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.math3.optimization.general;
018
019 import java.util.Arrays;
020
021 import org.apache.commons.math3.exception.ConvergenceException;
022 import org.apache.commons.math3.exception.util.LocalizedFormats;
023 import org.apache.commons.math3.optimization.PointVectorValuePair;
024 import org.apache.commons.math3.optimization.ConvergenceChecker;
025 import org.apache.commons.math3.linear.RealMatrix;
026 import org.apache.commons.math3.util.Precision;
027 import org.apache.commons.math3.util.FastMath;
028
029
030 /**
031 * This class solves a least squares problem using the Levenberg-Marquardt algorithm.
032 *
033 * <p>This implementation <em>should</em> work even for over-determined systems
034 * (i.e. systems having more point than equations). Over-determined systems
035 * are solved by ignoring the point which have the smallest impact according
036 * to their jacobian column norm. Only the rank of the matrix and some loop bounds
037 * are changed to implement this.</p>
038 *
039 * <p>The resolution engine is a simple translation of the MINPACK <a
040 * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
041 * changes. The changes include the over-determined resolution, the use of
042 * inherited convergence checker and the Q.R. decomposition which has been
043 * rewritten following the algorithm described in the
044 * P. Lascaux and R. Theodor book <i>Analyse numérique matricielle
045 * appliquée à l'art de l'ingénieur</i>, Masson 1986.</p>
046 * <p>The authors of the original fortran version are:
047 * <ul>
048 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
049 * <li>Burton S. Garbow</li>
050 * <li>Kenneth E. Hillstrom</li>
051 * <li>Jorge J. More</li>
052 * </ul>
053 * The redistribution policy for MINPACK is available <a
054 * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
055 * is reproduced below.</p>
056 *
057 * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
058 * <tr><td>
059 * Minpack Copyright Notice (1999) University of Chicago.
060 * All rights reserved
061 * </td></tr>
062 * <tr><td>
063 * Redistribution and use in source and binary forms, with or without
064 * modification, are permitted provided that the following conditions
065 * are met:
066 * <ol>
067 * <li>Redistributions of source code must retain the above copyright
068 * notice, this list of conditions and the following disclaimer.</li>
069 * <li>Redistributions in binary form must reproduce the above
070 * copyright notice, this list of conditions and the following
071 * disclaimer in the documentation and/or other materials provided
072 * with the distribution.</li>
073 * <li>The end-user documentation included with the redistribution, if any,
074 * must include the following acknowledgment:
075 * <code>This product includes software developed by the University of
076 * Chicago, as Operator of Argonne National Laboratory.</code>
077 * Alternately, this acknowledgment may appear in the software itself,
078 * if and wherever such third-party acknowledgments normally appear.</li>
079 * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
080 * WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
081 * UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
082 * THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
083 * IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
084 * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
085 * OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
086 * OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
087 * USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
088 * THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
089 * DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
090 * UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
091 * BE CORRECTED.</strong></li>
092 * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
093 * HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
094 * ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
095 * INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
096 * ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
097 * PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
098 * SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
099 * (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
100 * EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
101 * POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
102 * <ol></td></tr>
103 * </table>
104 * @version $Id: LevenbergMarquardtOptimizer.java 1423555 2012-12-18 18:02:52Z erans $
105 * @deprecated As of 3.1 (to be removed in 4.0).
106 * @since 2.0
107 *
108 */
109 @Deprecated
110 public class LevenbergMarquardtOptimizer extends AbstractLeastSquaresOptimizer {
111 /** Number of solved point. */
112 private int solvedCols;
113 /** Diagonal elements of the R matrix in the Q.R. decomposition. */
114 private double[] diagR;
115 /** Norms of the columns of the jacobian matrix. */
116 private double[] jacNorm;
117 /** Coefficients of the Householder transforms vectors. */
118 private double[] beta;
119 /** Columns permutation array. */
120 private int[] permutation;
121 /** Rank of the jacobian matrix. */
122 private int rank;
123 /** Levenberg-Marquardt parameter. */
124 private double lmPar;
125 /** Parameters evolution direction associated with lmPar. */
126 private double[] lmDir;
127 /** Positive input variable used in determining the initial step bound. */
128 private final double initialStepBoundFactor;
129 /** Desired relative error in the sum of squares. */
130 private final double costRelativeTolerance;
131 /** Desired relative error in the approximate solution parameters. */
132 private final double parRelativeTolerance;
133 /** Desired max cosine on the orthogonality between the function vector
134 * and the columns of the jacobian. */
135 private final double orthoTolerance;
136 /** Threshold for QR ranking. */
137 private final double qrRankingThreshold;
138 /** Weighted residuals. */
139 private double[] weightedResidual;
140 /** Weighted Jacobian. */
141 private double[][] weightedJacobian;
142
143 /**
144 * Build an optimizer for least squares problems with default values
145 * for all the tuning parameters (see the {@link
146 * #LevenbergMarquardtOptimizer(double,double,double,double,double)
147 * other contructor}.
148 * The default values for the algorithm settings are:
149 * <ul>
150 * <li>Initial step bound factor: 100</li>
151 * <li>Cost relative tolerance: 1e-10</li>
152 * <li>Parameters relative tolerance: 1e-10</li>
153 * <li>Orthogonality tolerance: 1e-10</li>
154 * <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
155 * </ul>
156 */
157 public LevenbergMarquardtOptimizer() {
158 this(100, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
159 }
160
161 /**
162 * Constructor that allows the specification of a custom convergence
163 * checker.
164 * Note that all the usual convergence checks will be <em>disabled</em>.
165 * The default values for the algorithm settings are:
166 * <ul>
167 * <li>Initial step bound factor: 100</li>
168 * <li>Cost relative tolerance: 1e-10</li>
169 * <li>Parameters relative tolerance: 1e-10</li>
170 * <li>Orthogonality tolerance: 1e-10</li>
171 * <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
172 * </ul>
173 *
174 * @param checker Convergence checker.
175 */
176 public LevenbergMarquardtOptimizer(ConvergenceChecker<PointVectorValuePair> checker) {
177 this(100, checker, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
178 }
179
180 /**
181 * Constructor that allows the specification of a custom convergence
182 * checker, in addition to the standard ones.
183 *
184 * @param initialStepBoundFactor Positive input variable used in
185 * determining the initial step bound. This bound is set to the
186 * product of initialStepBoundFactor and the euclidean norm of
187 * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
188 * itself. In most cases factor should lie in the interval
189 * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
190 * @param checker Convergence checker.
191 * @param costRelativeTolerance Desired relative error in the sum of
192 * squares.
193 * @param parRelativeTolerance Desired relative error in the approximate
194 * solution parameters.
195 * @param orthoTolerance Desired max cosine on the orthogonality between
196 * the function vector and the columns of the Jacobian.
197 * @param threshold Desired threshold for QR ranking. If the squared norm
198 * of a column vector is smaller or equal to this threshold during QR
199 * decomposition, it is considered to be a zero vector and hence the rank
200 * of the matrix is reduced.
201 */
202 public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
203 ConvergenceChecker<PointVectorValuePair> checker,
204 double costRelativeTolerance,
205 double parRelativeTolerance,
206 double orthoTolerance,
207 double threshold) {
208 super(checker);
209 this.initialStepBoundFactor = initialStepBoundFactor;
210 this.costRelativeTolerance = costRelativeTolerance;
211 this.parRelativeTolerance = parRelativeTolerance;
212 this.orthoTolerance = orthoTolerance;
213 this.qrRankingThreshold = threshold;
214 }
215
216 /**
217 * Build an optimizer for least squares problems with default values
218 * for some of the tuning parameters (see the {@link
219 * #LevenbergMarquardtOptimizer(double,double,double,double,double)
220 * other contructor}.
221 * The default values for the algorithm settings are:
222 * <ul>
223 * <li>Initial step bound factor}: 100</li>
224 * <li>QR ranking threshold}: {@link Precision#SAFE_MIN}</li>
225 * </ul>
226 *
227 * @param costRelativeTolerance Desired relative error in the sum of
228 * squares.
229 * @param parRelativeTolerance Desired relative error in the approximate
230 * solution parameters.
231 * @param orthoTolerance Desired max cosine on the orthogonality between
232 * the function vector and the columns of the Jacobian.
233 */
234 public LevenbergMarquardtOptimizer(double costRelativeTolerance,
235 double parRelativeTolerance,
236 double orthoTolerance) {
237 this(100,
238 costRelativeTolerance, parRelativeTolerance, orthoTolerance,
239 Precision.SAFE_MIN);
240 }
241
242 /**
243 * The arguments control the behaviour of the default convergence checking
244 * procedure.
245 * Additional criteria can defined through the setting of a {@link
246 * ConvergenceChecker}.
247 *
248 * @param initialStepBoundFactor Positive input variable used in
249 * determining the initial step bound. This bound is set to the
250 * product of initialStepBoundFactor and the euclidean norm of
251 * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
252 * itself. In most cases factor should lie in the interval
253 * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
254 * @param costRelativeTolerance Desired relative error in the sum of
255 * squares.
256 * @param parRelativeTolerance Desired relative error in the approximate
257 * solution parameters.
258 * @param orthoTolerance Desired max cosine on the orthogonality between
259 * the function vector and the columns of the Jacobian.
260 * @param threshold Desired threshold for QR ranking. If the squared norm
261 * of a column vector is smaller or equal to this threshold during QR
262 * decomposition, it is considered to be a zero vector and hence the rank
263 * of the matrix is reduced.
264 */
265 public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
266 double costRelativeTolerance,
267 double parRelativeTolerance,
268 double orthoTolerance,
269 double threshold) {
270 super(null); // No custom convergence criterion.
271 this.initialStepBoundFactor = initialStepBoundFactor;
272 this.costRelativeTolerance = costRelativeTolerance;
273 this.parRelativeTolerance = parRelativeTolerance;
274 this.orthoTolerance = orthoTolerance;
275 this.qrRankingThreshold = threshold;
276 }
277
278 /** {@inheritDoc} */
279 @Override
280 protected PointVectorValuePair doOptimize() {
281 final int nR = getTarget().length; // Number of observed data.
282 final double[] currentPoint = getStartPoint();
283 final int nC = currentPoint.length; // Number of parameters.
284
285 // arrays shared with the other private methods
286 solvedCols = FastMath.min(nR, nC);
287 diagR = new double[nC];
288 jacNorm = new double[nC];
289 beta = new double[nC];
290 permutation = new int[nC];
291 lmDir = new double[nC];
292
293 // local point
294 double delta = 0;
295 double xNorm = 0;
296 double[] diag = new double[nC];
297 double[] oldX = new double[nC];
298 double[] oldRes = new double[nR];
299 double[] oldObj = new double[nR];
300 double[] qtf = new double[nR];
301 double[] work1 = new double[nC];
302 double[] work2 = new double[nC];
303 double[] work3 = new double[nC];
304
305 final RealMatrix weightMatrixSqrt = getWeightSquareRoot();
306
307 // Evaluate the function at the starting point and calculate its norm.
308 double[] currentObjective = computeObjectiveValue(currentPoint);
309 double[] currentResiduals = computeResiduals(currentObjective);
310 PointVectorValuePair current = new PointVectorValuePair(currentPoint, currentObjective);
311 double currentCost = computeCost(currentResiduals);
312
313 // Outer loop.
314 lmPar = 0;
315 boolean firstIteration = true;
316 int iter = 0;
317 final ConvergenceChecker<PointVectorValuePair> checker = getConvergenceChecker();
318 while (true) {
319 ++iter;
320 final PointVectorValuePair previous = current;
321
322 // QR decomposition of the jacobian matrix
323 qrDecomposition(computeWeightedJacobian(currentPoint));
324
325 weightedResidual = weightMatrixSqrt.operate(currentResiduals);
326 for (int i = 0; i < nR; i++) {
327 qtf[i] = weightedResidual[i];
328 }
329
330 // compute Qt.res
331 qTy(qtf);
332
333 // now we don't need Q anymore,
334 // so let jacobian contain the R matrix with its diagonal elements
335 for (int k = 0; k < solvedCols; ++k) {
336 int pk = permutation[k];
337 weightedJacobian[k][pk] = diagR[pk];
338 }
339
340 if (firstIteration) {
341 // scale the point according to the norms of the columns
342 // of the initial jacobian
343 xNorm = 0;
344 for (int k = 0; k < nC; ++k) {
345 double dk = jacNorm[k];
346 if (dk == 0) {
347 dk = 1.0;
348 }
349 double xk = dk * currentPoint[k];
350 xNorm += xk * xk;
351 diag[k] = dk;
352 }
353 xNorm = FastMath.sqrt(xNorm);
354
355 // initialize the step bound delta
356 delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
357 }
358
359 // check orthogonality between function vector and jacobian columns
360 double maxCosine = 0;
361 if (currentCost != 0) {
362 for (int j = 0; j < solvedCols; ++j) {
363 int pj = permutation[j];
364 double s = jacNorm[pj];
365 if (s != 0) {
366 double sum = 0;
367 for (int i = 0; i <= j; ++i) {
368 sum += weightedJacobian[i][pj] * qtf[i];
369 }
370 maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * currentCost));
371 }
372 }
373 }
374 if (maxCosine <= orthoTolerance) {
375 // Convergence has been reached.
376 setCost(currentCost);
377 // Update (deprecated) "point" field.
378 point = current.getPoint();
379 return current;
380 }
381
382 // rescale if necessary
383 for (int j = 0; j < nC; ++j) {
384 diag[j] = FastMath.max(diag[j], jacNorm[j]);
385 }
386
387 // Inner loop.
388 for (double ratio = 0; ratio < 1.0e-4;) {
389
390 // save the state
391 for (int j = 0; j < solvedCols; ++j) {
392 int pj = permutation[j];
393 oldX[pj] = currentPoint[pj];
394 }
395 final double previousCost = currentCost;
396 double[] tmpVec = weightedResidual;
397 weightedResidual = oldRes;
398 oldRes = tmpVec;
399 tmpVec = currentObjective;
400 currentObjective = oldObj;
401 oldObj = tmpVec;
402
403 // determine the Levenberg-Marquardt parameter
404 determineLMParameter(qtf, delta, diag, work1, work2, work3);
405
406 // compute the new point and the norm of the evolution direction
407 double lmNorm = 0;
408 for (int j = 0; j < solvedCols; ++j) {
409 int pj = permutation[j];
410 lmDir[pj] = -lmDir[pj];
411 currentPoint[pj] = oldX[pj] + lmDir[pj];
412 double s = diag[pj] * lmDir[pj];
413 lmNorm += s * s;
414 }
415 lmNorm = FastMath.sqrt(lmNorm);
416 // on the first iteration, adjust the initial step bound.
417 if (firstIteration) {
418 delta = FastMath.min(delta, lmNorm);
419 }
420
421 // Evaluate the function at x + p and calculate its norm.
422 currentObjective = computeObjectiveValue(currentPoint);
423 currentResiduals = computeResiduals(currentObjective);
424 current = new PointVectorValuePair(currentPoint, currentObjective);
425 currentCost = computeCost(currentResiduals);
426
427 // compute the scaled actual reduction
428 double actRed = -1.0;
429 if (0.1 * currentCost < previousCost) {
430 double r = currentCost / previousCost;
431 actRed = 1.0 - r * r;
432 }
433
434 // compute the scaled predicted reduction
435 // and the scaled directional derivative
436 for (int j = 0; j < solvedCols; ++j) {
437 int pj = permutation[j];
438 double dirJ = lmDir[pj];
439 work1[j] = 0;
440 for (int i = 0; i <= j; ++i) {
441 work1[i] += weightedJacobian[i][pj] * dirJ;
442 }
443 }
444 double coeff1 = 0;
445 for (int j = 0; j < solvedCols; ++j) {
446 coeff1 += work1[j] * work1[j];
447 }
448 double pc2 = previousCost * previousCost;
449 coeff1 = coeff1 / pc2;
450 double coeff2 = lmPar * lmNorm * lmNorm / pc2;
451 double preRed = coeff1 + 2 * coeff2;
452 double dirDer = -(coeff1 + coeff2);
453
454 // ratio of the actual to the predicted reduction
455 ratio = (preRed == 0) ? 0 : (actRed / preRed);
456
457 // update the step bound
458 if (ratio <= 0.25) {
459 double tmp =
460 (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
461 if ((0.1 * currentCost >= previousCost) || (tmp < 0.1)) {
462 tmp = 0.1;
463 }
464 delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
465 lmPar /= tmp;
466 } else if ((lmPar == 0) || (ratio >= 0.75)) {
467 delta = 2 * lmNorm;
468 lmPar *= 0.5;
469 }
470
471 // test for successful iteration.
472 if (ratio >= 1.0e-4) {
473 // successful iteration, update the norm
474 firstIteration = false;
475 xNorm = 0;
476 for (int k = 0; k < nC; ++k) {
477 double xK = diag[k] * currentPoint[k];
478 xNorm += xK * xK;
479 }
480 xNorm = FastMath.sqrt(xNorm);
481
482 // tests for convergence.
483 if (checker != null) {
484 // we use the vectorial convergence checker
485 if (checker.converged(iter, previous, current)) {
486 setCost(currentCost);
487 // Update (deprecated) "point" field.
488 point = current.getPoint();
489 return current;
490 }
491 }
492 } else {
493 // failed iteration, reset the previous values
494 currentCost = previousCost;
495 for (int j = 0; j < solvedCols; ++j) {
496 int pj = permutation[j];
497 currentPoint[pj] = oldX[pj];
498 }
499 tmpVec = weightedResidual;
500 weightedResidual = oldRes;
501 oldRes = tmpVec;
502 tmpVec = currentObjective;
503 currentObjective = oldObj;
504 oldObj = tmpVec;
505 // Reset "current" to previous values.
506 current = new PointVectorValuePair(currentPoint, currentObjective);
507 }
508
509 // Default convergence criteria.
510 if ((FastMath.abs(actRed) <= costRelativeTolerance &&
511 preRed <= costRelativeTolerance &&
512 ratio <= 2.0) ||
513 delta <= parRelativeTolerance * xNorm) {
514 setCost(currentCost);
515 // Update (deprecated) "point" field.
516 point = current.getPoint();
517 return current;
518 }
519
520 // tests for termination and stringent tolerances
521 // (2.2204e-16 is the machine epsilon for IEEE754)
522 if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
523 throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
524 costRelativeTolerance);
525 } else if (delta <= 2.2204e-16 * xNorm) {
526 throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
527 parRelativeTolerance);
528 } else if (maxCosine <= 2.2204e-16) {
529 throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
530 orthoTolerance);
531 }
532 }
533 }
534 }
535
536 /**
537 * Determine the Levenberg-Marquardt parameter.
538 * <p>This implementation is a translation in Java of the MINPACK
539 * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
540 * routine.</p>
541 * <p>This method sets the lmPar and lmDir attributes.</p>
542 * <p>The authors of the original fortran function are:</p>
543 * <ul>
544 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
545 * <li>Burton S. Garbow</li>
546 * <li>Kenneth E. Hillstrom</li>
547 * <li>Jorge J. More</li>
548 * </ul>
549 * <p>Luc Maisonobe did the Java translation.</p>
550 *
551 * @param qy array containing qTy
552 * @param delta upper bound on the euclidean norm of diagR * lmDir
553 * @param diag diagonal matrix
554 * @param work1 work array
555 * @param work2 work array
556 * @param work3 work array
557 */
558 private void determineLMParameter(double[] qy, double delta, double[] diag,
559 double[] work1, double[] work2, double[] work3) {
560 final int nC = weightedJacobian[0].length;
561
562 // compute and store in x the gauss-newton direction, if the
563 // jacobian is rank-deficient, obtain a least squares solution
564 for (int j = 0; j < rank; ++j) {
565 lmDir[permutation[j]] = qy[j];
566 }
567 for (int j = rank; j < nC; ++j) {
568 lmDir[permutation[j]] = 0;
569 }
570 for (int k = rank - 1; k >= 0; --k) {
571 int pk = permutation[k];
572 double ypk = lmDir[pk] / diagR[pk];
573 for (int i = 0; i < k; ++i) {
574 lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk];
575 }
576 lmDir[pk] = ypk;
577 }
578
579 // evaluate the function at the origin, and test
580 // for acceptance of the Gauss-Newton direction
581 double dxNorm = 0;
582 for (int j = 0; j < solvedCols; ++j) {
583 int pj = permutation[j];
584 double s = diag[pj] * lmDir[pj];
585 work1[pj] = s;
586 dxNorm += s * s;
587 }
588 dxNorm = FastMath.sqrt(dxNorm);
589 double fp = dxNorm - delta;
590 if (fp <= 0.1 * delta) {
591 lmPar = 0;
592 return;
593 }
594
595 // if the jacobian is not rank deficient, the Newton step provides
596 // a lower bound, parl, for the zero of the function,
597 // otherwise set this bound to zero
598 double sum2;
599 double parl = 0;
600 if (rank == solvedCols) {
601 for (int j = 0; j < solvedCols; ++j) {
602 int pj = permutation[j];
603 work1[pj] *= diag[pj] / dxNorm;
604 }
605 sum2 = 0;
606 for (int j = 0; j < solvedCols; ++j) {
607 int pj = permutation[j];
608 double sum = 0;
609 for (int i = 0; i < j; ++i) {
610 sum += weightedJacobian[i][pj] * work1[permutation[i]];
611 }
612 double s = (work1[pj] - sum) / diagR[pj];
613 work1[pj] = s;
614 sum2 += s * s;
615 }
616 parl = fp / (delta * sum2);
617 }
618
619 // calculate an upper bound, paru, for the zero of the function
620 sum2 = 0;
621 for (int j = 0; j < solvedCols; ++j) {
622 int pj = permutation[j];
623 double sum = 0;
624 for (int i = 0; i <= j; ++i) {
625 sum += weightedJacobian[i][pj] * qy[i];
626 }
627 sum /= diag[pj];
628 sum2 += sum * sum;
629 }
630 double gNorm = FastMath.sqrt(sum2);
631 double paru = gNorm / delta;
632 if (paru == 0) {
633 // 2.2251e-308 is the smallest positive real for IEE754
634 paru = 2.2251e-308 / FastMath.min(delta, 0.1);
635 }
636
637 // if the input par lies outside of the interval (parl,paru),
638 // set par to the closer endpoint
639 lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
640 if (lmPar == 0) {
641 lmPar = gNorm / dxNorm;
642 }
643
644 for (int countdown = 10; countdown >= 0; --countdown) {
645
646 // evaluate the function at the current value of lmPar
647 if (lmPar == 0) {
648 lmPar = FastMath.max(2.2251e-308, 0.001 * paru);
649 }
650 double sPar = FastMath.sqrt(lmPar);
651 for (int j = 0; j < solvedCols; ++j) {
652 int pj = permutation[j];
653 work1[pj] = sPar * diag[pj];
654 }
655 determineLMDirection(qy, work1, work2, work3);
656
657 dxNorm = 0;
658 for (int j = 0; j < solvedCols; ++j) {
659 int pj = permutation[j];
660 double s = diag[pj] * lmDir[pj];
661 work3[pj] = s;
662 dxNorm += s * s;
663 }
664 dxNorm = FastMath.sqrt(dxNorm);
665 double previousFP = fp;
666 fp = dxNorm - delta;
667
668 // if the function is small enough, accept the current value
669 // of lmPar, also test for the exceptional cases where parl is zero
670 if ((FastMath.abs(fp) <= 0.1 * delta) ||
671 ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
672 return;
673 }
674
675 // compute the Newton correction
676 for (int j = 0; j < solvedCols; ++j) {
677 int pj = permutation[j];
678 work1[pj] = work3[pj] * diag[pj] / dxNorm;
679 }
680 for (int j = 0; j < solvedCols; ++j) {
681 int pj = permutation[j];
682 work1[pj] /= work2[j];
683 double tmp = work1[pj];
684 for (int i = j + 1; i < solvedCols; ++i) {
685 work1[permutation[i]] -= weightedJacobian[i][pj] * tmp;
686 }
687 }
688 sum2 = 0;
689 for (int j = 0; j < solvedCols; ++j) {
690 double s = work1[permutation[j]];
691 sum2 += s * s;
692 }
693 double correction = fp / (delta * sum2);
694
695 // depending on the sign of the function, update parl or paru.
696 if (fp > 0) {
697 parl = FastMath.max(parl, lmPar);
698 } else if (fp < 0) {
699 paru = FastMath.min(paru, lmPar);
700 }
701
702 // compute an improved estimate for lmPar
703 lmPar = FastMath.max(parl, lmPar + correction);
704
705 }
706 }
707
708 /**
709 * Solve a*x = b and d*x = 0 in the least squares sense.
710 * <p>This implementation is a translation in Java of the MINPACK
711 * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
712 * routine.</p>
713 * <p>This method sets the lmDir and lmDiag attributes.</p>
714 * <p>The authors of the original fortran function are:</p>
715 * <ul>
716 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
717 * <li>Burton S. Garbow</li>
718 * <li>Kenneth E. Hillstrom</li>
719 * <li>Jorge J. More</li>
720 * </ul>
721 * <p>Luc Maisonobe did the Java translation.</p>
722 *
723 * @param qy array containing qTy
724 * @param diag diagonal matrix
725 * @param lmDiag diagonal elements associated with lmDir
726 * @param work work array
727 */
728 private void determineLMDirection(double[] qy, double[] diag,
729 double[] lmDiag, double[] work) {
730
731 // copy R and Qty to preserve input and initialize s
732 // in particular, save the diagonal elements of R in lmDir
733 for (int j = 0; j < solvedCols; ++j) {
734 int pj = permutation[j];
735 for (int i = j + 1; i < solvedCols; ++i) {
736 weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]];
737 }
738 lmDir[j] = diagR[pj];
739 work[j] = qy[j];
740 }
741
742 // eliminate the diagonal matrix d using a Givens rotation
743 for (int j = 0; j < solvedCols; ++j) {
744
745 // prepare the row of d to be eliminated, locating the
746 // diagonal element using p from the Q.R. factorization
747 int pj = permutation[j];
748 double dpj = diag[pj];
749 if (dpj != 0) {
750 Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
751 }
752 lmDiag[j] = dpj;
753
754 // the transformations to eliminate the row of d
755 // modify only a single element of Qty
756 // beyond the first n, which is initially zero.
757 double qtbpj = 0;
758 for (int k = j; k < solvedCols; ++k) {
759 int pk = permutation[k];
760
761 // determine a Givens rotation which eliminates the
762 // appropriate element in the current row of d
763 if (lmDiag[k] != 0) {
764
765 final double sin;
766 final double cos;
767 double rkk = weightedJacobian[k][pk];
768 if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
769 final double cotan = rkk / lmDiag[k];
770 sin = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
771 cos = sin * cotan;
772 } else {
773 final double tan = lmDiag[k] / rkk;
774 cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
775 sin = cos * tan;
776 }
777
778 // compute the modified diagonal element of R and
779 // the modified element of (Qty,0)
780 weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
781 final double temp = cos * work[k] + sin * qtbpj;
782 qtbpj = -sin * work[k] + cos * qtbpj;
783 work[k] = temp;
784
785 // accumulate the tranformation in the row of s
786 for (int i = k + 1; i < solvedCols; ++i) {
787 double rik = weightedJacobian[i][pk];
788 final double temp2 = cos * rik + sin * lmDiag[i];
789 lmDiag[i] = -sin * rik + cos * lmDiag[i];
790 weightedJacobian[i][pk] = temp2;
791 }
792 }
793 }
794
795 // store the diagonal element of s and restore
796 // the corresponding diagonal element of R
797 lmDiag[j] = weightedJacobian[j][permutation[j]];
798 weightedJacobian[j][permutation[j]] = lmDir[j];
799 }
800
801 // solve the triangular system for z, if the system is
802 // singular, then obtain a least squares solution
803 int nSing = solvedCols;
804 for (int j = 0; j < solvedCols; ++j) {
805 if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
806 nSing = j;
807 }
808 if (nSing < solvedCols) {
809 work[j] = 0;
810 }
811 }
812 if (nSing > 0) {
813 for (int j = nSing - 1; j >= 0; --j) {
814 int pj = permutation[j];
815 double sum = 0;
816 for (int i = j + 1; i < nSing; ++i) {
817 sum += weightedJacobian[i][pj] * work[i];
818 }
819 work[j] = (work[j] - sum) / lmDiag[j];
820 }
821 }
822
823 // permute the components of z back to components of lmDir
824 for (int j = 0; j < lmDir.length; ++j) {
825 lmDir[permutation[j]] = work[j];
826 }
827 }
828
829 /**
830 * Decompose a matrix A as A.P = Q.R using Householder transforms.
831 * <p>As suggested in the P. Lascaux and R. Theodor book
832 * <i>Analyse numérique matricielle appliquée à
833 * l'art de l'ingénieur</i> (Masson, 1986), instead of representing
834 * the Householder transforms with u<sub>k</sub> unit vectors such that:
835 * <pre>
836 * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
837 * </pre>
838 * we use <sub>k</sub> non-unit vectors such that:
839 * <pre>
840 * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
841 * </pre>
842 * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
843 * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
844 * them from the v<sub>k</sub> vectors would be costly.</p>
845 * <p>This decomposition handles rank deficient cases since the tranformations
846 * are performed in non-increasing columns norms order thanks to columns
847 * pivoting. The diagonal elements of the R matrix are therefore also in
848 * non-increasing absolute values order.</p>
849 *
850 * @param jacobian Weighted Jacobian matrix at the current point.
851 * @exception ConvergenceException if the decomposition cannot be performed
852 */
853 private void qrDecomposition(RealMatrix jacobian) throws ConvergenceException {
854 // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
855 // hence the multiplication by -1.
856 weightedJacobian = jacobian.scalarMultiply(-1).getData();
857
858 final int nR = weightedJacobian.length;
859 final int nC = weightedJacobian[0].length;
860
861 // initializations
862 for (int k = 0; k < nC; ++k) {
863 permutation[k] = k;
864 double norm2 = 0;
865 for (int i = 0; i < nR; ++i) {
866 double akk = weightedJacobian[i][k];
867 norm2 += akk * akk;
868 }
869 jacNorm[k] = FastMath.sqrt(norm2);
870 }
871
872 // transform the matrix column after column
873 for (int k = 0; k < nC; ++k) {
874
875 // select the column with the greatest norm on active components
876 int nextColumn = -1;
877 double ak2 = Double.NEGATIVE_INFINITY;
878 for (int i = k; i < nC; ++i) {
879 double norm2 = 0;
880 for (int j = k; j < nR; ++j) {
881 double aki = weightedJacobian[j][permutation[i]];
882 norm2 += aki * aki;
883 }
884 if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
885 throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
886 nR, nC);
887 }
888 if (norm2 > ak2) {
889 nextColumn = i;
890 ak2 = norm2;
891 }
892 }
893 if (ak2 <= qrRankingThreshold) {
894 rank = k;
895 return;
896 }
897 int pk = permutation[nextColumn];
898 permutation[nextColumn] = permutation[k];
899 permutation[k] = pk;
900
901 // choose alpha such that Hk.u = alpha ek
902 double akk = weightedJacobian[k][pk];
903 double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
904 double betak = 1.0 / (ak2 - akk * alpha);
905 beta[pk] = betak;
906
907 // transform the current column
908 diagR[pk] = alpha;
909 weightedJacobian[k][pk] -= alpha;
910
911 // transform the remaining columns
912 for (int dk = nC - 1 - k; dk > 0; --dk) {
913 double gamma = 0;
914 for (int j = k; j < nR; ++j) {
915 gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]];
916 }
917 gamma *= betak;
918 for (int j = k; j < nR; ++j) {
919 weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk];
920 }
921 }
922 }
923 rank = solvedCols;
924 }
925
926 /**
927 * Compute the product Qt.y for some Q.R. decomposition.
928 *
929 * @param y vector to multiply (will be overwritten with the result)
930 */
931 private void qTy(double[] y) {
932 final int nR = weightedJacobian.length;
933 final int nC = weightedJacobian[0].length;
934
935 for (int k = 0; k < nC; ++k) {
936 int pk = permutation[k];
937 double gamma = 0;
938 for (int i = k; i < nR; ++i) {
939 gamma += weightedJacobian[i][pk] * y[i];
940 }
941 gamma *= beta[pk];
942 for (int i = k; i < nR; ++i) {
943 y[i] -= gamma * weightedJacobian[i][pk];
944 }
945 }
946 }
947 }