In numerical analysis, cubic splines are used for polynomial
interpolation.
A cubic spline is a spline constructed of piecewise third-order polynomials
which pass through a set of m control points." In the case of the natural
cubic spline, the second derivative of each polynomial is set to zero at the
endpoints of each interval of the piecewise function.
Cubic spline interpolation belongs to a collection of techniques that
interpolate a function or a set of values, producing a continuous polynomial.
In the case of the cubic spline, a piecewise function (polynomial) is produced.
We can either directly supply a set of inputs and their corresponding outputs
for said function, or if we explicitly know the function, we can define it as
a callback function and then generate a set of points by evaluating that
function at n points between a start and end point. We then use these values
to interpolate our piecewise polynomial.
https://en.wikipedia.org/wiki/Spline_interpolation
http://mathworld.wolfram.com/CubicSpline.html
public function testSolveNonzeroError()
{
// f(x) = x⁴ + 8x³ -13x² -92x + 96
$f = new Polynomial([1, 8, -13, -92, 96]);
// The error is bounded by:
// |f(x)-p(x)| = tol <= (1/4!) * h⁴ * max f⁽⁴⁾(x)
// where h = max hᵢ
// f'(x) = 4x³ +24x² -26x - 92
// f''(x) = 12x² - 48x - 26
// f'''(x) = 24x - 48
// f⁽⁴⁾(x) = 24
$a = 0;
$b = 10;
$n = 51;
// So, tol <= (1/24) * (1/5)⁴ * 24 = (1/5)⁴
$tol = 0.2 ** 4;
$roundoff = 1.0E-6;
// round off error
$p = NaturalCubicSpline::interpolate($f, $a, $b, $n);
// Check that p(x) agrees with f(x) at x = 0
$target = 0;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
// Check that p(x) agrees with f(x) at x = 2
$target = 2;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
// Check that p(x) agrees with f(x) at x = 4
$target = 4;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
// Check that p(x) agrees with f(x) at x = 6
$target = 6;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
// Check that p(x) agrees with f(x) at x = 8
$target = 8;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
// Check that p(x) agrees with f(x) at x = 10
$target = 10;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
// Check that p(x) agrees with f(x) at x = 7.32 (not a node)
$target = 7.32;
$expected = $f($target);
$x = $p($target);
$this->assertEquals($expected, $x, '', $tol + $roundoff);
}