public function testapproximatePolynomial()
{
// f(x) = x³ + 2x + 1
// Antiderivative F(x) = (1/4)x⁴ + x² + x
// Indefinite integral over [0, 4] = F(4) - F(0) = 84
$expected = 84;
// h denotes the size of subintervals, or equivalently, the
// distance between two points
// ζ₁, ζ₂, ... denotes the max of the fourth derivative of f(x) on
// interval 1, 2, ...
// f'(x) = 3x² + 2
// f''(x) = 6x
// f'''(x) = 6
// f⁽⁴⁾(x) = 0
// f⁽⁵⁾(x) = 0
// f⁽⁶⁾(x) = 0
// Error = O(h^5 * f⁽⁶⁾(x)) = 0
$tol = 0;
// Approximate with [0, 1], [1, 4], [2, 13], [3, 34], [4, 73]
$x = BoolesRule::approximate([[0, 1], [1, 4], [2, 13], [3, 34], [4, 73]]);
$this->assertEquals($expected, $x, '', $tol);
// Same test as above but with points not sorted to test sorting works
$x = BoolesRule::approximate([[0, 1], [3, 34], [2, 13], [1, 4], [4, 73]]);
$this->assertEquals($expected, $x, '', $tol);
// Similar test to above (same function, number of points, tolerance) but
// with a callback function to make sure this type of input is compatible
$func = function ($x) {
return $x ** 3 + 2 * $x + 1;
};
$start = 0;
$end = 4;
$n = 5;
$x = BoolesRule::approximate($func, $start, $end, $n);
$this->assertEquals($expected, $x, '', $tol);
}
