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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 }