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;
18
19 /**
20 * Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set.
21 * <p>
22 * The {@link #interpolate(double[], double[])} method returns a {@link PolynomialSplineFunction}
23 * consisting of n cubic polynomials, defined over the subintervals determined by the x values,
24 * x[0] < x[i] ... < x[n]. The x values are referred to as "knot points."</p>
25 * <p>
26 * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest
27 * knot point and strictly less than the largest knot point is computed by finding the subinterval to which
28 * x belongs and computing the value of the corresponding polynomial at <code>x - x[i] </code> where
29 * <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details.
30 * </p>
31 * <p>
32 * The interpolating polynomials satisfy: <ol>
33 * <li>The value of the PolynomialSplineFunction at each of the input x values equals the
34 * corresponding y value.</li>
35 * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials
36 * "match up" at the knot points, as do their first and second derivatives).</li>
37 * </ol></p>
38 * <p>
39 * The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires,
40 * <u>Numerical Analysis</u>, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131.
41 * </p>
42 *
43 * @version $Revision: 615734 $ $Date: 2008-01-27 23:10:03 -0700 (Sun, 27 Jan 2008) $
44 *
45 */
46 public class SplineInterpolator implements UnivariateRealInterpolator {
47
48 /**
49 * Computes an interpolating function for the data set.
50 * @param x the arguments for the interpolation points
51 * @param y the values for the interpolation points
52 * @return a function which interpolates the data set
53 */
54 public UnivariateRealFunction interpolate(double x[], double y[]) {
55 if (x.length != y.length) {
56 throw new IllegalArgumentException("Dataset arrays must have same length.");
57 }
58
59 if (x.length < 3) {
60 throw new IllegalArgumentException
61 ("At least 3 datapoints are required to compute a spline interpolant");
62 }
63
64 // Number of intervals. The number of data points is n + 1.
65 int n = x.length - 1;
66
67 for (int i = 0; i < n; i++) {
68 if (x[i] >= x[i + 1]) {
69 throw new IllegalArgumentException("Dataset x values must be strictly increasing.");
70 }
71 }
72
73 // Differences between knot points
74 double h[] = new double[n];
75 for (int i = 0; i < n; i++) {
76 h[i] = x[i + 1] - x[i];
77 }
78
79 double mu[] = new double[n];
80 double z[] = new double[n + 1];
81 mu[0] = 0d;
82 z[0] = 0d;
83 double g = 0;
84 for (int i = 1; i < n; i++) {
85 g = 2d * (x[i+1] - x[i - 1]) - h[i - 1] * mu[i -1];
86 mu[i] = h[i] / g;
87 z[i] = (3d * (y[i + 1] * h[i - 1] - y[i] * (x[i + 1] - x[i - 1])+ y[i - 1] * h[i]) /
88 (h[i - 1] * h[i]) - h[i - 1] * z[i - 1]) / g;
89 }
90
91 // cubic spline coefficients -- b is linear, c quadratic, d is cubic (original y's are constants)
92 double b[] = new double[n];
93 double c[] = new double[n + 1];
94 double d[] = new double[n];
95
96 z[n] = 0d;
97 c[n] = 0d;
98
99 for (int j = n -1; j >=0; j--) {
100 c[j] = z[j] - mu[j] * c[j + 1];
101 b[j] = (y[j + 1] - y[j]) / h[j] - h[j] * (c[j + 1] + 2d * c[j]) / 3d;
102 d[j] = (c[j + 1] - c[j]) / (3d * h[j]);
103 }
104
105 PolynomialFunction polynomials[] = new PolynomialFunction[n];
106 double coefficients[] = new double[4];
107 for (int i = 0; i < n; i++) {
108 coefficients[0] = y[i];
109 coefficients[1] = b[i];
110 coefficients[2] = c[i];
111 coefficients[3] = d[i];
112 polynomials[i] = new PolynomialFunction(coefficients);
113 }
114
115 return new PolynomialSplineFunction(x, polynomials);
116 }
117
118 }