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1   /*
2    * Copyright 2003-2004 The Apache Software Foundation.
3    *
4    * Licensed under the Apache License, Version 2.0 (the "License");
5    * you may not use this file except in compliance with the License.
6    * You may obtain a copy of the License at
7    *
8    *      http://www.apache.org/licenses/LICENSE-2.0
9    *
10   * Unless required by applicable law or agreed to in writing, software
11   * distributed under the License is distributed on an "AS IS" BASIS,
12   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
13   * See the License for the specific language governing permissions and
14   * limitations under the License.
15   */
16  package org.apache.commons.math.analysis;
17  
18  /**
19   * Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set.
20   * <p>
21   * The {@link #interpolate(double[], double[])} method returns a {@link PolynomialSplineFunction}
22   * consisting of n cubic polynomials, defined over the subintervals determined by the x values,  
23   * x[0] < x[i] ... < x[n].  The x values are referred to as "knot points."
24   * <p>
25   * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest
26   * knot point and strictly less than the largest knot point is computed by finding the subinterval to which
27   * x belongs and computing the value of the corresponding polynomial at <code>x - x[i] </code> where
28   * <code>i</code> is the index of the subinterval.  See {@link PolynomialSplineFunction} for more details.
29   * <p>
30   * The interpolating polynomials satisfy: <ol>
31   * <li>The value of the PolynomialSplineFunction at each of the input x values equals the 
32   *  corresponding y value.</li>
33   * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials 
34   *  "match up" at the knot points, as do their first and second derivatives).</li>
35   * </ol>
36   * <p>
37   * The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires, 
38   * <u>Numerical Analysis</u>, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131.
39   *
40   * @version $Revision: 355770 $ $Date: 2005-12-10 12:48:57 -0700 (Sat, 10 Dec 2005) $
41   *
42   */
43  public class SplineInterpolator implements UnivariateRealInterpolator {
44      
45      /**
46       * Computes an interpolating function for the data set.
47       * @param x the arguments for the interpolation points
48       * @param y the values for the interpolation points
49       * @return a function which interpolates the data set
50       */
51      public UnivariateRealFunction interpolate(double x[], double y[]) {
52          if (x.length != y.length) {
53              throw new IllegalArgumentException("Dataset arrays must have same length.");
54          }
55          
56          if (x.length < 3) {
57              throw new IllegalArgumentException
58                  ("At least 3 datapoints are required to compute a spline interpolant");
59          }
60          
61          // Number of intervals.  The number of data points is n + 1.
62          int n = x.length - 1;   
63          
64          for (int i = 0; i < n; i++) {
65              if (x[i]  >= x[i + 1]) {
66                  throw new IllegalArgumentException("Dataset x values must be strictly increasing.");
67              }
68          }
69          
70          // Differences between knot points
71          double h[] = new double[n];
72          for (int i = 0; i < n; i++) {
73              h[i] = x[i + 1] - x[i];
74          }
75          
76          double mu[] = new double[n];
77          double z[] = new double[n + 1];
78          mu[0] = 0d;
79          z[0] = 0d;
80          double g = 0;
81          for (int i = 1; i < n; i++) {
82              g = 2d * (x[i+1]  - x[i - 1]) - h[i - 1] * mu[i -1];
83              mu[i] = h[i] / g;
84              z[i] = (3d * (y[i + 1] * h[i - 1] - y[i] * (x[i + 1] - x[i - 1])+ y[i - 1] * h[i]) /
85                      (h[i - 1] * h[i]) - h[i - 1] * z[i - 1]) / g;
86          }
87         
88          // cubic spline coefficients --  b is linear, c quadratic, d is cubic (original y's are constants)
89          double b[] = new double[n];
90          double c[] = new double[n + 1];
91          double d[] = new double[n];
92          
93          z[n] = 0d;
94          c[n] = 0d;
95          
96          for (int j = n -1; j >=0; j--) {
97              c[j] = z[j] - mu[j] * c[j + 1];
98              b[j] = (y[j + 1] - y[j]) / h[j] - h[j] * (c[j + 1] + 2d * c[j]) / 3d;
99              d[j] = (c[j + 1] - c[j]) / (3d * h[j]);
100         }
101         
102         PolynomialFunction polynomials[] = new PolynomialFunction[n];
103         double coefficients[] = new double[4];
104         for (int i = 0; i < n; i++) {
105             coefficients[0] = y[i];
106             coefficients[1] = b[i];
107             coefficients[2] = c[i];
108             coefficients[3] = d[i];
109             polynomials[i] = new PolynomialFunction(coefficients);
110         }
111         
112         return new PolynomialSplineFunction(x, polynomials);
113     }
114 
115 }