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.interpolation;
18
19 import org.apache.commons.math.MathRuntimeException;
20 import org.apache.commons.math.analysis.polynomials.PolynomialFunction;
21 import org.apache.commons.math.analysis.polynomials.PolynomialSplineFunction;
22
23 /**
24 * Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set.
25 * <p>
26 * The {@link #interpolate(double[], double[])} method returns a {@link PolynomialSplineFunction}
27 * consisting of n cubic polynomials, defined over the subintervals determined by the x values,
28 * x[0] < x[i] ... < x[n]. The x values are referred to as "knot points."</p>
29 * <p>
30 * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest
31 * knot point and strictly less than the largest knot point is computed by finding the subinterval to which
32 * x belongs and computing the value of the corresponding polynomial at <code>x - x[i] </code> where
33 * <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details.
34 * </p>
35 * <p>
36 * The interpolating polynomials satisfy: <ol>
37 * <li>The value of the PolynomialSplineFunction at each of the input x values equals the
38 * corresponding y value.</li>
39 * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials
40 * "match up" at the knot points, as do their first and second derivatives).</li>
41 * </ol></p>
42 * <p>
43 * The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires,
44 * <u>Numerical Analysis</u>, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131.
45 * </p>
46 *
47 * @version $Revision: 799857 $ $Date: 2009-08-01 09:07:12 -0400 (Sat, 01 Aug 2009) $
48 *
49 */
50 public class SplineInterpolator implements UnivariateRealInterpolator {
51
52 /**
53 * Computes an interpolating function for the data set.
54 * @param x the arguments for the interpolation points
55 * @param y the values for the interpolation points
56 * @return a function which interpolates the data set
57 */
58 public PolynomialSplineFunction interpolate(double x[], double y[]) {
59 if (x.length != y.length) {
60 throw MathRuntimeException.createIllegalArgumentException(
61 "dimension mismatch {0} != {1}", x.length, y.length);
62 }
63
64 if (x.length < 3) {
65 throw MathRuntimeException.createIllegalArgumentException(
66 "{0} points are required, got only {1}", 3, x.length);
67 }
68
69 // Number of intervals. The number of data points is n + 1.
70 int n = x.length - 1;
71
72 for (int i = 0; i < n; i++) {
73 if (x[i] >= x[i + 1]) {
74 throw MathRuntimeException.createIllegalArgumentException(
75 "points {0} and {1} are not strictly increasing ({2} >= {3})",
76 i, i+1, x[i], x[i+1]);
77 }
78 }
79
80 // Differences between knot points
81 double h[] = new double[n];
82 for (int i = 0; i < n; i++) {
83 h[i] = x[i + 1] - x[i];
84 }
85
86 double mu[] = new double[n];
87 double z[] = new double[n + 1];
88 mu[0] = 0d;
89 z[0] = 0d;
90 double g = 0;
91 for (int i = 1; i < n; i++) {
92 g = 2d * (x[i+1] - x[i - 1]) - h[i - 1] * mu[i -1];
93 mu[i] = h[i] / g;
94 z[i] = (3d * (y[i + 1] * h[i - 1] - y[i] * (x[i + 1] - x[i - 1])+ y[i - 1] * h[i]) /
95 (h[i - 1] * h[i]) - h[i - 1] * z[i - 1]) / g;
96 }
97
98 // cubic spline coefficients -- b is linear, c quadratic, d is cubic (original y's are constants)
99 double b[] = new double[n];
100 double c[] = new double[n + 1];
101 double d[] = new double[n];
102
103 z[n] = 0d;
104 c[n] = 0d;
105
106 for (int j = n -1; j >=0; j--) {
107 c[j] = z[j] - mu[j] * c[j + 1];
108 b[j] = (y[j + 1] - y[j]) / h[j] - h[j] * (c[j + 1] + 2d * c[j]) / 3d;
109 d[j] = (c[j + 1] - c[j]) / (3d * h[j]);
110 }
111
112 PolynomialFunction polynomials[] = new PolynomialFunction[n];
113 double coefficients[] = new double[4];
114 for (int i = 0; i < n; i++) {
115 coefficients[0] = y[i];
116 coefficients[1] = b[i];
117 coefficients[2] = c[i];
118 coefficients[3] = d[i];
119 polynomials[i] = new PolynomialFunction(coefficients);
120 }
121
122 return new PolynomialSplineFunction(x, polynomials);
123 }
124
125 }