public function testNonzeroError()
{
// f(x) = x³ - 13x² -92x + 96
$f = function ($x) {
return $x ** 3 - 13 * $x ** 2 - 92 * $x + 96;
};
/*
* h²
* Error term for the Midpoint Formula: - f⁽³⁾(ζ₁)
* 6
*
* where ζ₁ lies between x₀ - h and x₀ + h
*
* h²
* Error term for the Endpoint Formula: - f⁽³⁾(ζ₀)
* 3
*
* where ζ₀ lies between x₀ and x₀ + 2h
*/
// f'(x) = 3x² - 26x - 92
// f''(x) = 6x - 26
// f⁽³⁾(x) = 6
// Error in Midpoint Formula on [0,2] (where h=1) < 1
// Error in Endpoint Formula on [0,2] (where h=1) < 2
$f’ = function ($x) {
return 3 * $x ** 2 - 26 * $x - 92;
};
$n = 3;
$a = 0;
$b = 2;
// Check that the endpoint formula agrees with f'(x) at x = 0
$target = 0;
$tol = 2;
$expected = $f’($target);
$actual = ThreePointFormula::differentiate($target, $f, $a, $b, $n);
$this->assertEquals($expected, $actual, '', $tol);
// Check that the midpoint formula agrees with f'(x) at x = 1
$target = 1;
$tol = 1;
$expected = $f’($target);
$actual = ThreePointFormula::differentiate($target, $f, $a, $b, $n);
$this->assertEquals($expected, $actual, '', $tol);
// Check that the (backward) endpoint formula agrees with f'(x) at x = 2
$target = 2;
$tol = 2;
$expected = $f’($target);
$actual = ThreePointFormula::differentiate($target, $f, $a, $b, $n);
$this->assertEquals($expected, $actual, '', $tol);
}
