1 /* 2 * Copyright 2003-2004 The Apache Software Foundation. 3 * 4 * Licensed under the Apache License, Version 2.0 (the "License"); 5 * you may not use this file except in compliance with the License. 6 * You may obtain a copy of the License at 7 * 8 * http://www.apache.org/licenses/LICENSE-2.0 9 * 10 * Unless required by applicable law or agreed to in writing, software 11 * distributed under the License is distributed on an "AS IS" BASIS, 12 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 13 * See the License for the specific language governing permissions and 14 * limitations under the License. 15 */ 16 package org.apache.commons.math.analysis; 17 18 /** 19 * Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set. 20 * <p> 21 * The {@link #interpolate(double[], double[])} method returns a {@link PolynomialSplineFunction} 22 * consisting of n cubic polynomials, defined over the subintervals determined by the x values, 23 * x[0] < x[i] ... < x[n]. The x values are referred to as "knot points." 24 * <p> 25 * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest 26 * knot point and strictly less than the largest knot point is computed by finding the subinterval to which 27 * x belongs and computing the value of the corresponding polynomial at <code>x - x[i] </code> where 28 * <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details. 29 * <p> 30 * The interpolating polynomials satisfy: <ol> 31 * <li>The value of the PolynomialSplineFunction at each of the input x values equals the 32 * corresponding y value.</li> 33 * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials 34 * "match up" at the knot points, as do their first and second derivatives).</li> 35 * </ol> 36 * <p> 37 * The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires, 38 * <u>Numerical Analysis</u>, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131. 39 * 40 * @version $Revision: 355770 $ $Date: 2005-12-10 12:48:57 -0700 (Sat, 10 Dec 2005) $ 41 * 42 */ 43 public class SplineInterpolator implements UnivariateRealInterpolator { 44 45 /** 46 * Computes an interpolating function for the data set. 47 * @param x the arguments for the interpolation points 48 * @param y the values for the interpolation points 49 * @return a function which interpolates the data set 50 */ 51 public UnivariateRealFunction interpolate(double x[], double y[]) { 52 if (x.length != y.length) { 53 throw new IllegalArgumentException("Dataset arrays must have same length."); 54 } 55 56 if (x.length < 3) { 57 throw new IllegalArgumentException 58 ("At least 3 datapoints are required to compute a spline interpolant"); 59 } 60 61 // Number of intervals. The number of data points is n + 1. 62 int n = x.length - 1; 63 64 for (int i = 0; i < n; i++) { 65 if (x[i] >= x[i + 1]) { 66 throw new IllegalArgumentException("Dataset x values must be strictly increasing."); 67 } 68 } 69 70 // Differences between knot points 71 double h[] = new double[n]; 72 for (int i = 0; i < n; i++) { 73 h[i] = x[i + 1] - x[i]; 74 } 75 76 double mu[] = new double[n]; 77 double z[] = new double[n + 1]; 78 mu[0] = 0d; 79 z[0] = 0d; 80 double g = 0; 81 for (int i = 1; i < n; i++) { 82 g = 2d * (x[i+1] - x[i - 1]) - h[i - 1] * mu[i -1]; 83 mu[i] = h[i] / g; 84 z[i] = (3d * (y[i + 1] * h[i - 1] - y[i] * (x[i + 1] - x[i - 1])+ y[i - 1] * h[i]) / 85 (h[i - 1] * h[i]) - h[i - 1] * z[i - 1]) / g; 86 } 87 88 // cubic spline coefficients -- b is linear, c quadratic, d is cubic (original y's are constants) 89 double b[] = new double[n]; 90 double c[] = new double[n + 1]; 91 double d[] = new double[n]; 92 93 z[n] = 0d; 94 c[n] = 0d; 95 96 for (int j = n -1; j >=0; j--) { 97 c[j] = z[j] - mu[j] * c[j + 1]; 98 b[j] = (y[j + 1] - y[j]) / h[j] - h[j] * (c[j + 1] + 2d * c[j]) / 3d; 99 d[j] = (c[j + 1] - c[j]) / (3d * h[j]); 100 } 101 102 PolynomialFunction polynomials[] = new PolynomialFunction[n]; 103 double coefficients[] = new double[4]; 104 for (int i = 0; i < n; i++) { 105 coefficients[0] = y[i]; 106 coefficients[1] = b[i]; 107 coefficients[2] = c[i]; 108 coefficients[3] = d[i]; 109 polynomials[i] = new PolynomialFunction(coefficients); 110 } 111 112 return new PolynomialSplineFunction(x, polynomials); 113 } 114 115 }