public static function differentiate($target, $source, ...$args)
{
// get an array of points from our $source argument
$points = self::getPoints($source, $args);
// Validate input, sort points, make sure spacing is constant, and make
// sure our target is contained in an interval supplied by our $source
self::validate($points, $degree = 5);
$sorted = self::sort($points);
self::isSpacingConstant($sorted);
self::isTargetInPoints($target, $sorted);
// Descriptive constants
$x = self::X;
$y = self::Y;
// Initialize
$n = count($sorted);
$h = ($sorted[4][$x] - $sorted[0][$x]) / 4;
/*
* Five Point Formula:
*
* - If the 3rd point is our $target, use the Midpoint Formula:
*
* 1 h⁴
* f′(x₀) = - [f(x₀-2h)-8f(x₀-h)+8f(x₀+h)-f(x₀+2h)] - - f⁽⁵⁾(ζ₁)
* 12h 30
*
* where ζ₁ lies between x₀ - 2h and x₀ + 2h
*
* - If the 1st or 5th point is our $target, use the Endpoint Formula:
* - Note that when the 3rd point is our $target, we use a negative h.
*
* 1 h⁴
* f′(x₀) = - [-25f(x₀)+48f(x₀+h)-36f(x₀+2h)+16f(x₀+3h)-3f(x₀+4h)] + - f⁽⁵⁾(ζ₀)
* 12h 5
*
* where ζ₀ lies between x₀ and x₀ + 4h
*/
// If the 3nd point is our $target, use the Midpoint Formula
if ($sorted[2][$x] == $target) {
$f⟮x₀⧿2h⟯ = $sorted[0][$y];
$f⟮x₀⧿h⟯ = $sorted[1][$y];
$f⟮x₀⧾h⟯ = $sorted[3][$y];
$f⟮x₀⧾2h⟯ = $sorted[4][$y];
$derivative = ($f⟮x₀⧿2h⟯ - 8 * $f⟮x₀⧿h⟯ + 8 * $f⟮x₀⧾h⟯ - $f⟮x₀⧾2h⟯) / (12 * $h);
// If the 1st or 5th point is our $target, use the Endpoint Formula
} else {
// The 1st point is our $target
if ($sorted[0][$x] == $target) {
$f⟮x₀⟯ = $sorted[0][$y];
$f⟮x₀⧾h⟯ = $sorted[1][$y];
$f⟮x₀⧾2h⟯ = $sorted[2][$y];
$f⟮x₀⧾3h⟯ = $sorted[3][$y];
$f⟮x₀⧾4h⟯ = $sorted[4][$y];
// If the 5th point is our $target, use negative h
} else {
$h = -$h;
$f⟮x₀⟯ = $sorted[4][$y];
$f⟮x₀⧾h⟯ = $sorted[3][$y];
$f⟮x₀⧾2h⟯ = $sorted[2][$y];
$f⟮x₀⧾3h⟯ = $sorted[1][$y];
$f⟮x₀⧾4h⟯ = $sorted[0][$y];
}
$derivative = (-25 * $f⟮x₀⟯ + 48 * $f⟮x₀⧾h⟯ - 36 * $f⟮x₀⧾2h⟯ + 16 * $f⟮x₀⧾3h⟯ - 3 * $f⟮x₀⧾4h⟯) / (12 * $h);
}
return $derivative;
}
