.ThermoSysPro.Functions.SplineInterpolation

Information

ThermoSysPro Version 3.1

Spline interpolation function. The resulting spline will be continuous and have a continuous first derivative.

Implementation

It uses a cardinal spline interpolation algorithm. Cardinal splines are a sub-set of cubic Hermite splines where each piece is a third-degree polynomial specified in Hermite form: i.e specified by its values and the first derivatives at the end points of the reference interval.

The derivatives are calculated based on the non-uniform cardinal grid approach according to:

y1_der=0.5(1-t)(y1-y0/(x1-x0)+(y2-y1/(x2-x1)

I.e. the derivative in a point is calculated as an average of the surrounding points with an extra input shape parameter t.

Inputs

• TabX: Vector containing x-table values
• TabY: Vector containing y-table values
• X: The x-value that the spline should be evaluated at
• t: Cardinal spline shape parameter. t = 0.5 is default and is generally a good choice. A value close to 1 will yield a stiff spline, t=0 corresponds to a Catmull-Rom spline and  t < 0 corresponds to a more “loose” spline. From testing, t=0.5  seems to be a good choice in general and is therefore chosen as a default value. A value of t = 1 corresponds to that the derivative in all data points is zero, which may result in strange curves . See Examples > TestSplineInterpolation for a demonstration.

Output

• Y: Interpolated value evaluated at X

Extrapolation

Linear extrapolation is employed if the reference value is not contained in the reference value table.

Example

• TabX = {1,2,3,4};
• TabY = {4,3,5,2};
  
  

References

http://people.cs.clemson.edu/~dhouse/courses/405/notes/splines.pdf

Interface

function SplineInterpolation
  input Real TabX[:] "References table";
  input Real TabY[:] "Results table";
  input Real X "Reference value";
  input Real t(max = 1) = 0.5 "Stiffness parameter";
  output Real Y "Interpolated result";
end SplineInterpolation;

Revisions