Spline Interpolation A Primer on the Basics by Don Allen

What are Splines? Splines interpolate data. Lower degree curves are used. Thus fewer points must be used. Consequently, we obtain a piecewise curve, valid only over a specific interval. Splines allow matching conditions – not unlike Hermite interpolation – but different

Linear Spline

The Linear Spline

Piecewise Polynomials

Linear Splines

Quadratic Spline

Cubic Spline To fit cubics to groups of four points. Rarely done

Cubic Spline To fit cubics to successive pairs of points, with matching conditions.

How it looks

Splines vs. Polynomials Remember the function → This function interpolated poorly at equally spaced points. Let’s compare with how well the natural cubic spline performs.

Function and Interpolant 12 equally spaced points Function in Red Interpolant in blue

Function and Cubic Spline 12 equally spaced points; Function in Red Spline in blue

Function, Interpolant, Cubic Spline 12 equally spaced points; Function in Green Spline in Red Interpolant in Blue

Cubic Spline – the equations

Cubic Spines Two types: both involve endpoint conditions – Natural S’’(endpoints) = 0 – Clamped S’(endpoints) = f’(endpoints) The error in interpolating a function with a clamped cubic spline depends on the fourth derivative.

Cubic Spline – derivation

Cubic Splines There are natural and clamped types They are very accurate in practice. They avoid most of the pitfalls of general polynomial interpolation. It is remarkably easy to program for the coefficients. The error depends on the second derivative. Full details are in the text and lecture notes.

Another Nasty Example