Arithmetic Ops Group Methods for the Weyl algebra

Allows arithmetic operators such as addition, multiplication, division, integer powers, etc. to be used for weyl objects. Idiom such as x^2 + y*z/5 should work as expected. Addition and subtraction, and scalar multiplication, are the same as those of the spray package; but “*” is interpreted as functional composition, and “^” is interpreted as repeated composition. A number of helper functions are documented here (which are not designed for the end-user).

Usage

# S3 method for class 'weyl'
Ops(e1, e2 = NULL)
weyl_prod_helper1(a,b,c,d)
weyl_prod_helper2(a,b,c,d)
weyl_prod_helper3(a,b,c,d)
weyl_prod_univariate_onerow(S1,S2,func)
weyl_prod_univariate_nrow(S1,S2)
weyl_prod_multivariate_onerow_singlecolumn(S1,S2,column)
weyl_prod_multivariate_onerow_allcolumns(S1,S2)
weyl_prod_multivariate_nrow_allcolumns(S1,S2)
weyl_power_scalar(S,n)

Arguments

S,S1,S2,e1,e2: Objects of class weyl, elements of a Weyl algebra
a,b,c,d: Integers, see details
column: column to be multiplied
n: Integer power (non-negative)
func: Function used for products

Details

All arithmetic is as for spray objects, apart from * and ^. Here, * is interpreted as operator concatenation: Thus, if \(w_1\) and \(w_2\) are Weyl objects, then \(w_1w_2\) is defined as the operator that takes \(f\) to \(w_1(w_2f)\).

Functions such as weyl_prod_multivariate_nrow_allcolumns() are low-level helper functions with self-explanatory names. In this context, “univariate” means the first Weyl algebra, generated by \(\left\lbrace x,\partial\right\rbrace\), subject to \(x\partial-\partial x=1\); and “multivariate” means the algebra generated by \(\left\lbrace x_1,x_2,\ldots,x_n,\partial_{x_1},\partial_{x_2},\ldots,\partial_{x_n}\right\rbrace\) where \(n>1\).

The product is somewhat user-customisable via option prodfunc, which affects function weyl_prod_univariate_onerow(). Currently the package offers three examples: weyl_prod_helper1(), weyl_prod_helper2(), and weyl_prod_helper3(). These are algebraically identical but occupy different positions on the efficiency-readability scale. The option defaults to weyl_prod_helper3(), which is the fastest but most opaque. The vignette provides further details, motivation, and examples.

Powers, as in x^n, are defined in the usual way. Negative powers will always return an error.

Value

Generally, return a weyl object

Author

Robin K. S. Hankin

Note

Function weyl_prod_univariate_nrow() is present for completeness, it is not used in the package

Examples

x <- rweyl(n=1,d=2)
y <- rweyl(n=1,d=2)
z <- rweyl(n=2,d=2)


x*(y+z) == x*y + x*z
#> [1] TRUE
is.zero(x*(y*z) - (x*y)*z)
#> [1] TRUE