Java tutorial
/* * 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.optim.nonlinear.vector.jacobian; import java.util.Arrays; import org.apache.commons.math3.exception.ConvergenceException; import org.apache.commons.math3.exception.util.LocalizedFormats; import org.apache.commons.math3.optim.PointVectorValuePair; import org.apache.commons.math3.optim.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> * * @version $Id: LevenbergMarquardtOptimizer.java 1416643 2012-12-03 19:37:14Z tn $ * @since 2.0 */ 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); 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 = 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) { // we use the vectorial convergence checker if (checker.converged(iter, previous, current)) { setCost(currentCost); 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); 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]; } } } }