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.math4.analysis.interpolation;
018
019import org.apache.commons.math4.analysis.polynomials.PolynomialFunction;
020import org.apache.commons.math4.analysis.polynomials.PolynomialSplineFunction;
021import org.apache.commons.math4.exception.DimensionMismatchException;
022import org.apache.commons.math4.exception.NonMonotonicSequenceException;
023import org.apache.commons.math4.exception.NumberIsTooSmallException;
024import org.apache.commons.math4.exception.util.LocalizedFormats;
025import org.apache.commons.math4.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 * {@code 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>
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    @Override
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        double g = 0;
093        int indexM1 = 0;
094        int index = 1;
095        int indexP1 = 2;
096        while (index < n) {
097            final double xIp1 = x[indexP1];
098            final double xIm1 = x[indexM1];
099            final double hIm1 = h[indexM1];
100            final double hI = h[index];
101            g = 2d * (xIp1 - xIm1) - hIm1 * mu[indexM1];
102            mu[index] = hI / g;
103            z[index] = (3d * (y[indexP1] * hIm1 - y[index] * (xIp1 - xIm1)+ y[indexM1] * hI) /
104                        (hIm1 * hI) - hIm1 * z[indexM1]) / g;
105
106            indexM1 = index;
107            index = indexP1;
108            indexP1 = indexP1 + 1;
109        }
110
111        // cubic spline coefficients --  b is linear, c quadratic, d is cubic (original y's are constants)
112        final double[] b = new double[n];
113        final double[] c = new double[n + 1];
114        final double[] d = new double[n];
115
116        for (int j = n - 1; j >= 0; j--) {
117            final double cJp1 = c[j + 1];
118            final double cJ = z[j] - mu[j] * cJp1;
119            final double hJ = h[j];
120            b[j] = (y[j + 1] - y[j]) / hJ - hJ * (cJp1 + 2d * cJ) / 3d;
121            c[j] = cJ;
122            d[j] = (cJp1 - cJ) / (3d * hJ);
123        }
124
125        final PolynomialFunction[] polynomials = new PolynomialFunction[n];
126        final double[] coefficients = new double[4];
127        for (int i = 0; i < n; i++) {
128            coefficients[0] = y[i];
129            coefficients[1] = b[i];
130            coefficients[2] = c[i];
131            coefficients[3] = d[i];
132            polynomials[i] = new PolynomialFunction(coefficients);
133        }
134
135        return new PolynomialSplineFunction(x, polynomials);
136    }
137}