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.math3.analysis.interpolation; 018 019 import org.apache.commons.math3.exception.DimensionMismatchException; 020 import org.apache.commons.math3.exception.util.LocalizedFormats; 021 import org.apache.commons.math3.exception.NumberIsTooSmallException; 022 import org.apache.commons.math3.exception.NonMonotonicSequenceException; 023 import org.apache.commons.math3.analysis.polynomials.PolynomialFunction; 024 import org.apache.commons.math3.analysis.polynomials.PolynomialSplineFunction; 025 import org.apache.commons.math3.util.MathArrays; 026 027 /** 028 * Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set. 029 * <p> 030 * The {@link #interpolate(double[], double[])} method returns a {@link PolynomialSplineFunction} 031 * consisting of n cubic polynomials, defined over the subintervals determined by the x values, 032 * x[0] < x[i] ... < x[n]. The x values are referred to as "knot points."</p> 033 * <p> 034 * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest 035 * knot point and strictly less than the largest knot point is computed by finding the subinterval to which 036 * x belongs and computing the value of the corresponding polynomial at <code>x - x[i] </code> where 037 * <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details. 038 * </p> 039 * <p> 040 * The interpolating polynomials satisfy: <ol> 041 * <li>The value of the PolynomialSplineFunction at each of the input x values equals the 042 * corresponding y value.</li> 043 * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials 044 * "match up" at the knot points, as do their first and second derivatives).</li> 045 * </ol></p> 046 * <p> 047 * The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires, 048 * <u>Numerical Analysis</u>, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131. 049 * </p> 050 * 051 * @version $Id: SplineInterpolator.java 1379905 2012-09-01 23:56:50Z erans $ 052 */ 053 public class SplineInterpolator implements UnivariateInterpolator { 054 /** 055 * Computes an interpolating function for the data set. 056 * @param x the arguments for the interpolation points 057 * @param y the values for the interpolation points 058 * @return a function which interpolates the data set 059 * @throws DimensionMismatchException if {@code x} and {@code y} 060 * have different sizes. 061 * @throws NonMonotonicSequenceException if {@code x} is not sorted in 062 * strict increasing order. 063 * @throws NumberIsTooSmallException if the size of {@code x} is smaller 064 * than 3. 065 */ 066 public PolynomialSplineFunction interpolate(double x[], double y[]) 067 throws DimensionMismatchException, 068 NumberIsTooSmallException, 069 NonMonotonicSequenceException { 070 if (x.length != y.length) { 071 throw new DimensionMismatchException(x.length, y.length); 072 } 073 074 if (x.length < 3) { 075 throw new NumberIsTooSmallException(LocalizedFormats.NUMBER_OF_POINTS, 076 x.length, 3, true); 077 } 078 079 // Number of intervals. The number of data points is n + 1. 080 final int n = x.length - 1; 081 082 MathArrays.checkOrder(x); 083 084 // Differences between knot points 085 final double h[] = new double[n]; 086 for (int i = 0; i < n; i++) { 087 h[i] = x[i + 1] - x[i]; 088 } 089 090 final double mu[] = new double[n]; 091 final double z[] = new double[n + 1]; 092 mu[0] = 0d; 093 z[0] = 0d; 094 double g = 0; 095 for (int i = 1; i < n; i++) { 096 g = 2d * (x[i+1] - x[i - 1]) - h[i - 1] * mu[i -1]; 097 mu[i] = h[i] / g; 098 z[i] = (3d * (y[i + 1] * h[i - 1] - y[i] * (x[i + 1] - x[i - 1])+ y[i - 1] * h[i]) / 099 (h[i - 1] * h[i]) - h[i - 1] * z[i - 1]) / g; 100 } 101 102 // cubic spline coefficients -- b is linear, c quadratic, d is cubic (original y's are constants) 103 final double b[] = new double[n]; 104 final double c[] = new double[n + 1]; 105 final double d[] = new double[n]; 106 107 z[n] = 0d; 108 c[n] = 0d; 109 110 for (int j = n -1; j >=0; j--) { 111 c[j] = z[j] - mu[j] * c[j + 1]; 112 b[j] = (y[j + 1] - y[j]) / h[j] - h[j] * (c[j + 1] + 2d * c[j]) / 3d; 113 d[j] = (c[j + 1] - c[j]) / (3d * h[j]); 114 } 115 116 final PolynomialFunction polynomials[] = new PolynomialFunction[n]; 117 final double coefficients[] = new double[4]; 118 for (int i = 0; i < n; i++) { 119 coefficients[0] = y[i]; 120 coefficients[1] = b[i]; 121 coefficients[2] = c[i]; 122 coefficients[3] = d[i]; 123 polynomials[i] = new PolynomialFunction(coefficients); 124 } 125 126 return new PolynomialSplineFunction(x, polynomials); 127 } 128 }