| /* |
| * Licensed to the Apache Software Foundation (ASF) under one or more |
| * contributor license agreements. See the NOTICE file distributed with |
| * this work for additional information regarding copyright ownership. |
| * The ASF licenses this file to You under the Apache License, Version 2.0 |
| * (the "License"); you may not use this file except in compliance with |
| * the License. You may obtain a copy of the License at |
| * |
| * http://www.apache.org/licenses/LICENSE-2.0 |
| * |
| * Unless required by applicable law or agreed to in writing, software |
| * distributed under the License is distributed on an "AS IS" BASIS, |
| * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. |
| * See the License for the specific language governing permissions and |
| * limitations under the License. |
| */ |
| package org.apache.commons.math3.optimization.general; |
| |
| import java.util.Arrays; |
| |
| import org.apache.commons.math3.exception.ConvergenceException; |
| import org.apache.commons.math3.exception.util.LocalizedFormats; |
| import org.apache.commons.math3.optimization.PointVectorValuePair; |
| import org.apache.commons.math3.optimization.ConvergenceChecker; |
| import org.apache.commons.math3.linear.RealMatrix; |
| import org.apache.commons.math3.util.Precision; |
| import org.apache.commons.math3.util.FastMath; |
| |
| |
| /** |
| * This class solves a least squares problem using the Levenberg-Marquardt algorithm. |
| * |
| * <p>This implementation <em>should</em> work even for over-determined systems |
| * (i.e. systems having more point than equations). Over-determined systems |
| * are solved by ignoring the point which have the smallest impact according |
| * to their jacobian column norm. Only the rank of the matrix and some loop bounds |
| * are changed to implement this.</p> |
| * |
| * <p>The resolution engine is a simple translation of the MINPACK <a |
| * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor |
| * changes. The changes include the over-determined resolution, the use of |
| * inherited convergence checker and the Q.R. decomposition which has been |
| * rewritten following the algorithm described in the |
| * P. Lascaux and R. Theodor book <i>Analyse numérique matricielle |
| * appliquée à l'art de l'ingénieur</i>, Masson 1986.</p> |
| * <p>The authors of the original fortran version are: |
| * <ul> |
| * <li>Argonne National Laboratory. MINPACK project. March 1980</li> |
| * <li>Burton S. Garbow</li> |
| * <li>Kenneth E. Hillstrom</li> |
| * <li>Jorge J. More</li> |
| * </ul> |
| * The redistribution policy for MINPACK is available <a |
| * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it |
| * is reproduced below.</p> |
| * |
| * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0"> |
| * <tr><td> |
| * Minpack Copyright Notice (1999) University of Chicago. |
| * All rights reserved |
| * </td></tr> |
| * <tr><td> |
| * Redistribution and use in source and binary forms, with or without |
| * modification, are permitted provided that the following conditions |
| * are met: |
| * <ol> |
| * <li>Redistributions of source code must retain the above copyright |
| * notice, this list of conditions and the following disclaimer.</li> |
| * <li>Redistributions in binary form must reproduce the above |
| * copyright notice, this list of conditions and the following |
| * disclaimer in the documentation and/or other materials provided |
| * with the distribution.</li> |
| * <li>The end-user documentation included with the redistribution, if any, |
| * must include the following acknowledgment: |
| * <code>This product includes software developed by the University of |
| * Chicago, as Operator of Argonne National Laboratory.</code> |
| * Alternately, this acknowledgment may appear in the software itself, |
| * if and wherever such third-party acknowledgments normally appear.</li> |
| * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS" |
| * WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE |
| * UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND |
| * THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR |
| * IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES |
| * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE |
| * OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY |
| * OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR |
| * USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF |
| * THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4) |
| * DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION |
| * UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL |
| * BE CORRECTED.</strong></li> |
| * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT |
| * HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF |
| * ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT, |
| * INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF |
| * ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF |
| * PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER |
| * SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT |
| * (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE, |
| * EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE |
| * POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li> |
| * <ol></td></tr> |
| * </table> |
| * @deprecated As of 3.1 (to be removed in 4.0). |
| * @since 2.0 |
| * |
| */ |
| @Deprecated |
| public class LevenbergMarquardtOptimizer extends AbstractLeastSquaresOptimizer { |
| /** Number of solved point. */ |
| private int solvedCols; |
| /** Diagonal elements of the R matrix in the Q.R. decomposition. */ |
| private double[] diagR; |
| /** Norms of the columns of the jacobian matrix. */ |
| private double[] jacNorm; |
| /** Coefficients of the Householder transforms vectors. */ |
| private double[] beta; |
| /** Columns permutation array. */ |
| private int[] permutation; |
| /** Rank of the jacobian matrix. */ |
| private int rank; |
| /** Levenberg-Marquardt parameter. */ |
| private double lmPar; |
| /** Parameters evolution direction associated with lmPar. */ |
| private double[] lmDir; |
| /** Positive input variable used in determining the initial step bound. */ |
| private final double initialStepBoundFactor; |
| /** Desired relative error in the sum of squares. */ |
| private final double costRelativeTolerance; |
| /** Desired relative error in the approximate solution parameters. */ |
| private final double parRelativeTolerance; |
| /** Desired max cosine on the orthogonality between the function vector |
| * and the columns of the jacobian. */ |
| private final double orthoTolerance; |
| /** Threshold for QR ranking. */ |
| private final double qrRankingThreshold; |
| /** Weighted residuals. */ |
| private double[] weightedResidual; |
| /** Weighted Jacobian. */ |
| private double[][] weightedJacobian; |
| |
| /** |
| * Build an optimizer for least squares problems with default values |
| * for all the tuning parameters (see the {@link |
| * #LevenbergMarquardtOptimizer(double,double,double,double,double) |
| * other contructor}. |
| * The default values for the algorithm settings are: |
| * <ul> |
| * <li>Initial step bound factor: 100</li> |
| * <li>Cost relative tolerance: 1e-10</li> |
| * <li>Parameters relative tolerance: 1e-10</li> |
| * <li>Orthogonality tolerance: 1e-10</li> |
| * <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li> |
| * </ul> |
| */ |
| public LevenbergMarquardtOptimizer() { |
| this(100, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN); |
| } |
| |
| /** |
| * Constructor that allows the specification of a custom convergence |
| * checker. |
| * Note that all the usual convergence checks will be <em>disabled</em>. |
| * The default values for the algorithm settings are: |
| * <ul> |
| * <li>Initial step bound factor: 100</li> |
| * <li>Cost relative tolerance: 1e-10</li> |
| * <li>Parameters relative tolerance: 1e-10</li> |
| * <li>Orthogonality tolerance: 1e-10</li> |
| * <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li> |
| * </ul> |
| * |
| * @param checker Convergence checker. |
| */ |
| public LevenbergMarquardtOptimizer(ConvergenceChecker<PointVectorValuePair> checker) { |
| this(100, checker, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN); |
| } |
| |
| /** |
| * Constructor that allows the specification of a custom convergence |
| * checker, in addition to the standard ones. |
| * |
| * @param initialStepBoundFactor Positive input variable used in |
| * determining the initial step bound. This bound is set to the |
| * product of initialStepBoundFactor and the euclidean norm of |
| * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor} |
| * itself. In most cases factor should lie in the interval |
| * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value. |
| * @param checker Convergence checker. |
| * @param costRelativeTolerance Desired relative error in the sum of |
| * squares. |
| * @param parRelativeTolerance Desired relative error in the approximate |
| * solution parameters. |
| * @param orthoTolerance Desired max cosine on the orthogonality between |
| * the function vector and the columns of the Jacobian. |
| * @param threshold Desired threshold for QR ranking. If the squared norm |
| * of a column vector is smaller or equal to this threshold during QR |
| * decomposition, it is considered to be a zero vector and hence the rank |
| * of the matrix is reduced. |
| */ |
| public LevenbergMarquardtOptimizer(double initialStepBoundFactor, |
| ConvergenceChecker<PointVectorValuePair> checker, |
| double costRelativeTolerance, |
| double parRelativeTolerance, |
| double orthoTolerance, |
| double threshold) { |
| super(checker); |
| this.initialStepBoundFactor = initialStepBoundFactor; |
| this.costRelativeTolerance = costRelativeTolerance; |
| this.parRelativeTolerance = parRelativeTolerance; |
| this.orthoTolerance = orthoTolerance; |
| this.qrRankingThreshold = threshold; |
| } |
| |
| /** |
| * Build an optimizer for least squares problems with default values |
| * for some of the tuning parameters (see the {@link |
| * #LevenbergMarquardtOptimizer(double,double,double,double,double) |
| * other contructor}. |
| * The default values for the algorithm settings are: |
| * <ul> |
| * <li>Initial step bound factor}: 100</li> |
| * <li>QR ranking threshold}: {@link Precision#SAFE_MIN}</li> |
| * </ul> |
| * |
| * @param costRelativeTolerance Desired relative error in the sum of |
| * squares. |
| * @param parRelativeTolerance Desired relative error in the approximate |
| * solution parameters. |
| * @param orthoTolerance Desired max cosine on the orthogonality between |
| * the function vector and the columns of the Jacobian. |
| */ |
| public LevenbergMarquardtOptimizer(double costRelativeTolerance, |
| double parRelativeTolerance, |
| double orthoTolerance) { |
| this(100, |
| costRelativeTolerance, parRelativeTolerance, orthoTolerance, |
| Precision.SAFE_MIN); |
| } |
| |
| /** |
| * The arguments control the behaviour of the default convergence checking |
| * procedure. |
| * Additional criteria can defined through the setting of a {@link |
| * ConvergenceChecker}. |
| * |
| * @param initialStepBoundFactor Positive input variable used in |
| * determining the initial step bound. This bound is set to the |
| * product of initialStepBoundFactor and the euclidean norm of |
| * {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor} |
| * itself. In most cases factor should lie in the interval |
| * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value. |
| * @param costRelativeTolerance Desired relative error in the sum of |
| * squares. |
| * @param parRelativeTolerance Desired relative error in the approximate |
| * solution parameters. |
| * @param orthoTolerance Desired max cosine on the orthogonality between |
| * the function vector and the columns of the Jacobian. |
| * @param threshold Desired threshold for QR ranking. If the squared norm |
| * of a column vector is smaller or equal to this threshold during QR |
| * decomposition, it is considered to be a zero vector and hence the rank |
| * of the matrix is reduced. |
| */ |
| public LevenbergMarquardtOptimizer(double initialStepBoundFactor, |
| double costRelativeTolerance, |
| double parRelativeTolerance, |
| double orthoTolerance, |
| double threshold) { |
| super(null); // No custom convergence criterion. |
| this.initialStepBoundFactor = initialStepBoundFactor; |
| this.costRelativeTolerance = costRelativeTolerance; |
| this.parRelativeTolerance = parRelativeTolerance; |
| this.orthoTolerance = orthoTolerance; |
| this.qrRankingThreshold = threshold; |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| protected PointVectorValuePair doOptimize() { |
| final int nR = getTarget().length; // Number of observed data. |
| final double[] currentPoint = getStartPoint(); |
| final int nC = currentPoint.length; // Number of parameters. |
| |
| // arrays shared with the other private methods |
| solvedCols = FastMath.min(nR, nC); |
| diagR = new double[nC]; |
| jacNorm = new double[nC]; |
| beta = new double[nC]; |
| permutation = new int[nC]; |
| lmDir = new double[nC]; |
| |
| // local point |
| double delta = 0; |
| double xNorm = 0; |
| double[] diag = new double[nC]; |
| double[] oldX = new double[nC]; |
| double[] oldRes = new double[nR]; |
| double[] oldObj = new double[nR]; |
| double[] qtf = new double[nR]; |
| double[] work1 = new double[nC]; |
| double[] work2 = new double[nC]; |
| double[] work3 = new double[nC]; |
| |
| final RealMatrix weightMatrixSqrt = getWeightSquareRoot(); |
| |
| // Evaluate the function at the starting point and calculate its norm. |
| double[] currentObjective = computeObjectiveValue(currentPoint); |
| double[] currentResiduals = computeResiduals(currentObjective); |
| PointVectorValuePair current = new PointVectorValuePair(currentPoint, currentObjective); |
| double currentCost = computeCost(currentResiduals); |
| |
| // Outer loop. |
| lmPar = 0; |
| boolean firstIteration = true; |
| int iter = 0; |
| final ConvergenceChecker<PointVectorValuePair> checker = getConvergenceChecker(); |
| while (true) { |
| ++iter; |
| final PointVectorValuePair previous = current; |
| |
| // QR decomposition of the jacobian matrix |
| qrDecomposition(computeWeightedJacobian(currentPoint)); |
| |
| weightedResidual = weightMatrixSqrt.operate(currentResiduals); |
| for (int i = 0; i < nR; i++) { |
| qtf[i] = weightedResidual[i]; |
| } |
| |
| // compute Qt.res |
| qTy(qtf); |
| |
| // now we don't need Q anymore, |
| // so let jacobian contain the R matrix with its diagonal elements |
| for (int k = 0; k < solvedCols; ++k) { |
| int pk = permutation[k]; |
| weightedJacobian[k][pk] = diagR[pk]; |
| } |
| |
| if (firstIteration) { |
| // scale the point according to the norms of the columns |
| // of the initial jacobian |
| xNorm = 0; |
| for (int k = 0; k < nC; ++k) { |
| double dk = jacNorm[k]; |
| if (dk == 0) { |
| dk = 1.0; |
| } |
| double xk = dk * currentPoint[k]; |
| xNorm += xk * xk; |
| diag[k] = dk; |
| } |
| xNorm = FastMath.sqrt(xNorm); |
| |
| // initialize the step bound delta |
| delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm); |
| } |
| |
| // check orthogonality between function vector and jacobian columns |
| double maxCosine = 0; |
| if (currentCost != 0) { |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| double s = jacNorm[pj]; |
| if (s != 0) { |
| double sum = 0; |
| for (int i = 0; i <= j; ++i) { |
| sum += weightedJacobian[i][pj] * qtf[i]; |
| } |
| maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * currentCost)); |
| } |
| } |
| } |
| if (maxCosine <= orthoTolerance) { |
| // Convergence has been reached. |
| setCost(currentCost); |
| // Update (deprecated) "point" field. |
| point = current.getPoint(); |
| return current; |
| } |
| |
| // rescale if necessary |
| for (int j = 0; j < nC; ++j) { |
| diag[j] = FastMath.max(diag[j], jacNorm[j]); |
| } |
| |
| // Inner loop. |
| for (double ratio = 0; ratio < 1.0e-4;) { |
| |
| // save the state |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| oldX[pj] = currentPoint[pj]; |
| } |
| final double previousCost = currentCost; |
| double[] tmpVec = weightedResidual; |
| weightedResidual = oldRes; |
| oldRes = tmpVec; |
| tmpVec = currentObjective; |
| currentObjective = oldObj; |
| oldObj = tmpVec; |
| |
| // determine the Levenberg-Marquardt parameter |
| determineLMParameter(qtf, delta, diag, work1, work2, work3); |
| |
| // compute the new point and the norm of the evolution direction |
| double lmNorm = 0; |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| lmDir[pj] = -lmDir[pj]; |
| currentPoint[pj] = oldX[pj] + lmDir[pj]; |
| double s = diag[pj] * lmDir[pj]; |
| lmNorm += s * s; |
| } |
| lmNorm = FastMath.sqrt(lmNorm); |
| // on the first iteration, adjust the initial step bound. |
| if (firstIteration) { |
| delta = FastMath.min(delta, lmNorm); |
| } |
| |
| // Evaluate the function at x + p and calculate its norm. |
| currentObjective = computeObjectiveValue(currentPoint); |
| currentResiduals = computeResiduals(currentObjective); |
| current = new PointVectorValuePair(currentPoint, currentObjective); |
| currentCost = computeCost(currentResiduals); |
| |
| // compute the scaled actual reduction |
| double actRed = -1.0; |
| if (0.1 * currentCost < previousCost) { |
| double r = currentCost / previousCost; |
| actRed = 1.0 - r * r; |
| } |
| |
| // compute the scaled predicted reduction |
| // and the scaled directional derivative |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| double dirJ = lmDir[pj]; |
| work1[j] = 0; |
| for (int i = 0; i <= j; ++i) { |
| work1[i] += weightedJacobian[i][pj] * dirJ; |
| } |
| } |
| double coeff1 = 0; |
| for (int j = 0; j < solvedCols; ++j) { |
| coeff1 += work1[j] * work1[j]; |
| } |
| double pc2 = previousCost * previousCost; |
| coeff1 /= pc2; |
| double coeff2 = lmPar * lmNorm * lmNorm / pc2; |
| double preRed = coeff1 + 2 * coeff2; |
| double dirDer = -(coeff1 + coeff2); |
| |
| // ratio of the actual to the predicted reduction |
| ratio = (preRed == 0) ? 0 : (actRed / preRed); |
| |
| // update the step bound |
| if (ratio <= 0.25) { |
| double tmp = |
| (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5; |
| if ((0.1 * currentCost >= previousCost) || (tmp < 0.1)) { |
| tmp = 0.1; |
| } |
| delta = tmp * FastMath.min(delta, 10.0 * lmNorm); |
| lmPar /= tmp; |
| } else if ((lmPar == 0) || (ratio >= 0.75)) { |
| delta = 2 * lmNorm; |
| lmPar *= 0.5; |
| } |
| |
| // test for successful iteration. |
| if (ratio >= 1.0e-4) { |
| // successful iteration, update the norm |
| firstIteration = false; |
| xNorm = 0; |
| for (int k = 0; k < nC; ++k) { |
| double xK = diag[k] * currentPoint[k]; |
| xNorm += xK * xK; |
| } |
| xNorm = FastMath.sqrt(xNorm); |
| |
| // tests for convergence. |
| if (checker != null && checker.converged(iter, previous, current)) { |
| setCost(currentCost); |
| // Update (deprecated) "point" field. |
| point = current.getPoint(); |
| return current; |
| } |
| } else { |
| // failed iteration, reset the previous values |
| currentCost = previousCost; |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| currentPoint[pj] = oldX[pj]; |
| } |
| tmpVec = weightedResidual; |
| weightedResidual = oldRes; |
| oldRes = tmpVec; |
| tmpVec = currentObjective; |
| currentObjective = oldObj; |
| oldObj = tmpVec; |
| // Reset "current" to previous values. |
| current = new PointVectorValuePair(currentPoint, currentObjective); |
| } |
| |
| // Default convergence criteria. |
| if ((FastMath.abs(actRed) <= costRelativeTolerance && |
| preRed <= costRelativeTolerance && |
| ratio <= 2.0) || |
| delta <= parRelativeTolerance * xNorm) { |
| setCost(currentCost); |
| // Update (deprecated) "point" field. |
| point = current.getPoint(); |
| return current; |
| } |
| |
| // tests for termination and stringent tolerances |
| // (2.2204e-16 is the machine epsilon for IEEE754) |
| if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) { |
| throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE, |
| costRelativeTolerance); |
| } else if (delta <= 2.2204e-16 * xNorm) { |
| throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE, |
| parRelativeTolerance); |
| } else if (maxCosine <= 2.2204e-16) { |
| throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE, |
| orthoTolerance); |
| } |
| } |
| } |
| } |
| |
| /** |
| * Determine the Levenberg-Marquardt parameter. |
| * <p>This implementation is a translation in Java of the MINPACK |
| * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a> |
| * routine.</p> |
| * <p>This method sets the lmPar and lmDir attributes.</p> |
| * <p>The authors of the original fortran function are:</p> |
| * <ul> |
| * <li>Argonne National Laboratory. MINPACK project. March 1980</li> |
| * <li>Burton S. Garbow</li> |
| * <li>Kenneth E. Hillstrom</li> |
| * <li>Jorge J. More</li> |
| * </ul> |
| * <p>Luc Maisonobe did the Java translation.</p> |
| * |
| * @param qy array containing qTy |
| * @param delta upper bound on the euclidean norm of diagR * lmDir |
| * @param diag diagonal matrix |
| * @param work1 work array |
| * @param work2 work array |
| * @param work3 work array |
| */ |
| private void determineLMParameter(double[] qy, double delta, double[] diag, |
| double[] work1, double[] work2, double[] work3) { |
| final int nC = weightedJacobian[0].length; |
| |
| // compute and store in x the gauss-newton direction, if the |
| // jacobian is rank-deficient, obtain a least squares solution |
| for (int j = 0; j < rank; ++j) { |
| lmDir[permutation[j]] = qy[j]; |
| } |
| for (int j = rank; j < nC; ++j) { |
| lmDir[permutation[j]] = 0; |
| } |
| for (int k = rank - 1; k >= 0; --k) { |
| int pk = permutation[k]; |
| double ypk = lmDir[pk] / diagR[pk]; |
| for (int i = 0; i < k; ++i) { |
| lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk]; |
| } |
| lmDir[pk] = ypk; |
| } |
| |
| // evaluate the function at the origin, and test |
| // for acceptance of the Gauss-Newton direction |
| double dxNorm = 0; |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| double s = diag[pj] * lmDir[pj]; |
| work1[pj] = s; |
| dxNorm += s * s; |
| } |
| dxNorm = FastMath.sqrt(dxNorm); |
| double fp = dxNorm - delta; |
| if (fp <= 0.1 * delta) { |
| lmPar = 0; |
| return; |
| } |
| |
| // if the jacobian is not rank deficient, the Newton step provides |
| // a lower bound, parl, for the zero of the function, |
| // otherwise set this bound to zero |
| double sum2; |
| double parl = 0; |
| if (rank == solvedCols) { |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| work1[pj] *= diag[pj] / dxNorm; |
| } |
| sum2 = 0; |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| double sum = 0; |
| for (int i = 0; i < j; ++i) { |
| sum += weightedJacobian[i][pj] * work1[permutation[i]]; |
| } |
| double s = (work1[pj] - sum) / diagR[pj]; |
| work1[pj] = s; |
| sum2 += s * s; |
| } |
| parl = fp / (delta * sum2); |
| } |
| |
| // calculate an upper bound, paru, for the zero of the function |
| sum2 = 0; |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| double sum = 0; |
| for (int i = 0; i <= j; ++i) { |
| sum += weightedJacobian[i][pj] * qy[i]; |
| } |
| sum /= diag[pj]; |
| sum2 += sum * sum; |
| } |
| double gNorm = FastMath.sqrt(sum2); |
| double paru = gNorm / delta; |
| if (paru == 0) { |
| // 2.2251e-308 is the smallest positive real for IEE754 |
| paru = 2.2251e-308 / FastMath.min(delta, 0.1); |
| } |
| |
| // if the input par lies outside of the interval (parl,paru), |
| // set par to the closer endpoint |
| lmPar = FastMath.min(paru, FastMath.max(lmPar, parl)); |
| if (lmPar == 0) { |
| lmPar = gNorm / dxNorm; |
| } |
| |
| for (int countdown = 10; countdown >= 0; --countdown) { |
| |
| // evaluate the function at the current value of lmPar |
| if (lmPar == 0) { |
| lmPar = FastMath.max(2.2251e-308, 0.001 * paru); |
| } |
| double sPar = FastMath.sqrt(lmPar); |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| work1[pj] = sPar * diag[pj]; |
| } |
| determineLMDirection(qy, work1, work2, work3); |
| |
| dxNorm = 0; |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| double s = diag[pj] * lmDir[pj]; |
| work3[pj] = s; |
| dxNorm += s * s; |
| } |
| dxNorm = FastMath.sqrt(dxNorm); |
| double previousFP = fp; |
| fp = dxNorm - delta; |
| |
| // if the function is small enough, accept the current value |
| // of lmPar, also test for the exceptional cases where parl is zero |
| if ((FastMath.abs(fp) <= 0.1 * delta) || |
| ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) { |
| return; |
| } |
| |
| // compute the Newton correction |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| work1[pj] = work3[pj] * diag[pj] / dxNorm; |
| } |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| work1[pj] /= work2[j]; |
| double tmp = work1[pj]; |
| for (int i = j + 1; i < solvedCols; ++i) { |
| work1[permutation[i]] -= weightedJacobian[i][pj] * tmp; |
| } |
| } |
| sum2 = 0; |
| for (int j = 0; j < solvedCols; ++j) { |
| double s = work1[permutation[j]]; |
| sum2 += s * s; |
| } |
| double correction = fp / (delta * sum2); |
| |
| // depending on the sign of the function, update parl or paru. |
| if (fp > 0) { |
| parl = FastMath.max(parl, lmPar); |
| } else if (fp < 0) { |
| paru = FastMath.min(paru, lmPar); |
| } |
| |
| // compute an improved estimate for lmPar |
| lmPar = FastMath.max(parl, lmPar + correction); |
| |
| } |
| } |
| |
| /** |
| * Solve a*x = b and d*x = 0 in the least squares sense. |
| * <p>This implementation is a translation in Java of the MINPACK |
| * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a> |
| * routine.</p> |
| * <p>This method sets the lmDir and lmDiag attributes.</p> |
| * <p>The authors of the original fortran function are:</p> |
| * <ul> |
| * <li>Argonne National Laboratory. MINPACK project. March 1980</li> |
| * <li>Burton S. Garbow</li> |
| * <li>Kenneth E. Hillstrom</li> |
| * <li>Jorge J. More</li> |
| * </ul> |
| * <p>Luc Maisonobe did the Java translation.</p> |
| * |
| * @param qy array containing qTy |
| * @param diag diagonal matrix |
| * @param lmDiag diagonal elements associated with lmDir |
| * @param work work array |
| */ |
| private void determineLMDirection(double[] qy, double[] diag, |
| double[] lmDiag, double[] work) { |
| |
| // copy R and Qty to preserve input and initialize s |
| // in particular, save the diagonal elements of R in lmDir |
| for (int j = 0; j < solvedCols; ++j) { |
| int pj = permutation[j]; |
| for (int i = j + 1; i < solvedCols; ++i) { |
| weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]]; |
| } |
| lmDir[j] = diagR[pj]; |
| work[j] = qy[j]; |
| } |
| |
| // eliminate the diagonal matrix d using a Givens rotation |
| for (int j = 0; j < solvedCols; ++j) { |
| |
| // prepare the row of d to be eliminated, locating the |
| // diagonal element using p from the Q.R. factorization |
| int pj = permutation[j]; |
| double dpj = diag[pj]; |
| if (dpj != 0) { |
| Arrays.fill(lmDiag, j + 1, lmDiag.length, 0); |
| } |
| lmDiag[j] = dpj; |
| |
| // the transformations to eliminate the row of d |
| // modify only a single element of Qty |
| // beyond the first n, which is initially zero. |
| double qtbpj = 0; |
| for (int k = j; k < solvedCols; ++k) { |
| int pk = permutation[k]; |
| |
| // determine a Givens rotation which eliminates the |
| // appropriate element in the current row of d |
| if (lmDiag[k] != 0) { |
| |
| final double sin; |
| final double cos; |
| double rkk = weightedJacobian[k][pk]; |
| if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) { |
| final double cotan = rkk / lmDiag[k]; |
| sin = 1.0 / FastMath.sqrt(1.0 + cotan * cotan); |
| cos = sin * cotan; |
| } else { |
| final double tan = lmDiag[k] / rkk; |
| cos = 1.0 / FastMath.sqrt(1.0 + tan * tan); |
| sin = cos * tan; |
| } |
| |
| // compute the modified diagonal element of R and |
| // the modified element of (Qty,0) |
| weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k]; |
| final double temp = cos * work[k] + sin * qtbpj; |
| qtbpj = -sin * work[k] + cos * qtbpj; |
| work[k] = temp; |
| |
| // accumulate the tranformation in the row of s |
| for (int i = k + 1; i < solvedCols; ++i) { |
| double rik = weightedJacobian[i][pk]; |
| final double temp2 = cos * rik + sin * lmDiag[i]; |
| lmDiag[i] = -sin * rik + cos * lmDiag[i]; |
| weightedJacobian[i][pk] = temp2; |
| } |
| } |
| } |
| |
| // store the diagonal element of s and restore |
| // the corresponding diagonal element of R |
| lmDiag[j] = weightedJacobian[j][permutation[j]]; |
| weightedJacobian[j][permutation[j]] = lmDir[j]; |
| } |
| |
| // solve the triangular system for z, if the system is |
| // singular, then obtain a least squares solution |
| int nSing = solvedCols; |
| for (int j = 0; j < solvedCols; ++j) { |
| if ((lmDiag[j] == 0) && (nSing == solvedCols)) { |
| nSing = j; |
| } |
| if (nSing < solvedCols) { |
| work[j] = 0; |
| } |
| } |
| if (nSing > 0) { |
| for (int j = nSing - 1; j >= 0; --j) { |
| int pj = permutation[j]; |
| double sum = 0; |
| for (int i = j + 1; i < nSing; ++i) { |
| sum += weightedJacobian[i][pj] * work[i]; |
| } |
| work[j] = (work[j] - sum) / lmDiag[j]; |
| } |
| } |
| |
| // permute the components of z back to components of lmDir |
| for (int j = 0; j < lmDir.length; ++j) { |
| lmDir[permutation[j]] = work[j]; |
| } |
| } |
| |
| /** |
| * Decompose a matrix A as A.P = Q.R using Householder transforms. |
| * <p>As suggested in the P. Lascaux and R. Theodor book |
| * <i>Analyse numérique matricielle appliquée à |
| * l'art de l'ingénieur</i> (Masson, 1986), instead of representing |
| * the Householder transforms with u<sub>k</sub> unit vectors such that: |
| * <pre> |
| * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup> |
| * </pre> |
| * we use <sub>k</sub> non-unit vectors such that: |
| * <pre> |
| * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup> |
| * </pre> |
| * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>. |
| * The beta<sub>k</sub> coefficients are provided upon exit as recomputing |
| * them from the v<sub>k</sub> vectors would be costly.</p> |
| * <p>This decomposition handles rank deficient cases since the tranformations |
| * are performed in non-increasing columns norms order thanks to columns |
| * pivoting. The diagonal elements of the R matrix are therefore also in |
| * non-increasing absolute values order.</p> |
| * |
| * @param jacobian Weighted Jacobian matrix at the current point. |
| * @exception ConvergenceException if the decomposition cannot be performed |
| */ |
| private void qrDecomposition(RealMatrix jacobian) throws ConvergenceException { |
| // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J), |
| // hence the multiplication by -1. |
| weightedJacobian = jacobian.scalarMultiply(-1).getData(); |
| |
| final int nR = weightedJacobian.length; |
| final int nC = weightedJacobian[0].length; |
| |
| // initializations |
| for (int k = 0; k < nC; ++k) { |
| permutation[k] = k; |
| double norm2 = 0; |
| for (int i = 0; i < nR; ++i) { |
| double akk = weightedJacobian[i][k]; |
| norm2 += akk * akk; |
| } |
| jacNorm[k] = FastMath.sqrt(norm2); |
| } |
| |
| // transform the matrix column after column |
| for (int k = 0; k < nC; ++k) { |
| |
| // select the column with the greatest norm on active components |
| int nextColumn = -1; |
| double ak2 = Double.NEGATIVE_INFINITY; |
| for (int i = k; i < nC; ++i) { |
| double norm2 = 0; |
| for (int j = k; j < nR; ++j) { |
| double aki = weightedJacobian[j][permutation[i]]; |
| norm2 += aki * aki; |
| } |
| if (Double.isInfinite(norm2) || Double.isNaN(norm2)) { |
| throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN, |
| nR, nC); |
| } |
| if (norm2 > ak2) { |
| nextColumn = i; |
| ak2 = norm2; |
| } |
| } |
| if (ak2 <= qrRankingThreshold) { |
| rank = k; |
| return; |
| } |
| int pk = permutation[nextColumn]; |
| permutation[nextColumn] = permutation[k]; |
| permutation[k] = pk; |
| |
| // choose alpha such that Hk.u = alpha ek |
| double akk = weightedJacobian[k][pk]; |
| double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2); |
| double betak = 1.0 / (ak2 - akk * alpha); |
| beta[pk] = betak; |
| |
| // transform the current column |
| diagR[pk] = alpha; |
| weightedJacobian[k][pk] -= alpha; |
| |
| // transform the remaining columns |
| for (int dk = nC - 1 - k; dk > 0; --dk) { |
| double gamma = 0; |
| for (int j = k; j < nR; ++j) { |
| gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]]; |
| } |
| gamma *= betak; |
| for (int j = k; j < nR; ++j) { |
| weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk]; |
| } |
| } |
| } |
| rank = solvedCols; |
| } |
| |
| /** |
| * Compute the product Qt.y for some Q.R. decomposition. |
| * |
| * @param y vector to multiply (will be overwritten with the result) |
| */ |
| private void qTy(double[] y) { |
| final int nR = weightedJacobian.length; |
| final int nC = weightedJacobian[0].length; |
| |
| for (int k = 0; k < nC; ++k) { |
| int pk = permutation[k]; |
| double gamma = 0; |
| for (int i = k; i < nR; ++i) { |
| gamma += weightedJacobian[i][pk] * y[i]; |
| } |
| gamma *= beta[pk]; |
| for (int i = k; i < nR; ++i) { |
| y[i] -= gamma * weightedJacobian[i][pk]; |
| } |
| } |
| } |
| } |
