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 */ 017package org.apache.commons.math4.legacy.analysis.interpolation; 018 019import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialFunctionLagrangeForm; 020import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialFunctionNewtonForm; 021import org.apache.commons.math4.legacy.exception.DimensionMismatchException; 022import org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException; 023import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException; 024 025/** 026 * Implements the <a href= 027 * "http://mathworld.wolfram.com/NewtonsDividedDifferenceInterpolationFormula.html"> 028 * Divided Difference Algorithm</a> for interpolation of real univariate 029 * functions. For reference, see <b>Introduction to Numerical Analysis</b>, 030 * ISBN 038795452X, chapter 2. 031 * <p> 032 * The actual code of Neville's evaluation is in PolynomialFunctionLagrangeForm, 033 * this class provides an easy-to-use interface to it.</p> 034 * 035 * @since 1.2 036 */ 037public class DividedDifferenceInterpolator 038 implements UnivariateInterpolator { 039 /** 040 * Compute an interpolating function for the dataset. 041 * 042 * @param x Interpolating points array. 043 * @param y Interpolating values array. 044 * @return a function which interpolates the dataset. 045 * @throws DimensionMismatchException if the array lengths are different. 046 * @throws NumberIsTooSmallException if the number of points is less than 2. 047 * @throws NonMonotonicSequenceException if {@code x} is not sorted in 048 * strictly increasing order. 049 */ 050 @Override 051 public PolynomialFunctionNewtonForm interpolate(double[] x, double[] y) 052 throws DimensionMismatchException, 053 NumberIsTooSmallException, 054 NonMonotonicSequenceException { 055 /* 056 * a[] and c[] are defined in the general formula of Newton form: 057 * p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... + 058 * a[n](x-c[0])(x-c[1])...(x-c[n-1]) 059 */ 060 PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y, true); 061 062 /* 063 * When used for interpolation, the Newton form formula becomes 064 * p(x) = f[x0] + f[x0,x1](x-x0) + f[x0,x1,x2](x-x0)(x-x1) + ... + 065 * f[x0,x1,...,x[n-1]](x-x0)(x-x1)...(x-x[n-2]) 066 * Therefore, a[k] = f[x0,x1,...,xk], c[k] = x[k]. 067 * <p> 068 * Note x[], y[], a[] have the same length but c[]'s size is one less.</p> 069 */ 070 final double[] c = new double[x.length-1]; 071 System.arraycopy(x, 0, c, 0, c.length); 072 073 final double[] a = computeDividedDifference(x, y); 074 return new PolynomialFunctionNewtonForm(a, c); 075 } 076 077 /** 078 * Return a copy of the divided difference array. 079 * <p> 080 * The divided difference array is defined recursively by <pre> 081 * f[x0] = f(x0) 082 * f[x0,x1,...,xk] = (f[x1,...,xk] - f[x0,...,x[k-1]]) / (xk - x0) 083 * </pre> 084 * <p> 085 * The computational complexity is \(O(n^2)\) where \(n\) is the common 086 * length of {@code x} and {@code y}.</p> 087 * 088 * @param x Interpolating points array. 089 * @param y Interpolating values array. 090 * @return a fresh copy of the divided difference array. 091 * @throws DimensionMismatchException if the array lengths are different. 092 * @throws NumberIsTooSmallException if the number of points is less than 2. 093 * @throws NonMonotonicSequenceException 094 * if {@code x} is not sorted in strictly increasing order. 095 */ 096 protected static double[] computeDividedDifference(final double[] x, final double[] y) 097 throws DimensionMismatchException, 098 NumberIsTooSmallException, 099 NonMonotonicSequenceException { 100 PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y, true); 101 102 final double[] divdiff = y.clone(); // initialization 103 104 final int n = x.length; 105 final double[] a = new double [n]; 106 a[0] = divdiff[0]; 107 for (int i = 1; i < n; i++) { 108 for (int j = 0; j < n-i; j++) { 109 final double denominator = x[j+i] - x[j]; 110 divdiff[j] = (divdiff[j+1] - divdiff[j]) / denominator; 111 } 112 a[i] = divdiff[0]; 113 } 114 115 return a; 116 } 117}