| 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, $degree = 3);
Validation::isSubintervalsMultiple($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;
}