1 /* 2 * Licensed to the Apache Software Foundation (ASF) under one or more 3 * contributor license agreements. See the NOTICE file distributed with 4 * this work for additional information regarding copyright ownership. 5 * The ASF licenses this file to You under the Apache License, Version 2.0 6 * (the "License"); you may not use this file except in compliance with 7 * the License. You may obtain a copy of the License at 8 * 9 * http://www.apache.org/licenses/LICENSE-2.0 10 * 11 * Unless required by applicable law or agreed to in writing, software 12 * distributed under the License is distributed on an "AS IS" BASIS, 13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 14 * See the License for the specific language governing permissions and 15 * limitations under the License. 16 */ 17 package org.apache.commons.math.analysis.interpolation; 18 19 import org.apache.commons.math.exception.DimensionMismatchException; 20 import org.apache.commons.math.exception.util.LocalizedFormats; 21 import org.apache.commons.math.exception.NumberIsTooSmallException; 22 import org.apache.commons.math.analysis.polynomials.PolynomialFunction; 23 import org.apache.commons.math.analysis.polynomials.PolynomialSplineFunction; 24 import org.apache.commons.math.util.MathUtils; 25 26 /** 27 * Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set. 28 * <p> 29 * The {@link #interpolate(double[], double[])} method returns a {@link PolynomialSplineFunction} 30 * consisting of n cubic polynomials, defined over the subintervals determined by the x values, 31 * x[0] < x[i] ... < x[n]. The x values are referred to as "knot points."</p> 32 * <p> 33 * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest 34 * knot point and strictly less than the largest knot point is computed by finding the subinterval to which 35 * x belongs and computing the value of the corresponding polynomial at <code>x - x[i] </code> where 36 * <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details. 37 * </p> 38 * <p> 39 * The interpolating polynomials satisfy: <ol> 40 * <li>The value of the PolynomialSplineFunction at each of the input x values equals the 41 * corresponding y value.</li> 42 * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials 43 * "match up" at the knot points, as do their first and second derivatives).</li> 44 * </ol></p> 45 * <p> 46 * The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires, 47 * <u>Numerical Analysis</u>, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131. 48 * </p> 49 * 50 * @version $Revision: 983921 $ $Date: 2010-08-10 12:46:06 +0200 (mar. 10 aot 2010) $ 51 * 52 */ 53 public class SplineInterpolator implements UnivariateRealInterpolator { 54 55 /** 56 * Computes an interpolating function for the data set. 57 * @param x the arguments for the interpolation points 58 * @param y the values for the interpolation points 59 * @return a function which interpolates the data set 60 * @throws DimensionMismatchException if {@code x} and {@code y} 61 * have different sizes. 62 * @throws org.apache.commons.math.exception.NonMonotonousSequenceException 63 * if {@code x} is not sorted in strict increasing order. 64 * @throws NumberIsTooSmallException if the size of {@code x} is smaller 65 * than 3. 66 */ 67 public PolynomialSplineFunction interpolate(double x[], double y[]) { 68 if (x.length != y.length) { 69 throw new DimensionMismatchException(x.length, y.length); 70 } 71 72 if (x.length < 3) { 73 throw new NumberIsTooSmallException(LocalizedFormats.NUMBER_OF_POINTS, 74 x.length, 3, true); 75 } 76 77 // Number of intervals. The number of data points is n + 1. 78 int n = x.length - 1; 79 80 MathUtils.checkOrder(x); 81 82 // Differences between knot points 83 double h[] = new double[n]; 84 for (int i = 0; i < n; i++) { 85 h[i] = x[i + 1] - x[i]; 86 } 87 88 double mu[] = new double[n]; 89 double z[] = new double[n + 1]; 90 mu[0] = 0d; 91 z[0] = 0d; 92 double g = 0; 93 for (int i = 1; i < n; i++) { 94 g = 2d * (x[i+1] - x[i - 1]) - h[i - 1] * mu[i -1]; 95 mu[i] = h[i] / g; 96 z[i] = (3d * (y[i + 1] * h[i - 1] - y[i] * (x[i + 1] - x[i - 1])+ y[i - 1] * h[i]) / 97 (h[i - 1] * h[i]) - h[i - 1] * z[i - 1]) / g; 98 } 99 100 // cubic spline coefficients -- b is linear, c quadratic, d is cubic (original y's are constants) 101 double b[] = new double[n]; 102 double c[] = new double[n + 1]; 103 double d[] = new double[n]; 104 105 z[n] = 0d; 106 c[n] = 0d; 107 108 for (int j = n -1; j >=0; j--) { 109 c[j] = z[j] - mu[j] * c[j + 1]; 110 b[j] = (y[j + 1] - y[j]) / h[j] - h[j] * (c[j + 1] + 2d * c[j]) / 3d; 111 d[j] = (c[j + 1] - c[j]) / (3d * h[j]); 112 } 113 114 PolynomialFunction polynomials[] = new PolynomialFunction[n]; 115 double coefficients[] = new double[4]; 116 for (int i = 0; i < n; i++) { 117 coefficients[0] = y[i]; 118 coefficients[1] = b[i]; 119 coefficients[2] = c[i]; 120 coefficients[3] = d[i]; 121 polynomials[i] = new PolynomialFunction(coefficients); 122 } 123 124 return new PolynomialSplineFunction(x, polynomials); 125 } 126 127 } 128
