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);
- }
- }