index += cols) {
792 gamma += jacobian[index] * jacobian[index + dkp];
793 }
794 gamma *= betak;
795 for (int index = kDiag; index < jacobian.length; index += cols) {
796 jacobian[index + dkp] -= gamma * jacobian[index];
797 }
798 }
799
800 }
801
802 rank = solvedCols;
803
804 }
805
806 /**
807 * Compute the product Qt.y for some Q.R. decomposition.
808 *
809 * @param y vector to multiply (will be overwritten with the result)
810 */
811 private void qTy(double[] y) {
812 for (int k = 0; k < cols; ++k) {
813 int pk = permutation[k];
814 int kDiag = k * cols + pk;
815 double gamma = 0;
816 for (int i = k, index = kDiag; i < rows; ++i, index += cols) {
817 gamma += jacobian[index] * y[i];
818 }
819 gamma *= beta[pk];
820 for (int i = k, index = kDiag; i < rows; ++i, index += cols) {
821 y[i] -= gamma * jacobian[index];
822 }
823 }
824 }
825
826 /** Number of solved variables. */
827 private int solvedCols;
828
829 /** Diagonal elements of the R matrix in the Q.R. decomposition. */
830 private double[] diagR;
831
832 /** Norms of the columns of the jacobian matrix. */
833 private double[] jacNorm;
834
835 /** Coefficients of the Householder transforms vectors. */
836 private double[] beta;
837
838 /** Columns permutation array. */
839 private int[] permutation;
840
841 /** Rank of the jacobian matrix. */
842 private int rank;
843
844 /** Levenberg-Marquardt parameter. */
845 private double lmPar;
846
847 /** Parameters evolution direction associated with lmPar. */
848 private double[] lmDir;
849
850 /** Positive input variable used in determining the initial step bound. */
851 private double initialStepBoundFactor;
852
853 /** Desired relative error in the sum of squares. */
854 private double costRelativeTolerance;
855
856 /** Desired relative error in the approximate solution parameters. */
857 private double parRelativeTolerance;
858
859 /** Desired max cosine on the orthogonality between the function vector
860 * and the columns of the jacobian. */
861 private double orthoTolerance;
862
863 /** Serializable version identifier */
864 private static final long serialVersionUID = -5705952631533171019L;
865
866 }