Stéphane Laurent 2024-08-04

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.

A ratioOfQsprays object represents a fraction of two multivariate polynomials 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.

Use 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] 166/79

It 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 are 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] -4.267767-2.56066i

Let us also mention the substituteSomeRatioOfQsprays function. This function allows to substitute the variables of a ratioOfQsprays fraction of polynomials with some ratioOfQsprays objects.

A 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] TRUE

As 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.

Let’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] TRUE

You 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