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    }