Scientific description of cubic spline (interpolation)

Dear all,

I am writing a scientific article (quantitative economics), and I need to describe the 'spline' method (cubic spline interpolation) as in griddedInterpolant. I can't find much, except the description:

"The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension. The interpolation is based on a cubic spline using not-a-knot end conditions."

Can anybody help me with a more scientific description? Is this some kind of standard algorithm? I found some formula for the one dimensional case, but I need a more general description for multiple dimensions (at least up to 4 dimensions).

[dpb]

Splines are a full branch of numerical methods of their own; the standard reference TMW lists is

[1] de Boor, Carl. A Practical Guide to Splines. Springer-Verlag, New York: 1978.

There's a discussion in the MATLAB doc on various end conditions at https://www.mathworks.com/help/curvefit/construct-cubic-spline-interpolants.html

Gargle has any number of hits; one that may be of interest for your purposes is at http://www.cs.tau.ac.il/~turkel/notes/numeng/spline_note.pdf

[Sargondjani]

Thanks dpb, especially the book is interesting!

More information is still very welcome, because the Matlab documentation is minimal... especially how it deals with multiple dimensions.

[Jan]

In the spline interpolation the dimensions are treated independently. It is like a bunch of 1D-interpolations.

Did you read this already: https://en.wikipedia.org/wiki/Spline_interpolation

[Sargondjani]

Thanks Jan, now I understand!! (I guess the ordering of the dimensions then matters?)

And yes, I had already checked wiki. Based on your comment I found this one, which clarifies quite a bit (even if it is a different method from what Matlab uses):

https://en.wikipedia.org/wiki/Tricubic_interpolation , which is based on Tricubic Interpolation in Three Dimensions (2005) by Lekien and Marsden.

[Matt J]

(I guess the ordering of the dimensions then matters?)

Nope. It doesn't.

[Sargondjani]

Ok, great!

[Matt J]

As an example,

A=rand(4);
x=3.7; y=1.2;

interp1( interp1(A,x,'spline') , y,'spline')

ans = 0.1894

interp1( interp1(A',y,'spline') , x,'spline')

ans = 0.1894

interpn(A,x,y,'spline')

ans = 0.1894

[Sargondjani]

I see. Nice! Thanks

[Sargondjani]

Another reference: Spline Functions: Basic Theory by Schumaker (3rd edition, 2007).

And by same author also "Spline Functions: Computational Methods" (SIAM, 2015)