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.exception.util.LocalizedFormats;
023 import org.apache.commons.math.optimization.OptimizationException;
024 import org.apache.commons.math.optimization.VectorialPointValuePair;
025 import org.apache.commons.math.util.FastMath;
026 import org.apache.commons.math.util.MathUtils;
027
028
029 /**
030 * This class solves a least squares problem using the Levenberg-Marquardt algorithm.
031 *
032 * <p>This implementation <em>should</em> work even for over-determined systems
033 * (i.e. systems having more point than equations). Over-determined systems
034 * are solved by ignoring the point which have the smallest impact according
035 * to their jacobian column norm. Only the rank of the matrix and some loop bounds
036 * are changed to implement this.</p>
037 *
038 * <p>The resolution engine is a simple translation of the MINPACK <a
039 * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
040 * changes. The changes include the over-determined resolution, the use of
041 * inherited convergence checker and the Q.R. decomposition which has been
042 * rewritten following the algorithm described in the
043 * P. Lascaux and R. Theodor book <i>Analyse numérique matricielle
044 * appliquée à l'art de l'ingénieur</i>, Masson 1986.</p>
045 * <p>The authors of the original fortran version are:
046 * <ul>
047 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
048 * <li>Burton S. Garbow</li>
049 * <li>Kenneth E. Hillstrom</li>
050 * <li>Jorge J. More</li>
051 * </ul>
052 * The redistribution policy for MINPACK is available <a
053 * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
054 * is reproduced below.</p>
055 *
056 * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
057 * <tr><td>
058 * Minpack Copyright Notice (1999) University of Chicago.
059 * All rights reserved
060 * </td></tr>
061 * <tr><td>
062 * Redistribution and use in source and binary forms, with or without
063 * modification, are permitted provided that the following conditions
064 * are met:
065 * <ol>
066 * <li>Redistributions of source code must retain the above copyright
067 * notice, this list of conditions and the following disclaimer.</li>
068 * <li>Redistributions in binary form must reproduce the above
069 * copyright notice, this list of conditions and the following
070 * disclaimer in the documentation and/or other materials provided
071 * with the distribution.</li>
072 * <li>The end-user documentation included with the redistribution, if any,
073 * must include the following acknowledgment:
074 * <code>This product includes software developed by the University of
075 * Chicago, as Operator of Argonne National Laboratory.</code>
076 * Alternately, this acknowledgment may appear in the software itself,
077 * if and wherever such third-party acknowledgments normally appear.</li>
078 * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
079 * WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
080 * UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
081 * THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
082 * IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
083 * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
084 * OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
085 * OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
086 * USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
087 * THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
088 * DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
089 * UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
090 * BE CORRECTED.</strong></li>
091 * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
092 * HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
093 * ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
094 * INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
095 * ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
096 * PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
097 * SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
098 * (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
099 * 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 $Revision: 1073272 $ $Date: 2011-02-22 10:22:25 +0100 (mar. 22 f??vr. 2011) $
104 * @since 2.0
105 *
106 */
107 public class LevenbergMarquardtOptimizer extends AbstractLeastSquaresOptimizer {
108
109 /** Number of solved point. */
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 /** Threshold for QR ranking. */
147 private double qrRankingThreshold;
148
149 /**
150 * Build an optimizer for least squares problems.
151 * <p>The default values for the algorithm settings are:
152 * <ul>
153 * <li>{@link #setConvergenceChecker(VectorialConvergenceChecker) vectorial convergence checker}: null</li>
154 * <li>{@link #setInitialStepBoundFactor(double) initial step bound factor}: 100.0</li>
155 * <li>{@link #setMaxIterations(int) maximal iterations}: 1000</li>
156 * <li>{@link #setCostRelativeTolerance(double) cost relative tolerance}: 1.0e-10</li>
157 * <li>{@link #setParRelativeTolerance(double) parameters relative tolerance}: 1.0e-10</li>
158 * <li>{@link #setOrthoTolerance(double) orthogonality tolerance}: 1.0e-10</li>
159 * <li>{@link #setQRRankingThreshold(double) QR ranking threshold}: {@link MathUtils#SAFE_MIN}</li>
160 * </ul>
161 * </p>
162 * <p>These default values may be overridden after construction. If the {@link
163 * #setConvergenceChecker vectorial convergence checker} is set to a non-null value, it
164 * will be used instead of the {@link #setCostRelativeTolerance cost relative tolerance}
165 * and {@link #setParRelativeTolerance parameters relative tolerance} settings.
166 */
167 public LevenbergMarquardtOptimizer() {
168
169 // set up the superclass with a default max cost evaluations setting
170 setMaxIterations(1000);
171
172 // default values for the tuning parameters
173 setConvergenceChecker(null);
174 setInitialStepBoundFactor(100.0);
175 setCostRelativeTolerance(1.0e-10);
176 setParRelativeTolerance(1.0e-10);
177 setOrthoTolerance(1.0e-10);
178 setQRRankingThreshold(MathUtils.SAFE_MIN);
179
180 }
181
182 /**
183 * Set the positive input variable used in determining the initial step bound.
184 * This bound is set to the product of initialStepBoundFactor and the euclidean
185 * norm of diag*x if nonzero, or else to initialStepBoundFactor itself. In most
186 * cases factor should lie in the interval (0.1, 100.0). 100.0 is a generally
187 * recommended value.
188 *
189 * @param initialStepBoundFactor initial step bound factor
190 */
191 public void setInitialStepBoundFactor(double initialStepBoundFactor) {
192 this.initialStepBoundFactor = initialStepBoundFactor;
193 }
194
195 /**
196 * Set the desired relative error in the sum of squares.
197 * <p>This setting is used only if the {@link #setConvergenceChecker vectorial
198 * convergence checker} is set to null.</p>
199 * @param costRelativeTolerance desired relative error in the sum of squares
200 */
201 public void setCostRelativeTolerance(double costRelativeTolerance) {
202 this.costRelativeTolerance = costRelativeTolerance;
203 }
204
205 /**
206 * Set the desired relative error in the approximate solution parameters.
207 * <p>This setting is used only if the {@link #setConvergenceChecker vectorial
208 * convergence checker} is set to null.</p>
209 * @param parRelativeTolerance desired relative error
210 * in the approximate solution parameters
211 */
212 public void setParRelativeTolerance(double parRelativeTolerance) {
213 this.parRelativeTolerance = parRelativeTolerance;
214 }
215
216 /**
217 * Set the desired max cosine on the orthogonality.
218 * <p>This setting is always used, regardless of the {@link #setConvergenceChecker
219 * vectorial convergence checker} being null or non-null.</p>
220 * @param orthoTolerance desired max cosine on the orthogonality
221 * between the function vector and the columns of the jacobian
222 */
223 public void setOrthoTolerance(double orthoTolerance) {
224 this.orthoTolerance = orthoTolerance;
225 }
226
227 /**
228 * Set the desired threshold for QR ranking.
229 * <p>
230 * If the squared norm of a column vector is smaller or equal to this threshold
231 * during QR decomposition, it is considered to be a zero vector and hence the
232 * rank of the matrix is reduced.
233 * </p>
234 * @param threshold threshold for QR ranking
235 * @since 2.2
236 */
237 public void setQRRankingThreshold(final double threshold) {
238 this.qrRankingThreshold = threshold;
239 }
240
241 /** {@inheritDoc} */
242 @Override
243 protected VectorialPointValuePair doOptimize()
244 throws FunctionEvaluationException, OptimizationException, IllegalArgumentException {
245
246 // arrays shared with the other private methods
247 solvedCols = Math.min(rows, cols);
248 diagR = new double[cols];
249 jacNorm = new double[cols];
250 beta = new double[cols];
251 permutation = new int[cols];
252 lmDir = new double[cols];
253
254 // local point
255 double delta = 0;
256 double xNorm = 0;
257 double[] diag = new double[cols];
258 double[] oldX = new double[cols];
259 double[] oldRes = new double[rows];
260 double[] oldObj = new double[rows];
261 double[] qtf = new double[rows];
262 double[] work1 = new double[cols];
263 double[] work2 = new double[cols];
264 double[] work3 = new double[cols];
265
266 // evaluate the function at the starting point and calculate its norm
267 updateResidualsAndCost();
268
269 // outer loop
270 lmPar = 0;
271 boolean firstIteration = true;
272 VectorialPointValuePair current = new VectorialPointValuePair(point, objective);
273 while (true) {
274 for (int i=0;i<rows;i++) {
275 qtf[i]=wresiduals[i];
276 }
277 incrementIterationsCounter();
278
279 // compute the Q.R. decomposition of the jacobian matrix
280 VectorialPointValuePair previous = current;
281 updateJacobian();
282 qrDecomposition();
283
284 // compute Qt.res
285 qTy(qtf);
286 // now we don't need Q anymore,
287 // so let jacobian contain the R matrix with its diagonal elements
288 for (int k = 0; k < solvedCols; ++k) {
289 int pk = permutation[k];
290 wjacobian[k][pk] = diagR[pk];
291 }
292
293 if (firstIteration) {
294
295 // scale the point according to the norms of the columns
296 // of the initial jacobian
297 xNorm = 0;
298 for (int k = 0; k < cols; ++k) {
299 double dk = jacNorm[k];
300 if (dk == 0) {
301 dk = 1.0;
302 }
303 double xk = dk * point[k];
304 xNorm += xk * xk;
305 diag[k] = dk;
306 }
307 xNorm = FastMath.sqrt(xNorm);
308
309 // initialize the step bound delta
310 delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
311
312 }
313
314 // check orthogonality between function vector and jacobian columns
315 double maxCosine = 0;
316 if (cost != 0) {
317 for (int j = 0; j < solvedCols; ++j) {
318 int pj = permutation[j];
319 double s = jacNorm[pj];
320 if (s != 0) {
321 double sum = 0;
322 for (int i = 0; i <= j; ++i) {
323 sum += wjacobian[i][pj] * qtf[i];
324 }
325 maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * cost));
326 }
327 }
328 }
329 if (maxCosine <= orthoTolerance) {
330 // convergence has been reached
331 updateResidualsAndCost();
332 current = new VectorialPointValuePair(point, objective);
333 return current;
334 }
335
336 // rescale if necessary
337 for (int j = 0; j < cols; ++j) {
338 diag[j] = FastMath.max(diag[j], jacNorm[j]);
339 }
340
341 // inner loop
342 for (double ratio = 0; ratio < 1.0e-4;) {
343
344 // save the state
345 for (int j = 0; j < solvedCols; ++j) {
346 int pj = permutation[j];
347 oldX[pj] = point[pj];
348 }
349 double previousCost = cost;
350 double[] tmpVec = residuals;
351 residuals = oldRes;
352 oldRes = tmpVec;
353 tmpVec = objective;
354 objective = oldObj;
355 oldObj = tmpVec;
356
357 // determine the Levenberg-Marquardt parameter
358 determineLMParameter(qtf, delta, diag, work1, work2, work3);
359
360 // compute the new point and the norm of the evolution direction
361 double lmNorm = 0;
362 for (int j = 0; j < solvedCols; ++j) {
363 int pj = permutation[j];
364 lmDir[pj] = -lmDir[pj];
365 point[pj] = oldX[pj] + lmDir[pj];
366 double s = diag[pj] * lmDir[pj];
367 lmNorm += s * s;
368 }
369 lmNorm = FastMath.sqrt(lmNorm);
370 // on the first iteration, adjust the initial step bound.
371 if (firstIteration) {
372 delta = FastMath.min(delta, lmNorm);
373 }
374
375 // evaluate the function at x + p and calculate its norm
376 updateResidualsAndCost();
377
378 // compute the scaled actual reduction
379 double actRed = -1.0;
380 if (0.1 * cost < previousCost) {
381 double r = cost / previousCost;
382 actRed = 1.0 - r * r;
383 }
384
385 // compute the scaled predicted reduction
386 // and the scaled directional derivative
387 for (int j = 0; j < solvedCols; ++j) {
388 int pj = permutation[j];
389 double dirJ = lmDir[pj];
390 work1[j] = 0;
391 for (int i = 0; i <= j; ++i) {
392 work1[i] += wjacobian[i][pj] * dirJ;
393 }
394 }
395 double coeff1 = 0;
396 for (int j = 0; j < solvedCols; ++j) {
397 coeff1 += work1[j] * work1[j];
398 }
399 double pc2 = previousCost * previousCost;
400 coeff1 = coeff1 / pc2;
401 double coeff2 = lmPar * lmNorm * lmNorm / pc2;
402 double preRed = coeff1 + 2 * coeff2;
403 double dirDer = -(coeff1 + coeff2);
404
405 // ratio of the actual to the predicted reduction
406 ratio = (preRed == 0) ? 0 : (actRed / preRed);
407
408 // update the step bound
409 if (ratio <= 0.25) {
410 double tmp =
411 (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
412 if ((0.1 * cost >= previousCost) || (tmp < 0.1)) {
413 tmp = 0.1;
414 }
415 delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
416 lmPar /= tmp;
417 } else if ((lmPar == 0) || (ratio >= 0.75)) {
418 delta = 2 * lmNorm;
419 lmPar *= 0.5;
420 }
421
422 // test for successful iteration.
423 if (ratio >= 1.0e-4) {
424 // successful iteration, update the norm
425 firstIteration = false;
426 xNorm = 0;
427 for (int k = 0; k < cols; ++k) {
428 double xK = diag[k] * point[k];
429 xNorm += xK * xK;
430 }
431 xNorm = FastMath.sqrt(xNorm);
432 current = new VectorialPointValuePair(point, objective);
433
434 // tests for convergence.
435 if (checker != null) {
436 // we use the vectorial convergence checker
437 if (checker.converged(getIterations(), previous, current)) {
438 return current;
439 }
440 }
441 } else {
442 // failed iteration, reset the previous values
443 cost = previousCost;
444 for (int j = 0; j < solvedCols; ++j) {
445 int pj = permutation[j];
446 point[pj] = oldX[pj];
447 }
448 tmpVec = residuals;
449 residuals = oldRes;
450 oldRes = tmpVec;
451 tmpVec = objective;
452 objective = oldObj;
453 oldObj = tmpVec;
454 }
455 if (checker==null) {
456 if (((FastMath.abs(actRed) <= costRelativeTolerance) &&
457 (preRed <= costRelativeTolerance) &&
458 (ratio <= 2.0)) ||
459 (delta <= parRelativeTolerance * xNorm)) {
460 return current;
461 }
462 }
463 // tests for termination and stringent tolerances
464 // (2.2204e-16 is the machine epsilon for IEEE754)
465 if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
466 throw new OptimizationException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
467 costRelativeTolerance);
468 } else if (delta <= 2.2204e-16 * xNorm) {
469 throw new OptimizationException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
470 parRelativeTolerance);
471 } else if (maxCosine <= 2.2204e-16) {
472 throw new OptimizationException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
473 orthoTolerance);
474 }
475
476 }
477
478 }
479
480 }
481
482 /**
483 * Determine the Levenberg-Marquardt parameter.
484 * <p>This implementation is a translation in Java of the MINPACK
485 * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
486 * routine.</p>
487 * <p>This method sets the lmPar and lmDir attributes.</p>
488 * <p>The authors of the original fortran function are:</p>
489 * <ul>
490 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
491 * <li>Burton S. Garbow</li>
492 * <li>Kenneth E. Hillstrom</li>
493 * <li>Jorge J. More</li>
494 * </ul>
495 * <p>Luc Maisonobe did the Java translation.</p>
496 *
497 * @param qy array containing qTy
498 * @param delta upper bound on the euclidean norm of diagR * lmDir
499 * @param diag diagonal matrix
500 * @param work1 work array
501 * @param work2 work array
502 * @param work3 work array
503 */
504 private void determineLMParameter(double[] qy, double delta, double[] diag,
505 double[] work1, double[] work2, double[] work3) {
506
507 // compute and store in x the gauss-newton direction, if the
508 // jacobian is rank-deficient, obtain a least squares solution
509 for (int j = 0; j < rank; ++j) {
510 lmDir[permutation[j]] = qy[j];
511 }
512 for (int j = rank; j < cols; ++j) {
513 lmDir[permutation[j]] = 0;
514 }
515 for (int k = rank - 1; k >= 0; --k) {
516 int pk = permutation[k];
517 double ypk = lmDir[pk] / diagR[pk];
518 for (int i = 0; i < k; ++i) {
519 lmDir[permutation[i]] -= ypk * wjacobian[i][pk];
520 }
521 lmDir[pk] = ypk;
522 }
523
524 // evaluate the function at the origin, and test
525 // for acceptance of the Gauss-Newton direction
526 double dxNorm = 0;
527 for (int j = 0; j < solvedCols; ++j) {
528 int pj = permutation[j];
529 double s = diag[pj] * lmDir[pj];
530 work1[pj] = s;
531 dxNorm += s * s;
532 }
533 dxNorm = FastMath.sqrt(dxNorm);
534 double fp = dxNorm - delta;
535 if (fp <= 0.1 * delta) {
536 lmPar = 0;
537 return;
538 }
539
540 // if the jacobian is not rank deficient, the Newton step provides
541 // a lower bound, parl, for the zero of the function,
542 // otherwise set this bound to zero
543 double sum2;
544 double parl = 0;
545 if (rank == solvedCols) {
546 for (int j = 0; j < solvedCols; ++j) {
547 int pj = permutation[j];
548 work1[pj] *= diag[pj] / dxNorm;
549 }
550 sum2 = 0;
551 for (int j = 0; j < solvedCols; ++j) {
552 int pj = permutation[j];
553 double sum = 0;
554 for (int i = 0; i < j; ++i) {
555 sum += wjacobian[i][pj] * work1[permutation[i]];
556 }
557 double s = (work1[pj] - sum) / diagR[pj];
558 work1[pj] = s;
559 sum2 += s * s;
560 }
561 parl = fp / (delta * sum2);
562 }
563
564 // calculate an upper bound, paru, for the zero of the function
565 sum2 = 0;
566 for (int j = 0; j < solvedCols; ++j) {
567 int pj = permutation[j];
568 double sum = 0;
569 for (int i = 0; i <= j; ++i) {
570 sum += wjacobian[i][pj] * qy[i];
571 }
572 sum /= diag[pj];
573 sum2 += sum * sum;
574 }
575 double gNorm = FastMath.sqrt(sum2);
576 double paru = gNorm / delta;
577 if (paru == 0) {
578 // 2.2251e-308 is the smallest positive real for IEE754
579 paru = 2.2251e-308 / FastMath.min(delta, 0.1);
580 }
581
582 // if the input par lies outside of the interval (parl,paru),
583 // set par to the closer endpoint
584 lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
585 if (lmPar == 0) {
586 lmPar = gNorm / dxNorm;
587 }
588
589 for (int countdown = 10; countdown >= 0; --countdown) {
590
591 // evaluate the function at the current value of lmPar
592 if (lmPar == 0) {
593 lmPar = FastMath.max(2.2251e-308, 0.001 * paru);
594 }
595 double sPar = FastMath.sqrt(lmPar);
596 for (int j = 0; j < solvedCols; ++j) {
597 int pj = permutation[j];
598 work1[pj] = sPar * diag[pj];
599 }
600 determineLMDirection(qy, work1, work2, work3);
601
602 dxNorm = 0;
603 for (int j = 0; j < solvedCols; ++j) {
604 int pj = permutation[j];
605 double s = diag[pj] * lmDir[pj];
606 work3[pj] = s;
607 dxNorm += s * s;
608 }
609 dxNorm = FastMath.sqrt(dxNorm);
610 double previousFP = fp;
611 fp = dxNorm - delta;
612
613 // if the function is small enough, accept the current value
614 // of lmPar, also test for the exceptional cases where parl is zero
615 if ((FastMath.abs(fp) <= 0.1 * delta) ||
616 ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
617 return;
618 }
619
620 // compute the Newton correction
621 for (int j = 0; j < solvedCols; ++j) {
622 int pj = permutation[j];
623 work1[pj] = work3[pj] * diag[pj] / dxNorm;
624 }
625 for (int j = 0; j < solvedCols; ++j) {
626 int pj = permutation[j];
627 work1[pj] /= work2[j];
628 double tmp = work1[pj];
629 for (int i = j + 1; i < solvedCols; ++i) {
630 work1[permutation[i]] -= wjacobian[i][pj] * tmp;
631 }
632 }
633 sum2 = 0;
634 for (int j = 0; j < solvedCols; ++j) {
635 double s = work1[permutation[j]];
636 sum2 += s * s;
637 }
638 double correction = fp / (delta * sum2);
639
640 // depending on the sign of the function, update parl or paru.
641 if (fp > 0) {
642 parl = FastMath.max(parl, lmPar);
643 } else if (fp < 0) {
644 paru = FastMath.min(paru, lmPar);
645 }
646
647 // compute an improved estimate for lmPar
648 lmPar = FastMath.max(parl, lmPar + correction);
649
650 }
651 }
652
653 /**
654 * Solve a*x = b and d*x = 0 in the least squares sense.
655 * <p>This implementation is a translation in Java of the MINPACK
656 * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
657 * routine.</p>
658 * <p>This method sets the lmDir and lmDiag attributes.</p>
659 * <p>The authors of the original fortran function are:</p>
660 * <ul>
661 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
662 * <li>Burton S. Garbow</li>
663 * <li>Kenneth E. Hillstrom</li>
664 * <li>Jorge J. More</li>
665 * </ul>
666 * <p>Luc Maisonobe did the Java translation.</p>
667 *
668 * @param qy array containing qTy
669 * @param diag diagonal matrix
670 * @param lmDiag diagonal elements associated with lmDir
671 * @param work work array
672 */
673 private void determineLMDirection(double[] qy, double[] diag,
674 double[] lmDiag, double[] work) {
675
676 // copy R and Qty to preserve input and initialize s
677 // in particular, save the diagonal elements of R in lmDir
678 for (int j = 0; j < solvedCols; ++j) {
679 int pj = permutation[j];
680 for (int i = j + 1; i < solvedCols; ++i) {
681 wjacobian[i][pj] = wjacobian[j][permutation[i]];
682 }
683 lmDir[j] = diagR[pj];
684 work[j] = qy[j];
685 }
686
687 // eliminate the diagonal matrix d using a Givens rotation
688 for (int j = 0; j < solvedCols; ++j) {
689
690 // prepare the row of d to be eliminated, locating the
691 // diagonal element using p from the Q.R. factorization
692 int pj = permutation[j];
693 double dpj = diag[pj];
694 if (dpj != 0) {
695 Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
696 }
697 lmDiag[j] = dpj;
698
699 // the transformations to eliminate the row of d
700 // modify only a single element of Qty
701 // beyond the first n, which is initially zero.
702 double qtbpj = 0;
703 for (int k = j; k < solvedCols; ++k) {
704 int pk = permutation[k];
705
706 // determine a Givens rotation which eliminates the
707 // appropriate element in the current row of d
708 if (lmDiag[k] != 0) {
709
710 final double sin;
711 final double cos;
712 double rkk = wjacobian[k][pk];
713 if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
714 final double cotan = rkk / lmDiag[k];
715 sin = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
716 cos = sin * cotan;
717 } else {
718 final double tan = lmDiag[k] / rkk;
719 cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
720 sin = cos * tan;
721 }
722
723 // compute the modified diagonal element of R and
724 // the modified element of (Qty,0)
725 wjacobian[k][pk] = cos * rkk + sin * lmDiag[k];
726 final double temp = cos * work[k] + sin * qtbpj;
727 qtbpj = -sin * work[k] + cos * qtbpj;
728 work[k] = temp;
729
730 // accumulate the tranformation in the row of s
731 for (int i = k + 1; i < solvedCols; ++i) {
732 double rik = wjacobian[i][pk];
733 final double temp2 = cos * rik + sin * lmDiag[i];
734 lmDiag[i] = -sin * rik + cos * lmDiag[i];
735 wjacobian[i][pk] = temp2;
736 }
737
738 }
739 }
740
741 // store the diagonal element of s and restore
742 // the corresponding diagonal element of R
743 lmDiag[j] = wjacobian[j][permutation[j]];
744 wjacobian[j][permutation[j]] = lmDir[j];
745
746 }
747
748 // solve the triangular system for z, if the system is
749 // singular, then obtain a least squares solution
750 int nSing = solvedCols;
751 for (int j = 0; j < solvedCols; ++j) {
752 if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
753 nSing = j;
754 }
755 if (nSing < solvedCols) {
756 work[j] = 0;
757 }
758 }
759 if (nSing > 0) {
760 for (int j = nSing - 1; j >= 0; --j) {
761 int pj = permutation[j];
762 double sum = 0;
763 for (int i = j + 1; i < nSing; ++i) {
764 sum += wjacobian[i][pj] * work[i];
765 }
766 work[j] = (work[j] - sum) / lmDiag[j];
767 }
768 }
769
770 // permute the components of z back to components of lmDir
771 for (int j = 0; j < lmDir.length; ++j) {
772 lmDir[permutation[j]] = work[j];
773 }
774
775 }
776
777 /**
778 * Decompose a matrix A as A.P = Q.R using Householder transforms.
779 * <p>As suggested in the P. Lascaux and R. Theodor book
780 * <i>Analyse numérique matricielle appliquée à
781 * l'art de l'ingénieur</i> (Masson, 1986), instead of representing
782 * the Householder transforms with u<sub>k</sub> unit vectors such that:
783 * <pre>
784 * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
785 * </pre>
786 * we use <sub>k</sub> non-unit vectors such that:
787 * <pre>
788 * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
789 * </pre>
790 * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
791 * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
792 * them from the v<sub>k</sub> vectors would be costly.</p>
793 * <p>This decomposition handles rank deficient cases since the tranformations
794 * are performed in non-increasing columns norms order thanks to columns
795 * pivoting. The diagonal elements of the R matrix are therefore also in
796 * non-increasing absolute values order.</p>
797 * @exception OptimizationException if the decomposition cannot be performed
798 */
799 private void qrDecomposition() throws OptimizationException {
800
801 // initializations
802 for (int k = 0; k < cols; ++k) {
803 permutation[k] = k;
804 double norm2 = 0;
805 for (int i = 0; i < wjacobian.length; ++i) {
806 double akk = wjacobian[i][k];
807 norm2 += akk * akk;
808 }
809 jacNorm[k] = FastMath.sqrt(norm2);
810 }
811
812 // transform the matrix column after column
813 for (int k = 0; k < cols; ++k) {
814
815 // select the column with the greatest norm on active components
816 int nextColumn = -1;
817 double ak2 = Double.NEGATIVE_INFINITY;
818 for (int i = k; i < cols; ++i) {
819 double norm2 = 0;
820 for (int j = k; j < wjacobian.length; ++j) {
821 double aki = wjacobian[j][permutation[i]];
822 norm2 += aki * aki;
823 }
824 if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
825 throw new OptimizationException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
826 rows, cols);
827 }
828 if (norm2 > ak2) {
829 nextColumn = i;
830 ak2 = norm2;
831 }
832 }
833 if (ak2 <= qrRankingThreshold) {
834 rank = k;
835 return;
836 }
837 int pk = permutation[nextColumn];
838 permutation[nextColumn] = permutation[k];
839 permutation[k] = pk;
840
841 // choose alpha such that Hk.u = alpha ek
842 double akk = wjacobian[k][pk];
843 double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.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 wjacobian[k][pk] -= alpha;
850
851 // transform the remaining columns
852 for (int dk = cols - 1 - k; dk > 0; --dk) {
853 double gamma = 0;
854 for (int j = k; j < wjacobian.length; ++j) {
855 gamma += wjacobian[j][pk] * wjacobian[j][permutation[k + dk]];
856 }
857 gamma *= betak;
858 for (int j = k; j < wjacobian.length; ++j) {
859 wjacobian[j][permutation[k + dk]] -= gamma * wjacobian[j][pk];
860 }
861 }
862
863 }
864
865 rank = solvedCols;
866
867 }
868
869 /**
870 * Compute the product Qt.y for some Q.R. decomposition.
871 *
872 * @param y vector to multiply (will be overwritten with the result)
873 */
874 private void qTy(double[] y) {
875 for (int k = 0; k < cols; ++k) {
876 int pk = permutation[k];
877 double gamma = 0;
878 for (int i = k; i < rows; ++i) {
879 gamma += wjacobian[i][pk] * y[i];
880 }
881 gamma *= beta[pk];
882 for (int i = k; i < rows; ++i) {
883 y[i] -= gamma * wjacobian[i][pk];
884 }
885 }
886 }
887
888 }