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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
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/*
* 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,
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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.
*
* This implementation should 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.
*
* The resolution engine is a simple translation of the MINPACK lmder 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 Analyse numérique matricielle
* appliquée à l'art de l'ingénieur, Masson 1986.
* The authors of the original fortran version are:
*
* - Argonne National Laboratory. MINPACK project. March 1980
* - Burton S. Garbow
* - Kenneth E. Hillstrom
* - Jorge J. More
*
* The redistribution policy for MINPACK is available here, for convenience, it
* is reproduced below.
*
*
*
* Minpack Copyright Notice (1999) University of Chicago.
* All rights reserved
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* - Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* - 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.
* - The end-user documentation included with the redistribution, if any,
* must include the following acknowledgment:
*
This product includes software developed by the University of
* Chicago, as Operator of Argonne National Laboratory.
* Alternately, this acknowledgment may appear in the software itself,
* if and wherever such third-party acknowledgments normally appear.
* - 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.
* - 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.
*
* @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:
*
* - Initial step bound factor: 100
* - Cost relative tolerance: 1e-10
* - Parameters relative tolerance: 1e-10
* - Orthogonality tolerance: 1e-10
* - QR ranking threshold: {@link Precision#SAFE_MIN}
*
*/
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 disabled.
* The default values for the algorithm settings are:
*
* - Initial step bound factor: 100
* - Cost relative tolerance: 1e-10
* - Parameters relative tolerance: 1e-10
* - Orthogonality tolerance: 1e-10
* - QR ranking threshold: {@link Precision#SAFE_MIN}
*
*
* @param checker Convergence checker.
*/
public LevenbergMarquardtOptimizer(ConvergenceChecker 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 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:
*
* - Initial step bound factor}: 100
* - QR ranking threshold}: {@link Precision#SAFE_MIN}
*
*
* @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 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.
* This implementation is a translation in Java of the MINPACK
* lmpar
* routine.
* This method sets the lmPar and lmDir attributes.
* The authors of the original fortran function are:
*
* - Argonne National Laboratory. MINPACK project. March 1980
* - Burton S. Garbow
* - Kenneth E. Hillstrom
* - Jorge J. More
*
* Luc Maisonobe did the Java translation.
*
* @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.
* This implementation is a translation in Java of the MINPACK
* qrsolv
* routine.
* This method sets the lmDir and lmDiag attributes.
* The authors of the original fortran function are:
*
* - Argonne National Laboratory. MINPACK project. March 1980
* - Burton S. Garbow
* - Kenneth E. Hillstrom
* - Jorge J. More
*
* Luc Maisonobe did the Java translation.
*
* @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.
* As suggested in the P. Lascaux and R. Theodor book
* Analyse numérique matricielle appliquée à
* l'art de l'ingénieur (Masson, 1986), instead of representing
* the Householder transforms with uk unit vectors such that:
*
* Hk = I - 2uk.ukt
*
* we use k non-unit vectors such that:
*
* Hk = I - betakvk.vkt
*
* where vk = ak - alphak ek.
* The betak coefficients are provided upon exit as recomputing
* them from the vk vectors would be costly.
* 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.
*
* @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];
}
}
}
}