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Multinomial Expansion

version (1.59 KB) by Isaac
Evaluate multinomial expansion and coefficients


Updated 21 Oct 2014

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MULTINOMIAL_EXPAND determines the matrix of powers for a multinomial expansion
of the form (x_1 + x_2 + x_3 + ... + x_ndim)^pow
Nmatrix - matrix of powers, each row representing a single term in the expansion
for example, the row [0,1,0,2] would represent (x_2)*(x_4)^2
Note, this is equivalent to finding all multiindices
k = [k_1,k_2,...,k_ndim] with |k|=sum(k)=pow
This can be useful for sparse-grid integration methods that use multiindices.

Ncoef - vector of coefficients (multinomial coefficient)

Ex. Evaluating the multinomial at a point x = [x_1, x_2, x_3, ... ]
-> sum(Ncoef .* repmat(x,size(Nmatrix,1),1).^Nmatrix)

Ex. Compute all multiindices of length 6 and order 4:
-> Nmatrix = multinomial_expand(4,6);

The method is recursive, so it can be a bit slow for smaller problems, but
it can work with relatively large size inputs (e.g. ndim = 50,100,...)
compared to previous versions (e.g. another implementation could
not handle ndim>15 due to memory constraints using a huge kron).
The savings in memory is because we only compute the
needed number of terms and don't use kron or factorial.

Cite As

Isaac (2022). Multinomial Expansion (, MATLAB Central File Exchange. Retrieved .

MATLAB Release Compatibility
Created with R2014b
Compatible with any release
Platform Compatibility
Windows macOS Linux

Inspired by: MULTINOMIAL, Multinomial Expansion

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