multipol (version 1.0-7)

ooom: One over one minus a multipol

Description

Uses Taylor's theorem to give one over one minus a multipol

Usage

ooom(n, a, maxorder=NULL)

Arguments

n

The order of the approximation; see details

a

A multipol

maxorder

A vector of integers giving the maximum order as per taylor()

Author

Robin K. S. Hankin

Details

The motivation for this function is the formal power series \((1-x)^{-1}=1+x+x^2+\ldots\). The way to think about it is to observe that \((1+x+x^2+\ldots+x^n)(1-x)=1-x^{n-1}\), even if \(x\) is a multivariate polynomial (one needs only power associativity and a distributivity law, so this works for polynomials). The right hand side is \(1\) if we neglect powers of \(x\) greater than the \(n\)-th, so the two terms on the left hand side are multiplicative inverses of one another.

Argument n specifies how many terms of the series to take.

The function uses an efficient array method when x has only a single non-zero entry. In other cases, a variant of Horner's method is used.

References

I. J. Good 1976. “On the application of symmetric Dirichlet distributions and their mixtures to contingency tables”. The Annals of Statistics, volume 4, number 6, pp1159-1189; equation 5.6, p1166

See Also

taylor

Examples

ooom(4,homog(3,1))


# How many 2x2 contingency tables of nonnegative integers with rowsums =
# c(2,2) and colsums = c(2,2) are there?  Good gives:

(
  ooom(2,lone(4,c(1,3))) *
  ooom(2,lone(4,c(1,4))) *
  ooom(2,lone(4,c(2,3))) *
  ooom(2,lone(4,c(2,4))) 
)[2,2,2,2]
  
# easier to use the aylmer package:

if (FALSE) {
library(aylmer)
no.of.boards(matrix(1,2,2))
}