Multivariate Polynomials with Rational Coefficients.

The ‘gmpoly’ package

R package for multivariate polynomials with rational coefficients.

library(gmpoly)

Define a polynomial with the gmpoly function:

pol <- gmpoly("4 x^(2, 1, 1) + 1/2 x^(0,1,0)")
pol
## gmpoly object algebraically equal to
## 1/2 x^(0,1,0) + 4 x^(2,1,1)

Some arithmetic on this polynomial:

-pol
## gmpoly object algebraically equal to
## -1/2 x^(0,1,0) - 4 x^(2,1,1)
2 * pol
## gmpoly object algebraically equal to
## x^(0,1,0) + 8 x^(2,1,1)
pol / 2
## gmpoly object algebraically equal to
## 1/4 x^(0,1,0) + 2 x^(2,1,1)
pol + 5
## gmpoly object algebraically equal to
## 5 x^(0,0,0) + 1/2 x^(0,1,0) + 4 x^(2,1,1)
pol - 5
## gmpoly object algebraically equal to
## -5 x^(0,0,0) + 1/2 x^(0,1,0) + 4 x^(2,1,1)
pol^2
## gmpoly object algebraically equal to
## 1/4 x^(0,2,0) + 4 x^(2,2,1) + 16 x^(4,2,2)

Note that you cannot directly use a bigq scalar, e.g. you can’t do gmp::as.bigq(5, 3) * pol. To perform such an operation, you have to use gmpolyConstant:

gmpolyConstant(3, "5/3") * pol
## gmpoly object algebraically equal to
## 5/6 x^(0,1,0) + 20/3 x^(2,1,1)

The gmpolyConstant function converts a scalar to a constant polynomial, and two polynomials can be added and multiplied:

pol1 <- gmpoly("2 x^(1,1) - 5/3 x^(0,1)")
pol2 <- gmpoly("-2 x^(1,1) + 3 x^(2,1)")
pol1 + pol2
## gmpoly object algebraically equal to
## -5/3 x^(0,1) + 3 x^(2,1)
pol1 * pol2
## gmpoly object algebraically equal to
## 10/3 x^(1,2) - 9 x^(2,2) + 6 x^(3,2)

Use gmpolyEval to evaluate a polynomial for some values of the variables:

library(gmp, warn.conflicts = FALSE)
pol <- gmpoly("5/2 x^(2,3) + 3 x^(1,1)")
gmpolyEval(pol, x = as.bigq(c(1, 1)))
## Big Rational ('bigq') :
## [1] 11/2

To evaluate the polynomial for several sets of variables, supply a matrix to the second argument:

X <- rbind(
  t(as.bigq(c(1, 1))),
  t(as.bigq(c(3, 4), c(4, 3)))
)
gmpolyEval(pol, x = X)
## Big Rational ('bigq') object of length 2:
## [1] 11/2 19/3