ClampedSplineInterpolator.java
/*
* 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.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.NonMonotonicSequenceException;
import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;
import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
/**
* Computes a clamped cubic spline interpolation for the data set.
* <p>
* The {@link #interpolate(double[], double[], double, double)} method returns a {@link PolynomialSplineFunction}
* consisting of n cubic polynomials, defined over the subintervals determined by the x values,
* {@code x[0] < x[i] ... < x[n]}. The x values are referred to as "knot points."</p>
* <p>
* The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest
* knot point and strictly less than the largest knot point is computed by finding the subinterval to which
* x belongs and computing the value of the corresponding polynomial at <code>x - x[i]</code> where
* <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details.
* </p>
* <p>
* The interpolating polynomials satisfy: <ol>
* <li>The value of the PolynomialSplineFunction at each of the input x values equals the
* corresponding 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 <i>clamped boundary condition</i>, i.e., the PolynomialSplineFunction 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>
* <p>
* The clamped cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires,
* <u>Numerical Analysis</u>, 9th Ed., 2010, Cengage Learning, ISBN 0-538-73351-9, pp 153-156.
* </p>
*
*/
public class ClampedSplineInterpolator extends SplineInterpolator {
/**
* Computes an interpolating function for the data set.
* @param x the arguments for the interpolation points
* @param y the values for the interpolation points
* @param fpo first derivative at the starting point of the returned spline function (starting slope), satisfying
* clamped boundary condition S′(x0) = f′(x0)
* @param fpn first derivative at the ending point of the returned spline function (ending slope), satisfying
* clamped boundary condition S′(xn) = f′(xn)
* @return a function which interpolates the data set
* @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 fpo, final double fpn)
throws DimensionMismatchException, NumberIsTooSmallException, NonMonotonicSequenceException {
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 alpha[] = new double[n + 1];
final double l[] = new double[n + 1];
alpha[0] = 3d * (y[1] - y[0]) / h[0] - 3d * fpo;
alpha[n] = 3d * fpn - 3d * (y[n] - y[n - 1]) / h[n - 1];
mu[0] = 0.5d;
l[0] = 2d * h[0];
z[0] = alpha[0] / l[0];
for (int i = 1; i < n; i++) {
alpha[i] = (3d / h[i]) * (y[i + 1] - y[i]) - (3d / h[i - 1]) * (y[i] - y[i - 1]);
l[i] = 2d * (x[i + 1] - x[i - 1]) - h[i - 1] * mu[i - 1];
mu[i] = h[i] / l[i];
z[i] = (alpha[i] - h[i - 1] * z[i - 1]) / l[i];
}
// cubic spline coefficients -- b is linear, c quadratic, d is cubic (original y's are constants)
final double b[] = new double[n];
final double c[] = new double[n + 1];
final double d[] = new double[n];
l[n] = h[n - 1] * (2d - mu[n - 1]);
z[n] = (alpha[n] - h[n - 1] * z[n - 1]) / l[n];
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] + 2d * c[j]) / 3d;
d[j] = (c[j + 1] - c[j]) / (3d * 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);
}
}