FAUL-GOODSEN-POWELL algorithm

This is a Python implementation of the Faul-Goodsen-Powell algorithm which produces an interpolant for d-dimensional data using the multiquadric radial basis functions. It works well for even very high dimensional data.

The interpolant, s(x) is of the form

$s(x) = \sum_i^n \lambda_i \phi(|x-x_i|) + \alpha$

where the $x_i$ are the data centers and $\phi(x) = (x^2+c^2)^{\frac{1}{2}}$ is the multiquadric radial basis function.

The algorithm returns the coefficients $\lambda_i$ and the value of $\alpha$.

INSTALLATION/REQUIREMENTS:

Python and Numpy

ALGORITHM DESCRIPTION - inputs - FGP(data, values, q, c, error)

• Data centers ($x_i$) and values at those points ($f_i$)

• Error

• Two parameters for the algorithm - $q$ and $c$:

• $c>0$, using a smaller value ($O(10^{-1})$ or smaller) is advised. This is the 'shape parameter' for the multiquadric.

• $q>0$, using a value of q=30 is standard - feel free to go between 5 and 50. A rule of thumb is that smaller q means each iteration is quicker, but we may need more iterations for convergence overally.

ALGORITHM DESCRIPTION - outputs

• Iteration count - k

• Interpolant coefficients - $\lambda_i$

• Interpolant constant - $\alpha$

• Interpolation error at the centers - err

You can try the DEMO version or use the FULL IMPLEMENTATION version as well.

The DEMO version lets you vary the distribution that the test data is drawn from:

• The unit d-ball

• The unit d-cube

• The unit d-Normal

• The integer grid in d-dimensions.