Matrix (Cross) Products (of Transpose)

The basic matrix product, %*% is implemented for all our Matrix and also for sparseVector classes, fully analogously to R's base matrix and vector objects.

The functions crossprod and tcrossprod are matrix products or “cross products”, ideally implemented efficiently without computing t(.)'s unnecessarily. They also return symmetricMatrix classed matrices when easily detectable, e.g., in crossprod(m), the one argument case.

tcrossprod() takes the cross-product of the transpose of a matrix. tcrossprod(x) is formally equivalent to, but faster than, the call x %*% t(x), and so is tcrossprod(x, y) instead of x %*% t(y).

Boolean matrix products are computed via either %&% or boolArith = TRUE.

# S4 method for class 'CsparseMatrix,diagonalMatrix'
x %*% y

# S4 method for class 'CsparseMatrix,diagonalMatrix'
crossprod(x, y = NULL, boolArith = NA, ...)
       ## .... and for many more signatures

# S4 method for class 'TsparseMatrix,missing'
tcrossprod(x, y = NULL, boolArith = NA, ...)
       ## .... and for many more signatures

Arguments

x

a matrix-like object

y

a matrix-like object, or for [t]crossprod() NULL (by default); the latter case is formally equivalent to y = x.

boolArith

logical, i.e., NA, TRUE, or FALSE. If true the result is (coerced to) a pattern matrix, i.e., "nMatrix", unless there are NA entries and the result will be a "lMatrix". If false the result is (coerced to) numeric. When NA, currently the default, the result is a pattern matrix when x and y are "nsparseMatrix" and numeric otherwise.

...

potentially more arguments passed to and from methods.

Details

For some classes in the Matrix package, such as dgCMatrix, it is much faster to calculate the cross-product of the transpose directly instead of calculating the transpose first and then its cross-product.

boolArith = TRUE for regular (“non cross”) matrix products, %*% cannot be specified. Instead, we provide the %&% operator for boolean matrix products.

Note

boolArith = TRUE, FALSE or NA has been newly introduced for Matrix 1.2.0 (March 2015). Its implementation has still not been tested extensively. Notably the behaviour for sparse matrices with x slots containing extra zeros had not been documented previously, see the %&% help page.

Currently, boolArith = TRUE is implemented via CsparseMatrix coercions which may be quite inefficient for dense matrices. Contributions for efficiency improvements are welcome.

Value

A Matrix object, in the one argument case of an appropriate symmetric matrix class, i.e., inheriting from symmetricMatrix.

Methods

%*%

signature(x = "dgeMatrix", y = "dgeMatrix"): Matrix multiplication; ditto for several other signature combinations, see showMethods("%*%", class = "dgeMatrix").

%*%

signature(x = "dtrMatrix", y = "matrix") and other signatures (use showMethods("%*%", class="dtrMatrix")): matrix multiplication. Multiplication of (matching) triangular matrices now should remain triangular (in the sense of class triangularMatrix).

crossprod

signature(x = "dgeMatrix", y = "dgeMatrix"): ditto for several other signatures, use showMethods("crossprod", class = "dgeMatrix"), matrix crossproduct, an efficient version of t(x) %*% y.

crossprod

signature(x = "CsparseMatrix", y = "missing") returns t(x) %*% x as an dsCMatrix object.

crossprod

signature(x = "TsparseMatrix", y = "missing") returns t(x) %*% x as an dsCMatrix object.

crossprod,tcrossprod

signature(x = "dtrMatrix", y = "matrix") and other signatures, see "%*%" above.

See also

tcrossprod in R's base, and crossprod and %*%. Matrix package %&% for boolean matrix product methods.

Examples

 ## A random sparse "incidence" matrix :
 m <- matrix(0, 400, 500)
 set.seed(12)
 m[runif(314, 0, length(m))] <- 1
 mm <- as(m, "CsparseMatrix")
 object.size(m) / object.size(mm) # smaller by a factor of > 200
#> 220.1 bytes

 ## tcrossprod() is very fast:
 system.time(tCmm <- tcrossprod(mm))# 0   (PIII, 933 MHz)
#>    user  system elapsed 
#>       0       0       0 
 system.time(cm <- crossprod(t(m))) # 0.16
#>    user  system elapsed 
#>   0.002   0.004   0.002 
 system.time(cm. <- tcrossprod(m))  # 0.02
#>    user  system elapsed 
#>   0.004   0.000   0.001 

 stopifnot(cm == as(tCmm, "matrix"))

 ## show sparse sub matrix
 tCmm[1:16, 1:30]
#> 16 x 30 sparse Matrix of class "dgCMatrix"
#>                                                                  
#>  [1,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
#>  [2,] . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
#>  [3,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
#>  [4,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
#>  [5,] . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . .
#>  [6,] . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . .
#>  [7,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
#>  [8,] . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . .
#>  [9,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
#> [10,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
#> [11,] . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . .
#> [12,] . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . .
#> [13,] . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . .
#> [14,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
#> [15,] . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . .
#> [16,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .