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.core.MathArrays;
22 import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
23 import org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException;
24 import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;
25 import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
26
27 /**
28 * Computes a clamped cubic spline interpolation for the data set.
29 * <p>
30 * The {@link #interpolate(double[], double[], double, double)} method returns a {@link PolynomialSplineFunction}
31 * consisting of n cubic polynomials, defined over the subintervals determined by the x values,
32 * {@code x[0] < x[i] ... < x[n]}. The x values are referred to as "knot points."</p>
33 * <p>
34 * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest
35 * knot point and strictly less than the largest knot point is computed by finding the subinterval to which
36 * x belongs and computing the value of the corresponding polynomial at <code>x - x[i]</code> where
37 * <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details.
38 * </p>
39 * <p>
40 * The interpolating polynomials satisfy: <ol>
41 * <li>The value of the PolynomialSplineFunction at each of the input x values equals the
42 * corresponding y value.</li>
43 * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials
44 * "match up" at the knot points, as do their first and second derivatives).</li>
45 * <li>The <i>clamped boundary condition</i>, i.e., the PolynomialSplineFunction takes "a specific direction" at both
46 * its start point and its end point by providing the desired first derivative values (slopes) as function parameters to
47 * {@link #interpolate(double[], double[], double, double)}.</li>
48 * </ol>
49 * <p>
50 * The clamped cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires,
51 * <u>Numerical Analysis</u>, 9th Ed., 2010, Cengage Learning, ISBN 0-538-73351-9, pp 153-156.
52 * </p>
53 *
54 */
55 public class ClampedSplineInterpolator extends SplineInterpolator {
56 /**
57 * Computes an interpolating function for the data set.
58 * @param x the arguments for the interpolation points
59 * @param y the values for the interpolation points
60 * @param fpo first derivative at the starting point of the returned spline function (starting slope), satisfying
61 * clamped boundary condition S′(x0) = f′(x0)
62 * @param fpn first derivative at the ending point of the returned spline function (ending slope), satisfying
63 * clamped boundary condition S′(xn) = f′(xn)
64 * @return a function which interpolates the data set
65 * @throws DimensionMismatchException if {@code x} and {@code y}
66 * have different sizes.
67 * @throws NumberIsTooSmallException if the size of {@code x < 3}.
68 * @throws org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException
69 * if {@code x} is not sorted in strict increasing order.
70 */
71 public PolynomialSplineFunction interpolate(final double[] x, final double[] y,
72 final double fpo, final double fpn)
73 throws DimensionMismatchException, NumberIsTooSmallException, NonMonotonicSequenceException {
74 if (x.length != y.length) {
75 throw new DimensionMismatchException(x.length, y.length);
76 }
77
78 if (x.length < 3) {
79 throw new NumberIsTooSmallException(LocalizedFormats.NUMBER_OF_POINTS,
80 x.length, 3, true);
81 }
82
83 // Number of intervals. The number of data points is n + 1.
84 final int n = x.length - 1;
85
86 MathArrays.checkOrder(x);
87
88 // Differences between knot points
89 final double h[] = new double[n];
90 for (int i = 0; i < n; i++) {
91 h[i] = x[i + 1] - x[i];
92 }
93
94 final double mu[] = new double[n];
95 final double z[] = new double[n + 1];
96 final double alpha[] = new double[n + 1];
97 final double l[] = new double[n + 1];
98
99 alpha[0] = 3d * (y[1] - y[0]) / h[0] - 3d * fpo;
100 alpha[n] = 3d * fpn - 3d * (y[n] - y[n - 1]) / h[n - 1];
101
102 mu[0] = 0.5d;
103 l[0] = 2d * h[0];
104 z[0] = alpha[0] / l[0];
105
106 for (int i = 1; i < n; i++) {
107
108 alpha[i] = (3d / h[i]) * (y[i + 1] - y[i]) - (3d / h[i - 1]) * (y[i] - y[i - 1]);
109 l[i] = 2d * (x[i + 1] - x[i - 1]) - h[i - 1] * mu[i - 1];
110 mu[i] = h[i] / l[i];
111 z[i] = (alpha[i] - h[i - 1] * z[i - 1]) / l[i];
112 }
113 // cubic spline coefficients -- b is linear, c quadratic, d is cubic (original y's are constants)
114 final double b[] = new double[n];
115 final double c[] = new double[n + 1];
116 final double d[] = new double[n];
117 l[n] = h[n - 1] * (2d - mu[n - 1]);
118 z[n] = (alpha[n] - h[n - 1] * z[n - 1]) / l[n];
119 c[n] = z[n];
120
121 for (int j = n - 1; j >= 0; j--) {
122 c[j] = z[j] - mu[j] * c[j + 1];
123 b[j] = ((y[j + 1] - y[j]) / h[j]) - h[j] * (c[j + 1] + 2d * c[j]) / 3d;
124 d[j] = (c[j + 1] - c[j]) / (3d * h[j]);
125 }
126
127 final PolynomialFunction polynomials[] = new PolynomialFunction[n];
128 final double coefficients[] = new double[4];
129 for (int i = 0; i < n; i++) {
130 coefficients[0] = y[i];
131 coefficients[1] = b[i];
132 coefficients[2] = c[i];
133 coefficients[3] = d[i];
134 polynomials[i] = new PolynomialFunction(coefficients);
135 }
136 return new PolynomialSplineFunction(x, polynomials);
137 }
138 }