<?php
namespace MathPHP\NumericalAnalysis\NumericalIntegration;

use MathPHP\NumericalAnalysis\Interpolation\LagrangePolynomial;
use MathPHP\Functions\Polynomial;

/**
 * Simpsons Rule
 *
 * In numerical analysis, Simpson's rule is a technique for approximating
 * the definite integral of a function.
 *
 * Simpson's rule belongs to the closed Newton-Cotes formulas, a group of methods
 * for numerical integration which approximate the integral of a function. We
 * can either directly supply a set of inputs and their corresponding outputs for
 * said function, or if we explicitly know the function, we can define it as a
 * callback function and then generate a set of points by evaluating that function
 * at n points between a start and end point. We then use these values to
 * interpolate a Lagrange polynomial. Finally, we integrate the Lagrange
 * polynomial to approximate the integral of our original function.
 *
 * Simpson's rule is produced by integrating the second Lagrange polynomial.
 *
 * https://en.wikipedia.org/wiki/Simpson%27s_rule
 * http://mathworld.wolfram.com/SimpsonsRule.html
 * http://www.efunda.com/math/num_integration/num_int_newton.cfm
 */
class SimpsonsRule extends NumericalIntegration
{
    /**
     * Use Simpson's Rule to aproximate the definite integral of a
     * function f(x). Our input can support either a set of arrays, or a callback
     * function with arguments (to produce a set of arrays). Each array in our
     * input contains two numbers which correspond to coordinates (x, y) or
     * equivalently, (x, f(x)), of the function f(x) whose definite integral we
     * are approximating.
     *
     * Note: Simpson's method requires that we have an even number of
     * subintervals (we must supply an odd number of points) and also that the
     * size of each subinterval is equal (spacing between each point is equal).
     *
     * The bounds of the definite integral to which we are approximating is
     * determined by the our inputs.
     *
     * Example: approximate([0, 10], [5, 5], [10, 7]) will approximate the definite
     * integral of the function that produces these coordinates with a lower
     * bound of 0, and an upper bound of 10.
     *
     * Example: approximate(function($x) {return $x**2;}, 0, 4 ,5) will produce
     * a set of arrays by evaluating the callback at 5 evenly spaced points
     * between 0 and 4. Then, this array will be used in our approximation.
     *
     * Simpson's Rule:
     *
     * xn        ⁿ⁻¹ xᵢ₊₁
     * ∫ f(x)dx = ∑   ∫ f(x)dx
     * x₁        ⁱ⁼¹  xᵢ
     *
     *         ⁽ⁿ⁻¹⁾/² h
     *          = ∑    - [f⟮x₂ᵢ₋₁⟯ + 4f⟮x₂ᵢ⟯ + f⟮x₂ᵢ₊₁⟯] + O(h⁵f⁗(x))
     *           ⁱ⁼¹   3
     * where h = (xn - x₁) / (n - 1)
     *
     * @param callable|array $source   The source of our approximation. Should be either
     *                           a callback function or a set of arrays. Each array
     *                           (point) contains precisely two numbers, an x and y.
     *                           Example array: [[1,2], [2,3], [3,4]].
     *                           Example callback: function($x) {return $x**2;}
     * @param number  ... $args The arguments of our callback function: start,
     *                           end, and n. Example: approximate($source, 0, 8, 5).
     *                           If $source is a set of points, do not input any
     *                           $args. Example: approximate($source).
     *
     * @return number            The approximation to the integral of f(x)
     */
    public static function approximate($source, ... $args)
    {
        // get an array of points from our $source argument
        $points = self::getPoints($source, $args);

        // Validate input and sort points
        self::validate($points, $degree = 3);
        Validation::isSubintervalsMultiple($points, $m = 2);
        $sorted = self::sort($points);
        Validation::isSpacingConstant($sorted);

        // Descriptive constants
        $x = self::X;
        $y = self::Y;

        // Initialize
        $n             = count($sorted);
        $subintervals  = $n - 1;
        $a             = $sorted[0][$x];
        $b             = $sorted[$n-1][$x];
        $h             = ($b - $a) / $subintervals;
        $approximation = 0;

        /*
         * Summation
         * ⁽ⁿ⁻¹⁾/² h
         *    ∑    - [f⟮x₂ᵢ₋₁⟯ + 4f⟮x₂ᵢ⟯ + f⟮x₂ᵢ₊₁⟯] + O(h⁵f⁗(x))
         *   ⁱ⁼¹   3
         *  where h = (xn - x₁) / (n - 1)
         */
        for ($i = 1; $i < ($subintervals/2) + 1; $i++) {
            $x₂ᵢ₋₁          = $sorted[(2*$i)-2][$x];
            $x₂ᵢ            = $sorted[(2*$i)-1][$x];
            $x₂ᵢ₊₁          = $sorted[(2*$i)][$x];
            $f⟮x₂ᵢ₋₁⟯        = $sorted[(2*$i)-2][$y];  // y₂ᵢ₋₁
            $f⟮x₂ᵢ⟯          = $sorted[(2*$i)-1][$y];  // y₂ᵢ
            $f⟮x₂ᵢ₊₁⟯        = $sorted[(2*$i)][$y];    // y₂ᵢ₊₁
            $lagrange       = LagrangePolynomial::interpolate([[$x₂ᵢ₋₁, $f⟮x₂ᵢ₋₁⟯], [$x₂ᵢ, $f⟮x₂ᵢ⟯], [$x₂ᵢ₊₁, $f⟮x₂ᵢ₊₁⟯]]);
            $integral       = $lagrange->integrate();
            $approximation += $integral($x₂ᵢ₊₁) - $integral($x₂ᵢ₋₁); // definite integral of lagrange polynomial
        }

        return $approximation;
    }
}
