1 | /*
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2 | * Licensed to the Apache Software Foundation (ASF) under one or more
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3 | * contributor license agreements. See the NOTICE file distributed with
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4 | * this work for additional information regarding copyright ownership.
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5 | * The ASF licenses this file to You under the Apache License, Version 2.0
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6 | * (the "License"); you may not use this file except in compliance with
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7 | * the License. You may obtain a copy of the License at
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8 | *
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9 | * http://www.apache.org/licenses/LICENSE-2.0
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10 | *
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11 | * Unless required by applicable law or agreed to in writing, software
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12 | * distributed under the License is distributed on an "AS IS" BASIS,
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13 | * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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14 | * See the License for the specific language governing permissions and
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15 | * limitations under the License.
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16 | */
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17 | package agents.anac.y2019.harddealer.math3.fitting.leastsquares;
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18 |
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19 | import java.util.Arrays;
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20 |
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21 | import agents.anac.y2019.harddealer.math3.fitting.leastsquares.LeastSquaresProblem.Evaluation;
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22 | import agents.anac.y2019.harddealer.math3.linear.ArrayRealVector;
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23 | import agents.anac.y2019.harddealer.math3.linear.RealMatrix;
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24 | import agents.anac.y2019.harddealer.math3.exception.ConvergenceException;
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25 | import agents.anac.y2019.harddealer.math3.exception.util.LocalizedFormats;
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26 | import agents.anac.y2019.harddealer.math3.optim.ConvergenceChecker;
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27 | import agents.anac.y2019.harddealer.math3.util.Incrementor;
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28 | import agents.anac.y2019.harddealer.math3.util.Precision;
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29 | import agents.anac.y2019.harddealer.math3.util.FastMath;
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30 |
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31 |
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32 | /**
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33 | * This class solves a least-squares problem using the Levenberg-Marquardt
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34 | * algorithm.
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35 | *
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36 | * <p>This implementation <em>should</em> work even for over-determined systems
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37 | * (i.e. systems having more point than equations). Over-determined systems
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38 | * are solved by ignoring the point which have the smallest impact according
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39 | * to their jacobian column norm. Only the rank of the matrix and some loop bounds
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40 | * are changed to implement this.</p>
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41 | *
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42 | * <p>The resolution engine is a simple translation of the MINPACK <a
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43 | * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
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44 | * changes. The changes include the over-determined resolution, the use of
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45 | * inherited convergence checker and the Q.R. decomposition which has been
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46 | * rewritten following the algorithm described in the
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47 | * P. Lascaux and R. Theodor book <i>Analyse numérique matricielle
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48 | * appliquée à l'art de l'ingénieur</i>, Masson 1986.</p>
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49 | * <p>The authors of the original fortran version are:
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50 | * <ul>
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51 | * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
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52 | * <li>Burton S. Garbow</li>
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53 | * <li>Kenneth E. Hillstrom</li>
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54 | * <li>Jorge J. More</li>
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55 | * </ul>
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56 | * The redistribution policy for MINPACK is available <a
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57 | * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
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58 | * is reproduced below.</p>
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59 | *
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60 | * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
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61 | * <tr><td>
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62 | * Minpack Copyright Notice (1999) University of Chicago.
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63 | * All rights reserved
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64 | * </td></tr>
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65 | * <tr><td>
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66 | * Redistribution and use in source and binary forms, with or without
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67 | * modification, are permitted provided that the following conditions
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68 | * are met:
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69 | * <ol>
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70 | * <li>Redistributions of source code must retain the above copyright
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71 | * notice, this list of conditions and the following disclaimer.</li>
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72 | * <li>Redistributions in binary form must reproduce the above
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73 | * copyright notice, this list of conditions and the following
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74 | * disclaimer in the documentation and/or other materials provided
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75 | * with the distribution.</li>
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76 | * <li>The end-user documentation included with the redistribution, if any,
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77 | * must include the following acknowledgment:
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78 | * <code>This product includes software developed by the University of
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79 | * Chicago, as Operator of Argonne National Laboratory.</code>
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80 | * Alternately, this acknowledgment may appear in the software itself,
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81 | * if and wherever such third-party acknowledgments normally appear.</li>
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82 | * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
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83 | * WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
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84 | * UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
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85 | * THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
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86 | * IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
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87 | * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
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88 | * OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
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89 | * OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
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90 | * USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
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91 | * THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
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92 | * DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
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93 | * UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
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94 | * BE CORRECTED.</strong></li>
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95 | * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
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96 | * HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
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97 | * ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
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98 | * INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
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99 | * ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
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100 | * PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
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101 | * SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
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102 | * (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
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103 | * EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
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104 | * POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
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105 | * <ol></td></tr>
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106 | * </table>
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107 | *
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108 | * @since 3.3
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109 | */
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110 | public class LevenbergMarquardtOptimizer implements LeastSquaresOptimizer {
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111 |
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112 | /** Twice the "epsilon machine". */
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113 | private static final double TWO_EPS = 2 * Precision.EPSILON;
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114 |
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115 | /* configuration parameters */
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116 | /** Positive input variable used in determining the initial step bound. */
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117 | private final double initialStepBoundFactor;
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118 | /** Desired relative error in the sum of squares. */
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119 | private final double costRelativeTolerance;
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120 | /** Desired relative error in the approximate solution parameters. */
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121 | private final double parRelativeTolerance;
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122 | /** Desired max cosine on the orthogonality between the function vector
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123 | * and the columns of the jacobian. */
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124 | private final double orthoTolerance;
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125 | /** Threshold for QR ranking. */
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126 | private final double qrRankingThreshold;
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127 |
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128 | /** Default constructor.
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129 | * <p>
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130 | * The default values for the algorithm settings are:
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131 | * <ul>
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132 | * <li>Initial step bound factor: 100</li>
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133 | * <li>Cost relative tolerance: 1e-10</li>
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134 | * <li>Parameters relative tolerance: 1e-10</li>
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135 | * <li>Orthogonality tolerance: 1e-10</li>
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136 | * <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
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137 | * </ul>
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138 | **/
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139 | public LevenbergMarquardtOptimizer() {
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140 | this(100, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
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141 | }
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142 |
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143 | /**
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144 | * Construct an instance with all parameters specified.
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145 | *
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146 | * @param initialStepBoundFactor initial step bound factor
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147 | * @param costRelativeTolerance cost relative tolerance
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148 | * @param parRelativeTolerance parameters relative tolerance
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149 | * @param orthoTolerance orthogonality tolerance
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150 | * @param qrRankingThreshold threshold in the QR decomposition. Columns with a 2
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151 | * norm less than this threshold are considered to be
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152 | * all 0s.
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153 | */
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154 | public LevenbergMarquardtOptimizer(
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155 | final double initialStepBoundFactor,
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156 | final double costRelativeTolerance,
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157 | final double parRelativeTolerance,
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158 | final double orthoTolerance,
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159 | final double qrRankingThreshold) {
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160 | this.initialStepBoundFactor = initialStepBoundFactor;
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161 | this.costRelativeTolerance = costRelativeTolerance;
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162 | this.parRelativeTolerance = parRelativeTolerance;
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163 | this.orthoTolerance = orthoTolerance;
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164 | this.qrRankingThreshold = qrRankingThreshold;
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165 | }
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166 |
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167 | /**
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168 | * @param newInitialStepBoundFactor Positive input variable used in
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169 | * determining the initial step bound. This bound is set to the
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170 | * product of initialStepBoundFactor and the euclidean norm of
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171 | * {@code diag * x} if non-zero, or else to {@code newInitialStepBoundFactor}
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172 | * itself. In most cases factor should lie in the interval
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173 | * {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
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174 | * of the matrix is reduced.
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175 | * @return a new instance.
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176 | */
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177 | public LevenbergMarquardtOptimizer withInitialStepBoundFactor(double newInitialStepBoundFactor) {
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178 | return new LevenbergMarquardtOptimizer(
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179 | newInitialStepBoundFactor,
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180 | costRelativeTolerance,
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181 | parRelativeTolerance,
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182 | orthoTolerance,
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183 | qrRankingThreshold);
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184 | }
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185 |
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186 | /**
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187 | * @param newCostRelativeTolerance Desired relative error in the sum of squares.
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188 | * @return a new instance.
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189 | */
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190 | public LevenbergMarquardtOptimizer withCostRelativeTolerance(double newCostRelativeTolerance) {
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191 | return new LevenbergMarquardtOptimizer(
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192 | initialStepBoundFactor,
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193 | newCostRelativeTolerance,
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194 | parRelativeTolerance,
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195 | orthoTolerance,
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196 | qrRankingThreshold);
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197 | }
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198 |
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199 | /**
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200 | * @param newParRelativeTolerance Desired relative error in the approximate solution
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201 | * parameters.
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202 | * @return a new instance.
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203 | */
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204 | public LevenbergMarquardtOptimizer withParameterRelativeTolerance(double newParRelativeTolerance) {
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205 | return new LevenbergMarquardtOptimizer(
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206 | initialStepBoundFactor,
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207 | costRelativeTolerance,
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208 | newParRelativeTolerance,
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209 | orthoTolerance,
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210 | qrRankingThreshold);
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211 | }
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212 |
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213 | /**
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214 | * Modifies the given parameter.
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215 | *
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216 | * @param newOrthoTolerance Desired max cosine on the orthogonality between
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217 | * the function vector and the columns of the Jacobian.
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218 | * @return a new instance.
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219 | */
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220 | public LevenbergMarquardtOptimizer withOrthoTolerance(double newOrthoTolerance) {
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221 | return new LevenbergMarquardtOptimizer(
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222 | initialStepBoundFactor,
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223 | costRelativeTolerance,
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224 | parRelativeTolerance,
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225 | newOrthoTolerance,
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226 | qrRankingThreshold);
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227 | }
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228 |
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229 | /**
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230 | * @param newQRRankingThreshold Desired threshold for QR ranking.
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231 | * If the squared norm of a column vector is smaller or equal to this
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232 | * threshold during QR decomposition, it is considered to be a zero vector
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233 | * and hence the rank of the matrix is reduced.
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234 | * @return a new instance.
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235 | */
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236 | public LevenbergMarquardtOptimizer withRankingThreshold(double newQRRankingThreshold) {
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237 | return new LevenbergMarquardtOptimizer(
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238 | initialStepBoundFactor,
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239 | costRelativeTolerance,
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240 | parRelativeTolerance,
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241 | orthoTolerance,
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242 | newQRRankingThreshold);
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243 | }
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244 |
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245 | /**
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246 | * Gets the value of a tuning parameter.
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247 | * @see #withInitialStepBoundFactor(double)
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248 | *
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249 | * @return the parameter's value.
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250 | */
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251 | public double getInitialStepBoundFactor() {
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252 | return initialStepBoundFactor;
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253 | }
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254 |
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255 | /**
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256 | * Gets the value of a tuning parameter.
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257 | * @see #withCostRelativeTolerance(double)
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258 | *
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259 | * @return the parameter's value.
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260 | */
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261 | public double getCostRelativeTolerance() {
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262 | return costRelativeTolerance;
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263 | }
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264 |
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265 | /**
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266 | * Gets the value of a tuning parameter.
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267 | * @see #withParameterRelativeTolerance(double)
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268 | *
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269 | * @return the parameter's value.
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270 | */
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271 | public double getParameterRelativeTolerance() {
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272 | return parRelativeTolerance;
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273 | }
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274 |
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275 | /**
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276 | * Gets the value of a tuning parameter.
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277 | * @see #withOrthoTolerance(double)
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278 | *
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279 | * @return the parameter's value.
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280 | */
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281 | public double getOrthoTolerance() {
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282 | return orthoTolerance;
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283 | }
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284 |
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285 | /**
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286 | * Gets the value of a tuning parameter.
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287 | * @see #withRankingThreshold(double)
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288 | *
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289 | * @return the parameter's value.
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290 | */
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291 | public double getRankingThreshold() {
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292 | return qrRankingThreshold;
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293 | }
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294 |
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295 | /** {@inheritDoc} */
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296 | public Optimum optimize(final LeastSquaresProblem problem) {
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297 | // Pull in relevant data from the problem as locals.
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298 | final int nR = problem.getObservationSize(); // Number of observed data.
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299 | final int nC = problem.getParameterSize(); // Number of parameters.
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300 | // Counters.
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301 | final Incrementor iterationCounter = problem.getIterationCounter();
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302 | final Incrementor evaluationCounter = problem.getEvaluationCounter();
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303 | // Convergence criterion.
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304 | final ConvergenceChecker<Evaluation> checker = problem.getConvergenceChecker();
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305 |
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306 | // arrays shared with the other private methods
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307 | final int solvedCols = FastMath.min(nR, nC);
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308 | /* Parameters evolution direction associated with lmPar. */
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309 | double[] lmDir = new double[nC];
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310 | /* Levenberg-Marquardt parameter. */
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311 | double lmPar = 0;
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312 |
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313 | // local point
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314 | double delta = 0;
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315 | double xNorm = 0;
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316 | double[] diag = new double[nC];
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317 | double[] oldX = new double[nC];
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318 | double[] oldRes = new double[nR];
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319 | double[] qtf = new double[nR];
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320 | double[] work1 = new double[nC];
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321 | double[] work2 = new double[nC];
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322 | double[] work3 = new double[nC];
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323 |
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324 |
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325 | // Evaluate the function at the starting point and calculate its norm.
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326 | evaluationCounter.incrementCount();
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327 | //value will be reassigned in the loop
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328 | Evaluation current = problem.evaluate(problem.getStart());
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329 | double[] currentResiduals = current.getResiduals().toArray();
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330 | double currentCost = current.getCost();
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331 | double[] currentPoint = current.getPoint().toArray();
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332 |
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333 | // Outer loop.
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334 | boolean firstIteration = true;
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335 | while (true) {
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336 | iterationCounter.incrementCount();
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337 |
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338 | final Evaluation previous = current;
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339 |
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340 | // QR decomposition of the jacobian matrix
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341 | final InternalData internalData
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342 | = qrDecomposition(current.getJacobian(), solvedCols);
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343 | final double[][] weightedJacobian = internalData.weightedJacobian;
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344 | final int[] permutation = internalData.permutation;
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345 | final double[] diagR = internalData.diagR;
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346 | final double[] jacNorm = internalData.jacNorm;
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347 |
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348 | //residuals already have weights applied
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349 | double[] weightedResidual = currentResiduals;
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350 | for (int i = 0; i < nR; i++) {
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351 | qtf[i] = weightedResidual[i];
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352 | }
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353 |
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354 | // compute Qt.res
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355 | qTy(qtf, internalData);
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356 |
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357 | // now we don't need Q anymore,
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358 | // so let jacobian contain the R matrix with its diagonal elements
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359 | for (int k = 0; k < solvedCols; ++k) {
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360 | int pk = permutation[k];
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361 | weightedJacobian[k][pk] = diagR[pk];
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362 | }
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363 |
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364 | if (firstIteration) {
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365 | // scale the point according to the norms of the columns
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366 | // of the initial jacobian
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367 | xNorm = 0;
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368 | for (int k = 0; k < nC; ++k) {
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369 | double dk = jacNorm[k];
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370 | if (dk == 0) {
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371 | dk = 1.0;
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372 | }
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373 | double xk = dk * currentPoint[k];
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374 | xNorm += xk * xk;
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375 | diag[k] = dk;
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376 | }
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377 | xNorm = FastMath.sqrt(xNorm);
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378 |
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379 | // initialize the step bound delta
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380 | delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
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381 | }
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382 |
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383 | // check orthogonality between function vector and jacobian columns
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384 | double maxCosine = 0;
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385 | if (currentCost != 0) {
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386 | for (int j = 0; j < solvedCols; ++j) {
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387 | int pj = permutation[j];
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388 | double s = jacNorm[pj];
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389 | if (s != 0) {
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390 | double sum = 0;
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391 | for (int i = 0; i <= j; ++i) {
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392 | sum += weightedJacobian[i][pj] * qtf[i];
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393 | }
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394 | maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * currentCost));
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395 | }
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396 | }
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397 | }
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398 | if (maxCosine <= orthoTolerance) {
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399 | // Convergence has been reached.
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400 | return new OptimumImpl(
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401 | current,
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402 | evaluationCounter.getCount(),
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403 | iterationCounter.getCount());
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404 | }
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405 |
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406 | // rescale if necessary
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407 | for (int j = 0; j < nC; ++j) {
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408 | diag[j] = FastMath.max(diag[j], jacNorm[j]);
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409 | }
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410 |
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411 | // Inner loop.
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412 | for (double ratio = 0; ratio < 1.0e-4;) {
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413 |
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414 | // save the state
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415 | for (int j = 0; j < solvedCols; ++j) {
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416 | int pj = permutation[j];
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417 | oldX[pj] = currentPoint[pj];
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418 | }
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419 | final double previousCost = currentCost;
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420 | double[] tmpVec = weightedResidual;
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421 | weightedResidual = oldRes;
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422 | oldRes = tmpVec;
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423 |
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424 | // determine the Levenberg-Marquardt parameter
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425 | lmPar = determineLMParameter(qtf, delta, diag,
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426 | internalData, solvedCols,
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427 | work1, work2, work3, lmDir, lmPar);
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428 |
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429 | // compute the new point and the norm of the evolution direction
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430 | double lmNorm = 0;
|
---|
431 | for (int j = 0; j < solvedCols; ++j) {
|
---|
432 | int pj = permutation[j];
|
---|
433 | lmDir[pj] = -lmDir[pj];
|
---|
434 | currentPoint[pj] = oldX[pj] + lmDir[pj];
|
---|
435 | double s = diag[pj] * lmDir[pj];
|
---|
436 | lmNorm += s * s;
|
---|
437 | }
|
---|
438 | lmNorm = FastMath.sqrt(lmNorm);
|
---|
439 | // on the first iteration, adjust the initial step bound.
|
---|
440 | if (firstIteration) {
|
---|
441 | delta = FastMath.min(delta, lmNorm);
|
---|
442 | }
|
---|
443 |
|
---|
444 | // Evaluate the function at x + p and calculate its norm.
|
---|
445 | evaluationCounter.incrementCount();
|
---|
446 | current = problem.evaluate(new ArrayRealVector(currentPoint));
|
---|
447 | currentResiduals = current.getResiduals().toArray();
|
---|
448 | currentCost = current.getCost();
|
---|
449 | currentPoint = current.getPoint().toArray();
|
---|
450 |
|
---|
451 | // compute the scaled actual reduction
|
---|
452 | double actRed = -1.0;
|
---|
453 | if (0.1 * currentCost < previousCost) {
|
---|
454 | double r = currentCost / previousCost;
|
---|
455 | actRed = 1.0 - r * r;
|
---|
456 | }
|
---|
457 |
|
---|
458 | // compute the scaled predicted reduction
|
---|
459 | // and the scaled directional derivative
|
---|
460 | for (int j = 0; j < solvedCols; ++j) {
|
---|
461 | int pj = permutation[j];
|
---|
462 | double dirJ = lmDir[pj];
|
---|
463 | work1[j] = 0;
|
---|
464 | for (int i = 0; i <= j; ++i) {
|
---|
465 | work1[i] += weightedJacobian[i][pj] * dirJ;
|
---|
466 | }
|
---|
467 | }
|
---|
468 | double coeff1 = 0;
|
---|
469 | for (int j = 0; j < solvedCols; ++j) {
|
---|
470 | coeff1 += work1[j] * work1[j];
|
---|
471 | }
|
---|
472 | double pc2 = previousCost * previousCost;
|
---|
473 | coeff1 /= pc2;
|
---|
474 | double coeff2 = lmPar * lmNorm * lmNorm / pc2;
|
---|
475 | double preRed = coeff1 + 2 * coeff2;
|
---|
476 | double dirDer = -(coeff1 + coeff2);
|
---|
477 |
|
---|
478 | // ratio of the actual to the predicted reduction
|
---|
479 | ratio = (preRed == 0) ? 0 : (actRed / preRed);
|
---|
480 |
|
---|
481 | // update the step bound
|
---|
482 | if (ratio <= 0.25) {
|
---|
483 | double tmp =
|
---|
484 | (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
|
---|
485 | if ((0.1 * currentCost >= previousCost) || (tmp < 0.1)) {
|
---|
486 | tmp = 0.1;
|
---|
487 | }
|
---|
488 | delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
|
---|
489 | lmPar /= tmp;
|
---|
490 | } else if ((lmPar == 0) || (ratio >= 0.75)) {
|
---|
491 | delta = 2 * lmNorm;
|
---|
492 | lmPar *= 0.5;
|
---|
493 | }
|
---|
494 |
|
---|
495 | // test for successful iteration.
|
---|
496 | if (ratio >= 1.0e-4) {
|
---|
497 | // successful iteration, update the norm
|
---|
498 | firstIteration = false;
|
---|
499 | xNorm = 0;
|
---|
500 | for (int k = 0; k < nC; ++k) {
|
---|
501 | double xK = diag[k] * currentPoint[k];
|
---|
502 | xNorm += xK * xK;
|
---|
503 | }
|
---|
504 | xNorm = FastMath.sqrt(xNorm);
|
---|
505 |
|
---|
506 | // tests for convergence.
|
---|
507 | if (checker != null && checker.converged(iterationCounter.getCount(), previous, current)) {
|
---|
508 | return new OptimumImpl(current, evaluationCounter.getCount(), iterationCounter.getCount());
|
---|
509 | }
|
---|
510 | } else {
|
---|
511 | // failed iteration, reset the previous values
|
---|
512 | currentCost = previousCost;
|
---|
513 | for (int j = 0; j < solvedCols; ++j) {
|
---|
514 | int pj = permutation[j];
|
---|
515 | currentPoint[pj] = oldX[pj];
|
---|
516 | }
|
---|
517 | tmpVec = weightedResidual;
|
---|
518 | weightedResidual = oldRes;
|
---|
519 | oldRes = tmpVec;
|
---|
520 | // Reset "current" to previous values.
|
---|
521 | current = previous;
|
---|
522 | }
|
---|
523 |
|
---|
524 | // Default convergence criteria.
|
---|
525 | if ((FastMath.abs(actRed) <= costRelativeTolerance &&
|
---|
526 | preRed <= costRelativeTolerance &&
|
---|
527 | ratio <= 2.0) ||
|
---|
528 | delta <= parRelativeTolerance * xNorm) {
|
---|
529 | return new OptimumImpl(current, evaluationCounter.getCount(), iterationCounter.getCount());
|
---|
530 | }
|
---|
531 |
|
---|
532 | // tests for termination and stringent tolerances
|
---|
533 | if (FastMath.abs(actRed) <= TWO_EPS &&
|
---|
534 | preRed <= TWO_EPS &&
|
---|
535 | ratio <= 2.0) {
|
---|
536 | throw new ConvergenceException(LocalizedFormats.TOO_SMALL_COST_RELATIVE_TOLERANCE,
|
---|
537 | costRelativeTolerance);
|
---|
538 | } else if (delta <= TWO_EPS * xNorm) {
|
---|
539 | throw new ConvergenceException(LocalizedFormats.TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE,
|
---|
540 | parRelativeTolerance);
|
---|
541 | } else if (maxCosine <= TWO_EPS) {
|
---|
542 | throw new ConvergenceException(LocalizedFormats.TOO_SMALL_ORTHOGONALITY_TOLERANCE,
|
---|
543 | orthoTolerance);
|
---|
544 | }
|
---|
545 | }
|
---|
546 | }
|
---|
547 | }
|
---|
548 |
|
---|
549 | /**
|
---|
550 | * Holds internal data.
|
---|
551 | * This structure was created so that all optimizer fields can be "final".
|
---|
552 | * Code should be further refactored in order to not pass around arguments
|
---|
553 | * that will modified in-place (cf. "work" arrays).
|
---|
554 | */
|
---|
555 | private static class InternalData {
|
---|
556 | /** Weighted Jacobian. */
|
---|
557 | private final double[][] weightedJacobian;
|
---|
558 | /** Columns permutation array. */
|
---|
559 | private final int[] permutation;
|
---|
560 | /** Rank of the Jacobian matrix. */
|
---|
561 | private final int rank;
|
---|
562 | /** Diagonal elements of the R matrix in the QR decomposition. */
|
---|
563 | private final double[] diagR;
|
---|
564 | /** Norms of the columns of the jacobian matrix. */
|
---|
565 | private final double[] jacNorm;
|
---|
566 | /** Coefficients of the Householder transforms vectors. */
|
---|
567 | private final double[] beta;
|
---|
568 |
|
---|
569 | /**
|
---|
570 | * @param weightedJacobian Weighted Jacobian.
|
---|
571 | * @param permutation Columns permutation array.
|
---|
572 | * @param rank Rank of the Jacobian matrix.
|
---|
573 | * @param diagR Diagonal elements of the R matrix in the QR decomposition.
|
---|
574 | * @param jacNorm Norms of the columns of the jacobian matrix.
|
---|
575 | * @param beta Coefficients of the Householder transforms vectors.
|
---|
576 | */
|
---|
577 | InternalData(double[][] weightedJacobian,
|
---|
578 | int[] permutation,
|
---|
579 | int rank,
|
---|
580 | double[] diagR,
|
---|
581 | double[] jacNorm,
|
---|
582 | double[] beta) {
|
---|
583 | this.weightedJacobian = weightedJacobian;
|
---|
584 | this.permutation = permutation;
|
---|
585 | this.rank = rank;
|
---|
586 | this.diagR = diagR;
|
---|
587 | this.jacNorm = jacNorm;
|
---|
588 | this.beta = beta;
|
---|
589 | }
|
---|
590 | }
|
---|
591 |
|
---|
592 | /**
|
---|
593 | * Determines the Levenberg-Marquardt parameter.
|
---|
594 | *
|
---|
595 | * <p>This implementation is a translation in Java of the MINPACK
|
---|
596 | * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
|
---|
597 | * routine.</p>
|
---|
598 | * <p>This method sets the lmPar and lmDir attributes.</p>
|
---|
599 | * <p>The authors of the original fortran function are:</p>
|
---|
600 | * <ul>
|
---|
601 | * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
|
---|
602 | * <li>Burton S. Garbow</li>
|
---|
603 | * <li>Kenneth E. Hillstrom</li>
|
---|
604 | * <li>Jorge J. More</li>
|
---|
605 | * </ul>
|
---|
606 | * <p>Luc Maisonobe did the Java translation.</p>
|
---|
607 | *
|
---|
608 | * @param qy Array containing qTy.
|
---|
609 | * @param delta Upper bound on the euclidean norm of diagR * lmDir.
|
---|
610 | * @param diag Diagonal matrix.
|
---|
611 | * @param internalData Data (modified in-place in this method).
|
---|
612 | * @param solvedCols Number of solved point.
|
---|
613 | * @param work1 work array
|
---|
614 | * @param work2 work array
|
---|
615 | * @param work3 work array
|
---|
616 | * @param lmDir the "returned" LM direction will be stored in this array.
|
---|
617 | * @param lmPar the value of the LM parameter from the previous iteration.
|
---|
618 | * @return the new LM parameter
|
---|
619 | */
|
---|
620 | private double determineLMParameter(double[] qy, double delta, double[] diag,
|
---|
621 | InternalData internalData, int solvedCols,
|
---|
622 | double[] work1, double[] work2, double[] work3,
|
---|
623 | double[] lmDir, double lmPar) {
|
---|
624 | final double[][] weightedJacobian = internalData.weightedJacobian;
|
---|
625 | final int[] permutation = internalData.permutation;
|
---|
626 | final int rank = internalData.rank;
|
---|
627 | final double[] diagR = internalData.diagR;
|
---|
628 |
|
---|
629 | final int nC = weightedJacobian[0].length;
|
---|
630 |
|
---|
631 | // compute and store in x the gauss-newton direction, if the
|
---|
632 | // jacobian is rank-deficient, obtain a least squares solution
|
---|
633 | for (int j = 0; j < rank; ++j) {
|
---|
634 | lmDir[permutation[j]] = qy[j];
|
---|
635 | }
|
---|
636 | for (int j = rank; j < nC; ++j) {
|
---|
637 | lmDir[permutation[j]] = 0;
|
---|
638 | }
|
---|
639 | for (int k = rank - 1; k >= 0; --k) {
|
---|
640 | int pk = permutation[k];
|
---|
641 | double ypk = lmDir[pk] / diagR[pk];
|
---|
642 | for (int i = 0; i < k; ++i) {
|
---|
643 | lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk];
|
---|
644 | }
|
---|
645 | lmDir[pk] = ypk;
|
---|
646 | }
|
---|
647 |
|
---|
648 | // evaluate the function at the origin, and test
|
---|
649 | // for acceptance of the Gauss-Newton direction
|
---|
650 | double dxNorm = 0;
|
---|
651 | for (int j = 0; j < solvedCols; ++j) {
|
---|
652 | int pj = permutation[j];
|
---|
653 | double s = diag[pj] * lmDir[pj];
|
---|
654 | work1[pj] = s;
|
---|
655 | dxNorm += s * s;
|
---|
656 | }
|
---|
657 | dxNorm = FastMath.sqrt(dxNorm);
|
---|
658 | double fp = dxNorm - delta;
|
---|
659 | if (fp <= 0.1 * delta) {
|
---|
660 | lmPar = 0;
|
---|
661 | return lmPar;
|
---|
662 | }
|
---|
663 |
|
---|
664 | // if the jacobian is not rank deficient, the Newton step provides
|
---|
665 | // a lower bound, parl, for the zero of the function,
|
---|
666 | // otherwise set this bound to zero
|
---|
667 | double sum2;
|
---|
668 | double parl = 0;
|
---|
669 | if (rank == solvedCols) {
|
---|
670 | for (int j = 0; j < solvedCols; ++j) {
|
---|
671 | int pj = permutation[j];
|
---|
672 | work1[pj] *= diag[pj] / dxNorm;
|
---|
673 | }
|
---|
674 | sum2 = 0;
|
---|
675 | for (int j = 0; j < solvedCols; ++j) {
|
---|
676 | int pj = permutation[j];
|
---|
677 | double sum = 0;
|
---|
678 | for (int i = 0; i < j; ++i) {
|
---|
679 | sum += weightedJacobian[i][pj] * work1[permutation[i]];
|
---|
680 | }
|
---|
681 | double s = (work1[pj] - sum) / diagR[pj];
|
---|
682 | work1[pj] = s;
|
---|
683 | sum2 += s * s;
|
---|
684 | }
|
---|
685 | parl = fp / (delta * sum2);
|
---|
686 | }
|
---|
687 |
|
---|
688 | // calculate an upper bound, paru, for the zero of the function
|
---|
689 | sum2 = 0;
|
---|
690 | for (int j = 0; j < solvedCols; ++j) {
|
---|
691 | int pj = permutation[j];
|
---|
692 | double sum = 0;
|
---|
693 | for (int i = 0; i <= j; ++i) {
|
---|
694 | sum += weightedJacobian[i][pj] * qy[i];
|
---|
695 | }
|
---|
696 | sum /= diag[pj];
|
---|
697 | sum2 += sum * sum;
|
---|
698 | }
|
---|
699 | double gNorm = FastMath.sqrt(sum2);
|
---|
700 | double paru = gNorm / delta;
|
---|
701 | if (paru == 0) {
|
---|
702 | paru = Precision.SAFE_MIN / FastMath.min(delta, 0.1);
|
---|
703 | }
|
---|
704 |
|
---|
705 | // if the input par lies outside of the interval (parl,paru),
|
---|
706 | // set par to the closer endpoint
|
---|
707 | lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
|
---|
708 | if (lmPar == 0) {
|
---|
709 | lmPar = gNorm / dxNorm;
|
---|
710 | }
|
---|
711 |
|
---|
712 | for (int countdown = 10; countdown >= 0; --countdown) {
|
---|
713 |
|
---|
714 | // evaluate the function at the current value of lmPar
|
---|
715 | if (lmPar == 0) {
|
---|
716 | lmPar = FastMath.max(Precision.SAFE_MIN, 0.001 * paru);
|
---|
717 | }
|
---|
718 | double sPar = FastMath.sqrt(lmPar);
|
---|
719 | for (int j = 0; j < solvedCols; ++j) {
|
---|
720 | int pj = permutation[j];
|
---|
721 | work1[pj] = sPar * diag[pj];
|
---|
722 | }
|
---|
723 | determineLMDirection(qy, work1, work2, internalData, solvedCols, work3, lmDir);
|
---|
724 |
|
---|
725 | dxNorm = 0;
|
---|
726 | for (int j = 0; j < solvedCols; ++j) {
|
---|
727 | int pj = permutation[j];
|
---|
728 | double s = diag[pj] * lmDir[pj];
|
---|
729 | work3[pj] = s;
|
---|
730 | dxNorm += s * s;
|
---|
731 | }
|
---|
732 | dxNorm = FastMath.sqrt(dxNorm);
|
---|
733 | double previousFP = fp;
|
---|
734 | fp = dxNorm - delta;
|
---|
735 |
|
---|
736 | // if the function is small enough, accept the current value
|
---|
737 | // of lmPar, also test for the exceptional cases where parl is zero
|
---|
738 | if (FastMath.abs(fp) <= 0.1 * delta ||
|
---|
739 | (parl == 0 &&
|
---|
740 | fp <= previousFP &&
|
---|
741 | previousFP < 0)) {
|
---|
742 | return lmPar;
|
---|
743 | }
|
---|
744 |
|
---|
745 | // compute the Newton correction
|
---|
746 | for (int j = 0; j < solvedCols; ++j) {
|
---|
747 | int pj = permutation[j];
|
---|
748 | work1[pj] = work3[pj] * diag[pj] / dxNorm;
|
---|
749 | }
|
---|
750 | for (int j = 0; j < solvedCols; ++j) {
|
---|
751 | int pj = permutation[j];
|
---|
752 | work1[pj] /= work2[j];
|
---|
753 | double tmp = work1[pj];
|
---|
754 | for (int i = j + 1; i < solvedCols; ++i) {
|
---|
755 | work1[permutation[i]] -= weightedJacobian[i][pj] * tmp;
|
---|
756 | }
|
---|
757 | }
|
---|
758 | sum2 = 0;
|
---|
759 | for (int j = 0; j < solvedCols; ++j) {
|
---|
760 | double s = work1[permutation[j]];
|
---|
761 | sum2 += s * s;
|
---|
762 | }
|
---|
763 | double correction = fp / (delta * sum2);
|
---|
764 |
|
---|
765 | // depending on the sign of the function, update parl or paru.
|
---|
766 | if (fp > 0) {
|
---|
767 | parl = FastMath.max(parl, lmPar);
|
---|
768 | } else if (fp < 0) {
|
---|
769 | paru = FastMath.min(paru, lmPar);
|
---|
770 | }
|
---|
771 |
|
---|
772 | // compute an improved estimate for lmPar
|
---|
773 | lmPar = FastMath.max(parl, lmPar + correction);
|
---|
774 | }
|
---|
775 |
|
---|
776 | return lmPar;
|
---|
777 | }
|
---|
778 |
|
---|
779 | /**
|
---|
780 | * Solve a*x = b and d*x = 0 in the least squares sense.
|
---|
781 | * <p>This implementation is a translation in Java of the MINPACK
|
---|
782 | * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
|
---|
783 | * routine.</p>
|
---|
784 | * <p>This method sets the lmDir and lmDiag attributes.</p>
|
---|
785 | * <p>The authors of the original fortran function are:</p>
|
---|
786 | * <ul>
|
---|
787 | * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
|
---|
788 | * <li>Burton S. Garbow</li>
|
---|
789 | * <li>Kenneth E. Hillstrom</li>
|
---|
790 | * <li>Jorge J. More</li>
|
---|
791 | * </ul>
|
---|
792 | * <p>Luc Maisonobe did the Java translation.</p>
|
---|
793 | *
|
---|
794 | * @param qy array containing qTy
|
---|
795 | * @param diag diagonal matrix
|
---|
796 | * @param lmDiag diagonal elements associated with lmDir
|
---|
797 | * @param internalData Data (modified in-place in this method).
|
---|
798 | * @param solvedCols Number of sloved point.
|
---|
799 | * @param work work array
|
---|
800 | * @param lmDir the "returned" LM direction is stored in this array
|
---|
801 | */
|
---|
802 | private void determineLMDirection(double[] qy, double[] diag,
|
---|
803 | double[] lmDiag,
|
---|
804 | InternalData internalData,
|
---|
805 | int solvedCols,
|
---|
806 | double[] work,
|
---|
807 | double[] lmDir) {
|
---|
808 | final int[] permutation = internalData.permutation;
|
---|
809 | final double[][] weightedJacobian = internalData.weightedJacobian;
|
---|
810 | final double[] diagR = internalData.diagR;
|
---|
811 |
|
---|
812 | // copy R and Qty to preserve input and initialize s
|
---|
813 | // in particular, save the diagonal elements of R in lmDir
|
---|
814 | for (int j = 0; j < solvedCols; ++j) {
|
---|
815 | int pj = permutation[j];
|
---|
816 | for (int i = j + 1; i < solvedCols; ++i) {
|
---|
817 | weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]];
|
---|
818 | }
|
---|
819 | lmDir[j] = diagR[pj];
|
---|
820 | work[j] = qy[j];
|
---|
821 | }
|
---|
822 |
|
---|
823 | // eliminate the diagonal matrix d using a Givens rotation
|
---|
824 | for (int j = 0; j < solvedCols; ++j) {
|
---|
825 |
|
---|
826 | // prepare the row of d to be eliminated, locating the
|
---|
827 | // diagonal element using p from the Q.R. factorization
|
---|
828 | int pj = permutation[j];
|
---|
829 | double dpj = diag[pj];
|
---|
830 | if (dpj != 0) {
|
---|
831 | Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
|
---|
832 | }
|
---|
833 | lmDiag[j] = dpj;
|
---|
834 |
|
---|
835 | // the transformations to eliminate the row of d
|
---|
836 | // modify only a single element of Qty
|
---|
837 | // beyond the first n, which is initially zero.
|
---|
838 | double qtbpj = 0;
|
---|
839 | for (int k = j; k < solvedCols; ++k) {
|
---|
840 | int pk = permutation[k];
|
---|
841 |
|
---|
842 | // determine a Givens rotation which eliminates the
|
---|
843 | // appropriate element in the current row of d
|
---|
844 | if (lmDiag[k] != 0) {
|
---|
845 |
|
---|
846 | final double sin;
|
---|
847 | final double cos;
|
---|
848 | double rkk = weightedJacobian[k][pk];
|
---|
849 | if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
|
---|
850 | final double cotan = rkk / lmDiag[k];
|
---|
851 | sin = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
|
---|
852 | cos = sin * cotan;
|
---|
853 | } else {
|
---|
854 | final double tan = lmDiag[k] / rkk;
|
---|
855 | cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
|
---|
856 | sin = cos * tan;
|
---|
857 | }
|
---|
858 |
|
---|
859 | // compute the modified diagonal element of R and
|
---|
860 | // the modified element of (Qty,0)
|
---|
861 | weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
|
---|
862 | final double temp = cos * work[k] + sin * qtbpj;
|
---|
863 | qtbpj = -sin * work[k] + cos * qtbpj;
|
---|
864 | work[k] = temp;
|
---|
865 |
|
---|
866 | // accumulate the tranformation in the row of s
|
---|
867 | for (int i = k + 1; i < solvedCols; ++i) {
|
---|
868 | double rik = weightedJacobian[i][pk];
|
---|
869 | final double temp2 = cos * rik + sin * lmDiag[i];
|
---|
870 | lmDiag[i] = -sin * rik + cos * lmDiag[i];
|
---|
871 | weightedJacobian[i][pk] = temp2;
|
---|
872 | }
|
---|
873 | }
|
---|
874 | }
|
---|
875 |
|
---|
876 | // store the diagonal element of s and restore
|
---|
877 | // the corresponding diagonal element of R
|
---|
878 | lmDiag[j] = weightedJacobian[j][permutation[j]];
|
---|
879 | weightedJacobian[j][permutation[j]] = lmDir[j];
|
---|
880 | }
|
---|
881 |
|
---|
882 | // solve the triangular system for z, if the system is
|
---|
883 | // singular, then obtain a least squares solution
|
---|
884 | int nSing = solvedCols;
|
---|
885 | for (int j = 0; j < solvedCols; ++j) {
|
---|
886 | if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
|
---|
887 | nSing = j;
|
---|
888 | }
|
---|
889 | if (nSing < solvedCols) {
|
---|
890 | work[j] = 0;
|
---|
891 | }
|
---|
892 | }
|
---|
893 | if (nSing > 0) {
|
---|
894 | for (int j = nSing - 1; j >= 0; --j) {
|
---|
895 | int pj = permutation[j];
|
---|
896 | double sum = 0;
|
---|
897 | for (int i = j + 1; i < nSing; ++i) {
|
---|
898 | sum += weightedJacobian[i][pj] * work[i];
|
---|
899 | }
|
---|
900 | work[j] = (work[j] - sum) / lmDiag[j];
|
---|
901 | }
|
---|
902 | }
|
---|
903 |
|
---|
904 | // permute the components of z back to components of lmDir
|
---|
905 | for (int j = 0; j < lmDir.length; ++j) {
|
---|
906 | lmDir[permutation[j]] = work[j];
|
---|
907 | }
|
---|
908 | }
|
---|
909 |
|
---|
910 | /**
|
---|
911 | * Decompose a matrix A as A.P = Q.R using Householder transforms.
|
---|
912 | * <p>As suggested in the P. Lascaux and R. Theodor book
|
---|
913 | * <i>Analyse numérique matricielle appliquée à
|
---|
914 | * l'art de l'ingénieur</i> (Masson, 1986), instead of representing
|
---|
915 | * the Householder transforms with u<sub>k</sub> unit vectors such that:
|
---|
916 | * <pre>
|
---|
917 | * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
|
---|
918 | * </pre>
|
---|
919 | * we use <sub>k</sub> non-unit vectors such that:
|
---|
920 | * <pre>
|
---|
921 | * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
|
---|
922 | * </pre>
|
---|
923 | * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
|
---|
924 | * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
|
---|
925 | * them from the v<sub>k</sub> vectors would be costly.</p>
|
---|
926 | * <p>This decomposition handles rank deficient cases since the tranformations
|
---|
927 | * are performed in non-increasing columns norms order thanks to columns
|
---|
928 | * pivoting. The diagonal elements of the R matrix are therefore also in
|
---|
929 | * non-increasing absolute values order.</p>
|
---|
930 | *
|
---|
931 | * @param jacobian Weighted Jacobian matrix at the current point.
|
---|
932 | * @param solvedCols Number of solved point.
|
---|
933 | * @return data used in other methods of this class.
|
---|
934 | * @throws ConvergenceException if the decomposition cannot be performed.
|
---|
935 | */
|
---|
936 | private InternalData qrDecomposition(RealMatrix jacobian,
|
---|
937 | int solvedCols) throws ConvergenceException {
|
---|
938 | // Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
|
---|
939 | // hence the multiplication by -1.
|
---|
940 | final double[][] weightedJacobian = jacobian.scalarMultiply(-1).getData();
|
---|
941 |
|
---|
942 | final int nR = weightedJacobian.length;
|
---|
943 | final int nC = weightedJacobian[0].length;
|
---|
944 |
|
---|
945 | final int[] permutation = new int[nC];
|
---|
946 | final double[] diagR = new double[nC];
|
---|
947 | final double[] jacNorm = new double[nC];
|
---|
948 | final double[] beta = new double[nC];
|
---|
949 |
|
---|
950 | // initializations
|
---|
951 | for (int k = 0; k < nC; ++k) {
|
---|
952 | permutation[k] = k;
|
---|
953 | double norm2 = 0;
|
---|
954 | for (int i = 0; i < nR; ++i) {
|
---|
955 | double akk = weightedJacobian[i][k];
|
---|
956 | norm2 += akk * akk;
|
---|
957 | }
|
---|
958 | jacNorm[k] = FastMath.sqrt(norm2);
|
---|
959 | }
|
---|
960 |
|
---|
961 | // transform the matrix column after column
|
---|
962 | for (int k = 0; k < nC; ++k) {
|
---|
963 |
|
---|
964 | // select the column with the greatest norm on active components
|
---|
965 | int nextColumn = -1;
|
---|
966 | double ak2 = Double.NEGATIVE_INFINITY;
|
---|
967 | for (int i = k; i < nC; ++i) {
|
---|
968 | double norm2 = 0;
|
---|
969 | for (int j = k; j < nR; ++j) {
|
---|
970 | double aki = weightedJacobian[j][permutation[i]];
|
---|
971 | norm2 += aki * aki;
|
---|
972 | }
|
---|
973 | if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
|
---|
974 | throw new ConvergenceException(LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN,
|
---|
975 | nR, nC);
|
---|
976 | }
|
---|
977 | if (norm2 > ak2) {
|
---|
978 | nextColumn = i;
|
---|
979 | ak2 = norm2;
|
---|
980 | }
|
---|
981 | }
|
---|
982 | if (ak2 <= qrRankingThreshold) {
|
---|
983 | return new InternalData(weightedJacobian, permutation, k, diagR, jacNorm, beta);
|
---|
984 | }
|
---|
985 | int pk = permutation[nextColumn];
|
---|
986 | permutation[nextColumn] = permutation[k];
|
---|
987 | permutation[k] = pk;
|
---|
988 |
|
---|
989 | // choose alpha such that Hk.u = alpha ek
|
---|
990 | double akk = weightedJacobian[k][pk];
|
---|
991 | double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
|
---|
992 | double betak = 1.0 / (ak2 - akk * alpha);
|
---|
993 | beta[pk] = betak;
|
---|
994 |
|
---|
995 | // transform the current column
|
---|
996 | diagR[pk] = alpha;
|
---|
997 | weightedJacobian[k][pk] -= alpha;
|
---|
998 |
|
---|
999 | // transform the remaining columns
|
---|
1000 | for (int dk = nC - 1 - k; dk > 0; --dk) {
|
---|
1001 | double gamma = 0;
|
---|
1002 | for (int j = k; j < nR; ++j) {
|
---|
1003 | gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]];
|
---|
1004 | }
|
---|
1005 | gamma *= betak;
|
---|
1006 | for (int j = k; j < nR; ++j) {
|
---|
1007 | weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk];
|
---|
1008 | }
|
---|
1009 | }
|
---|
1010 | }
|
---|
1011 |
|
---|
1012 | return new InternalData(weightedJacobian, permutation, solvedCols, diagR, jacNorm, beta);
|
---|
1013 | }
|
---|
1014 |
|
---|
1015 | /**
|
---|
1016 | * Compute the product Qt.y for some Q.R. decomposition.
|
---|
1017 | *
|
---|
1018 | * @param y vector to multiply (will be overwritten with the result)
|
---|
1019 | * @param internalData Data.
|
---|
1020 | */
|
---|
1021 | private void qTy(double[] y,
|
---|
1022 | InternalData internalData) {
|
---|
1023 | final double[][] weightedJacobian = internalData.weightedJacobian;
|
---|
1024 | final int[] permutation = internalData.permutation;
|
---|
1025 | final double[] beta = internalData.beta;
|
---|
1026 |
|
---|
1027 | final int nR = weightedJacobian.length;
|
---|
1028 | final int nC = weightedJacobian[0].length;
|
---|
1029 |
|
---|
1030 | for (int k = 0; k < nC; ++k) {
|
---|
1031 | int pk = permutation[k];
|
---|
1032 | double gamma = 0;
|
---|
1033 | for (int i = k; i < nR; ++i) {
|
---|
1034 | gamma += weightedJacobian[i][pk] * y[i];
|
---|
1035 | }
|
---|
1036 | gamma *= beta[pk];
|
---|
1037 | for (int i = k; i < nR; ++i) {
|
---|
1038 | y[i] -= gamma * weightedJacobian[i][pk];
|
---|
1039 | }
|
---|
1040 | }
|
---|
1041 | }
|
---|
1042 | }
|
---|