ClampedSplineInterpolator.java
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* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
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* See the License for the specific language governing permissions and
* limitations under the License.
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package org.apache.commons.math4.legacy.analysis.interpolation;
import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialFunction;
import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialSplineFunction;
import org.apache.commons.math4.legacy.core.MathArrays;
import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;
import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
/**
* Clamped cubic spline interpolator.
* The interpolating function consists in cubic polynomial functions defined over the
* subintervals determined by the "knot points".
*
* The interpolating polynomials satisfy:
* <ol>
* <li>The value of the interpolating function at each of the input {@code x} values
* equals the corresponding input {@code y} value.</li>
* <li>Adjacent polynomials are equal through two derivatives at the knot points
* (i.e., adjacent polynomials "match up" at the knot points, as do their first and
* second derivatives).</li>
* <li>The clamped boundary condition, i.e. the interpolating function takes "a
* specific direction" at both its start point and its end point by providing the
* desired first derivative values (slopes) as function parameters to
* {@link #interpolate(double[], double[], double, double)}.</li>
* </ol>
*
* The algorithm is implemented as described in
* <blockquote>
* R.L. Burden, J.D. Faires,
* <em>Numerical Analysis</em>, 9th Ed., 2010, Cengage Learning, ISBN 0-538-73351-9, pp 153-156.
* </blockquote>
*
*/
public class ClampedSplineInterpolator implements UnivariateInterpolator {
/**
*
* The first derivatives evaluated at the first and last knot points are
* approximated from a natural/unclamped spline that passes through the same
* set of points.
*
* @param x Arguments for the interpolation points.
* @param y Values for the interpolation points.
* @return the interpolating function.
* @throws DimensionMismatchException if {@code x} and {@code y} have different sizes.
* @throws NumberIsTooSmallException if the size of {@code x < 3}.
* @throws org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException if {@code x} is not sorted in strict increasing order.
*/
@Override
public PolynomialSplineFunction interpolate(final double[] x,
final double[] y) {
final SplineInterpolator spliner = new SplineInterpolator();
final PolynomialSplineFunction spline = spliner.interpolate(x, y);
final PolynomialSplineFunction derivativeSpline = spline.polynomialSplineDerivative();
final double fpStart = derivativeSpline.value(x[0]);
final double fpEnd = derivativeSpline.value(x[x.length - 1]);
return this.interpolate(x, y, fpStart, fpEnd);
}
/**
* Computes an interpolating function for the data set with defined
* boundary conditions.
*
* @param x Arguments for the interpolation points.
* @param y Values for the interpolation points.
* @param fpStart First derivative at the starting point of the returned
* spline function (starting slope).
* @param fpEnd First derivative at the ending point of the returned
* spline function (ending slope).
* @return the interpolating function.
* @throws DimensionMismatchException if {@code x} and {@code y} have different sizes.
* @throws NumberIsTooSmallException if the size of {@code x < 3}.
* @throws org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException if {@code x} is not sorted in strict increasing order.
*/
public PolynomialSplineFunction interpolate(final double[] x,
final double[] y,
final double fpStart,
final double fpEnd) {
if (x.length != y.length) {
throw new DimensionMismatchException(x.length, y.length);
}
if (x.length < 3) {
throw new NumberIsTooSmallException(LocalizedFormats.NUMBER_OF_POINTS,
x.length, 3, true);
}
// Number of intervals. The number of data points is n + 1.
final int n = x.length - 1;
MathArrays.checkOrder(x);
// Differences between knot points.
final double[] h = new double[n];
for (int i = 0; i < n; i++) {
h[i] = x[i + 1] - x[i];
}
final double[] mu = new double[n];
final double[] z = new double[n + 1];
final double alpha0 = 3 * ((y[1] - y[0]) / h[0] - fpStart);
final double alphaN = 3 * (fpEnd - (y[n] - y[n - 1]) / h[n - 1]);
mu[0] = 0.5;
final double ell0 = 2 * h[0];
z[0] = alpha0 / ell0;
for (int i = 1; i < n; i++) {
final double alpha = 3 * ((y[i + 1] - y[i]) / h[i] - (y[i] - y[i - 1]) / h[i - 1]);
final double ell = 2 * (x[i + 1] - x[i - 1]) - h[i - 1] * mu[i - 1];
mu[i] = h[i] / ell;
z[i] = (alpha - h[i - 1] * z[i - 1]) / ell;
}
// Cubic spline coefficients.
final double[] b = new double[n]; // Linear.
final double[] c = new double[n + 1]; // Quadratic.
final double[] d = new double[n]; // Cubic.
final double ellN = h[n - 1] * (2 - mu[n - 1]);
z[n] = (alphaN - h[n - 1] * z[n - 1]) / ellN;
c[n] = z[n];
for (int j = n - 1; j >= 0; j--) {
c[j] = z[j] - mu[j] * c[j + 1];
b[j] = ((y[j + 1] - y[j]) / h[j]) - h[j] * (c[j + 1] + 2 * c[j]) / 3;
d[j] = (c[j + 1] - c[j]) / (3 * h[j]);
}
final PolynomialFunction[] polynomials = new PolynomialFunction[n];
final double[] coefficients = new double[4];
for (int i = 0; i < n; i++) {
coefficients[0] = y[i];
coefficients[1] = b[i];
coefficients[2] = c[i];
coefficients[3] = d[i];
polynomials[i] = new PolynomialFunction(coefficients);
}
return new PolynomialSplineFunction(x, polynomials);
}
}