--- Rachel Alcraft, Bioinformatics ---

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Psu Multivarse

A multivariate interpolation library

Library Description

This library provides a simple interface to multivariate interpolation for values and partial derivatives. Two methods are employed:

• Multivariate fitting which has a bottleneck in matrix size.
• Tri-directional tri-polynomial interpolation, this has no bottleneck in degree but at very high degree, as a recursive function, can be slow

The problem

Interpolating sequences and data in 3d cubes eg electron density. Creating an f(x,y,z). Finding the partial derivatives.

Theorem 1: There is an invariant solution for a polynomial fitting a given degree and dimension

Conclusion: The constant matrices can be precalculated making a time efficient solution to interpolation.

Partial differentiation

The coefficients matrix can be differentiated by a simple matrix manipulation

Theorem 2: Partial differentation of the multivariate function is equivalent to differentiation of a tri-polynomial solution

Conclusion: The dimensional problem inherent in matrices can be solved by using a tri-directional polynomial fit.

Some of the invariant matrices

Code documentation

References

• Auckland Uni Maths Dept "Multivariate polynomial interpolation"
• Olver 2016 "On Multivariate Interpolation"
• Gasca and Sauer 1999, "On the history of multivariate polynomial interpolation"
• Hecht et al 2010, "Multivariate Interpolation - lifting the curse of dimensionality"

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