/*
* 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.math.optimization.general;
import java.util.Arrays;
import org.apache.commons.math.FunctionEvaluationException;
import org.apache.commons.math.exception.util.LocalizedFormats;
import org.apache.commons.math.optimization.OptimizationException;
import org.apache.commons.math.optimization.VectorialPointValuePair;
import org.apache.commons.math.util.FastMath;
import org.apache.commons.math.util.MathUtils;
/**
* 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 $Revision: 1073272 $ $Date: 2011-02-22 10:22:25 +0100 (mar. 22 févr. 2011) $
* @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 double initialStepBoundFactor;
/** Desired relative error in the sum of squares. */
private double costRelativeTolerance;
/** Desired relative error in the approximate solution parameters. */
private double parRelativeTolerance;
/** Desired max cosine on the orthogonality between the function vector
* and the columns of the jacobian. */
private double orthoTolerance;
/** Threshold for QR ranking. */
private double qrRankingThreshold;
/**
* Build an optimizer for least squares problems.
* <p>The default values for the algorithm settings are:
* <ul>
* <li>{@link #setConvergenceChecker(VectorialConvergenceChecker) vectorial convergence checker}: null</li>
* <li>{@link #setInitialStepBoundFactor(double) initial step bound factor}: 100.0</li>
* <li>{@link #setMaxIterations(int) maximal iterations}: 1000</li>
* <li>{@link #setCostRelativeTolerance(double) cost relative tolerance}: 1.0e-10</li>
* <li>{@link #setParRelativeTolerance(double) parameters relative tolerance}: 1.0e-10</li>
* <li>{@link #setOrthoTolerance(double) orthogonality tolerance}: 1.0e-10</li>
* <li>{@link #setQRRankingThreshold(double) QR ranking threshold}: {@link MathUtils#SAFE_MIN}</li>
* </ul>
* </p>
* <p>These default values may be overridden after construction. If the {@link
* #setConvergenceChecker vectorial convergence checker} is set to a non-null value, it
* will be used instead of the {@link #setCostRelativeTolerance cost relative tolerance}
* and {@link #setParRelativeTolerance parameters relative tolerance} settings.
*/
public LevenbergMarquardtOptimizer() {
// set up the superclass with a default max cost evaluations setting
setMaxIterations(1000);
// default values for the tuning parameters
setConvergenceChecker(null);
setInitialStepBoundFactor(100.0);
setCostRelativeTolerance(1.0e-10);
setParRelativeTolerance(1.0e-10);
setOrthoTolerance(1.0e-10);
setQRRankingThreshold(MathUtils.SAFE_MIN);
}
/**
* Set the positive input variable used in determining the initial step bound.
* This bound is set to the product of initialStepBoundFactor and the euclidean
* norm of diag*x if nonzero, or else to initialStepBoundFactor itself. In most
* cases factor should lie in the interval (0.1, 100.0). 100.0 is a generally
* recommended value.
*
* @param initialStepBoundFactor initial step bound factor
*/
public void setInitialStepBoundFactor(double initialStepBoundFactor) {
this.initialStepBoundFactor = initialStepBoundFactor;
}
/**
* Set the desired relative error in the sum of squares.
* <p>This setting is used only if the {@link #setConvergenceChecker vectorial
* convergence checker} is set to null.</p>
* @param costRelativeTolerance desired relative error in the sum of squares
*/
public void setCostRelativeTolerance(double costRelativeTolerance) {
this.costRelativeTolerance = costRelativeTolerance;
}
/**
* Set the desired relative error in the approximate solution parameters.
* <p>This setting is used only if the {@link #setConvergenceChecker vectorial
* convergence checker} is set to null.</p>
* @param parRelativeTolerance desired relative error
* in the approximate solution parameters
*/
public void setParRelativeTolerance(double parRelativeTolerance) {
this.parRelativeTolerance = parRelativeTolerance;
}
/**
* Set the desired max cosine on the orthogonality.
* <p>This setting is always used, regardless of the {@link #setConvergenceChecker
* vectorial convergence checker} being null or non-null.</p>
* @param orthoTolerance desired max cosine on the orthogonality
* between the function vector and the columns of the jacobian
*/
public void setOrthoTolerance(double orthoTolerance) {
this.orthoTolerance = orthoTolerance;
}
/**
* Set the desired threshold for QR ranking.
* <p>
* 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.
* </p>
* @param threshold threshold for QR ranking
* @since 2.2
*/
public void setQRRankingThreshold(final double threshold) {
this.qrRankingThreshold = threshold;
}
/** {@inheritDoc} */
@Override
protected VectorialPointValuePair doOptimize()
throws FunctionEvaluationException, OptimizationException, IllegalArgumentException {
// arrays shared with the other private methods
solvedCols = Math.min(rows, cols);
diagR = new double[cols];
jacNorm = new double[cols];
beta = new double[cols];
permutation = new int[cols];
lmDir = new double[cols];
// local point
double delta = 0;
double xNorm = 0;
double[] diag = new double[cols];
double[] oldX = new double[cols];
double[] oldRes = new double[rows];
double[] oldObj = new double[rows];
double[] qtf = new double[rows];
double[] work1 = new double[cols];
double[] work2 = new double[cols];
double[] work3 = new double[cols];
// evaluate the function at the starting point and calculate its norm
updateResidualsAndCost();
// outer loop
lmPar = 0;
boolean firstIteration = true;
VectorialPointValuePair current = new VectorialPointValuePair(point, objective);
while (true) {
for (int i=0;i<rows;i++) {
qtf[i]=wresiduals[i];
}
incrementIterationsCounter();
// compute the Q.R. decomposition of the jacobian matrix
VectorialPointValuePair previous = current;
updateJacobian();
qrDecomposition();
// 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];
wjacobian[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 < cols; ++k) {
double dk = jacNorm[k];
if (dk == 0) {
dk = 1.0;
}
double xk = dk * point[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 (cost != 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 += wjacobian[i][pj] * qtf[i];
}
maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * cost));
}
}
}
if (maxCosine <= orthoTolerance) {
// convergence has been reached
updateResidualsAndCost();
current = new VectorialPointValuePair(point, objective);
return current;
}
// rescale if necessary
for (int j = 0; j < cols; ++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] = point[pj];
}
double previousCost = cost;
double[] tmpVec = residuals;
residuals = oldRes;
oldRes = tmpVec;
tmpVec = objective;
objective = 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];
point[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
updateResidualsAndCost();
// compute the scaled actual reduction
double actRed = -1.0;
if (0.1 * cost < previousCost) {
double r = cost / 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] += wjacobian[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 * cost >= 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 < cols; ++k) {
double xK = diag[k] * point[k];
xNorm += xK * xK;
}
xNorm = FastMath.sqrt(xNorm);
current = new VectorialPointValuePair(point, objective);
// tests for convergence.
if (checker != null) {
// we use the vectorial convergence checker
if (checker.converged(getIterations(), previous, current)) {
return current;
}
}
} else {
// failed iteration, reset the previous values
cost = previousCost;
for (int j = 0; j < solvedCols; ++j) {
int pj = permutation[j];
point[pj] = oldX[pj];
}
tmpVec = residuals;
residuals = oldRes;
oldRes = tmpVec;
tmpVec = objective;
objective = oldObj;
oldObj = tmpVec;
}
if (checker==null) {
if (((FastMath.abs(actRed) <= costRelativeTolerance) &&
(preRed <= costRelativeTolerance) &&
(ratio <= 2.0)) ||
(delta <= parRelativeTolerance * xNorm)) {
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 OptimizationException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
costRelativeTolerance);
} else if (delta <= 2.2204e-16 * xNorm) {
throw new OptimizationException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
parRelativeTolerance);
} else if (maxCosine <= 2.2204e-16) {
throw new OptimizationException(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) {
// 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 < cols; ++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 * wjacobian[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 += wjacobian[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 += wjacobian[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]] -= wjacobian[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) {
wjacobian[i][pj] = wjacobian[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 = wjacobian[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)
wjacobian[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 = wjacobian[i][pk];
final double temp2 = cos * rik + sin * lmDiag[i];
lmDiag[i] = -sin * rik + cos * lmDiag[i];
wjacobian[i][pk] = temp2;
}
}
}
// store the diagonal element of s and restore
// the corresponding diagonal element of R
lmDiag[j] = wjacobian[j][permutation[j]];
wjacobian[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 += wjacobian[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>
* @exception OptimizationException if the decomposition cannot be performed
*/
private void qrDecomposition() throws OptimizationException {
// initializations
for (int k = 0; k < cols; ++k) {
permutation[k] = k;
double norm2 = 0;
for (int i = 0; i < wjacobian.length; ++i) {
double akk = wjacobian[i][k];
norm2 += akk * akk;
}
jacNorm[k] = FastMath.sqrt(norm2);
}
// transform the matrix column after column
for (int k = 0; k < cols; ++k) {
// select the column with the greatest norm on active components
int nextColumn = -1;
double ak2 = Double.NEGATIVE_INFINITY;
for (int i = k; i < cols; ++i) {
double norm2 = 0;
for (int j = k; j < wjacobian.length; ++j) {
double aki = wjacobian[j][permutation[i]];
norm2 += aki * aki;
}
if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
throw new OptimizationException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
rows, cols);
}
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 = wjacobian[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;
wjacobian[k][pk] -= alpha;
// transform the remaining columns
for (int dk = cols - 1 - k; dk > 0; --dk) {
double gamma = 0;
for (int j = k; j < wjacobian.length; ++j) {
gamma += wjacobian[j][pk] * wjacobian[j][permutation[k + dk]];
}
gamma *= betak;
for (int j = k; j < wjacobian.length; ++j) {
wjacobian[j][permutation[k + dk]] -= gamma * wjacobian[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) {
for (int k = 0; k < cols; ++k) {
int pk = permutation[k];
double gamma = 0;
for (int i = k; i < rows; ++i) {
gamma += wjacobian[i][pk] * y[i];
}
gamma *= beta[pk];
for (int i = k; i < rows; ++i) {
y[i] -= gamma * wjacobian[i][pk];
}
}
}
}