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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
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
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
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 }