/*
 * 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.interpolation;

import java.io.Serializable;
import org.apache.commons.math3.analysis.polynomials.PolynomialFunctionLagrangeForm;
import org.apache.commons.math3.analysis.polynomials.PolynomialFunctionNewtonForm;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.NonMonotonicSequenceException;

Implements the 
Divided Difference Algorithm for interpolation of real univariate
functions. For reference, see Introduction to Numerical Analysis,
ISBN 038795452X, chapter 2.

The actual code of Neville's evaluation is in PolynomialFunctionLagrangeForm,
this class provides an easy-to-use interface to it.
Since:
1.2/**
 * Implements the <a href=
 * "http://mathworld.wolfram.com/NewtonsDividedDifferenceInterpolationFormula.html">
 * Divided Difference Algorithm</a> for interpolation of real univariate
 * functions. For reference, see <b>Introduction to Numerical Analysis</b>,
 * ISBN 038795452X, chapter 2.
 * <p>
 * The actual code of Neville's evaluation is in PolynomialFunctionLagrangeForm,
 * this class provides an easy-to-use interface to it.</p>
 *
 * @since 1.2
 */
public class DividedDifferenceInterpolator
    implements UnivariateInterpolator, Serializable {
    
serializable version identifier /** serializable version identifier */
    private static final long serialVersionUID = 107049519551235069L;

    
Compute an interpolating function for the dataset.
Params:
x – Interpolating points array.
y – Interpolating values array.
Throws:
DimensionMismatchException – if the array lengths are different.
NumberIsTooSmallException – if the number of points is less than 2.
NonMonotonicSequenceException – if x is not sorted in strictly increasing order.
Returns:
a function which interpolates the dataset./**
     * Compute an interpolating function for the dataset.
     *
     * @param x Interpolating points array.
     * @param y Interpolating values array.
     * @return a function which interpolates the dataset.
     * @throws DimensionMismatchException if the array lengths are different.
     * @throws NumberIsTooSmallException if the number of points is less than 2.
     * @throws NonMonotonicSequenceException if {@code x} is not sorted in
     * strictly increasing order.
     */
    public PolynomialFunctionNewtonForm interpolate(double x[], double y[])
        throws DimensionMismatchException,
               NumberIsTooSmallException,
               NonMonotonicSequenceException {
        /**
         * a[] and c[] are defined in the general formula of Newton form:
         * p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... +
         *        a[n](x-c[0])(x-c[1])...(x-c[n-1])
         */
        PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y, true);

        /**
         * When used for interpolation, the Newton form formula becomes
         * p(x) = f[x0] + f[x0,x1](x-x0) + f[x0,x1,x2](x-x0)(x-x1) + ... +
         *        f[x0,x1,...,x[n-1]](x-x0)(x-x1)...(x-x[n-2])
         * Therefore, a[k] = f[x0,x1,...,xk], c[k] = x[k].
         * <p>
         * Note x[], y[], a[] have the same length but c[]'s size is one less.</p>
         */
        final double[] c = new double[x.length-1];
        System.arraycopy(x, 0, c, 0, c.length);

        final double[] a = computeDividedDifference(x, y);
        return new PolynomialFunctionNewtonForm(a, c);
    }

    
Return a copy of the divided difference array.

The divided difference array is defined recursively by 
f[x0] = f(x0)
f[x0,x1,...,xk] = (f[x1,...,xk] - f[x0,...,x[k-1]]) / (xk - x0)

 The computational complexity is \(O(n^2)\) where \(n\) is the common length of x and y.
Params:
x – Interpolating points array.
y – Interpolating values array.
Throws:
DimensionMismatchException – if the array lengths are different.
NumberIsTooSmallException – if the number of points is less than 2.
NonMonotonicSequenceException –  if x is not sorted in strictly increasing order.
Returns:
a fresh copy of the divided difference array./**
     * Return a copy of the divided difference array.
     * <p>
     * The divided difference array is defined recursively by <pre>
     * f[x0] = f(x0)
     * f[x0,x1,...,xk] = (f[x1,...,xk] - f[x0,...,x[k-1]]) / (xk - x0)
     * </pre>
     * <p>
     * The computational complexity is \(O(n^2)\) where \(n\) is the common
     * length of {@code x} and {@code y}.</p>
     *
     * @param x Interpolating points array.
     * @param y Interpolating values array.
     * @return a fresh copy of the divided difference array.
     * @throws DimensionMismatchException if the array lengths are different.
     * @throws NumberIsTooSmallException if the number of points is less than 2.
     * @throws NonMonotonicSequenceException
     * if {@code x} is not sorted in strictly increasing order.
     */
    protected static double[] computeDividedDifference(final double x[], final double y[])
        throws DimensionMismatchException,
               NumberIsTooSmallException,
               NonMonotonicSequenceException {
        PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y, true);

        final double[] divdiff = y.clone(); // initialization

        final int n = x.length;
        final double[] a = new double [n];
        a[0] = divdiff[0];
        for (int i = 1; i < n; i++) {
            for (int j = 0; j < n-i; j++) {
                final double denominator = x[j+i] - x[j];
                divdiff[j] = (divdiff[j+1] - divdiff[j]) / denominator;
            }
            a[i] = divdiff[0];
        }

        return a;
    }
}