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.math3.analysis.interpolation;
018
019 import org.apache.commons.math3.exception.DimensionMismatchException;
020 import org.apache.commons.math3.exception.util.LocalizedFormats;
021 import org.apache.commons.math3.exception.NumberIsTooSmallException;
022 import org.apache.commons.math3.exception.NonMonotonicSequenceException;
023 import org.apache.commons.math3.analysis.polynomials.PolynomialFunction;
024 import org.apache.commons.math3.analysis.polynomials.PolynomialSplineFunction;
025 import org.apache.commons.math3.util.MathArrays;
026
027 /**
028 * Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set.
029 * <p>
030 * The {@link #interpolate(double[], double[])} method returns a {@link PolynomialSplineFunction}
031 * consisting of n cubic polynomials, defined over the subintervals determined by the x values,
032 * x[0] < x[i] ... < x[n]. The x values are referred to as "knot points."</p>
033 * <p>
034 * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest
035 * knot point and strictly less than the largest knot point is computed by finding the subinterval to which
036 * x belongs and computing the value of the corresponding polynomial at <code>x - x[i] </code> where
037 * <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details.
038 * </p>
039 * <p>
040 * The interpolating polynomials satisfy: <ol>
041 * <li>The value of the PolynomialSplineFunction at each of the input x values equals the
042 * corresponding y value.</li>
043 * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials
044 * "match up" at the knot points, as do their first and second derivatives).</li>
045 * </ol></p>
046 * <p>
047 * The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires,
048 * <u>Numerical Analysis</u>, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131.
049 * </p>
050 *
051 * @version $Id: SplineInterpolator.java 1379905 2012-09-01 23:56:50Z erans $
052 */
053 public class SplineInterpolator implements UnivariateInterpolator {
054 /**
055 * Computes an interpolating function for the data set.
056 * @param x the arguments for the interpolation points
057 * @param y the values for the interpolation points
058 * @return a function which interpolates the data set
059 * @throws DimensionMismatchException if {@code x} and {@code y}
060 * have different sizes.
061 * @throws NonMonotonicSequenceException if {@code x} is not sorted in
062 * strict increasing order.
063 * @throws NumberIsTooSmallException if the size of {@code x} is smaller
064 * than 3.
065 */
066 public PolynomialSplineFunction interpolate(double x[], double y[])
067 throws DimensionMismatchException,
068 NumberIsTooSmallException,
069 NonMonotonicSequenceException {
070 if (x.length != y.length) {
071 throw new DimensionMismatchException(x.length, y.length);
072 }
073
074 if (x.length < 3) {
075 throw new NumberIsTooSmallException(LocalizedFormats.NUMBER_OF_POINTS,
076 x.length, 3, true);
077 }
078
079 // Number of intervals. The number of data points is n + 1.
080 final int n = x.length - 1;
081
082 MathArrays.checkOrder(x);
083
084 // Differences between knot points
085 final double h[] = new double[n];
086 for (int i = 0; i < n; i++) {
087 h[i] = x[i + 1] - x[i];
088 }
089
090 final double mu[] = new double[n];
091 final double z[] = new double[n + 1];
092 mu[0] = 0d;
093 z[0] = 0d;
094 double g = 0;
095 for (int i = 1; i < n; i++) {
096 g = 2d * (x[i+1] - x[i - 1]) - h[i - 1] * mu[i -1];
097 mu[i] = h[i] / g;
098 z[i] = (3d * (y[i + 1] * h[i - 1] - y[i] * (x[i + 1] - x[i - 1])+ y[i - 1] * h[i]) /
099 (h[i - 1] * h[i]) - h[i - 1] * z[i - 1]) / g;
100 }
101
102 // cubic spline coefficients -- b is linear, c quadratic, d is cubic (original y's are constants)
103 final double b[] = new double[n];
104 final double c[] = new double[n + 1];
105 final double d[] = new double[n];
106
107 z[n] = 0d;
108 c[n] = 0d;
109
110 for (int j = n -1; j >=0; j--) {
111 c[j] = z[j] - mu[j] * c[j + 1];
112 b[j] = (y[j + 1] - y[j]) / h[j] - h[j] * (c[j + 1] + 2d * c[j]) / 3d;
113 d[j] = (c[j + 1] - c[j]) / (3d * h[j]);
114 }
115
116 final PolynomialFunction polynomials[] = new PolynomialFunction[n];
117 final double coefficients[] = new double[4];
118 for (int i = 0; i < n; i++) {
119 coefficients[0] = y[i];
120 coefficients[1] = b[i];
121 coefficients[2] = c[i];
122 coefficients[3] = d[i];
123 polynomials[i] = new PolynomialFunction(coefficients);
124 }
125
126 return new PolynomialSplineFunction(x, polynomials);
127 }
128 }