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* Licensed to the Apache Software Foundation (ASF) under one or more
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* 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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* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.analysis.interpolation;
import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.analysis.function.Expm1;
import org.apache.commons.math3.analysis.function.Sin;
import org.apache.commons.math3.exception.NonMonotonicSequenceException;
import org.apache.commons.math3.util.FastMath;
import org.junit.Assert;
import org.junit.Test;
/**
* Test case for Neville interpolator.
* <p>
* The error of polynomial interpolation is
* f(z) - p(z) = f^(n)(zeta) * (z-x[0])(z-x[1])...(z-x[n-1]) / n!
* where f^(n) is the n-th derivative of the approximated function and
* zeta is some point in the interval determined by x[] and z.
* <p>
* Since zeta is unknown, f^(n)(zeta) cannot be calculated. But we can bound
* it and use the absolute value upper bound for estimates. For reference,
* see <b>Introduction to Numerical Analysis</b>, ISBN 038795452X, chapter 2.
*
*/
public final class NevilleInterpolatorTest {
/**
* Test of interpolator for the sine function.
* <p>
* |sin^(n)(zeta)| <= 1.0, zeta in [0, 2*PI]
*/
@Test
public void testSinFunction() {
UnivariateFunction f = new Sin();
UnivariateInterpolator interpolator = new NevilleInterpolator();
double x[], y[], z, expected, result, tolerance;
// 6 interpolating points on interval [0, 2*PI]
int n = 6;
double min = 0.0, max = 2 * FastMath.PI;
x = new double[n];
y = new double[n];
for (int i = 0; i < n; i++) {
x[i] = min + i * (max - min) / n;
y[i] = f.value(x[i]);
}
double derivativebound = 1.0;
UnivariateFunction p = interpolator.interpolate(x, y);
z = FastMath.PI / 4; expected = f.value(z); result = p.value(z);
tolerance = FastMath.abs(derivativebound * partialerror(x, z));
Assert.assertEquals(expected, result, tolerance);
z = FastMath.PI * 1.5; expected = f.value(z); result = p.value(z);
tolerance = FastMath.abs(derivativebound * partialerror(x, z));
Assert.assertEquals(expected, result, tolerance);
}
/**
* Test of interpolator for the exponential function.
* <p>
* |expm1^(n)(zeta)| <= e, zeta in [-1, 1]
*/
@Test
public void testExpm1Function() {
UnivariateFunction f = new Expm1();
UnivariateInterpolator interpolator = new NevilleInterpolator();
double x[], y[], z, expected, result, tolerance;
// 5 interpolating points on interval [-1, 1]
int n = 5;
double min = -1.0, max = 1.0;
x = new double[n];
y = new double[n];
for (int i = 0; i < n; i++) {
x[i] = min + i * (max - min) / n;
y[i] = f.value(x[i]);
}
double derivativebound = FastMath.E;
UnivariateFunction p = interpolator.interpolate(x, y);
z = 0.0; expected = f.value(z); result = p.value(z);
tolerance = FastMath.abs(derivativebound * partialerror(x, z));
Assert.assertEquals(expected, result, tolerance);
z = 0.5; expected = f.value(z); result = p.value(z);
tolerance = FastMath.abs(derivativebound * partialerror(x, z));
Assert.assertEquals(expected, result, tolerance);
z = -0.5; expected = f.value(z); result = p.value(z);
tolerance = FastMath.abs(derivativebound * partialerror(x, z));
Assert.assertEquals(expected, result, tolerance);
}
/**
* Test of parameters for the interpolator.
*/
@Test
public void testParameters() {
UnivariateInterpolator interpolator = new NevilleInterpolator();
try {
// bad abscissas array
double x[] = { 1.0, 2.0, 2.0, 4.0 };
double y[] = { 0.0, 4.0, 4.0, 2.5 };
UnivariateFunction p = interpolator.interpolate(x, y);
p.value(0.0);
Assert.fail("Expecting NonMonotonicSequenceException - bad abscissas array");
} catch (NonMonotonicSequenceException ex) {
// expected
}
}
/**
* Returns the partial error term (z-x[0])(z-x[1])...(z-x[n-1])/n!
*/
protected double partialerror(double x[], double z) throws
IllegalArgumentException {
if (x.length < 1) {
throw new IllegalArgumentException
("Interpolation array cannot be empty.");
}
double out = 1;
for (int i = 0; i < x.length; i++) {
out *= (z - x[i]) / (i + 1);
}
return out;
}
}