According to the documentation
gemm!(tA, tB, alpha, A, B, beta, C)
Update C as alpha * A * B + beta*C or the other three variants according to tA (transpose A) and tB. Returns the updated C.
Note: here, alpha and beta must be float type scalars. A, B and C are all matrices. It's up to you to make sure the matrix dimensions match.
Thus, the tA and tB parameters refer to whether you want to apply the transpose operation to A or to B before multiplying. Note that this will cost you some computation time and allocations - the transpose isn't free. (thus, if you were going to apply the multiplication many times, each time with the same transpose specification, you'd be better off storing your matrix as the transposed version from the beginning). Select N for no transpose, T for transpose. You must select one or the other.
The difference between gemm!() and gemv!() is that for gemm!() you already need to have allocated the matrix C. The ! is a "modify in place" signal. Consider the following illustration of their different uses:
A = rand(5,5)
B = rand(5,5)
C = Array(Float64, 5, 5)
BLAS.gemm!('N', 'T', 1.0, A, B, 0.0, C)
D = BLAS.gemm('N', 'T', 1.0, A, B)
julia> C == D
true
Each of these, in essence, perform the calculation C = A * B'. (Technically, gemm!() performs C = (0.0)*C + (1.0)*A * B'.)
Thus, the syntax for the modify in place gemm!() is a bit unusual in some respects (unless you've already worked with a language like C in which case it seems very intuitive). You don't have the explicit = sign like you frequently do when calling functions in assigning values in a high level object oriented language like Julia.
As the illustration above shows, the outcome of gemm!() and gemm() in this case is identical, even though the syntax and procedure to achieve that outcome is a bit different. Practically speaking, however, performance differences between the two can be significant, depending on your use case. In particular, if you are going to be performing that multiplication operation many times, replacing/updating the value of C each time, then gemm!() can be a decent bit quicker because you don't need to keep re-allocating new memory each time, which does have time costs, both in the initial memory allocation and then in the garbage collection later on.
