Ops {weyl} | R Documentation |
Arithmetic Ops Group Methods for the Weyl algebra
Description
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 |
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.
Value
Generally, return a weyl object
Note
Function weyl_prod_univariate_nrow()
is present for
completeness, it is not used in the package
Author(s)
Robin K. S. Hankin
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
is.zero(x*(y*z) - (x*y)*z)