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.math4.legacy.analysis.interpolation;
18
19 import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialFunction;
20 import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialSplineFunction;
21 import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
22 import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;
23 import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
24 import org.apache.commons.math4.legacy.core.MathArrays;
25
26 /**
27 * Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set.
28 * <p>
29 * The {@link #interpolate(double[], double[])} method returns a {@link PolynomialSplineFunction}
30 * consisting of n cubic polynomials, defined over the subintervals determined by the x values,
31 * {@code x[0] < x[i] ... < x[n].} The x values are referred to as "knot points."</p>
32 * <p>
33 * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest
34 * knot point and strictly less than the largest knot point is computed by finding the subinterval to which
35 * x belongs and computing the value of the corresponding polynomial at <code>x - x[i] </code> where
36 * <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details.
37 * </p>
38 * <p>
39 * The interpolating polynomials satisfy: <ol>
40 * <li>The value of the PolynomialSplineFunction at each of the input x values equals the
41 * corresponding y value.</li>
42 * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials
43 * "match up" at the knot points, as do their first and second derivatives).</li>
44 * </ol>
45 * <p>
46 * The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires,
47 * <u>Numerical Analysis</u>, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131.
48 * </p>
49 *
50 */
51 public class SplineInterpolator implements UnivariateInterpolator {
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 * @throws DimensionMismatchException if {@code x} and {@code y}
58 * have different sizes.
59 * @throws NumberIsTooSmallException if the size of {@code x < 3}.
60 * @throws org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException
61 * if {@code x} is not sorted in strict increasing order.
62 */
63 @Override
64 public PolynomialSplineFunction interpolate(double[] x, double[] y) {
65 if (x.length != y.length) {
66 throw new DimensionMismatchException(x.length, y.length);
67 }
68
69 if (x.length < 3) {
70 throw new NumberIsTooSmallException(LocalizedFormats.NUMBER_OF_POINTS,
71 x.length, 3, true);
72 }
73
74 // Number of intervals. The number of data points is n + 1.
75 final int n = x.length - 1;
76
77 MathArrays.checkOrder(x);
78
79 // Differences between knot points
80 final double[] h = new double[n];
81 for (int i = 0; i < n; i++) {
82 h[i] = x[i + 1] - x[i];
83 }
84
85 final double[] mu = new double[n];
86 final double[] z = new double[n + 1];
87 double g = 0;
88 int indexM1 = 0;
89 int index = 1;
90 int indexP1 = 2;
91 while (index < n) {
92 final double xIp1 = x[indexP1];
93 final double xIm1 = x[indexM1];
94 final double hIm1 = h[indexM1];
95 final double hI = h[index];
96 g = 2d * (xIp1 - xIm1) - hIm1 * mu[indexM1];
97 mu[index] = hI / g;
98 z[index] = (3d * (y[indexP1] * hIm1 - y[index] * (xIp1 - xIm1)+ y[indexM1] * hI) /
99 (hIm1 * hI) - hIm1 * z[indexM1]) / g;
100
101 indexM1 = index;
102 index = indexP1;
103 indexP1 = indexP1 + 1;
104 }
105
106 // cubic spline coefficients -- b is linear, c quadratic, d is cubic (original y's are constants)
107 final double[] b = new double[n];
108 final double[] c = new double[n + 1];
109 final double[] d = new double[n];
110
111 for (int j = n - 1; j >= 0; j--) {
112 final double cJp1 = c[j + 1];
113 final double cJ = z[j] - mu[j] * cJp1;
114 final double hJ = h[j];
115 b[j] = (y[j + 1] - y[j]) / hJ - hJ * (cJp1 + 2d * cJ) / 3d;
116 c[j] = cJ;
117 d[j] = (cJp1 - cJ) / (3d * hJ);
118 }
119
120 final PolynomialFunction[] polynomials = new PolynomialFunction[n];
121 final double[] coefficients = new double[4];
122 for (int i = 0; i < n; i++) {
123 coefficients[0] = y[i];
124 coefficients[1] = b[i];
125 coefficients[2] = c[i];
126 coefficients[3] = d[i];
127 polynomials[i] = new PolynomialFunction(coefficients);
128 }
129
130 return new PolynomialSplineFunction(x, polynomials);
131 }
132 }