How to represent polynomials with numeric vectors in R

Question

In R how would one represent polynomial expressions and do polynomial math with the numeric vector objects? For example:

x1 <- c(2,1)  # 2 + x
x2 <- c(-1,3)  # -1 + 3*x

And want:

x1 * x2 # to return -2 + 5*x + 3*x^2 

Note: I answered a question this morning and then the poster apparently deleted it (making me wonder if it were homework.) So I am re-posting the question from memory.

Answer 1

One could multiply the coefficients directly using outer and then aggregate the results

x1 <- c(2,1)  # 2 + x
x2 <- c(-1,3)  # -1 + 3*x
tmp <- outer(x1, x2)
tapply(tmp, row(tmp) + col(tmp) - 1, sum)
# 1  2  3 
#-2  5  3

x1 <- c(2, 1) # 2 + x
x2 <- c(-1, 3, 2) # -1 + 3*x + 2*x^2
tmp <- outer(x1, x2)
tapply(tmp, row(tmp) + col(tmp) - 1, sum) # should give -2 + 5*x + 7*x^2 + 2*x^3
# 1  2  3  4 
#-2  5  7  2

as discussed in the comments the '-1' in the code isn't necessary. When coming up with the solution that helped me because it allowed me to map each location in the output of outer to where it would end up in the final vector. If we did a '-2' instead then it would map to the exponent on x in the resulting polynomial. But we really don't need it so something like the following would work just as well:

tmp <- outer(x1, x2)
tapply(tmp, row(tmp) + col(tmp), sum)

Answer 2

Use the polynom package:

 require(polynom)
# Loading required package: polynom
# From the example for as.polynomial
 p <- as.polynomial(c(1,0,3,0))
 p
# 1 + 3*x^2 

 x1 <- c(2,1)
 x2 <- c(-1,3)
 px1 <- as.polynomial(x1)
 px2 <- as.polynomial(x2)

 px1*px2
# -2 + 5*x + 3*x^2 
 prod.p <- .Last.value
 str(prod.p)
# Class 'polynomial'  num [1:3] -2 5 3
 unclass(prod.p)
# [1] -2  5  3

Answer 3

Polynomial multiplication is the convolution of the coefficients

convolve(c(2,1),rev(c(-1,3)),type="open")
#[1] -2  5  3