Boolean Arithmetic Matrix Products: %&% and Methods

For boolean or “pattern” matrices, i.e., R objects of class nMatrix, it is natural to allow matrix products using boolean instead of numerical arithmetic.

In package Matrix, we use the binary operator %&% (aka “infix”) function) for this and provide methods for all our matrices and the traditional R matrices (see matrix).

Methods

We provide methods for both the “traditional” (R base) matrices and numeric vectors and conceptually all matrices and sparseVectors in package Matrix.

signature(x = "ANY", y = "ANY")

signature(x = "ANY", y = "Matrix")

signature(x = "Matrix", y = "ANY")

signature(x = "nMatrix", y = "nMatrix")

signature(x = "nMatrix", y = "nsparseMatrix")

signature(x = "nsparseMatrix", y = "nMatrix")

signature(x = "nsparseMatrix", y = "nsparseMatrix")

signature(x = "sparseVector", y = "sparseVector")

Note

These boolean arithmetic matrix products had been newly introduced for Matrix 1.2.0 (March 2015). Its implementation has still not been tested extensively.

Originally, it was left unspecified how non-structural zeros, i.e., 0's as part of the M@x slot should be treated for numeric ("dMatrix") and logical ("lMatrix") sparse matrices. We now specify that boolean matrix products should behave as if applied to drop0(M), i.e., as if dropping such zeros from the matrix before using it.
Equivalently, for all matrices M, boolean arithmetic should work as if applied to M != 0 (or M != FALSE).

The current implementation ends up coercing both x and y to (virtual) class nsparseMatrix which may be quite inefficient for dense matrices. A future implementation may well return a matrix with different class, but the “same” content, i.e., the same matrix entries \(m_ij\).

See also

%*%, crossprod(), or tcrossprod(), for (regular) matrix product methods.

Value

a pattern matrix, i.e., inheriting from "nMatrix", or an "ldiMatrix" in case of a diagonal matrix.

Examples

set.seed(7)
L <- Matrix(rnorm(20) > 1,    4,5)
(N <- as(L, "nMatrix"))
#> 4 x 5 sparse Matrix of class "ngCMatrix"
#>               
#> [1,] | . . | .
#> [2,] . . | . .
#> [3,] . . . | .
#> [4,] . . | . .
L. <- L; L.[1:2,1] <- TRUE; L.@x[1:2] <- FALSE; L. # has "zeros" to drop0()
#> 4 x 5 sparse Matrix of class "lgCMatrix"
#>               
#> [1,] : . . | .
#> [2,] : . | . .
#> [3,] . . . | .
#> [4,] . . | . .
D <- Matrix(round(rnorm(30)), 5,6) # -> values in -1:1 (for this seed)
L %&% D
#> 4 x 6 sparse Matrix of class "ngCMatrix"
#>                 
#> [1,] | | | | | .
#> [2,] | | . | . |
#> [3,] | | . . | .
#> [4,] | | . | . |
stopifnot(identical(L %&% D, N %&% D),
          all(L %&% D == as((L %*% abs(D)) > 0, "sparseMatrix")))

## cross products , possibly with  boolArith = TRUE :
crossprod(N)     # -> sparse patter'n' (TRUE/FALSE : boolean arithmetic)
#> 5 x 5 sparse Matrix of class "nsCMatrix"
#>               
#> [1,] | . . | .
#> [2,] . . . . .
#> [3,] . . | . .
#> [4,] | . . | .
#> [5,] . . . . .
crossprod(N  +0) # -> numeric Matrix (with same "pattern")
#> 5 x 5 sparse Matrix of class "dsCMatrix"
#>               
#> [1,] 1 . . 1 .
#> [2,] . . . . .
#> [3,] . . 2 . .
#> [4,] 1 . . 2 .
#> [5,] . . . . .
stopifnot(all(crossprod(N) == t(N) %&% N),
          identical(crossprod(N), crossprod(N +0, boolArith=TRUE)),
          identical(crossprod(L), crossprod(N   , boolArith=FALSE)))
crossprod(D, boolArith =  TRUE) # pattern: "nsCMatrix"
#> 6 x 6 sparse Matrix of class "nsCMatrix"
#>                 
#> [1,] | | | | | |
#> [2,] | | | | | |
#> [3,] | | | | | |
#> [4,] | | | | | |
#> [5,] | | | | | |
#> [6,] | | | | | |
crossprod(L, boolArith =  TRUE) #  ditto
#> 5 x 5 sparse Matrix of class "nsCMatrix"
#>               
#> [1,] | . . | .
#> [2,] . . . . .
#> [3,] . . | . .
#> [4,] | . . | .
#> [5,] . . . . .
crossprod(L, boolArith = FALSE) # numeric: "dsCMatrix"
#> 5 x 5 sparse Matrix of class "dsCMatrix"
#>               
#> [1,] 1 . . 1 .
#> [2,] . . . . .
#> [3,] . . 2 . .
#> [4,] 1 . . 2 .
#> [5,] . . . . .