Locally Determined Splines

Instructor Note: Here you can read about a trick used to avoid propagation of variations, which is done by 'cutting' each segment from the others.

The shape-preserving local spline features artificially computed 1^st-order derivatives at all nodes (d_j, j = 1, N +1 ). Hence, it does not propagate variations as a global spline does. Indeed, the boundary conditions on the coordinates and the 1^st-order derivatives (4 per each interval) determine each third-order polynomial segment completely, so there is no need to look for any extra equations.

Let's start by deriving d_j, j = 2,N first. To compute these 1^st-order derivatives at the intermediate nodes, very simple natural heuristic rules are employed:

If _j-1_j ≤ 0, where , then we set d_j = 0;

Otherwise, d_i is a weighted harmonic mean expressed as

with the weight w_j determined by lengths of two adjacent intervals as in this example,

Now, assuming that we have terminal derivatives y'₁ and y'_N+1, the conditions for deriving 4N unknown coefficients become

Instructor Note: Here you can read more about the so-called 'osculatory' interpolation.

For easy reference, the conditions for deriving unknown coefficients relying on y'₁ and y'_N+1 are repeated here:

You might ask if it is possible to avoid assigning the 1^st-order derivatives at the endpoints. The answer is yes! We may use some heuristic rules to compute the 1^st-order derivatives at the endpoints as well.

To this end, the piecewise cubic Hermite or 'osculatory' interpolation (another name for the local spline) uses one-sided (non-centered) three-point formulas to approximate these derivatives:

After computing these endpoint derivatives, two additional checks apply. First, if the sign of d₁ happens to differ from the sign of d₂, then d₁ is set to zero (the same with d_N+1 and d_N). Second, if the node 2(N) is found to be a discrete local extremum, then the derivative d₁ (d_N+1) is limited to 3delta₁ (3delta_N).

Instructor Note: Explore the original 'osculatory' and slope-varying local splines using this interactive media. You are welcome to compare these to the splines determined globally.

Open interactive graph full screen.

Click on the 'MATLAB Interpolating Functions' button in the left navigation area to see a list of MATLAB commands relevant to the subject of this tutorial.