Spline Interpolation

A Primer on the Basicsby

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

Data: x i, y i, i 0, 1, , n

Linear SplineData: x i, y i, i 0, 1, , n

The Linear Spline

Data: x i, y i, i 0, 1, , n

Piecewise Polynomials

Linear Splines

Quadratic SplineData: x i, y i, i 0, 1, , n

Cubic SplineTo fit cubics to groups of four points.

Rarely done

Data: x i, y i, i 0, 1, , n

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

Data: x i, y i, i 0, 1, , n

How it looks

Data: x i, y i, i 0, 1, , n

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.

fx 1

1 5x 2

Function and Interpolant

12 equally spaced pointsFunction in RedInterpolant in blue

Function and Cubic Spline

12 equally spaced points;Function in RedSpline in blue

Function, Interpolant, Cubic Spline

12 equally spaced points;Function in GreenSpline in RedInterpolant 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

Another Nasty Example

Another Nasty Example

Another Nasty Example