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 */
017package org.apache.commons.math3.analysis.interpolation;
018
019import org.apache.commons.math3.exception.DimensionMismatchException;
020import org.apache.commons.math3.exception.util.LocalizedFormats;
021import org.apache.commons.math3.exception.NumberIsTooSmallException;
022import org.apache.commons.math3.exception.NonMonotonicSequenceException;
023import org.apache.commons.math3.analysis.polynomials.PolynomialFunction;
024import org.apache.commons.math3.analysis.polynomials.PolynomialSplineFunction;
025import 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 */
052public class SplineInterpolator implements UnivariateInterpolator {
053    /**
054     * Computes an interpolating function for the data set.
055     * @param x the arguments for the interpolation points
056     * @param y the values for the interpolation points
057     * @return a function which interpolates the data set
058     * @throws DimensionMismatchException if {@code x} and {@code y}
059     * have different sizes.
060     * @throws NonMonotonicSequenceException if {@code x} is not sorted in
061     * strict increasing order.
062     * @throws NumberIsTooSmallException if the size of {@code x} is smaller
063     * than 3.
064     */
065    public PolynomialSplineFunction interpolate(double x[], double y[])
066        throws DimensionMismatchException,
067               NumberIsTooSmallException,
068               NonMonotonicSequenceException {
069        if (x.length != y.length) {
070            throw new DimensionMismatchException(x.length, y.length);
071        }
072
073        if (x.length < 3) {
074            throw new NumberIsTooSmallException(LocalizedFormats.NUMBER_OF_POINTS,
075                                                x.length, 3, true);
076        }
077
078        // Number of intervals.  The number of data points is n + 1.
079        final int n = x.length - 1;
080
081        MathArrays.checkOrder(x);
082
083        // Differences between knot points
084        final double h[] = new double[n];
085        for (int i = 0; i < n; i++) {
086            h[i] = x[i + 1] - x[i];
087        }
088
089        final double mu[] = new double[n];
090        final double z[] = new double[n + 1];
091        mu[0] = 0d;
092        z[0] = 0d;
093        double g = 0;
094        for (int i = 1; i < n; i++) {
095            g = 2d * (x[i+1]  - x[i - 1]) - h[i - 1] * mu[i -1];
096            mu[i] = h[i] / g;
097            z[i] = (3d * (y[i + 1] * h[i - 1] - y[i] * (x[i + 1] - x[i - 1])+ y[i - 1] * h[i]) /
098                    (h[i - 1] * h[i]) - h[i - 1] * z[i - 1]) / g;
099        }
100
101        // cubic spline coefficients --  b is linear, c quadratic, d is cubic (original y's are constants)
102        final double b[] = new double[n];
103        final double c[] = new double[n + 1];
104        final double d[] = new double[n];
105
106        z[n] = 0d;
107        c[n] = 0d;
108
109        for (int j = n -1; j >=0; j--) {
110            c[j] = z[j] - mu[j] * c[j + 1];
111            b[j] = (y[j + 1] - y[j]) / h[j] - h[j] * (c[j + 1] + 2d * c[j]) / 3d;
112            d[j] = (c[j + 1] - c[j]) / (3d * h[j]);
113        }
114
115        final PolynomialFunction polynomials[] = new PolynomialFunction[n];
116        final double coefficients[] = new double[4];
117        for (int i = 0; i < n; i++) {
118            coefficients[0] = y[i];
119            coefficients[1] = b[i];
120            coefficients[2] = c[i];
121            coefficients[3] = d[i];
122            polynomials[i] = new PolynomialFunction(coefficients);
123        }
124
125        return new PolynomialSplineFunction(x, polynomials);
126    }
127}