public function testNonzeroError()
{
// f(x) = x⁵ - 13x² -92x + 96
$f = function ($x) {
return $x ** 5 - 13 * $x ** 2 - 92 * $x + 96;
};
/*
* h⁴
* Error term for the Midpoint Formula: - f⁽⁵⁾(ζ₁)
* 30
*
* where ζ₁ lies between x₀ - 2h and x₀ + 2h
*
* h⁴
* Error term for the Endpoint Formula: - f⁽⁵⁾(ζ₀)
* 5
*
* where ζ₀ lies between x₀ and x₀ + 4h
*/
// f'(x) = 5x⁴ - 26x - 92
// f''(x) = 20x³ - 26
// f⁽³⁾(x) = 60x²
// f⁽⁴⁾(x) = 120x
// f⁽⁵⁾(x) = 120
// Error in Midpoint Formula on [0,4] (where h=1) < 4
// Error in Endpoint Formula on [0,4] (where h=1) < 24
$f’ = function ($x) {
return 5 * $x ** 4 - 26 * $x - 92;
};
$n = 5;
$a = 0;
$b = 4;
// Check that the endpoint formula agrees with f'(x) at x = 0
$target = 0;
$tol = 24;
$expected = $f’($target);
$actual = FivePointFormula::differentiate($target, $f, $a, $b, $n);
$this->assertEquals($expected, $actual, '', $tol);
// Check that the midpoint formula agrees with f'(x) at x = 2
$target = 2;
$tol = 4;
$expected = $f’($target);
$actual = FivePointFormula::differentiate($target, $f, $a, $b, $n);
$this->assertEquals($expected, $actual, '', $tol);
// Check that the (backward) endpoint formula agrees with f'(x) at x = 4
$target = 4;
$tol = 24;
$expected = $f’($target);
$actual = FivePointFormula::differentiate($target, $f, $a, $b, $n);
$this->assertEquals($expected, $actual, '', $tol);
}
