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.optim.nonlinear.vector.jacobian;
018
019 import java.util.Arrays;
020 import org.apache.commons.math3.exception.ConvergenceException;
021 import org.apache.commons.math3.exception.util.LocalizedFormats;
022 import org.apache.commons.math3.optim.PointVectorValuePair;
023 import org.apache.commons.math3.optim.ConvergenceChecker;
024 import org.apache.commons.math3.linear.RealMatrix;
025 import org.apache.commons.math3.util.Precision;
026 import org.apache.commons.math3.util.FastMath;
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 *
104 * @version $Id: LevenbergMarquardtOptimizer.java 1416643 2012-12-03 19:37:14Z tn $
105 * @since 2.0
106 */
107 public class LevenbergMarquardtOptimizer
108 extends AbstractLeastSquaresOptimizer {
109 /** Number of solved point. */
110 private int solvedCols;
111 /** Diagonal elements of the R matrix in the Q.R. decomposition. */
112 private double[] diagR;
113 /** Norms of the columns of the jacobian matrix. */
114 private double[] jacNorm;
115 /** Coefficients of the Householder transforms vectors. */
116 private double[] beta;
117 /** Columns permutation array. */
118 private int[] permutation;
119 /** Rank of the jacobian matrix. */
120 private int rank;
121 /** Levenberg-Marquardt parameter. */
122 private double lmPar;
123 /** Parameters evolution direction associated with lmPar. */
124 private double[] lmDir;
125 /** Positive input variable used in determining the initial step bound. */
126 private final double initialStepBoundFactor;
127 /** Desired relative error in the sum of squares. */
128 private final double costRelativeTolerance;
129 /** Desired relative error in the approximate solution parameters. */
130 private final double parRelativeTolerance;
131 /** Desired max cosine on the orthogonality between the function vector
132 * and the columns of the jacobian. */
133 private final double orthoTolerance;
134 /** Threshold for QR ranking. */
135 private final double qrRankingThreshold;
136 /** Weighted residuals. */
137 private double[] weightedResidual;
138 /** Weighted Jacobian. */
139 private double[][] weightedJacobian;
140
141 /**
142 * Build an optimizer for least squares problems with default values
143 * for all the tuning parameters (see the {@link
144 * #LevenbergMarquardtOptimizer(double,double,double,double,double)
145 * other contructor}.
146 * The default values for the algorithm settings are:
147 * <ul>
148 * <li>Initial step bound factor: 100</li>
149 * <li>Cost relative tolerance: 1e-10</li>
150 * <li>Parameters relative tolerance: 1e-10</li>
151 * <li>Orthogonality tolerance: 1e-10</li>
152 * <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
153 * </ul>
154 */
155 public LevenbergMarquardtOptimizer() {
156 this(100, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
157 }
158
159 /**
160 * Constructor that allows the specification of a custom convergence
161 * checker.
162 * Note that all the usual convergence checks will be <em>disabled</em>.
163 * The default values for the algorithm settings are:
164 * <ul>
165 * <li>Initial step bound factor: 100</li>
166 * <li>Cost relative tolerance: 1e-10</li>
167 * <li>Parameters relative tolerance: 1e-10</li>
168 * <li>Orthogonality tolerance: 1e-10</li>
169 * <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
170 * </ul>
171 *
172 * @param checker Convergence checker.
173 */
174 public LevenbergMarquardtOptimizer(ConvergenceChecker<PointVectorValuePair> checker) {
175 this(100, checker, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
176 }
177
178 /**
179 * Constructor that allows the specification of a custom convergence
180 * checker, in addition to the standard ones.
181 *
182 * @param initialStepBoundFactor Positive input variable used in
183 * determining the initial step bound. This bound is set to the
184 * product of initialStepBoundFactor and the euclidean norm of
185 * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
186 * itself. In most cases factor should lie in the interval
187 * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
188 * @param checker Convergence checker.
189 * @param costRelativeTolerance Desired relative error in the sum of
190 * squares.
191 * @param parRelativeTolerance Desired relative error in the approximate
192 * solution parameters.
193 * @param orthoTolerance Desired max cosine on the orthogonality between
194 * the function vector and the columns of the Jacobian.
195 * @param threshold Desired threshold for QR ranking. If the squared norm
196 * of a column vector is smaller or equal to this threshold during QR
197 * decomposition, it is considered to be a zero vector and hence the rank
198 * of the matrix is reduced.
199 */
200 public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
201 ConvergenceChecker<PointVectorValuePair> checker,
202 double costRelativeTolerance,
203 double parRelativeTolerance,
204 double orthoTolerance,
205 double threshold) {
206 super(checker);
207 this.initialStepBoundFactor = initialStepBoundFactor;
208 this.costRelativeTolerance = costRelativeTolerance;
209 this.parRelativeTolerance = parRelativeTolerance;
210 this.orthoTolerance = orthoTolerance;
211 this.qrRankingThreshold = threshold;
212 }
213
214 /**
215 * Build an optimizer for least squares problems with default values
216 * for some of the tuning parameters (see the {@link
217 * #LevenbergMarquardtOptimizer(double,double,double,double,double)
218 * other contructor}.
219 * The default values for the algorithm settings are:
220 * <ul>
221 * <li>Initial step bound factor}: 100</li>
222 * <li>QR ranking threshold}: {@link Precision#SAFE_MIN}</li>
223 * </ul>
224 *
225 * @param costRelativeTolerance Desired relative error in the sum of
226 * squares.
227 * @param parRelativeTolerance Desired relative error in the approximate
228 * solution parameters.
229 * @param orthoTolerance Desired max cosine on the orthogonality between
230 * the function vector and the columns of the Jacobian.
231 */
232 public LevenbergMarquardtOptimizer(double costRelativeTolerance,
233 double parRelativeTolerance,
234 double orthoTolerance) {
235 this(100,
236 costRelativeTolerance, parRelativeTolerance, orthoTolerance,
237 Precision.SAFE_MIN);
238 }
239
240 /**
241 * The arguments control the behaviour of the default convergence checking
242 * procedure.
243 * Additional criteria can defined through the setting of a {@link
244 * ConvergenceChecker}.
245 *
246 * @param initialStepBoundFactor Positive input variable used in
247 * determining the initial step bound. This bound is set to the
248 * product of initialStepBoundFactor and the euclidean norm of
249 * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
250 * itself. In most cases factor should lie in the interval
251 * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
252 * @param costRelativeTolerance Desired relative error in the sum of
253 * squares.
254 * @param parRelativeTolerance Desired relative error in the approximate
255 * solution parameters.
256 * @param orthoTolerance Desired max cosine on the orthogonality between
257 * the function vector and the columns of the Jacobian.
258 * @param threshold Desired threshold for QR ranking. If the squared norm
259 * of a column vector is smaller or equal to this threshold during QR
260 * decomposition, it is considered to be a zero vector and hence the rank
261 * of the matrix is reduced.
262 */
263 public LevenbergMarquardtOptimizer(double initialStepBoundFactor,
264 double costRelativeTolerance,
265 double parRelativeTolerance,
266 double orthoTolerance,
267 double threshold) {
268 super(null); // No custom convergence criterion.
269 this.initialStepBoundFactor = initialStepBoundFactor;
270 this.costRelativeTolerance = costRelativeTolerance;
271 this.parRelativeTolerance = parRelativeTolerance;
272 this.orthoTolerance = orthoTolerance;
273 this.qrRankingThreshold = threshold;
274 }
275
276 /** {@inheritDoc} */
277 @Override
278 protected PointVectorValuePair doOptimize() {
279 final int nR = getTarget().length; // Number of observed data.
280 final double[] currentPoint = getStartPoint();
281 final int nC = currentPoint.length; // Number of parameters.
282
283 // arrays shared with the other private methods
284 solvedCols = FastMath.min(nR, nC);
285 diagR = new double[nC];
286 jacNorm = new double[nC];
287 beta = new double[nC];
288 permutation = new int[nC];
289 lmDir = new double[nC];
290
291 // local point
292 double delta = 0;
293 double xNorm = 0;
294 double[] diag = new double[nC];
295 double[] oldX = new double[nC];
296 double[] oldRes = new double[nR];
297 double[] oldObj = new double[nR];
298 double[] qtf = new double[nR];
299 double[] work1 = new double[nC];
300 double[] work2 = new double[nC];
301 double[] work3 = new double[nC];
302
303 final RealMatrix weightMatrixSqrt = getWeightSquareRoot();
304
305 // Evaluate the function at the starting point and calculate its norm.
306 double[] currentObjective = computeObjectiveValue(currentPoint);
307 double[] currentResiduals = computeResiduals(currentObjective);
308 PointVectorValuePair current = new PointVectorValuePair(currentPoint, currentObjective);
309 double currentCost = computeCost(currentResiduals);
310
311 // Outer loop.
312 lmPar = 0;
313 boolean firstIteration = true;
314 int iter = 0;
315 final ConvergenceChecker<PointVectorValuePair> checker = getConvergenceChecker();
316 while (true) {
317 ++iter;
318 final PointVectorValuePair previous = current;
319
320 // QR decomposition of the jacobian matrix
321 qrDecomposition(computeWeightedJacobian(currentPoint));
322
323 weightedResidual = weightMatrixSqrt.operate(currentResiduals);
324 for (int i = 0; i < nR; i++) {
325 qtf[i] = weightedResidual[i];
326 }
327
328 // compute Qt.res
329 qTy(qtf);
330
331 // now we don't need Q anymore,
332 // so let jacobian contain the R matrix with its diagonal elements
333 for (int k = 0; k < solvedCols; ++k) {
334 int pk = permutation[k];
335 weightedJacobian[k][pk] = diagR[pk];
336 }
337
338 if (firstIteration) {
339 // scale the point according to the norms of the columns
340 // of the initial jacobian
341 xNorm = 0;
342 for (int k = 0; k < nC; ++k) {
343 double dk = jacNorm[k];
344 if (dk == 0) {
345 dk = 1.0;
346 }
347 double xk = dk * currentPoint[k];
348 xNorm += xk * xk;
349 diag[k] = dk;
350 }
351 xNorm = FastMath.sqrt(xNorm);
352
353 // initialize the step bound delta
354 delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
355 }
356
357 // check orthogonality between function vector and jacobian columns
358 double maxCosine = 0;
359 if (currentCost != 0) {
360 for (int j = 0; j < solvedCols; ++j) {
361 int pj = permutation[j];
362 double s = jacNorm[pj];
363 if (s != 0) {
364 double sum = 0;
365 for (int i = 0; i <= j; ++i) {
366 sum += weightedJacobian[i][pj] * qtf[i];
367 }
368 maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * currentCost));
369 }
370 }
371 }
372 if (maxCosine <= orthoTolerance) {
373 // Convergence has been reached.
374 setCost(currentCost);
375 return current;
376 }
377
378 // rescale if necessary
379 for (int j = 0; j < nC; ++j) {
380 diag[j] = FastMath.max(diag[j], jacNorm[j]);
381 }
382
383 // Inner loop.
384 for (double ratio = 0; ratio < 1.0e-4;) {
385
386 // save the state
387 for (int j = 0; j < solvedCols; ++j) {
388 int pj = permutation[j];
389 oldX[pj] = currentPoint[pj];
390 }
391 final double previousCost = currentCost;
392 double[] tmpVec = weightedResidual;
393 weightedResidual = oldRes;
394 oldRes = tmpVec;
395 tmpVec = currentObjective;
396 currentObjective = oldObj;
397 oldObj = tmpVec;
398
399 // determine the Levenberg-Marquardt parameter
400 determineLMParameter(qtf, delta, diag, work1, work2, work3);
401
402 // compute the new point and the norm of the evolution direction
403 double lmNorm = 0;
404 for (int j = 0; j < solvedCols; ++j) {
405 int pj = permutation[j];
406 lmDir[pj] = -lmDir[pj];
407 currentPoint[pj] = oldX[pj] + lmDir[pj];
408 double s = diag[pj] * lmDir[pj];
409 lmNorm += s * s;
410 }
411 lmNorm = FastMath.sqrt(lmNorm);
412 // on the first iteration, adjust the initial step bound.
413 if (firstIteration) {
414 delta = FastMath.min(delta, lmNorm);
415 }
416
417 // Evaluate the function at x + p and calculate its norm.
418 currentObjective = computeObjectiveValue(currentPoint);
419 currentResiduals = computeResiduals(currentObjective);
420 current = new PointVectorValuePair(currentPoint, currentObjective);
421 currentCost = computeCost(currentResiduals);
422
423 // compute the scaled actual reduction
424 double actRed = -1.0;
425 if (0.1 * currentCost < previousCost) {
426 double r = currentCost / previousCost;
427 actRed = 1.0 - r * r;
428 }
429
430 // compute the scaled predicted reduction
431 // and the scaled directional derivative
432 for (int j = 0; j < solvedCols; ++j) {
433 int pj = permutation[j];
434 double dirJ = lmDir[pj];
435 work1[j] = 0;
436 for (int i = 0; i <= j; ++i) {
437 work1[i] += weightedJacobian[i][pj] * dirJ;
438 }
439 }
440 double coeff1 = 0;
441 for (int j = 0; j < solvedCols; ++j) {
442 coeff1 += work1[j] * work1[j];
443 }
444 double pc2 = previousCost * previousCost;
445 coeff1 = coeff1 / pc2;
446 double coeff2 = lmPar * lmNorm * lmNorm / pc2;
447 double preRed = coeff1 + 2 * coeff2;
448 double dirDer = -(coeff1 + coeff2);
449
450 // ratio of the actual to the predicted reduction
451 ratio = (preRed == 0) ? 0 : (actRed / preRed);
452
453 // update the step bound
454 if (ratio <= 0.25) {
455 double tmp =
456 (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
457 if ((0.1 * currentCost >= previousCost) || (tmp < 0.1)) {
458 tmp = 0.1;
459 }
460 delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
461 lmPar /= tmp;
462 } else if ((lmPar == 0) || (ratio >= 0.75)) {
463 delta = 2 * lmNorm;
464 lmPar *= 0.5;
465 }
466
467 // test for successful iteration.
468 if (ratio >= 1.0e-4) {
469 // successful iteration, update the norm
470 firstIteration = false;
471 xNorm = 0;
472 for (int k = 0; k < nC; ++k) {
473 double xK = diag[k] * currentPoint[k];
474 xNorm += xK * xK;
475 }
476 xNorm = FastMath.sqrt(xNorm);
477
478 // tests for convergence.
479 if (checker != null) {
480 // we use the vectorial convergence checker
481 if (checker.converged(iter, previous, current)) {
482 setCost(currentCost);
483 return current;
484 }
485 }
486 } else {
487 // failed iteration, reset the previous values
488 currentCost = previousCost;
489 for (int j = 0; j < solvedCols; ++j) {
490 int pj = permutation[j];
491 currentPoint[pj] = oldX[pj];
492 }
493 tmpVec = weightedResidual;
494 weightedResidual = oldRes;
495 oldRes = tmpVec;
496 tmpVec = currentObjective;
497 currentObjective = oldObj;
498 oldObj = tmpVec;
499 // Reset "current" to previous values.
500 current = new PointVectorValuePair(currentPoint, currentObjective);
501 }
502
503 // Default convergence criteria.
504 if ((FastMath.abs(actRed) <= costRelativeTolerance &&
505 preRed <= costRelativeTolerance &&
506 ratio <= 2.0) ||
507 delta <= parRelativeTolerance * xNorm) {
508 setCost(currentCost);
509 return current;
510 }
511
512 // tests for termination and stringent tolerances
513 // (2.2204e-16 is the machine epsilon for IEEE754)
514 if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
515 throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
516 costRelativeTolerance);
517 } else if (delta <= 2.2204e-16 * xNorm) {
518 throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
519 parRelativeTolerance);
520 } else if (maxCosine <= 2.2204e-16) {
521 throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
522 orthoTolerance);
523 }
524 }
525 }
526 }
527
528 /**
529 * Determine the Levenberg-Marquardt parameter.
530 * <p>This implementation is a translation in Java of the MINPACK
531 * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
532 * routine.</p>
533 * <p>This method sets the lmPar and lmDir attributes.</p>
534 * <p>The authors of the original fortran function are:</p>
535 * <ul>
536 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
537 * <li>Burton S. Garbow</li>
538 * <li>Kenneth E. Hillstrom</li>
539 * <li>Jorge J. More</li>
540 * </ul>
541 * <p>Luc Maisonobe did the Java translation.</p>
542 *
543 * @param qy array containing qTy
544 * @param delta upper bound on the euclidean norm of diagR * lmDir
545 * @param diag diagonal matrix
546 * @param work1 work array
547 * @param work2 work array
548 * @param work3 work array
549 */
550 private void determineLMParameter(double[] qy, double delta, double[] diag,
551 double[] work1, double[] work2, double[] work3) {
552 final int nC = weightedJacobian[0].length;
553
554 // compute and store in x the gauss-newton direction, if the
555 // jacobian is rank-deficient, obtain a least squares solution
556 for (int j = 0; j < rank; ++j) {
557 lmDir[permutation[j]] = qy[j];
558 }
559 for (int j = rank; j < nC; ++j) {
560 lmDir[permutation[j]] = 0;
561 }
562 for (int k = rank - 1; k >= 0; --k) {
563 int pk = permutation[k];
564 double ypk = lmDir[pk] / diagR[pk];
565 for (int i = 0; i < k; ++i) {
566 lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk];
567 }
568 lmDir[pk] = ypk;
569 }
570
571 // evaluate the function at the origin, and test
572 // for acceptance of the Gauss-Newton direction
573 double dxNorm = 0;
574 for (int j = 0; j < solvedCols; ++j) {
575 int pj = permutation[j];
576 double s = diag[pj] * lmDir[pj];
577 work1[pj] = s;
578 dxNorm += s * s;
579 }
580 dxNorm = FastMath.sqrt(dxNorm);
581 double fp = dxNorm - delta;
582 if (fp <= 0.1 * delta) {
583 lmPar = 0;
584 return;
585 }
586
587 // if the jacobian is not rank deficient, the Newton step provides
588 // a lower bound, parl, for the zero of the function,
589 // otherwise set this bound to zero
590 double sum2;
591 double parl = 0;
592 if (rank == solvedCols) {
593 for (int j = 0; j < solvedCols; ++j) {
594 int pj = permutation[j];
595 work1[pj] *= diag[pj] / dxNorm;
596 }
597 sum2 = 0;
598 for (int j = 0; j < solvedCols; ++j) {
599 int pj = permutation[j];
600 double sum = 0;
601 for (int i = 0; i < j; ++i) {
602 sum += weightedJacobian[i][pj] * work1[permutation[i]];
603 }
604 double s = (work1[pj] - sum) / diagR[pj];
605 work1[pj] = s;
606 sum2 += s * s;
607 }
608 parl = fp / (delta * sum2);
609 }
610
611 // calculate an upper bound, paru, for the zero of the function
612 sum2 = 0;
613 for (int j = 0; j < solvedCols; ++j) {
614 int pj = permutation[j];
615 double sum = 0;
616 for (int i = 0; i <= j; ++i) {
617 sum += weightedJacobian[i][pj] * qy[i];
618 }
619 sum /= diag[pj];
620 sum2 += sum * sum;
621 }
622 double gNorm = FastMath.sqrt(sum2);
623 double paru = gNorm / delta;
624 if (paru == 0) {
625 // 2.2251e-308 is the smallest positive real for IEE754
626 paru = 2.2251e-308 / FastMath.min(delta, 0.1);
627 }
628
629 // if the input par lies outside of the interval (parl,paru),
630 // set par to the closer endpoint
631 lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
632 if (lmPar == 0) {
633 lmPar = gNorm / dxNorm;
634 }
635
636 for (int countdown = 10; countdown >= 0; --countdown) {
637
638 // evaluate the function at the current value of lmPar
639 if (lmPar == 0) {
640 lmPar = FastMath.max(2.2251e-308, 0.001 * paru);
641 }
642 double sPar = FastMath.sqrt(lmPar);
643 for (int j = 0; j < solvedCols; ++j) {
644 int pj = permutation[j];
645 work1[pj] = sPar * diag[pj];
646 }
647 determineLMDirection(qy, work1, work2, work3);
648
649 dxNorm = 0;
650 for (int j = 0; j < solvedCols; ++j) {
651 int pj = permutation[j];
652 double s = diag[pj] * lmDir[pj];
653 work3[pj] = s;
654 dxNorm += s * s;
655 }
656 dxNorm = FastMath.sqrt(dxNorm);
657 double previousFP = fp;
658 fp = dxNorm - delta;
659
660 // if the function is small enough, accept the current value
661 // of lmPar, also test for the exceptional cases where parl is zero
662 if ((FastMath.abs(fp) <= 0.1 * delta) ||
663 ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
664 return;
665 }
666
667 // compute the Newton correction
668 for (int j = 0; j < solvedCols; ++j) {
669 int pj = permutation[j];
670 work1[pj] = work3[pj] * diag[pj] / dxNorm;
671 }
672 for (int j = 0; j < solvedCols; ++j) {
673 int pj = permutation[j];
674 work1[pj] /= work2[j];
675 double tmp = work1[pj];
676 for (int i = j + 1; i < solvedCols; ++i) {
677 work1[permutation[i]] -= weightedJacobian[i][pj] * tmp;
678 }
679 }
680 sum2 = 0;
681 for (int j = 0; j < solvedCols; ++j) {
682 double s = work1[permutation[j]];
683 sum2 += s * s;
684 }
685 double correction = fp / (delta * sum2);
686
687 // depending on the sign of the function, update parl or paru.
688 if (fp > 0) {
689 parl = FastMath.max(parl, lmPar);
690 } else if (fp < 0) {
691 paru = FastMath.min(paru, lmPar);
692 }
693
694 // compute an improved estimate for lmPar
695 lmPar = FastMath.max(parl, lmPar + correction);
696
697 }
698 }
699
700 /**
701 * Solve a*x = b and d*x = 0 in the least squares sense.
702 * <p>This implementation is a translation in Java of the MINPACK
703 * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
704 * routine.</p>
705 * <p>This method sets the lmDir and lmDiag attributes.</p>
706 * <p>The authors of the original fortran function are:</p>
707 * <ul>
708 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
709 * <li>Burton S. Garbow</li>
710 * <li>Kenneth E. Hillstrom</li>
711 * <li>Jorge J. More</li>
712 * </ul>
713 * <p>Luc Maisonobe did the Java translation.</p>
714 *
715 * @param qy array containing qTy
716 * @param diag diagonal matrix
717 * @param lmDiag diagonal elements associated with lmDir
718 * @param work work array
719 */
720 private void determineLMDirection(double[] qy, double[] diag,
721 double[] lmDiag, double[] work) {
722
723 // copy R and Qty to preserve input and initialize s
724 // in particular, save the diagonal elements of R in lmDir
725 for (int j = 0; j < solvedCols; ++j) {
726 int pj = permutation[j];
727 for (int i = j + 1; i < solvedCols; ++i) {
728 weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]];
729 }
730 lmDir[j] = diagR[pj];
731 work[j] = qy[j];
732 }
733
734 // eliminate the diagonal matrix d using a Givens rotation
735 for (int j = 0; j < solvedCols; ++j) {
736
737 // prepare the row of d to be eliminated, locating the
738 // diagonal element using p from the Q.R. factorization
739 int pj = permutation[j];
740 double dpj = diag[pj];
741 if (dpj != 0) {
742 Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
743 }
744 lmDiag[j] = dpj;
745
746 // the transformations to eliminate the row of d
747 // modify only a single element of Qty
748 // beyond the first n, which is initially zero.
749 double qtbpj = 0;
750 for (int k = j; k < solvedCols; ++k) {
751 int pk = permutation[k];
752
753 // determine a Givens rotation which eliminates the
754 // appropriate element in the current row of d
755 if (lmDiag[k] != 0) {
756
757 final double sin;
758 final double cos;
759 double rkk = weightedJacobian[k][pk];
760 if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
761 final double cotan = rkk / lmDiag[k];
762 sin = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
763 cos = sin * cotan;
764 } else {
765 final double tan = lmDiag[k] / rkk;
766 cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
767 sin = cos * tan;
768 }
769
770 // compute the modified diagonal element of R and
771 // the modified element of (Qty,0)
772 weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
773 final double temp = cos * work[k] + sin * qtbpj;
774 qtbpj = -sin * work[k] + cos * qtbpj;
775 work[k] = temp;
776
777 // accumulate the tranformation in the row of s
778 for (int i = k + 1; i < solvedCols; ++i) {
779 double rik = weightedJacobian[i][pk];
780 final double temp2 = cos * rik + sin * lmDiag[i];
781 lmDiag[i] = -sin * rik + cos * lmDiag[i];
782 weightedJacobian[i][pk] = temp2;
783 }
784 }
785 }
786
787 // store the diagonal element of s and restore
788 // the corresponding diagonal element of R
789 lmDiag[j] = weightedJacobian[j][permutation[j]];
790 weightedJacobian[j][permutation[j]] = lmDir[j];
791 }
792
793 // solve the triangular system for z, if the system is
794 // singular, then obtain a least squares solution
795 int nSing = solvedCols;
796 for (int j = 0; j < solvedCols; ++j) {
797 if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
798 nSing = j;
799 }
800 if (nSing < solvedCols) {
801 work[j] = 0;
802 }
803 }
804 if (nSing > 0) {
805 for (int j = nSing - 1; j >= 0; --j) {
806 int pj = permutation[j];
807 double sum = 0;
808 for (int i = j + 1; i < nSing; ++i) {
809 sum += weightedJacobian[i][pj] * work[i];
810 }
811 work[j] = (work[j] - sum) / lmDiag[j];
812 }
813 }
814
815 // permute the components of z back to components of lmDir
816 for (int j = 0; j < lmDir.length; ++j) {
817 lmDir[permutation[j]] = work[j];
818 }
819 }
820
821 /**
822 * Decompose a matrix A as A.P = Q.R using Householder transforms.
823 * <p>As suggested in the P. Lascaux and R. Theodor book
824 * <i>Analyse numérique matricielle appliquée à
825 * l'art de l'ingénieur</i> (Masson, 1986), instead of representing
826 * the Householder transforms with u<sub>k</sub> unit vectors such that:
827 * <pre>
828 * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
829 * </pre>
830 * we use <sub>k</sub> non-unit vectors such that:
831 * <pre>
832 * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
833 * </pre>
834 * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
835 * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
836 * them from the v<sub>k</sub> vectors would be costly.</p>
837 * <p>This decomposition handles rank deficient cases since the tranformations
838 * are performed in non-increasing columns norms order thanks to columns
839 * pivoting. The diagonal elements of the R matrix are therefore also in
840 * non-increasing absolute values order.</p>
841 *
842 * @param jacobian Weighted Jacobian matrix at the current point.
843 * @exception ConvergenceException if the decomposition cannot be performed
844 */
845 private void qrDecomposition(RealMatrix jacobian) throws ConvergenceException {
846 // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
847 // hence the multiplication by -1.
848 weightedJacobian = jacobian.scalarMultiply(-1).getData();
849
850 final int nR = weightedJacobian.length;
851 final int nC = weightedJacobian[0].length;
852
853 // initializations
854 for (int k = 0; k < nC; ++k) {
855 permutation[k] = k;
856 double norm2 = 0;
857 for (int i = 0; i < nR; ++i) {
858 double akk = weightedJacobian[i][k];
859 norm2 += akk * akk;
860 }
861 jacNorm[k] = FastMath.sqrt(norm2);
862 }
863
864 // transform the matrix column after column
865 for (int k = 0; k < nC; ++k) {
866
867 // select the column with the greatest norm on active components
868 int nextColumn = -1;
869 double ak2 = Double.NEGATIVE_INFINITY;
870 for (int i = k; i < nC; ++i) {
871 double norm2 = 0;
872 for (int j = k; j < nR; ++j) {
873 double aki = weightedJacobian[j][permutation[i]];
874 norm2 += aki * aki;
875 }
876 if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
877 throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
878 nR, nC);
879 }
880 if (norm2 > ak2) {
881 nextColumn = i;
882 ak2 = norm2;
883 }
884 }
885 if (ak2 <= qrRankingThreshold) {
886 rank = k;
887 return;
888 }
889 int pk = permutation[nextColumn];
890 permutation[nextColumn] = permutation[k];
891 permutation[k] = pk;
892
893 // choose alpha such that Hk.u = alpha ek
894 double akk = weightedJacobian[k][pk];
895 double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
896 double betak = 1.0 / (ak2 - akk * alpha);
897 beta[pk] = betak;
898
899 // transform the current column
900 diagR[pk] = alpha;
901 weightedJacobian[k][pk] -= alpha;
902
903 // transform the remaining columns
904 for (int dk = nC - 1 - k; dk > 0; --dk) {
905 double gamma = 0;
906 for (int j = k; j < nR; ++j) {
907 gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]];
908 }
909 gamma *= betak;
910 for (int j = k; j < nR; ++j) {
911 weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk];
912 }
913 }
914 }
915 rank = solvedCols;
916 }
917
918 /**
919 * Compute the product Qt.y for some Q.R. decomposition.
920 *
921 * @param y vector to multiply (will be overwritten with the result)
922 */
923 private void qTy(double[] y) {
924 final int nR = weightedJacobian.length;
925 final int nC = weightedJacobian[0].length;
926
927 for (int k = 0; k < nC; ++k) {
928 int pk = permutation[k];
929 double gamma = 0;
930 for (int i = k; i < nR; ++i) {
931 gamma += weightedJacobian[i][pk] * y[i];
932 }
933 gamma *= beta[pk];
934 for (int i = k; i < nR; ++i) {
935 y[i] -= gamma * weightedJacobian[i][pk];
936 }
937 }
938 }
939 }