001 /*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements. See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License. You may obtain a copy of the License at
008 *
009 * http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017 package org.apache.commons.math.analysis.interpolation;
018
019 import org.apache.commons.math.MathRuntimeException;
020 import org.apache.commons.math.analysis.polynomials.PolynomialFunction;
021 import org.apache.commons.math.analysis.polynomials.PolynomialSplineFunction;
022
023 /**
024 * Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set.
025 * <p>
026 * The {@link #interpolate(double[], double[])} method returns a {@link PolynomialSplineFunction}
027 * consisting of n cubic polynomials, defined over the subintervals determined by the x values,
028 * x[0] < x[i] ... < x[n]. The x values are referred to as "knot points."</p>
029 * <p>
030 * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest
031 * knot point and strictly less than the largest knot point is computed by finding the subinterval to which
032 * x belongs and computing the value of the corresponding polynomial at <code>x - x[i] </code> where
033 * <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details.
034 * </p>
035 * <p>
036 * The interpolating polynomials satisfy: <ol>
037 * <li>The value of the PolynomialSplineFunction at each of the input x values equals the
038 * corresponding y value.</li>
039 * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials
040 * "match up" at the knot points, as do their first and second derivatives).</li>
041 * </ol></p>
042 * <p>
043 * The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires,
044 * <u>Numerical Analysis</u>, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131.
045 * </p>
046 *
047 * @version $Revision: 811685 $ $Date: 2009-09-05 13:36:48 -0400 (Sat, 05 Sep 2009) $
048 *
049 */
050 public class SplineInterpolator implements UnivariateRealInterpolator {
051
052 /**
053 * Computes an interpolating function for the data set.
054 * @param x the arguments for the interpolation points
055 * @param y the values for the interpolation points
056 * @return a function which interpolates the data set
057 */
058 public PolynomialSplineFunction interpolate(double x[], double y[]) {
059 if (x.length != y.length) {
060 throw MathRuntimeException.createIllegalArgumentException(
061 "dimension mismatch {0} != {1}", x.length, y.length);
062 }
063
064 if (x.length < 3) {
065 throw MathRuntimeException.createIllegalArgumentException(
066 "{0} points are required, got only {1}", 3, x.length);
067 }
068
069 // Number of intervals. The number of data points is n + 1.
070 int n = x.length - 1;
071
072 for (int i = 0; i < n; i++) {
073 if (x[i] >= x[i + 1]) {
074 throw MathRuntimeException.createIllegalArgumentException(
075 "points {0} and {1} are not strictly increasing ({2} >= {3})",
076 i, i+1, x[i], x[i+1]);
077 }
078 }
079
080 // Differences between knot points
081 double h[] = new double[n];
082 for (int i = 0; i < n; i++) {
083 h[i] = x[i + 1] - x[i];
084 }
085
086 double mu[] = new double[n];
087 double z[] = new double[n + 1];
088 mu[0] = 0d;
089 z[0] = 0d;
090 double g = 0;
091 for (int i = 1; i < n; i++) {
092 g = 2d * (x[i+1] - x[i - 1]) - h[i - 1] * mu[i -1];
093 mu[i] = h[i] / g;
094 z[i] = (3d * (y[i + 1] * h[i - 1] - y[i] * (x[i + 1] - x[i - 1])+ y[i - 1] * h[i]) /
095 (h[i - 1] * h[i]) - h[i - 1] * z[i - 1]) / g;
096 }
097
098 // cubic spline coefficients -- b is linear, c quadratic, d is cubic (original y's are constants)
099 double b[] = new double[n];
100 double c[] = new double[n + 1];
101 double d[] = new double[n];
102
103 z[n] = 0d;
104 c[n] = 0d;
105
106 for (int j = n -1; j >=0; j--) {
107 c[j] = z[j] - mu[j] * c[j + 1];
108 b[j] = (y[j + 1] - y[j]) / h[j] - h[j] * (c[j + 1] + 2d * c[j]) / 3d;
109 d[j] = (c[j + 1] - c[j]) / (3d * h[j]);
110 }
111
112 PolynomialFunction polynomials[] = new PolynomialFunction[n];
113 double coefficients[] = new double[4];
114 for (int i = 0; i < n; i++) {
115 coefficients[0] = y[i];
116 coefficients[1] = b[i];
117 coefficients[2] = c[i];
118 coefficients[3] = d[i];
119 polynomials[i] = new PolynomialFunction(coefficients);
120 }
121
122 return new PolynomialSplineFunction(x, polynomials);
123 }
124
125 }