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.optimization.general;
18
19 import java.util.Arrays;
20
21 import org.apache.commons.math.exception.ConvergenceException;
22 import org.apache.commons.math.exception.util.LocalizedFormats;
23 import org.apache.commons.math.optimization.VectorialPointValuePair;
24 import org.apache.commons.math.optimization.ConvergenceChecker;
25 import org.apache.commons.math.util.MathUtils;
26 import org.apache.commons.math.util.FastMath;
27
28
29 /**
30 * This class solves a least squares problem using the Levenberg-Marquardt algorithm.
31 *
32 * <p>This implementation <em>should</em> work even for over-determined systems
33 * (i.e. systems having more point than equations). Over-determined systems
34 * are solved by ignoring the point which have the smallest impact according
35 * to their jacobian column norm. Only the rank of the matrix and some loop bounds
36 * are changed to implement this.</p>
37 *
38 * <p>The resolution engine is a simple translation of the MINPACK <a
39 * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
40 * changes. The changes include the over-determined resolution, the use of
41 * inherited convergence checker and the Q.R. decomposition which has been
42 * rewritten following the algorithm described in the
43 * P. Lascaux and R. Theodor book <i>Analyse numérique matricielle
44 * appliquée à l'art de l'ingénieur</i>, Masson 1986.</p>
45 * <p>The authors of the original fortran version are:
46 * <ul>
47 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
48 * <li>Burton S. Garbow</li>
49 * <li>Kenneth E. Hillstrom</li>
50 * <li>Jorge J. More</li>
51 * </ul>
52 * The redistribution policy for MINPACK is available <a
53 * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
54 * is reproduced below.</p>
55 *
56 * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
57 * <tr><td>
58 * Minpack Copyright Notice (1999) University of Chicago.
59 * All rights reserved
60 * </td></tr>
61 * <tr><td>
62 * Redistribution and use in source and binary forms, with or without
63 * modification, are permitted provided that the following conditions
64 * are met:
65 * <ol>
66 * <li>Redistributions of source code must retain the above copyright
67 * notice, this list of conditions and the following disclaimer.</li>
68 * <li>Redistributions in binary form must reproduce the above
69 * copyright notice, this list of conditions and the following
70 * disclaimer in the documentation and/or other materials provided
71 * with the distribution.</li>
72 * <li>The end-user documentation included with the redistribution, if any,
73 * must include the following acknowledgment:
74 * <code>This product includes software developed by the University of
75 * Chicago, as Operator of Argonne National Laboratory.</code>
76 * Alternately, this acknowledgment may appear in the software itself,
77 * if and wherever such third-party acknowledgments normally appear.</li>
78 * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
79 * WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
80 * UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
81 * THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
82 * IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
83 * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
84 * OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
85 * OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
86 * USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
87 * THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
88 * DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
89 * UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
90 * BE CORRECTED.</strong></li>
91 * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
92 * HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
93 * ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
94 * INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
95 * ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
96 * PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
97 * SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
98 * (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
99 * EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
100 * POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
101 * <ol></td></tr>
102 * </table>
103 * @version $Id: LevenbergMarquardtOptimizer.java 1166311 2011-09-07 18:48:06Z luc $
104 * @since 2.0
105 *
106 */
107 public class LevenbergMarquardtOptimizer extends AbstractLeastSquaresOptimizer {
108 /** Number of solved point. */
109 private int solvedCols;
110 /** Diagonal elements of the R matrix in the Q.R. decomposition. */
111 private double[] diagR;
112 /** Norms of the columns of the jacobian matrix. */
113 private double[] jacNorm;
114 /** Coefficients of the Householder transforms vectors. */
115 private double[] beta;
116 /** Columns permutation array. */
117 private int[] permutation;
118 /** Rank of the jacobian matrix. */
119 private int rank;
120 /** Levenberg-Marquardt parameter. */
121 private double lmPar;
122 /** Parameters evolution direction associated with lmPar. */
123 private double[] lmDir;
124 /** Positive input variable used in determining the initial step bound. */
125 private final double initialStepBoundFactor;
126 /** Desired relative error in the sum of squares. */
127 private final double costRelativeTolerance;
128 /** Desired relative error in the approximate solution parameters. */
129 private final double parRelativeTolerance;
130 /** Desired max cosine on the orthogonality between the function vector
131 * and the columns of the jacobian. */
132 private final double orthoTolerance;
133 /** Threshold for QR ranking. */
134 private final double qrRankingThreshold;
135
136 /**
137 * Build an optimizer for least squares problems with default values
138 * for all the tuning parameters (see the {@link
139 * #LevenbergMarquardtOptimizer(double,double,double,double,double)
140 * other contructor}.
141 * The default values for the algorithm settings are:
142 * <ul>
143 * <li>Initial step bound factor: 100</li>
144 * <li>Cost relative tolerance: 1e-10</li>
145 * <li>Parameters relative tolerance: 1e-10</li>
146 * <li>Orthogonality tolerance: 1e-10</li>
147 * <li>QR ranking threshold: {@link MathUtils#SAFE_MIN}</li>
148 * </ul>
149 */
150 public LevenbergMarquardtOptimizer() {
151 this(100, 1e-10, 1e-10, 1e-10, MathUtils.SAFE_MIN);
152 }
153
154 /**
155 * Constructor that allows the specification of a custom convergence
156 * checker.
157 * Note that all the usual convergence checks will be <em>disabled</em>.
158 * The default values for the algorithm settings are:
159 * <ul>
160 * <li>Initial step bound factor: 100</li>
161 * <li>Cost relative tolerance: 1e-10</li>
162 * <li>Parameters relative tolerance: 1e-10</li>
163 * <li>Orthogonality tolerance: 1e-10</li>
164 * <li>QR ranking threshold: {@link MathUtils#SAFE_MIN}</li>
165 * </ul>
166 *
167 * @param checker Convergence checker.
168 */
169 public LevenbergMarquardtOptimizer(ConvergenceChecker<VectorialPointValuePair> checker) {
170 this(100, checker, 1e-10, 1e-10, 1e-10, MathUtils.SAFE_MIN);
171 }
172
173 /**
174 * Constructor that allows the specification of a custom convergence
175 * checker, in addition to the standard ones.
176 *
177 * @param initialStepBoundFactor Positive input variable used in
178 * determining the initial step bound. This bound is set to the
179 * product of initialStepBoundFactor and the euclidean norm of
180 * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
181 * itself. In most cases factor should lie in the interval
182 * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
183 * @param checker Convergence checker.
184 * @param costRelativeTolerance Desired relative error in the sum of
185 * squares.
186 * @param parRelativeTolerance Desired relative error in the approximate
187 * solution parameters.
188 * @param orthoTolerance Desired max cosine on the orthogonality between
189 * the function vector and the columns of the Jacobian.
190 * @param threshold Desired threshold for QR ranking. If the squared norm
191 * of a column vector is smaller or equal to this threshold during QR
192 * decomposition, it is considered to be a zero vector and hence the rank
193 * of the matrix is reduced.
194 */
195 public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
196 ConvergenceChecker<VectorialPointValuePair> checker,
197 double costRelativeTolerance,
198 double parRelativeTolerance,
199 double orthoTolerance,
200 double threshold) {
201 super(checker);
202 this.initialStepBoundFactor = initialStepBoundFactor;
203 this.costRelativeTolerance = costRelativeTolerance;
204 this.parRelativeTolerance = parRelativeTolerance;
205 this.orthoTolerance = orthoTolerance;
206 this.qrRankingThreshold = threshold;
207 }
208
209 /**
210 * Build an optimizer for least squares problems with default values
211 * for some of the tuning parameters (see the {@link
212 * #LevenbergMarquardtOptimizer(double,double,double,double,double)
213 * other contructor}.
214 * The default values for the algorithm settings are:
215 * <ul>
216 * <li>Initial step bound factor}: 100</li>
217 * <li>QR ranking threshold}: {@link MathUtils#SAFE_MIN}</li>
218 * </ul>
219 *
220 * @param costRelativeTolerance Desired relative error in the sum of
221 * squares.
222 * @param parRelativeTolerance Desired relative error in the approximate
223 * solution parameters.
224 * @param orthoTolerance Desired max cosine on the orthogonality between
225 * the function vector and the columns of the Jacobian.
226 */
227 public LevenbergMarquardtOptimizer(double costRelativeTolerance,
228 double parRelativeTolerance,
229 double orthoTolerance) {
230 this(100,
231 costRelativeTolerance, parRelativeTolerance, orthoTolerance,
232 MathUtils.SAFE_MIN);
233 }
234
235 /**
236 * The arguments control the behaviour of the default convergence checking
237 * procedure.
238 * Additional criteria can defined through the setting of a {@link
239 * ConvergenceChecker}.
240 *
241 * @param initialStepBoundFactor Positive input variable used in
242 * determining the initial step bound. This bound is set to the
243 * product of initialStepBoundFactor and the euclidean norm of
244 * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
245 * itself. In most cases factor should lie in the interval
246 * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
247 * @param costRelativeTolerance Desired relative error in the sum of
248 * squares.
249 * @param parRelativeTolerance Desired relative error in the approximate
250 * solution parameters.
251 * @param orthoTolerance Desired max cosine on the orthogonality between
252 * the function vector and the columns of the Jacobian.
253 * @param threshold Desired threshold for QR ranking. If the squared norm
254 * of a column vector is smaller or equal to this threshold during QR
255 * decomposition, it is considered to be a zero vector and hence the rank
256 * of the matrix is reduced.
257 */
258 public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
259 double costRelativeTolerance,
260 double parRelativeTolerance,
261 double orthoTolerance,
262 double threshold) {
263 this.initialStepBoundFactor = initialStepBoundFactor;
264 this.costRelativeTolerance = costRelativeTolerance;
265 this.parRelativeTolerance = parRelativeTolerance;
266 this.orthoTolerance = orthoTolerance;
267 this.qrRankingThreshold = threshold;
268 }
269
270 /** {@inheritDoc} */
271 @Override
272 protected VectorialPointValuePair doOptimize() {
273 // arrays shared with the other private methods
274 solvedCols = FastMath.min(rows, cols);
275 diagR = new double[cols];
276 jacNorm = new double[cols];
277 beta = new double[cols];
278 permutation = new int[cols];
279 lmDir = new double[cols];
280
281 // local point
282 double delta = 0;
283 double xNorm = 0;
284 double[] diag = new double[cols];
285 double[] oldX = new double[cols];
286 double[] oldRes = new double[rows];
287 double[] oldObj = new double[rows];
288 double[] qtf = new double[rows];
289 double[] work1 = new double[cols];
290 double[] work2 = new double[cols];
291 double[] work3 = new double[cols];
292
293 // evaluate the function at the starting point and calculate its norm
294 updateResidualsAndCost();
295
296 // outer loop
297 lmPar = 0;
298 boolean firstIteration = true;
299 VectorialPointValuePair current = new VectorialPointValuePair(point, objective);
300 int iter = 0;
301 final ConvergenceChecker<VectorialPointValuePair> checker = getConvergenceChecker();
302 while (true) {
303 ++iter;
304
305 for (int i=0;i<rows;i++) {
306 qtf[i]=weightedResiduals[i];
307 }
308
309 // compute the Q.R. decomposition of the jacobian matrix
310 VectorialPointValuePair previous = current;
311 updateJacobian();
312 qrDecomposition();
313
314 // compute Qt.res
315 qTy(qtf);
316 // now we don't need Q anymore,
317 // so let jacobian contain the R matrix with its diagonal elements
318 for (int k = 0; k < solvedCols; ++k) {
319 int pk = permutation[k];
320 weightedResidualJacobian[k][pk] = diagR[pk];
321 }
322
323 if (firstIteration) {
324 // scale the point according to the norms of the columns
325 // of the initial jacobian
326 xNorm = 0;
327 for (int k = 0; k < cols; ++k) {
328 double dk = jacNorm[k];
329 if (dk == 0) {
330 dk = 1.0;
331 }
332 double xk = dk * point[k];
333 xNorm += xk * xk;
334 diag[k] = dk;
335 }
336 xNorm = FastMath.sqrt(xNorm);
337
338 // initialize the step bound delta
339 delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
340 }
341
342 // check orthogonality between function vector and jacobian columns
343 double maxCosine = 0;
344 if (cost != 0) {
345 for (int j = 0; j < solvedCols; ++j) {
346 int pj = permutation[j];
347 double s = jacNorm[pj];
348 if (s != 0) {
349 double sum = 0;
350 for (int i = 0; i <= j; ++i) {
351 sum += weightedResidualJacobian[i][pj] * qtf[i];
352 }
353 maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * cost));
354 }
355 }
356 }
357 if (maxCosine <= orthoTolerance) {
358 // convergence has been reached
359 updateResidualsAndCost();
360 current = new VectorialPointValuePair(point, objective);
361 return current;
362 }
363
364 // rescale if necessary
365 for (int j = 0; j < cols; ++j) {
366 diag[j] = FastMath.max(diag[j], jacNorm[j]);
367 }
368
369 // inner loop
370 for (double ratio = 0; ratio < 1.0e-4;) {
371
372 // save the state
373 for (int j = 0; j < solvedCols; ++j) {
374 int pj = permutation[j];
375 oldX[pj] = point[pj];
376 }
377 double previousCost = cost;
378 double[] tmpVec = weightedResiduals;
379 weightedResiduals = oldRes;
380 oldRes = tmpVec;
381 tmpVec = objective;
382 objective = oldObj;
383 oldObj = tmpVec;
384
385 // determine the Levenberg-Marquardt parameter
386 determineLMParameter(qtf, delta, diag, work1, work2, work3);
387
388 // compute the new point and the norm of the evolution direction
389 double lmNorm = 0;
390 for (int j = 0; j < solvedCols; ++j) {
391 int pj = permutation[j];
392 lmDir[pj] = -lmDir[pj];
393 point[pj] = oldX[pj] + lmDir[pj];
394 double s = diag[pj] * lmDir[pj];
395 lmNorm += s * s;
396 }
397 lmNorm = FastMath.sqrt(lmNorm);
398 // on the first iteration, adjust the initial step bound.
399 if (firstIteration) {
400 delta = FastMath.min(delta, lmNorm);
401 }
402
403 // evaluate the function at x + p and calculate its norm
404 updateResidualsAndCost();
405
406 // compute the scaled actual reduction
407 double actRed = -1.0;
408 if (0.1 * cost < previousCost) {
409 double r = cost / previousCost;
410 actRed = 1.0 - r * r;
411 }
412
413 // compute the scaled predicted reduction
414 // and the scaled directional derivative
415 for (int j = 0; j < solvedCols; ++j) {
416 int pj = permutation[j];
417 double dirJ = lmDir[pj];
418 work1[j] = 0;
419 for (int i = 0; i <= j; ++i) {
420 work1[i] += weightedResidualJacobian[i][pj] * dirJ;
421 }
422 }
423 double coeff1 = 0;
424 for (int j = 0; j < solvedCols; ++j) {
425 coeff1 += work1[j] * work1[j];
426 }
427 double pc2 = previousCost * previousCost;
428 coeff1 = coeff1 / pc2;
429 double coeff2 = lmPar * lmNorm * lmNorm / pc2;
430 double preRed = coeff1 + 2 * coeff2;
431 double dirDer = -(coeff1 + coeff2);
432
433 // ratio of the actual to the predicted reduction
434 ratio = (preRed == 0) ? 0 : (actRed / preRed);
435
436 // update the step bound
437 if (ratio <= 0.25) {
438 double tmp =
439 (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
440 if ((0.1 * cost >= previousCost) || (tmp < 0.1)) {
441 tmp = 0.1;
442 }
443 delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
444 lmPar /= tmp;
445 } else if ((lmPar == 0) || (ratio >= 0.75)) {
446 delta = 2 * lmNorm;
447 lmPar *= 0.5;
448 }
449
450 // test for successful iteration.
451 if (ratio >= 1.0e-4) {
452 // successful iteration, update the norm
453 firstIteration = false;
454 xNorm = 0;
455 for (int k = 0; k < cols; ++k) {
456 double xK = diag[k] * point[k];
457 xNorm += xK * xK;
458 }
459 xNorm = FastMath.sqrt(xNorm);
460 current = new VectorialPointValuePair(point, objective);
461
462 // tests for convergence.
463 if (checker != null) {
464 // we use the vectorial convergence checker
465 if (checker.converged(iter, previous, current)) {
466 return current;
467 }
468 }
469 } else {
470 // failed iteration, reset the previous values
471 cost = previousCost;
472 for (int j = 0; j < solvedCols; ++j) {
473 int pj = permutation[j];
474 point[pj] = oldX[pj];
475 }
476 tmpVec = weightedResiduals;
477 weightedResiduals = oldRes;
478 oldRes = tmpVec;
479 tmpVec = objective;
480 objective = oldObj;
481 oldObj = tmpVec;
482 }
483
484 // Default convergence criteria.
485 if ((FastMath.abs(actRed) <= costRelativeTolerance &&
486 preRed <= costRelativeTolerance &&
487 ratio <= 2.0) ||
488 delta <= parRelativeTolerance * xNorm) {
489 return current;
490 }
491
492 // tests for termination and stringent tolerances
493 // (2.2204e-16 is the machine epsilon for IEEE754)
494 if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
495 throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
496 costRelativeTolerance);
497 } else if (delta <= 2.2204e-16 * xNorm) {
498 throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
499 parRelativeTolerance);
500 } else if (maxCosine <= 2.2204e-16) {
501 throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
502 orthoTolerance);
503 }
504 }
505 }
506 }
507
508 /**
509 * Determine the Levenberg-Marquardt parameter.
510 * <p>This implementation is a translation in Java of the MINPACK
511 * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
512 * routine.</p>
513 * <p>This method sets the lmPar and lmDir attributes.</p>
514 * <p>The authors of the original fortran function are:</p>
515 * <ul>
516 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
517 * <li>Burton S. Garbow</li>
518 * <li>Kenneth E. Hillstrom</li>
519 * <li>Jorge J. More</li>
520 * </ul>
521 * <p>Luc Maisonobe did the Java translation.</p>
522 *
523 * @param qy array containing qTy
524 * @param delta upper bound on the euclidean norm of diagR * lmDir
525 * @param diag diagonal matrix
526 * @param work1 work array
527 * @param work2 work array
528 * @param work3 work array
529 */
530 private void determineLMParameter(double[] qy, double delta, double[] diag,
531 double[] work1, double[] work2, double[] work3) {
532
533 // compute and store in x the gauss-newton direction, if the
534 // jacobian is rank-deficient, obtain a least squares solution
535 for (int j = 0; j < rank; ++j) {
536 lmDir[permutation[j]] = qy[j];
537 }
538 for (int j = rank; j < cols; ++j) {
539 lmDir[permutation[j]] = 0;
540 }
541 for (int k = rank - 1; k >= 0; --k) {
542 int pk = permutation[k];
543 double ypk = lmDir[pk] / diagR[pk];
544 for (int i = 0; i < k; ++i) {
545 lmDir[permutation[i]] -= ypk * weightedResidualJacobian[i][pk];
546 }
547 lmDir[pk] = ypk;
548 }
549
550 // evaluate the function at the origin, and test
551 // for acceptance of the Gauss-Newton direction
552 double dxNorm = 0;
553 for (int j = 0; j < solvedCols; ++j) {
554 int pj = permutation[j];
555 double s = diag[pj] * lmDir[pj];
556 work1[pj] = s;
557 dxNorm += s * s;
558 }
559 dxNorm = FastMath.sqrt(dxNorm);
560 double fp = dxNorm - delta;
561 if (fp <= 0.1 * delta) {
562 lmPar = 0;
563 return;
564 }
565
566 // if the jacobian is not rank deficient, the Newton step provides
567 // a lower bound, parl, for the zero of the function,
568 // otherwise set this bound to zero
569 double sum2;
570 double parl = 0;
571 if (rank == solvedCols) {
572 for (int j = 0; j < solvedCols; ++j) {
573 int pj = permutation[j];
574 work1[pj] *= diag[pj] / dxNorm;
575 }
576 sum2 = 0;
577 for (int j = 0; j < solvedCols; ++j) {
578 int pj = permutation[j];
579 double sum = 0;
580 for (int i = 0; i < j; ++i) {
581 sum += weightedResidualJacobian[i][pj] * work1[permutation[i]];
582 }
583 double s = (work1[pj] - sum) / diagR[pj];
584 work1[pj] = s;
585 sum2 += s * s;
586 }
587 parl = fp / (delta * sum2);
588 }
589
590 // calculate an upper bound, paru, for the zero of the function
591 sum2 = 0;
592 for (int j = 0; j < solvedCols; ++j) {
593 int pj = permutation[j];
594 double sum = 0;
595 for (int i = 0; i <= j; ++i) {
596 sum += weightedResidualJacobian[i][pj] * qy[i];
597 }
598 sum /= diag[pj];
599 sum2 += sum * sum;
600 }
601 double gNorm = FastMath.sqrt(sum2);
602 double paru = gNorm / delta;
603 if (paru == 0) {
604 // 2.2251e-308 is the smallest positive real for IEE754
605 paru = 2.2251e-308 / FastMath.min(delta, 0.1);
606 }
607
608 // if the input par lies outside of the interval (parl,paru),
609 // set par to the closer endpoint
610 lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
611 if (lmPar == 0) {
612 lmPar = gNorm / dxNorm;
613 }
614
615 for (int countdown = 10; countdown >= 0; --countdown) {
616
617 // evaluate the function at the current value of lmPar
618 if (lmPar == 0) {
619 lmPar = FastMath.max(2.2251e-308, 0.001 * paru);
620 }
621 double sPar = FastMath.sqrt(lmPar);
622 for (int j = 0; j < solvedCols; ++j) {
623 int pj = permutation[j];
624 work1[pj] = sPar * diag[pj];
625 }
626 determineLMDirection(qy, work1, work2, work3);
627
628 dxNorm = 0;
629 for (int j = 0; j < solvedCols; ++j) {
630 int pj = permutation[j];
631 double s = diag[pj] * lmDir[pj];
632 work3[pj] = s;
633 dxNorm += s * s;
634 }
635 dxNorm = FastMath.sqrt(dxNorm);
636 double previousFP = fp;
637 fp = dxNorm - delta;
638
639 // if the function is small enough, accept the current value
640 // of lmPar, also test for the exceptional cases where parl is zero
641 if ((FastMath.abs(fp) <= 0.1 * delta) ||
642 ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
643 return;
644 }
645
646 // compute the Newton correction
647 for (int j = 0; j < solvedCols; ++j) {
648 int pj = permutation[j];
649 work1[pj] = work3[pj] * diag[pj] / dxNorm;
650 }
651 for (int j = 0; j < solvedCols; ++j) {
652 int pj = permutation[j];
653 work1[pj] /= work2[j];
654 double tmp = work1[pj];
655 for (int i = j + 1; i < solvedCols; ++i) {
656 work1[permutation[i]] -= weightedResidualJacobian[i][pj] * tmp;
657 }
658 }
659 sum2 = 0;
660 for (int j = 0; j < solvedCols; ++j) {
661 double s = work1[permutation[j]];
662 sum2 += s * s;
663 }
664 double correction = fp / (delta * sum2);
665
666 // depending on the sign of the function, update parl or paru.
667 if (fp > 0) {
668 parl = FastMath.max(parl, lmPar);
669 } else if (fp < 0) {
670 paru = FastMath.min(paru, lmPar);
671 }
672
673 // compute an improved estimate for lmPar
674 lmPar = FastMath.max(parl, lmPar + correction);
675
676 }
677 }
678
679 /**
680 * Solve a*x = b and d*x = 0 in the least squares sense.
681 * <p>This implementation is a translation in Java of the MINPACK
682 * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
683 * routine.</p>
684 * <p>This method sets the lmDir and lmDiag attributes.</p>
685 * <p>The authors of the original fortran function are:</p>
686 * <ul>
687 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
688 * <li>Burton S. Garbow</li>
689 * <li>Kenneth E. Hillstrom</li>
690 * <li>Jorge J. More</li>
691 * </ul>
692 * <p>Luc Maisonobe did the Java translation.</p>
693 *
694 * @param qy array containing qTy
695 * @param diag diagonal matrix
696 * @param lmDiag diagonal elements associated with lmDir
697 * @param work work array
698 */
699 private void determineLMDirection(double[] qy, double[] diag,
700 double[] lmDiag, double[] work) {
701
702 // copy R and Qty to preserve input and initialize s
703 // in particular, save the diagonal elements of R in lmDir
704 for (int j = 0; j < solvedCols; ++j) {
705 int pj = permutation[j];
706 for (int i = j + 1; i < solvedCols; ++i) {
707 weightedResidualJacobian[i][pj] = weightedResidualJacobian[j][permutation[i]];
708 }
709 lmDir[j] = diagR[pj];
710 work[j] = qy[j];
711 }
712
713 // eliminate the diagonal matrix d using a Givens rotation
714 for (int j = 0; j < solvedCols; ++j) {
715
716 // prepare the row of d to be eliminated, locating the
717 // diagonal element using p from the Q.R. factorization
718 int pj = permutation[j];
719 double dpj = diag[pj];
720 if (dpj != 0) {
721 Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
722 }
723 lmDiag[j] = dpj;
724
725 // the transformations to eliminate the row of d
726 // modify only a single element of Qty
727 // beyond the first n, which is initially zero.
728 double qtbpj = 0;
729 for (int k = j; k < solvedCols; ++k) {
730 int pk = permutation[k];
731
732 // determine a Givens rotation which eliminates the
733 // appropriate element in the current row of d
734 if (lmDiag[k] != 0) {
735
736 final double sin;
737 final double cos;
738 double rkk = weightedResidualJacobian[k][pk];
739 if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
740 final double cotan = rkk / lmDiag[k];
741 sin = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
742 cos = sin * cotan;
743 } else {
744 final double tan = lmDiag[k] / rkk;
745 cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
746 sin = cos * tan;
747 }
748
749 // compute the modified diagonal element of R and
750 // the modified element of (Qty,0)
751 weightedResidualJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
752 final double temp = cos * work[k] + sin * qtbpj;
753 qtbpj = -sin * work[k] + cos * qtbpj;
754 work[k] = temp;
755
756 // accumulate the tranformation in the row of s
757 for (int i = k + 1; i < solvedCols; ++i) {
758 double rik = weightedResidualJacobian[i][pk];
759 final double temp2 = cos * rik + sin * lmDiag[i];
760 lmDiag[i] = -sin * rik + cos * lmDiag[i];
761 weightedResidualJacobian[i][pk] = temp2;
762 }
763 }
764 }
765
766 // store the diagonal element of s and restore
767 // the corresponding diagonal element of R
768 lmDiag[j] = weightedResidualJacobian[j][permutation[j]];
769 weightedResidualJacobian[j][permutation[j]] = lmDir[j];
770 }
771
772 // solve the triangular system for z, if the system is
773 // singular, then obtain a least squares solution
774 int nSing = solvedCols;
775 for (int j = 0; j < solvedCols; ++j) {
776 if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
777 nSing = j;
778 }
779 if (nSing < solvedCols) {
780 work[j] = 0;
781 }
782 }
783 if (nSing > 0) {
784 for (int j = nSing - 1; j >= 0; --j) {
785 int pj = permutation[j];
786 double sum = 0;
787 for (int i = j + 1; i < nSing; ++i) {
788 sum += weightedResidualJacobian[i][pj] * work[i];
789 }
790 work[j] = (work[j] - sum) / lmDiag[j];
791 }
792 }
793
794 // permute the components of z back to components of lmDir
795 for (int j = 0; j < lmDir.length; ++j) {
796 lmDir[permutation[j]] = work[j];
797 }
798 }
799
800 /**
801 * Decompose a matrix A as A.P = Q.R using Householder transforms.
802 * <p>As suggested in the P. Lascaux and R. Theodor book
803 * <i>Analyse numérique matricielle appliquée à
804 * l'art de l'ingénieur</i> (Masson, 1986), instead of representing
805 * the Householder transforms with u<sub>k</sub> unit vectors such that:
806 * <pre>
807 * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
808 * </pre>
809 * we use <sub>k</sub> non-unit vectors such that:
810 * <pre>
811 * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
812 * </pre>
813 * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
814 * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
815 * them from the v<sub>k</sub> vectors would be costly.</p>
816 * <p>This decomposition handles rank deficient cases since the tranformations
817 * are performed in non-increasing columns norms order thanks to columns
818 * pivoting. The diagonal elements of the R matrix are therefore also in
819 * non-increasing absolute values order.</p>
820 * @exception ConvergenceException if the decomposition cannot be performed
821 */
822 private void qrDecomposition() throws ConvergenceException {
823
824 // initializations
825 for (int k = 0; k < cols; ++k) {
826 permutation[k] = k;
827 double norm2 = 0;
828 for (int i = 0; i < weightedResidualJacobian.length; ++i) {
829 double akk = weightedResidualJacobian[i][k];
830 norm2 += akk * akk;
831 }
832 jacNorm[k] = FastMath.sqrt(norm2);
833 }
834
835 // transform the matrix column after column
836 for (int k = 0; k < cols; ++k) {
837
838 // select the column with the greatest norm on active components
839 int nextColumn = -1;
840 double ak2 = Double.NEGATIVE_INFINITY;
841 for (int i = k; i < cols; ++i) {
842 double norm2 = 0;
843 for (int j = k; j < weightedResidualJacobian.length; ++j) {
844 double aki = weightedResidualJacobian[j][permutation[i]];
845 norm2 += aki * aki;
846 }
847 if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
848 throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
849 rows, cols);
850 }
851 if (norm2 > ak2) {
852 nextColumn = i;
853 ak2 = norm2;
854 }
855 }
856 if (ak2 <= qrRankingThreshold) {
857 rank = k;
858 return;
859 }
860 int pk = permutation[nextColumn];
861 permutation[nextColumn] = permutation[k];
862 permutation[k] = pk;
863
864 // choose alpha such that Hk.u = alpha ek
865 double akk = weightedResidualJacobian[k][pk];
866 double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
867 double betak = 1.0 / (ak2 - akk * alpha);
868 beta[pk] = betak;
869
870 // transform the current column
871 diagR[pk] = alpha;
872 weightedResidualJacobian[k][pk] -= alpha;
873
874 // transform the remaining columns
875 for (int dk = cols - 1 - k; dk > 0; --dk) {
876 double gamma = 0;
877 for (int j = k; j < weightedResidualJacobian.length; ++j) {
878 gamma += weightedResidualJacobian[j][pk] * weightedResidualJacobian[j][permutation[k + dk]];
879 }
880 gamma *= betak;
881 for (int j = k; j < weightedResidualJacobian.length; ++j) {
882 weightedResidualJacobian[j][permutation[k + dk]] -= gamma * weightedResidualJacobian[j][pk];
883 }
884 }
885 }
886 rank = solvedCols;
887 }
888
889 /**
890 * Compute the product Qt.y for some Q.R. decomposition.
891 *
892 * @param y vector to multiply (will be overwritten with the result)
893 */
894 private void qTy(double[] y) {
895 for (int k = 0; k < cols; ++k) {
896 int pk = permutation[k];
897 double gamma = 0;
898 for (int i = k; i < rows; ++i) {
899 gamma += weightedResidualJacobian[i][pk] * y[i];
900 }
901 gamma *= beta[pk];
902 for (int i = k; i < rows; ++i) {
903 y[i] -= gamma * weightedResidualJacobian[i][pk];
904 }
905 }
906 }
907 }