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* <tr> * </table> * @author Argonne National Laboratory. MINPACK project. March 1980 (original fortran minpack tests) * @author Burton S. Garbow (original fortran minpack tests) * @author Kenneth E. Hillstrom (original fortran minpack tests) * @author Jorge J. More (original fortran minpack tests) * @author Luc Maisonobe (non-minpack tests and minpack tests Java translation) */ public class GaussNewtonOptimizerTest extends TestCase { public GaussNewtonOptimizerTest(String name) { super(name); } public void testTrivial() throws FunctionEvaluationException, OptimizationException { LinearProblem problem = new LinearProblem(new double[][] { { 2 } }, new double[] { 3 }); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); VectorialPointValuePair optimum = optimizer.optimize(problem, problem.target, new double[] { 1 }, new double[] { 0 }); assertEquals(0, optimizer.getRMS(), 1.0e-10); assertEquals(1.5, optimum.getPoint()[0], 1.0e-10); assertEquals(3.0, optimum.getValue()[0], 1.0e-10); } public void testColumnsPermutation() throws FunctionEvaluationException, OptimizationException { LinearProblem problem = new LinearProblem(new double[][] { { 1.0, -1.0 }, { 0.0, 2.0 }, { 1.0, -2.0 } }, new double[] { 4.0, 6.0, 1.0 }); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); VectorialPointValuePair optimum = optimizer.optimize(problem, problem.target, new double[] { 1, 1, 1 }, new double[] { 0, 0 }); assertEquals(0, optimizer.getRMS(), 1.0e-10); assertEquals(7.0, optimum.getPoint()[0], 1.0e-10); assertEquals(3.0, optimum.getPoint()[1], 1.0e-10); assertEquals(4.0, optimum.getValue()[0], 1.0e-10); assertEquals(6.0, optimum.getValue()[1], 1.0e-10); assertEquals(1.0, optimum.getValue()[2], 1.0e-10); } public void testNoDependency() throws FunctionEvaluationException, OptimizationException { LinearProblem problem = new LinearProblem(new double[][] { { 2, 0, 0, 0, 0, 0 }, { 0, 2, 0, 0, 0, 0 }, { 0, 0, 2, 0, 0, 0 }, { 0, 0, 0, 2, 0, 0 }, { 0, 0, 0, 0, 2, 0 }, { 0, 0, 0, 0, 0, 2 } }, new double[] { 0.0, 1.1, 2.2, 3.3, 4.4, 5.5 }); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); VectorialPointValuePair optimum = optimizer.optimize(problem, problem.target, new double[] { 1, 1, 1, 1, 1, 1 }, new double[] { 0, 0, 0, 0, 0, 0 }); assertEquals(0, optimizer.getRMS(), 1.0e-10); for (int i = 0; i < problem.target.length; ++i) { assertEquals(0.55 * i, optimum.getPoint()[i], 1.0e-10); } } public void testOneSet() throws FunctionEvaluationException, OptimizationException { LinearProblem problem = new LinearProblem(new double[][] { { 1, 0, 0 }, { -1, 1, 0 }, { 0, -1, 1 } }, new double[] { 1, 1, 1}); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); VectorialPointValuePair optimum = optimizer.optimize(problem, problem.target, new double[] { 1, 1, 1 }, new double[] { 0, 0, 0 }); assertEquals(0, optimizer.getRMS(), 1.0e-10); assertEquals(1.0, optimum.getPoint()[0], 1.0e-10); assertEquals(2.0, optimum.getPoint()[1], 1.0e-10); assertEquals(3.0, optimum.getPoint()[2], 1.0e-10); } public void testTwoSets() throws FunctionEvaluationException, OptimizationException { double epsilon = 1.0e-7; LinearProblem problem = new LinearProblem(new double[][] { { 2, 1, 0, 4, 0, 0 }, { -4, -2, 3, -7, 0, 0 }, { 4, 1, -2, 8, 0, 0 }, { 0, -3, -12, -1, 0, 0 }, { 0, 0, 0, 0, epsilon, 1 }, { 0, 0, 0, 0, 1, 1 } }, new double[] { 2, -9, 2, 2, 1 + epsilon * epsilon, 2}); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); VectorialPointValuePair optimum = optimizer.optimize(problem, problem.target, new double[] { 1, 1, 1, 1, 1, 1 }, new double[] { 0, 0, 0, 0, 0, 0 }); assertEquals(0, optimizer.getRMS(), 1.0e-10); assertEquals( 3.0, optimum.getPoint()[0], 1.0e-10); assertEquals( 4.0, optimum.getPoint()[1], 1.0e-10); assertEquals(-1.0, optimum.getPoint()[2], 1.0e-10); assertEquals(-2.0, optimum.getPoint()[3], 1.0e-10); assertEquals( 1.0 + epsilon, optimum.getPoint()[4], 1.0e-10); assertEquals( 1.0 - epsilon, optimum.getPoint()[5], 1.0e-10); } public void testNonInversible() { LinearProblem problem = new LinearProblem(new double[][] { { 1, 2, -3 }, { 2, 1, 3 }, { -3, 0, -9 } }, new double[] { 1, 1, 1 }); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); try { optimizer.optimize(problem, problem.target, new double[] { 1, 1, 1 }, new double[] { 0, 0, 0 }); fail("an exception should have been caught"); } catch (OptimizationException ee) { // expected behavior } catch (Exception e) { fail("wrong exception type caught"); } } public void testIllConditioned() throws FunctionEvaluationException, OptimizationException { LinearProblem problem1 = new LinearProblem(new double[][] { { 10.0, 7.0, 8.0, 7.0 }, { 7.0, 5.0, 6.0, 5.0 }, { 8.0, 6.0, 10.0, 9.0 }, { 7.0, 5.0, 9.0, 10.0 } }, new double[] { 32, 23, 33, 31 }); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); VectorialPointValuePair optimum1 = optimizer.optimize(problem1, problem1.target, new double[] { 1, 1, 1, 1 }, new double[] { 0, 1, 2, 3 }); assertEquals(0, optimizer.getRMS(), 1.0e-10); assertEquals(1.0, optimum1.getPoint()[0], 1.0e-10); assertEquals(1.0, optimum1.getPoint()[1], 1.0e-10); assertEquals(1.0, optimum1.getPoint()[2], 1.0e-10); assertEquals(1.0, optimum1.getPoint()[3], 1.0e-10); LinearProblem problem2 = new LinearProblem(new double[][] { { 10.00, 7.00, 8.10, 7.20 }, { 7.08, 5.04, 6.00, 5.00 }, { 8.00, 5.98, 9.89, 9.00 }, { 6.99, 4.99, 9.00, 9.98 } }, new double[] { 32, 23, 33, 31 }); VectorialPointValuePair optimum2 = optimizer.optimize(problem2, problem2.target, new double[] { 1, 1, 1, 1 }, new double[] { 0, 1, 2, 3 }); assertEquals(0, optimizer.getRMS(), 1.0e-10); assertEquals(-81.0, optimum2.getPoint()[0], 1.0e-8); assertEquals(137.0, optimum2.getPoint()[1], 1.0e-8); assertEquals(-34.0, optimum2.getPoint()[2], 1.0e-8); assertEquals( 22.0, optimum2.getPoint()[3], 1.0e-8); } public void testMoreEstimatedParametersSimple() { LinearProblem problem = new LinearProblem(new double[][] { { 3.0, 2.0, 0.0, 0.0 }, { 0.0, 1.0, -1.0, 1.0 }, { 2.0, 0.0, 1.0, 0.0 } }, new double[] { 7.0, 3.0, 5.0 }); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); try { optimizer.optimize(problem, problem.target, new double[] { 1, 1, 1 }, new double[] { 7, 6, 5, 4 }); fail("an exception should have been caught"); } catch (OptimizationException ee) { // expected behavior } catch (Exception e) { fail("wrong exception type caught"); } } public void testMoreEstimatedParametersUnsorted() { LinearProblem problem = new LinearProblem(new double[][] { { 1.0, 1.0, 0.0, 0.0, 0.0, 0.0 }, { 0.0, 0.0, 1.0, 1.0, 1.0, 0.0 }, { 0.0, 0.0, 0.0, 0.0, 1.0, -1.0 }, { 0.0, 0.0, -1.0, 1.0, 0.0, 1.0 }, { 0.0, 0.0, 0.0, -1.0, 1.0, 0.0 } }, new double[] { 3.0, 12.0, -1.0, 7.0, 1.0 }); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); try { optimizer.optimize(problem, problem.target, new double[] { 1, 1, 1, 1, 1 }, new double[] { 2, 2, 2, 2, 2, 2 }); fail("an exception should have been caught"); } catch (OptimizationException ee) { // expected behavior } catch (Exception e) { fail("wrong exception type caught"); } } public void testRedundantEquations() throws FunctionEvaluationException, OptimizationException { LinearProblem problem = new LinearProblem(new double[][] { { 1.0, 1.0 }, { 1.0, -1.0 }, { 1.0, 3.0 } }, new double[] { 3.0, 1.0, 5.0 }); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); VectorialPointValuePair optimum = optimizer.optimize(problem, problem.target, new double[] { 1, 1, 1 }, new double[] { 1, 1 }); assertEquals(0, optimizer.getRMS(), 1.0e-10); assertEquals(2.0, optimum.getPoint()[0], 1.0e-8); assertEquals(1.0, optimum.getPoint()[1], 1.0e-8); } public void testInconsistentEquations() throws FunctionEvaluationException, OptimizationException { LinearProblem problem = new LinearProblem(new double[][] { { 1.0, 1.0 }, { 1.0, -1.0 }, { 1.0, 3.0 } }, new double[] { 3.0, 1.0, 4.0 }); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); optimizer.optimize(problem, problem.target, new double[] { 1, 1, 1 }, new double[] { 1, 1 }); assertTrue(optimizer.getRMS() > 0.1); } public void testInconsistentSizes() throws FunctionEvaluationException, OptimizationException { LinearProblem problem = new LinearProblem(new double[][] { { 1, 0 }, { 0, 1 } }, new double[] { -1, 1 }); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); VectorialPointValuePair optimum = optimizer.optimize(problem, problem.target, new double[] { 1, 1 }, new double[] { 0, 0 }); assertEquals(0, optimizer.getRMS(), 1.0e-10); assertEquals(-1, optimum.getPoint()[0], 1.0e-10); assertEquals(+1, optimum.getPoint()[1], 1.0e-10); try { optimizer.optimize(problem, problem.target, new double[] { 1 }, new double[] { 0, 0 }); fail("an exception should have been thrown"); } catch (OptimizationException oe) { // expected behavior } catch (Exception e) { fail("wrong exception caught"); } try { optimizer.optimize(problem, new double[] { 1 }, new double[] { 1 }, new double[] { 0, 0 }); fail("an exception should have been thrown"); } catch (FunctionEvaluationException oe) { // expected behavior } catch (Exception e) { fail("wrong exception caught"); } } public void testMaxIterations() { Circle circle = new Circle(); circle.addPoint( 30.0, 68.0); circle.addPoint( 50.0, -6.0); circle.addPoint(110.0, -20.0); circle.addPoint( 35.0, 15.0); circle.addPoint( 45.0, 97.0); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialPointChecker(1.0e-30, 1.0e-30)); try { optimizer.optimize(circle, new double[] { 0, 0, 0, 0, 0 }, new double[] { 1, 1, 1, 1, 1 }, new double[] { 98.680, 47.345 }); fail("an exception should have been caught"); } catch (OptimizationException ee) { // expected behavior } catch (Exception e) { fail("wrong exception type caught"); } } public void testCircleFitting() throws FunctionEvaluationException, OptimizationException { Circle circle = new Circle(); circle.addPoint( 30.0, 68.0); circle.addPoint( 50.0, -6.0); circle.addPoint(110.0, -20.0); circle.addPoint( 35.0, 15.0); circle.addPoint( 45.0, 97.0); GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-13, 1.0e-13)); VectorialPointValuePair optimum = optimizer.optimize(circle, new double[] { 0, 0, 0, 0, 0 }, new double[] { 1, 1, 1, 1, 1 }, new double[] { 98.680, 47.345 }); assertEquals(1.768262623567235, Math.sqrt(circle.getN()) * optimizer.getRMS(), 1.0e-10); Point2D.Double center = new Point2D.Double(optimum.getPointRef()[0], optimum.getPointRef()[1]); assertEquals(69.96016175359975, circle.getRadius(center), 1.0e-10); assertEquals(96.07590209601095, center.x, 1.0e-10); assertEquals(48.135167894714, center.y, 1.0e-10); } public void testCircleFittingBadInit() throws FunctionEvaluationException, OptimizationException { Circle circle = new Circle(); double[][] points = new double[][] { {-0.312967, 0.072366}, {-0.339248, 0.132965}, {-0.379780, 0.202724}, {-0.390426, 0.260487}, {-0.361212, 0.328325}, {-0.346039, 0.392619}, {-0.280579, 0.444306}, {-0.216035, 0.470009}, {-0.149127, 0.493832}, {-0.075133, 0.483271}, {-0.007759, 0.452680}, { 0.060071, 0.410235}, { 0.103037, 0.341076}, { 0.118438, 0.273884}, { 0.131293, 0.192201}, { 0.115869, 0.129797}, { 0.072223, 0.058396}, { 0.022884, 0.000718}, {-0.053355, -0.020405}, {-0.123584, -0.032451}, {-0.216248, -0.032862}, {-0.278592, -0.005008}, {-0.337655, 0.056658}, {-0.385899, 0.112526}, {-0.405517, 0.186957}, {-0.415374, 0.262071}, {-0.387482, 0.343398}, {-0.347322, 0.397943}, {-0.287623, 0.458425}, {-0.223502, 0.475513}, {-0.135352, 0.478186}, {-0.061221, 0.483371}, { 0.003711, 0.422737}, { 0.065054, 0.375830}, { 0.108108, 0.297099}, { 0.123882, 0.222850}, { 0.117729, 0.134382}, { 0.085195, 0.056820}, { 0.029800, -0.019138}, {-0.027520, -0.072374}, {-0.102268, -0.091555}, {-0.200299, -0.106578}, {-0.292731, -0.091473}, {-0.356288, -0.051108}, {-0.420561, 0.014926}, {-0.471036, 0.074716}, {-0.488638, 0.182508}, {-0.485990, 0.254068}, {-0.463943, 0.338438}, {-0.406453, 0.404704}, {-0.334287, 0.466119}, {-0.254244, 0.503188}, {-0.161548, 0.495769}, {-0.075733, 0.495560}, { 0.001375, 0.434937}, { 0.082787, 0.385806}, { 0.115490, 0.323807}, { 0.141089, 0.223450}, { 0.138693, 0.131703}, { 0.126415, 0.049174}, { 0.066518, -0.010217}, {-0.005184, -0.070647}, {-0.080985, -0.103635}, {-0.177377, -0.116887}, {-0.260628, -0.100258}, {-0.335756, -0.056251}, {-0.405195, -0.000895}, {-0.444937, 0.085456}, {-0.484357, 0.175597}, {-0.472453, 0.248681}, {-0.438580, 0.347463}, {-0.402304, 0.422428}, {-0.326777, 0.479438}, {-0.247797, 0.505581}, {-0.152676, 0.519380}, {-0.071754, 0.516264}, { 0.015942, 0.472802}, { 0.076608, 0.419077}, { 0.127673, 0.330264}, { 0.159951, 0.262150}, { 0.153530, 0.172681}, { 0.140653, 0.089229}, { 0.078666, 0.024981}, { 0.023807, -0.037022}, {-0.048837, -0.077056}, {-0.127729, -0.075338}, {-0.221271, -0.067526} }; double[] target = new double[points.length]; Arrays.fill(target, 0.0); double[] weights = new double[points.length]; Arrays.fill(weights, 2.0); for (int i = 0; i < points.length; ++i) { circle.addPoint(points[i][0], points[i][1]); } GaussNewtonOptimizer optimizer = new GaussNewtonOptimizer(true); optimizer.setMaxIterations(100); optimizer.setConvergenceChecker(new SimpleVectorialValueChecker(1.0e-6, 1.0e-6)); try { optimizer.optimize(circle, target, weights, new double[] { -12, -12 }); fail("an exception should have been caught"); } catch (OptimizationException ee) { // expected behavior } catch (Exception e) { fail("wrong exception type caught"); } VectorialPointValuePair optimum = optimizer.optimize(circle, target, weights, new double[] { 0, 0 }); assertEquals(-0.1517383071957963, optimum.getPointRef()[0], 1.0e-8); assertEquals(0.2074999736353867, optimum.getPointRef()[1], 1.0e-8); assertEquals(0.04268731682389561, optimizer.getRMS(), 1.0e-8); } private static class LinearProblem implements DifferentiableMultivariateVectorialFunction, Serializable { private static final long serialVersionUID = -8804268799379350190L; final RealMatrix factors; final double[] target; public LinearProblem(double[][] factors, double[] target) { this.factors = new BlockRealMatrix(factors); this.target = target; } public double[] value(double[] variables) { return factors.operate(variables); } public MultivariateMatrixFunction jacobian() { return new MultivariateMatrixFunction() { private static final long serialVersionUID = -8387467946663627585L; public double[][] value(double[] point) { return factors.getData(); } }; } } private static class Circle implements DifferentiableMultivariateVectorialFunction, Serializable { private static final long serialVersionUID = -7165774454925027042L; private ArrayList<Point2D.Double> points; public Circle() { points = new ArrayList<Point2D.Double>(); } public void addPoint(double px, double py) { points.add(new Point2D.Double(px, py)); } public int getN() { return points.size(); } public double getRadius(Point2D.Double center) { double r = 0; for (Point2D.Double point : points) { r += point.distance(center); } return r / points.size(); } private double[][] jacobian(double[] variables) { int n = points.size(); Point2D.Double center = new Point2D.Double(variables[0], variables[1]); // gradient of the optimal radius double dRdX = 0; double dRdY = 0; for (Point2D.Double pk : points) { double dk = pk.distance(center); dRdX += (center.x - pk.x) / dk; dRdY += (center.y - pk.y) / dk; } dRdX /= n; dRdY /= n; // jacobian of the radius residuals double[][] jacobian = new double[n][2]; for (int i = 0; i < n; ++i) { Point2D.Double pi = points.get(i); double di = pi.distance(center); jacobian[i][0] = (center.x - pi.x) / di - dRdX; jacobian[i][1] = (center.y - pi.y) / di - dRdY; } return jacobian; } public double[] value(double[] variables) { Point2D.Double center = new Point2D.Double(variables[0], variables[1]); double radius = getRadius(center); double[] residuals = new double[points.size()]; for (int i = 0; i < residuals.length; ++i) { residuals[i] = points.get(i).distance(center) - radius; } return residuals; } public MultivariateMatrixFunction jacobian() { return new MultivariateMatrixFunction() { private static final long serialVersionUID = -4340046230875165095L; public double[][] value(double[] point) { return jacobian(point); } }; } } }

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Commons Math example source code file (GaussNewtonOptimizerTest.java)

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Java - Commons Math tags/keywords

awt, circle, exception, functionevaluationexception, functionevaluationexception, gaussnewtonoptimizer, gaussnewtonoptimizer, gaussnewtonoptimizertest, geometry, io, linearproblem, linearproblem, multivariatematrixfunction, optimizationexception, optimizationexception, serializable, util, vectorialpointvaluepair

The Commons Math GaussNewtonOptimizerTest.java source code

/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

package org.apache.commons.math.optimization.general;

import java.awt.geom.Point2D;
import java.io.Serializable;
import java.util.ArrayList;
import java.util.Arrays;

import junit.framework.TestCase;

import org.apache.commons.math.FunctionEvaluationException;
import org.apache.commons.math.analysis.DifferentiableMultivariateVectorialFunction;
import org.apache.commons.math.analysis.MultivariateMatrixFunction;
import org.apache.commons.math.linear.BlockRealMatrix;
import org.apache.commons.math.linear.RealMatrix;
import org.apache.commons.math.optimization.OptimizationException;
import org.apache.commons.math.optimization.SimpleVectorialPointChecker;
import org.apache.commons.math.optimization.SimpleVectorialValueChecker;
import org.apache.commons.math.optimization.VectorialPointValuePair;

/**
 * <p>Some of the unit tests are re-implementations of the MINPACK  and  test files.
 * 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>
* Minpack Copyright Notice (1999) University of Chicago. * All rights reserved * </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>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>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> * <ol>
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