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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
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/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */
package org.apache.commons.math3.analysis.polynomials;

import java.util.Arrays;

import org.apache.commons.math3.analysis.DifferentiableUnivariateFunction;
import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;
import org.apache.commons.math3.analysis.differentiation.UnivariateDifferentiableFunction;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.NonMonotonicSequenceException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.MathArrays;

/**
 * Represents a polynomial spline function.
 * 
 * A polynomial spline function consists of a set of
 * interpolating polynomials and an ascending array of domain
 * knot points, determining the intervals over which the spline function
 * is defined by the constituent polynomials.  The polynomials are assumed to
 * have been computed to match the values of another function at the knot
 * points.  The value consistency constraints are not currently enforced by
 * PolynomialSplineFunction itself, but are assumed to hold among
 * the polynomials and knot points passed to the constructor.
 * 
 * N.B.:  The polynomials in the polynomials property must be
 * centered on the knot points to compute the spline function values.
 * See below.
 * 
 * The domain of the polynomial spline function is
 * [smallest knot, largest knot].  Attempts to evaluate the
 * function at values outside of this range generate IllegalArgumentExceptions.
 * 
 * 
 * The value of the polynomial spline function for an argument x
 * is computed as follows:
 * 

 * The knot array is searched to find the segment to which x
 * belongs.  If x is less than the smallest knot point or greater
 * than the largest one, an IllegalArgumentException
 * is thrown.
 *  Let j be the index of the largest knot point that is less
 * than or equal to x.  The value returned is
 * {@code polynomials[j](x - knot[j])}
 *
 */
public class PolynomialSplineFunction implements UnivariateDifferentiableFunction, DifferentiableUnivariateFunction {
    /**
     * Spline segment interval delimiters (knots).
     * Size is n + 1 for n segments.
     */
    private final double knots[];
    /**
     * The polynomial functions that make up the spline.  The first element
     * determines the value of the spline over the first subinterval, the
     * second over the second, etc.   Spline function values are determined by
     * evaluating these functions at {@code (x - knot[i])} where i is the
     * knot segment to which x belongs.
     */
    private final PolynomialFunction polynomials[];
    /**
     * Number of spline segments. It is equal to the number of polynomials and
     * to the number of partition points - 1.
     */
    private final int n;


    /**
     * Construct a polynomial spline function with the given segment delimiters
     * and interpolating polynomials.
     * The constructor copies both arrays and assigns the copies to the knots
     * and polynomials properties, respectively.
     *
     * @param knots Spline segment interval delimiters.
     * @param polynomials Polynomial functions that make up the spline.
     * @throws NullArgumentException if either of the input arrays is {@code null}.
     * @throws NumberIsTooSmallException if knots has length less than 2.
     * @throws DimensionMismatchException if {@code polynomials.length != knots.length - 1}.
     * @throws NonMonotonicSequenceException if the {@code knots} array is not strictly increasing.
     *
     */
    public PolynomialSplineFunction(double knots[], PolynomialFunction polynomials[])
        throws NullArgumentException, NumberIsTooSmallException,
               DimensionMismatchException, NonMonotonicSequenceException{
        if (knots == null ||
            polynomials == null) {
            throw new NullArgumentException();
        }
        if (knots.length < 2) {
            throw new NumberIsTooSmallException(LocalizedFormats.NOT_ENOUGH_POINTS_IN_SPLINE_PARTITION,
                                                2, knots.length, false);
        }
        if (knots.length - 1 != polynomials.length) {
            throw new DimensionMismatchException(polynomials.length, knots.length);
        }
        MathArrays.checkOrder(knots);

        this.n = knots.length -1;
        this.knots = new double[n + 1];
        System.arraycopy(knots, 0, this.knots, 0, n + 1);
        this.polynomials = new PolynomialFunction[n];
        System.arraycopy(polynomials, 0, this.polynomials, 0, n);
    }

    /**
     * Compute the value for the function.
     * See {@link PolynomialSplineFunction} for details on the algorithm for
     * computing the value of the function.
     *
     * @param v Point for which the function value should be computed.
     * @return the value.
     * @throws OutOfRangeException if {@code v} is outside of the domain of the
     * spline function (smaller than the smallest knot point or larger than the
     * largest knot point).
     */
    public double value(double v) {
        if (v < knots[0] || v > knots[n]) {
            throw new OutOfRangeException(v, knots[0], knots[n]);
        }
        int i = Arrays.binarySearch(knots, v);
        if (i < 0) {
            i = -i - 2;
        }
        // This will handle the case where v is the last knot value
        // There are only n-1 polynomials, so if v is the last knot
        // then we will use the last polynomial to calculate the value.
        if ( i >= polynomials.length ) {
            i--;
        }
        return polynomials[i].value(v - knots[i]);
    }

    /**
     * Get the derivative of the polynomial spline function.
     *
     * @return the derivative function.
     */
    public UnivariateFunction derivative() {
        return polynomialSplineDerivative();
    }

    /**
     * Get the derivative of the polynomial spline function.
     *
     * @return the derivative function.
     */
    public PolynomialSplineFunction polynomialSplineDerivative() {
        PolynomialFunction derivativePolynomials[] = new PolynomialFunction[n];
        for (int i = 0; i < n; i++) {
            derivativePolynomials[i] = polynomials[i].polynomialDerivative();
        }
        return new PolynomialSplineFunction(knots, derivativePolynomials);
    }


    /** {@inheritDoc}
     * @since 3.1
     */
    public DerivativeStructure value(final DerivativeStructure t) {
        final double t0 = t.getValue();
        if (t0 < knots[0] || t0 > knots[n]) {
            throw new OutOfRangeException(t0, knots[0], knots[n]);
        }
        int i = Arrays.binarySearch(knots, t0);
        if (i < 0) {
            i = -i - 2;
        }
        // This will handle the case where t is the last knot value
        // There are only n-1 polynomials, so if t is the last knot
        // then we will use the last polynomial to calculate the value.
        if ( i >= polynomials.length ) {
            i--;
        }
        return polynomials[i].value(t.subtract(knots[i]));
    }

    /**
     * Get the number of spline segments.
     * It is also the number of polynomials and the number of knot points - 1.
     *
     * @return the number of spline segments.
     */
    public int getN() {
        return n;
    }

    /**
     * Get a copy of the interpolating polynomials array.
     * It returns a fresh copy of the array. Changes made to the copy will
     * not affect the polynomials property.
     *
     * @return the interpolating polynomials.
     */
    public PolynomialFunction[] getPolynomials() {
        PolynomialFunction p[] = new PolynomialFunction[n];
        System.arraycopy(polynomials, 0, p, 0, n);
        return p;
    }

    /**
     * Get an array copy of the knot points.
     * It returns a fresh copy of the array. Changes made to the copy
     * will not affect the knots property.
     *
     * @return the knot points.
     */
    public double[] getKnots() {
        double out[] = new double[n + 1];
        System.arraycopy(knots, 0, out, 0, n + 1);
        return out;
    }

    /**
     * Indicates whether a point is within the interpolation range.
     *
     * @param x Point.
     * @return {@code true} if {@code x} is a valid point.
     */
    public boolean isValidPoint(double x) {
        if (x < knots[0] ||
            x > knots[n]) {
            return false;
        } else {
            return true;
        }
    }
}