| public static approximate ( $source, $args ) : number | ||
| $source | The source of our approximation. Should be either a callback function or a set of arrays. Each array (point) contains precisely two numbers, an x and y. Example array: [[1,2], [2,3], [3,4]]. Example callback: function($x) {return $x**2;} | |
| return | number | The approximation to the integral of f(x) |
public static function approximate($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 = 2);
$sorted = self::sort($points);
// Descriptive constants
$x = self::X;
$y = self::Y;
// Initialize
$n = count($sorted);
$steps = $n - 1;
$approximation = 0;
/*
* Summation
* ⁿ⁻¹ h
* ∑ - [f(xᵢ₊₁) + f(xᵢ)]
* ⁱ⁼¹ 2
* where h = xᵢ₊₁ - xᵢ
*/
for ($i = 0; $i < $steps; $i++) {
$xᵢ = $sorted[$i][$x];
$xᵢ₊₁ = $sorted[$i + 1][$x];
$f⟮xᵢ⟯ = $sorted[$i][$y];
// yᵢ
$f⟮xᵢ₊₁⟯ = $sorted[$i + 1][$y];
// yᵢ₊₁
$lagrange = LagrangePolynomial::interpolate([[$xᵢ, $f⟮xᵢ⟯], [$xᵢ₊₁, $f⟮xᵢ₊₁⟯]]);
$integral = $lagrange->integrate();
$approximation += $integral($xᵢ₊₁) - $integral($xᵢ);
// definite integral of lagrange polynomial
}
return $approximation;
}