Stéphane Laurent 2024-07-26
Fractions of multivariate polynomials with rational coefficients.
The qspray package allows arithmetic (and more) on multivariate polynomials with rational coefficients. Based on this one, the ratioOfQsprays package allows to manipulate fractions of multivariate polynomials with rational coefficients.
These notes about the ratioOfQsprays package assume that the reader is a bit familiar with the qspray package.
ratioOfQspraysA ratioOfQsprays object represents a fraction of two
multivariate polynomial with rational coefficients. Such polynomials are
represented by qspray objects. The easiest way to create a
ratioOfQsprays is to introduce the variables of the
polynomials with the qlone function (from the
qspray package), and then to build a
qspray numerator and a qspray denominator with
the arithmetic operations. For example:
library(ratioOfQsprays)
f <- function(x1, x2, x3) {
(2*x1^2 + x2*x3) / (4*x1 - 3*x3 + 1)
}
# variables:
x1 <- qlone(1)
x2 <- qlone(2)
x3 <- qlone(3)
# the 'ratioOfQsprays':
( roq <- f(x1, x2, x3) )
## [ 1/2*x^2 + 1/4*y.z ] %//% [ x - 3/4*z + 1/4 ]The denominator of a ratioOfQsprays fraction of
polynomials is always monic. That means it is a polynomial
whose leading coefficient is 1.
Arithmetic on ratioOfQsprays objects is available:
roq^2
## [ 1/4*x^4 + 1/4*x^2.y.z + 1/16*y^2.z^2 ] %//% [ x^2 - 3/2*x.z + 1/2*x + 9/16*z^2 - 3/8*z + 1/16 ]
roq - roq
## [ 0 ]
1 / roq
## [ 2*x - 3/2*z + 1/2 ] %//% [ x^2 + 1/2*y.z ]
2*roq + (x2 + x3)/x1
## [ x^3 + 1/2*x.y.z + x.y + x.z - 3/4*y.z + 1/4*y - 3/4*z^2 + 1/4*z ] %//% [ x^2 - 3/4*x.z + 1/4*x ]You don’t like my quotient bar %//%? Be patient, we will
see how to change it later. I adopted this large quotient bar because it
is more easy to find it than a single slash / in a
ratioOfQsprays having a long expression.
Rational numbers and qspray polynomials are coercible to
ratioOfQsprays objects, and then you can also perform
arithmetic operations between a ratioOfQsprays and such an
object:
2 * roq
## [ x^2 + 1/2*y.z ] %//% [ x - 3/4*z + 1/4 ]
"1/2" * roq
## [ 1/4*x^2 + 1/8*y.z ] %//% [ x - 3/4*z + 1/4 ]
roq + gmp::as.bigq("7/3")
## [ 1/2*x^2 + 7/3*x + 1/4*y.z - 7/4*z + 7/12 ] %//% [ x - 3/4*z + 1/4 ]
x1 + roq + x3^2
## [ 3/2*x^2 + x.z^2 - 3/4*x.z + 1/4*x + 1/4*y.z - 3/4*z^3 + 1/4*z^2 ] %//% [ x - 3/4*z + 1/4 ]The result of an arithmetic operation is always an irreducible fraction. To perform this step, the C++ library CGAL is used to compute a greatest common divisor of the numerator and the denominator of the possibly non-reduced fraction resulting from the arithmetic operation, and then to divide both of them by this greatest common divisor. This is very efficient in general.
ratioOfQspraysUse evalRatioOfQsprays to evaluate a
ratioOfQsprays. This function returns a bigq
number:
library(gmp) # rational numbers
x <- c("4", "3", "2/5")
evalRatioOfQsprays(roq, x)
## Big Rational ('bigq') :
## [1] 166/79
x <- as.bigq(x)
evalRatioOfQsprays(roq, x)
## Big Rational ('bigq') :
## [1] 166/79
f(x[1], x[2], x[3])
## Big Rational ('bigq') :
## [1] 166/79It is also possible to substitute some values to only a subset of the
variables, with the help of the function
substituteRatioOfQsprays. You have to indicate the
variables you don’t want to replace with NA:
x <- c(NA, "3", "2/5")
substituteRatioOfQsprays(roq, x)
## [ 1/2*x^2 + 3/10 ] %//% [ x - 1/20 ]
x <- as.bigq(x)
f(x1, x[2], x[3])
## [ 1/2*x^2 + 3/10 ] %//% [ x - 1/20 ]And it is possible to convert a ratioOfQsprays to a
function which is evaluated by Ryacas:
fyac <- as.function(roq)
fyac("4", "3", "2/5") # = evalRatioOfQsprays(roq, c("4", "3", "2/5"))
## [1] "166/79"Actually you can pass some literal variables to this function:
fyac("x", "3", "2/5") # = substituteRatioOfQsprays(roq, c(NA, "3", "2/5"))
## [1] "(2*(5*x^2+3))/(20*x-1)"
fyac("x", "y", "z") # = roq
## [1] "(z*y+2*x^2)/(4*x-3*z+1)"
fyac("x", "x", "x")
## [1] "(3*x^2)/(x+1)"Complex numbers and allowed; the imaginary unit is denoted by
I. See the Yacas
documentation for more information.
fyac("Sqrt(2)", "2 + 2*I", "3")
## [1] "Complex(10/(Sqrt(32)-8),6/(Sqrt(32)-8))"You can get numerical approximations by setting the option
N=TRUE in as.function:
fyacN <- as.function(roq, N = TRUE)
fyacN("4", "3", "2/5")
## [1] 2.101266
fyacN("x", "3", "2/5")
## expression((2 * (5 * x^2 + 3))/(20 * x - 1))
fyacN("Sqrt(2)", "2 + 2*I", "3")
## [1] -4.267767-2.56066iratioOfQspraysA couple of functions to query a ratioOfQsprays are
available:
getNumerator(roq)
## 1/2*x^2 + 1/4*y.z
getDenominator(roq)
## x - 3/4*z + 1/4
numberOfVariables(roq)
## [1] 3
isConstant(roq)
## [1] FALSE
isConstant(roq / roq)
## [1] TRUE
isUnivariate(roq)
## [1] FALSE
isUnivariate(x1 / (x1^2 + 1))
## [1] TRUE
isPolynomial(roq)
## [1] FALSE
isPolynomial((x1^2 - x2^2) / (x1 - x2))
## [1] TRUEratioOfQspraysAs you have seen, the variables of roq are denoted by
x, y, z. This is the default way
of printing a ratioOfQsprays which has no more than three
variables. If it has more than three variables, then they are denoted by
x1, x2, x3, …:
x4 <- qlone(4)
roq / x4
## [ 1/2*x1^2 + 1/4*x2.x3 ] %//% [ x1.x4 - 3/4*x3.x4 + 1/4*x4 ]It is possible to control the way a ratioOfQsprays is
printed. For example, let’s say you want to print roq by
using a1, a2, a3 for the
variables and you want to change the symbol for the quotient bar:
showRatioOfQspraysOption(roq, "x") <- "a"
showRatioOfQspraysOption(roq, "quotientBar") <- " / "
roq
## [ 1/2*a1^2 + 1/4*a2.a3 ] / [ a1 - 3/4*a3 + 1/4 ]Now, if you perform an arithmetic operation between roq
at first position and an another ratioOfQsprays,
these show options are passed to the result if possible:
roq + (x1 + 1)/x2
## [ 1/2*a1^2.a2 + a1^2 - 3/4*a1.a3 + 5/4*a1 + 1/4*a2^2.a3 - 3/4*a3 + 1/4 ] / [ a1.a2 - 3/4*a2.a3 + 1/4*a2 ]If you perform an arithmetic operation between roq and
an object coercible to a ratioOfQsprays object but which is
not a ratioOfQsprays object, such as a bigq
number or a qspray object, the show options of
roq are passed to the result, even if roq is
not at the first position:
x1 * roq
## [ 1/2*a1^3 + 1/4*a1.a2.a3 ] / [ a1 - 3/4*a3 + 1/4 ]An obvious example of a situation in which it is not always possible to transfer the show options is when you use three letters for the variables, e.g.
showRatioOfQspraysOption(roq, "showQspray") <- showQsprayXYZ(c("A", "B", "C"))
roq
## [ 1/2*A^2 + 1/4*B.C ] / [ A - 3/4*C + 1/4 ]but then you add to roq a ratioOfQsprays
containing the fourth variable:
roq + x4/(x4 + 1)
## [ 1/2*A1^2.A4 + 1/2*A1^2 + A1.A4 + 1/4*A2.A3.A4 + 1/4*A2.A3 - 3/4*A3.A4 + 1/4*A4 ] / [ A1.A4 + A1 - 3/4*A3.A4 - 3/4*A3 + 1/4*A4 + 1/4 ]Obviously it is not possible to denote the resulting fraction of
polynomials with the letters A, B and
C. The solution I adopted consists in taking the first of
these letters and to index it. The same method is used for the
qspray polynomials.
ratioOfQspraysLet’s take a ratioOfQsprays fraction of polynomials:
f <- function(x, y, z) {
(2*x^2 + y*z) / (4*x - 3*z + 1)
}
x <- qlone(1); y <- qlone(2); z <- qlone(3)
roq <- f(x, y, z)You can differentiate it:
derivRatioOfQsprays(roq, 2) # derivative w.r.t. y
## [ 1/4*z ] %//% [ x - 3/4*z + 1/4 ]You can permute its variables:
swapVariables(roq, 2, 3) == f(x, z, y)
## [1] TRUEYou can perform some polynomial changes of its variables:
changeVariables(roq, list(x+1, y^2, x+y+z)) == f(x+1, y^2, x+y+z)
## [1] TRUE