Arithmetic Ops Group Methods for mvp objects

Allows arithmetic operators to be used for multivariate polynomials such as addition, multiplication, integer powers, etc.

Usage

# S3 method for class 'mvp'
Ops(e1, e2)
mvp_negative(S)
mvp_times_mvp(S1,S2)
mvp_times_scalar(S,x)
mvp_plus_mvp(S1,S2)
mvp_plus_numeric(S,x)
mvp_eq_mvp(S1,S2)
mvp_modulo(S1,S2)

Arguments

e1,e2,S,S1,S2: Objects of class mvp
x: Scalar, length one numeric vector

Details

The function Ops.mvp() passes unary and binary arithmetic operators “+”, “-”, “*” and “^” to the appropriate specialist function.

The most interesting operator is “*”, which is passed to mvp_times_mvp(). I guess “+” is quite interesting too.

The caret “^” denotes arithmetic exponentiation, as in x^3==x*x*x. As an experimental feature, this is (sort of) vectorised: if n is a vector, then a^n returns the sum of a raised to the power of each element of n. For example, a^c(n1,n2,n3) is a^n1 + a^n2 + a^n3. Internally, n is tabulated in the interests of efficiency, so a^c(0,2,5,5,5) = 1 + a^2 + 3a^5 is evaluated with only a single fifth power. Similar functionality is implemented in the freealg package.

Value

The high-level functions documented here return an object of mvp, the low-level functions documented at lowlevel.Rd return lists. But don't use the low-level functions.

Author

Robin K. S. Hankin

Note

Function mvp_modulo() is distinctly sub-optimal and inst/mvp_modulo.Rmd details ideas for better implementation.

Examples

(p1 <- rmvp(3))
#> mvp object algebraically equal to
#> b^2 d e f^2 + b^2 d^2 e^2 + d^2 e^2 f^2
(p2 <- rmvp(3))
#> mvp object algebraically equal to
#> a b^2 d^2 f + 3 a^2 e^2 f + 2 c f^3

p1*p2
#> mvp object algebraically equal to
#> a b^2 d^4 e^2 f^3 + a b^4 d^3 e f^3 + a b^4 d^4 e^2 f + 3 a^2 b^2 d e^3 f^3 + 3
#> a^2 b^2 d^2 e^4 f + 3 a^2 d^2 e^4 f^3 + 2 b^2 c d e f^5 + 2 b^2 c d^2 e^2 f^3 +
#> 2 c d^2 e^2 f^5

p1+p2
#> mvp object algebraically equal to
#> a b^2 d^2 f + 3 a^2 e^2 f + b^2 d e f^2 + b^2 d^2 e^2 + 2 c f^3 + d^2 e^2 f^2

p1^3
#> mvp object algebraically equal to
#> 3 b^2 d^5 e^5 f^6 + 3 b^2 d^6 e^6 f^4 + 3 b^4 d^4 e^4 f^6 + 6 b^4 d^5 e^5 f^4 +
#> 3 b^4 d^6 e^6 f^2 + b^6 d^3 e^3 f^6 + 3 b^6 d^4 e^4 f^4 + 3 b^6 d^5 e^5 f^2 +
#> b^6 d^6 e^6 + d^6 e^6 f^6


p1*(p1+p2) == p1^2+p1*p2  # should be TRUE
#> [1] TRUE

Arithmetic Ops Group Methods for `mvp` objects