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.exception.DimensionMismatchException;
020 import org.apache.commons.math.exception.util.LocalizedFormats;
021 import org.apache.commons.math.exception.NumberIsTooSmallException;
022 import org.apache.commons.math.analysis.polynomials.PolynomialFunction;
023 import org.apache.commons.math.analysis.polynomials.PolynomialSplineFunction;
024 import org.apache.commons.math.util.MathUtils;
025
026 /**
027 * Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set.
028 * <p>
029 * The {@link #interpolate(double[], double[])} method returns a {@link PolynomialSplineFunction}
030 * consisting of n cubic polynomials, defined over the subintervals determined by the x values,
031 * x[0] < x[i] ... < x[n]. The x values are referred to as "knot points."</p>
032 * <p>
033 * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest
034 * knot point and strictly less than the largest knot point is computed by finding the subinterval to which
035 * x belongs and computing the value of the corresponding polynomial at <code>x - x[i] </code> where
036 * <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details.
037 * </p>
038 * <p>
039 * The interpolating polynomials satisfy: <ol>
040 * <li>The value of the PolynomialSplineFunction at each of the input x values equals the
041 * corresponding y value.</li>
042 * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials
043 * "match up" at the knot points, as do their first and second derivatives).</li>
044 * </ol></p>
045 * <p>
046 * The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires,
047 * <u>Numerical Analysis</u>, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131.
048 * </p>
049 *
050 * @version $Revision: 983921 $ $Date: 2010-08-10 12:46:06 +0200 (mar. 10 ao??t 2010) $
051 *
052 */
053 public class SplineInterpolator implements UnivariateRealInterpolator {
054
055 /**
056 * Computes an interpolating function for the data set.
057 * @param x the arguments for the interpolation points
058 * @param y the values for the interpolation points
059 * @return a function which interpolates the data set
060 * @throws DimensionMismatchException if {@code x} and {@code y}
061 * have different sizes.
062 * @throws org.apache.commons.math.exception.NonMonotonousSequenceException
063 * if {@code x} is not sorted in strict increasing order.
064 * @throws NumberIsTooSmallException if the size of {@code x} is smaller
065 * than 3.
066 */
067 public PolynomialSplineFunction interpolate(double x[], double y[]) {
068 if (x.length != y.length) {
069 throw new DimensionMismatchException(x.length, y.length);
070 }
071
072 if (x.length < 3) {
073 throw new NumberIsTooSmallException(LocalizedFormats.NUMBER_OF_POINTS,
074 x.length, 3, true);
075 }
076
077 // Number of intervals. The number of data points is n + 1.
078 int n = x.length - 1;
079
080 MathUtils.checkOrder(x);
081
082 // Differences between knot points
083 double h[] = new double[n];
084 for (int i = 0; i < n; i++) {
085 h[i] = x[i + 1] - x[i];
086 }
087
088 double mu[] = new double[n];
089 double z[] = new double[n + 1];
090 mu[0] = 0d;
091 z[0] = 0d;
092 double g = 0;
093 for (int i = 1; i < n; i++) {
094 g = 2d * (x[i+1] - x[i - 1]) - h[i - 1] * mu[i -1];
095 mu[i] = h[i] / g;
096 z[i] = (3d * (y[i + 1] * h[i - 1] - y[i] * (x[i + 1] - x[i - 1])+ y[i - 1] * h[i]) /
097 (h[i - 1] * h[i]) - h[i - 1] * z[i - 1]) / g;
098 }
099
100 // cubic spline coefficients -- b is linear, c quadratic, d is cubic (original y's are constants)
101 double b[] = new double[n];
102 double c[] = new double[n + 1];
103 double d[] = new double[n];
104
105 z[n] = 0d;
106 c[n] = 0d;
107
108 for (int j = n -1; j >=0; j--) {
109 c[j] = z[j] - mu[j] * c[j + 1];
110 b[j] = (y[j + 1] - y[j]) / h[j] - h[j] * (c[j + 1] + 2d * c[j]) / 3d;
111 d[j] = (c[j + 1] - c[j]) / (3d * h[j]);
112 }
113
114 PolynomialFunction polynomials[] = new PolynomialFunction[n];
115 double coefficients[] = new double[4];
116 for (int i = 0; i < n; i++) {
117 coefficients[0] = y[i];
118 coefficients[1] = b[i];
119 coefficients[2] = c[i];
120 coefficients[3] = d[i];
121 polynomials[i] = new PolynomialFunction(coefficients);
122 }
123
124 return new PolynomialSplineFunction(x, polynomials);
125 }
126
127 }