public function testSolveNonzeroError()
{
// f(x) = x⁴ + 8x³ -13x² -92x + 96
$f = function ($x) {
return $x ** 4 + 8 * $x ** 3 - 13 * $x ** 2 - 92 * $x + 96;
};
// The error is bounded by:
// |f(x)-p(x)| = tol <= (max f⁽ⁿ⁺¹⁾(x))*(x-x₀)*...*(x-xn)/(n+1)!
// f'(x) = 4x³ +24x² -26x - 92
// f''(x) = 12x² - 48x - 26
// f'''(x) = 24x - 48
// f⁽⁴⁾(x) = 24
$a = 0;
$b = 9;
$n = 4;
// So, tol <= 24*(x-x₀)*...*(x-xn)/(4!) = (x-x₀)*...*(x-xn) where
$x₀ = 0;
$x₁ = 3;
$x₂ = 6;
$x₃ = 9;
$p = NewtonPolynomialForward::interpolate($f, $a, $b, $n);
// Check that p(x) agrees with f(x) at x = 0
$target = 0;
$tol = ($target - $x₀) * ($target - $x₁) * ($target - $x₂) * ($target - $x₃);
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol);
// Check that p(x) agrees with f(x) at x = 2
$target = 2;
$tol = abs(($target - $x₀) * ($target - $x₁) * ($target - $x₂) * ($target - $x₃));
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol);
// Check that p(x) agrees with f(x) at x = 4
$target = 4;
$tol = abs(($target - $x₀) * ($target - $x₁) * ($target - $x₂) * ($target - $x₃));
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol);
// Check that p(x) agrees with f(x) at x = 6
$target = 6;
$tol = abs(($target - $x₀) * ($target - $x₁) * ($target - $x₂) * ($target - $x₃));
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol);
// Check that p(x) agrees with f(x) at x = 8
$target = 8;
$tol = abs(($target - $x₀) * ($target - $x₁) * ($target - $x₂) * ($target - $x₃));
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol);
// Check that p(x) agrees with f(x) at x = 10
$target = 10;
$tol = abs(($target - $x₀) * ($target - $x₁) * ($target - $x₂) * ($target - $x₃));
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol);
// Check that p(x) agrees with f(x) at x = -99
$target = -99;
$tol = abs(($target - $x₀) * ($target - $x₁) * ($target - $x₂) * ($target - $x₃));
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol);
}
