001 /* 002 * Licensed to the Apache Software Foundation (ASF) under one or more 003 * contributor license agreements. See the NOTICE file distributed with 004 * this work for additional information regarding copyright ownership. 005 * The ASF licenses this file to You under the Apache License, Version 2.0 006 * (the "License"); you may not use this file except in compliance with 007 * the License. You may obtain a copy of the License at 008 * 009 * http://www.apache.org/licenses/LICENSE-2.0 010 * 011 * Unless required by applicable law or agreed to in writing, software 012 * distributed under the License is distributed on an "AS IS" BASIS, 013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 014 * See the License for the specific language governing permissions and 015 * limitations under the License. 016 */ 017 package org.apache.commons.math.analysis.interpolation; 018 019 import org.apache.commons.math.exception.DimensionMismatchException; 020 import org.apache.commons.math.exception.util.LocalizedFormats; 021 import org.apache.commons.math.exception.NumberIsTooSmallException; 022 import org.apache.commons.math.analysis.polynomials.PolynomialFunction; 023 import org.apache.commons.math.analysis.polynomials.PolynomialSplineFunction; 024 import org.apache.commons.math.util.MathUtils; 025 026 /** 027 * Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set. 028 * <p> 029 * The {@link #interpolate(double[], double[])} method returns a {@link PolynomialSplineFunction} 030 * consisting of n cubic polynomials, defined over the subintervals determined by the x values, 031 * x[0] < x[i] ... < x[n]. The x values are referred to as "knot points."</p> 032 * <p> 033 * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest 034 * knot point and strictly less than the largest knot point is computed by finding the subinterval to which 035 * x belongs and computing the value of the corresponding polynomial at <code>x - x[i] </code> where 036 * <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details. 037 * </p> 038 * <p> 039 * The interpolating polynomials satisfy: <ol> 040 * <li>The value of the PolynomialSplineFunction at each of the input x values equals the 041 * corresponding y value.</li> 042 * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials 043 * "match up" at the knot points, as do their first and second derivatives).</li> 044 * </ol></p> 045 * <p> 046 * The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires, 047 * <u>Numerical Analysis</u>, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131. 048 * </p> 049 * 050 * @version $Revision: 983921 $ $Date: 2010-08-10 12:46:06 +0200 (mar. 10 ao??t 2010) $ 051 * 052 */ 053 public class SplineInterpolator implements UnivariateRealInterpolator { 054 055 /** 056 * Computes an interpolating function for the data set. 057 * @param x the arguments for the interpolation points 058 * @param y the values for the interpolation points 059 * @return a function which interpolates the data set 060 * @throws DimensionMismatchException if {@code x} and {@code y} 061 * have different sizes. 062 * @throws org.apache.commons.math.exception.NonMonotonousSequenceException 063 * if {@code x} is not sorted in strict increasing order. 064 * @throws NumberIsTooSmallException if the size of {@code x} is smaller 065 * than 3. 066 */ 067 public PolynomialSplineFunction interpolate(double x[], double y[]) { 068 if (x.length != y.length) { 069 throw new DimensionMismatchException(x.length, y.length); 070 } 071 072 if (x.length < 3) { 073 throw new NumberIsTooSmallException(LocalizedFormats.NUMBER_OF_POINTS, 074 x.length, 3, true); 075 } 076 077 // Number of intervals. The number of data points is n + 1. 078 int n = x.length - 1; 079 080 MathUtils.checkOrder(x); 081 082 // Differences between knot points 083 double h[] = new double[n]; 084 for (int i = 0; i < n; i++) { 085 h[i] = x[i + 1] - x[i]; 086 } 087 088 double mu[] = new double[n]; 089 double z[] = new double[n + 1]; 090 mu[0] = 0d; 091 z[0] = 0d; 092 double g = 0; 093 for (int i = 1; i < n; i++) { 094 g = 2d * (x[i+1] - x[i - 1]) - h[i - 1] * mu[i -1]; 095 mu[i] = h[i] / g; 096 z[i] = (3d * (y[i + 1] * h[i - 1] - y[i] * (x[i + 1] - x[i - 1])+ y[i - 1] * h[i]) / 097 (h[i - 1] * h[i]) - h[i - 1] * z[i - 1]) / g; 098 } 099 100 // 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 }