public function testSolvePolynomial()
{
// f(x) = x² + 2x + 1
// Antiderivative F(x) = (1/3)x³ + x² + x
// Indefinite integral over [0, 3] = F(3) - F(0) = 21
$expected = 21;
// h₁, h₂, ... denotes the size on interval 1, 2, ...
// ζ₁, ζ₂, ... denotes the max of the second derivative of f(x) on
// interval 1, 2, ...
// f'(x) = 2x + 2
// f''(x) = 2
// ζ = f''(x) = 2
// Error = Sum(ζ₁h₁³ + ζ₂h₂³ + ...) = 2 * Sum(h₁³ + h₂³ + ...)
// Approximate with endpoints: (0, 1) and (3, 16)
// Error = 2 * ((3 - 0)²) = 18
$tol = 18;
$x = TrapezoidalRule::approximate([[0, 1], [3, 16]]);
$this->assertEquals($expected, $x, '', $tol);
// Approximate with endpoints and one interior point: (0, 1), (1, 4),
// and (3, 16)
// Error = 2 * ((1 - 0)² + (3 - 1)²) = 10
$tol = 10;
$x = TrapezoidalRule::approximate([[0, 1], [1, 4], [3, 16]]);
$this->assertEquals($expected, $x, '', $tol);
// Approximate with endpoints and two interior points: (0, 1), (1, 4),
// (2, 9), and (3, 16)
// Error = 2 * ((1 - 0)² + (2 - 1)² + (3 - 2)²) = 6
$tol = 6;
$x = TrapezoidalRule::approximate([[0, 1], [1, 4], [2, 9], [3, 16]]);
$this->assertEquals($expected, $x, '', $tol);
// Same test as above but with points not sorted to test sorting works
$tol = 6;
$x = TrapezoidalRule::approximate([[1, 4], [3, 16], [0, 1], [2, 9]]);
$this->assertEquals($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 ** 2 + 2 * $x + 1;
};
$start = 0;
$end = 3;
$n = 4;
$tol = 6;
$x = TrapezoidalRule::approximate($func, $start, $end, $n);
$this->assertEquals($expected, $x, '', $tol);
}
