The zero polynomial — zero • spray

Test for the zero, or empty, polynomial

Usage

zero(d)
is.zero(x)
is.empty(L)

Arguments

L,x: A two-element list of indices and values, possibly a spray object or numeric vector
d: Integer specifying dimensionality of the spray (the arity)

Details

Functions is.empty() and is.zero() are synonyms. If spray objects are interpreted as multivariate polynomials, “is.zero()” is more intuitive, if sprays are interpreted as sparse arrays, “is.empty()” is better (for me).

Passing a zero-row index matrix can have unexpected effects:


> dput(spray(matrix(0,0,5),9))
structure(list(structure(numeric(0), .Dim = c(0L, 5L)), numeric(0)), class = "spray")

Above, the index matrix has zero rows (and no elements) but the fact that it has five columns is retained. Arguably spray() should return an error here, as the number of rows of the index matrix should match the length of the coefficient vector and they do not: the index has zero rows and the coefficient vector has length 1 (although they match in the returned value). The returned spray object has no coefficients [specifically, numeric(0)]; this is consistent with the index matrix having zero rows.

Zero coefficients are discarded by the back end:


> spray(matrix(1,1,5),0)
empty sparse array with 5 columns
> dput(spray(matrix(1,1,5),0))
structure(list(structure(numeric(0), dim = c(0L, 5L)), numeric(0)), class = "spray")

Above, the index matrix given to spray() has one row but the coefficient is a length-one vector with element zero. The resulting spray object has a NULL index matrix [because rows with zero coefficients are removed] and a NULL coefficient. It is also permissible to pass a a zero-row matrix:


   spray(matrix(0,0,5),0)
empty sparse array with 5 columns

   dput(spray(matrix(0,0,5),0))
structure(list(structure(numeric(0), dim = c(0L, 5L)), numeric(0)), class = "spray")

In previous versions of the package, the index matrix in the returned spray object could be NULL under some circumstances. If so, the arity of the spray object is lost. It is probably worth noting that spray(), given a zero-row index matrix, loses a length one coefficients vector, but complains about a length-two coefficient vector:


> dput(spray(matrix(0,0,5),0))
structure(list(structure(numeric(0), dim = c(0L, 5L)), numeric(0)), class = "spray")
> dput(spray(matrix(0,0,5),3))
structure(list(structure(numeric(0), dim = c(0L, 5L)), numeric(0)), class = "spray")
> dput(spray(matrix(0,0,5),1:2))
Error in is_valid_spray(L) : nrow(L[[1]]) == length(L[[2]]) is not TRUE
>
> identical(spray(matrix(0,0,5),0),spray(matrix(0,0,5),3))
[1] TRUE

Examples


(a <- lone(1,3))
#>  a b c     val
#>  1 0 0  =    1

is.zero(a-a)  # should be TRUE
#> [1] TRUE

is.zero(zero(6))
#> [1] TRUE

x <- spray(t(0:1))
y <- spray(t(1:0))

is.zero((x+y)*(x-y)-(x^2-y^2)) # TRUE
#> [1] TRUE