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 * https://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
18 /*
19 * This is not the original file distributed by the Apache Software Foundation
20 * It has been modified by the Hipparchus project
21 */
22 package org.hipparchus.analysis.interpolation;
23
24 import java.io.Serializable;
25
26 import org.hipparchus.analysis.polynomials.PolynomialFunctionLagrangeForm;
27 import org.hipparchus.analysis.polynomials.PolynomialFunctionNewtonForm;
28 import org.hipparchus.exception.MathIllegalArgumentException;
29
30 /**
31 * Implements the <a href=
32 * "http://mathworld.wolfram.com/NewtonsDividedDifferenceInterpolationFormula.html">
33 * Divided Difference Algorithm</a> for interpolation of real univariate
34 * functions. For reference, see <b>Introduction to Numerical Analysis</b>,
35 * ISBN 038795452X, chapter 2.
36 * <p>
37 * The actual code of Neville's evaluation is in PolynomialFunctionLagrangeForm,
38 * this class provides an easy-to-use interface to it.</p>
39 *
40 */
41 public class DividedDifferenceInterpolator
42 implements UnivariateInterpolator, Serializable {
43 /** serializable version identifier */
44 private static final long serialVersionUID = 107049519551235069L;
45
46 /** Empty constructor.
47 * <p>
48 * This constructor is not strictly necessary, but it prevents spurious
49 * javadoc warnings with JDK 18 and later.
50 * </p>
51 * @since 3.0
52 */
53 public DividedDifferenceInterpolator() { // NOPMD - unnecessary constructor added intentionally to make javadoc happy
54 // nothing to do
55 }
56
57 /**
58 * Compute an interpolating function for the dataset.
59 *
60 * @param x Interpolating points array.
61 * @param y Interpolating values array.
62 * @return a function which interpolates the dataset.
63 * @throws MathIllegalArgumentException if the array lengths are different.
64 * @throws MathIllegalArgumentException if the number of points is less than 2.
65 * @throws MathIllegalArgumentException if {@code x} is not sorted in
66 * strictly increasing order.
67 */
68 @Override
69 public PolynomialFunctionNewtonForm interpolate(double[] x, double[] y)
70 throws MathIllegalArgumentException {
71 /**
72 * a[] and c[] are defined in the general formula of Newton form:
73 * p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... +
74 * a[n](x-c[0])(x-c[1])...(x-c[n-1])
75 */
76 PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y, true);
77
78 /**
79 * When used for interpolation, the Newton form formula becomes
80 * p(x) = f[x0] + f[x0,x1](x-x0) + f[x0,x1,x2](x-x0)(x-x1) + ... +
81 * f[x0,x1,...,x[n-1]](x-x0)(x-x1)...(x-x[n-2])
82 * Therefore, a[k] = f[x0,x1,...,xk], c[k] = x[k].
83 * <p>
84 * Note x[], y[], a[] have the same length but c[]'s size is one less.</p>
85 */
86 final double[] c = new double[x.length-1];
87 System.arraycopy(x, 0, c, 0, c.length);
88
89 final double[] a = computeDividedDifference(x, y);
90 return new PolynomialFunctionNewtonForm(a, c);
91 }
92
93 /**
94 * Return a copy of the divided difference array.
95 * <p>
96 * The divided difference array is defined recursively by <pre>
97 * f[x0] = f(x0)
98 * f[x0,x1,...,xk] = (f[x1,...,xk] - f[x0,...,x[k-1]]) / (xk - x0)
99 * </pre>
100 * <p>
101 * The computational complexity is \(O(n^2)\) where \(n\) is the common
102 * length of {@code x} and {@code y}.</p>
103 *
104 * @param x Interpolating points array.
105 * @param y Interpolating values array.
106 * @return a fresh copy of the divided difference array.
107 * @throws MathIllegalArgumentException if the array lengths are different.
108 * @throws MathIllegalArgumentException if the number of points is less than 2.
109 * @throws MathIllegalArgumentException
110 * if {@code x} is not sorted in strictly increasing order.
111 */
112 protected static double[] computeDividedDifference(final double[] x, final double[] y)
113 throws MathIllegalArgumentException {
114 PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y, true);
115
116 final double[] divdiff = y.clone(); // initialization
117
118 final int n = x.length;
119 final double[] a = new double [n];
120 a[0] = divdiff[0];
121 for (int i = 1; i < n; i++) {
122 for (int j = 0; j < n-i; j++) {
123 final double denominator = x[j+i] - x[j];
124 divdiff[j] = (divdiff[j+1] - divdiff[j]) / denominator;
125 }
126 a[i] = divdiff[0];
127 }
128
129 return a;
130 }
131 }