[Rd] accelerating matrix multiply

[Martin Maechler]

>>>>> Cohn, Robert S <robert.s.cohn at intel.com>
>>>>>     on Sat, 7 Jan 2017 16:41:42 +0000 writes:

> I am using R to multiply some large (30k x 30k double)
> matrices on a 64 core machine (xeon phi).  I added some timers
> to src/main/array.c to see where the time is going. All of the
> time is being spent in the matprod function, most of that time
> is spent in dgemm. 15 seconds is in matprod in some code that
> is checking if there are NaNs.

> > system.time (C <- B %*% A)
> nancheck: wall time 15.240282s
>    dgemm: wall time 43.111064s
>  matprod: wall time 58.351572s
>     user   system  elapsed 
> 2710.154   20.999   58.398
> 
> The NaN checking code is not being vectorized because of the
> early exit when NaN is detected:
> 
> 	/* Don't trust the BLAS to handle NA/NaNs correctly: PR#4582
> 	 * The test is only O(n) here.
> 	 */
> 	for (R_xlen_t i = 0; i < NRX*ncx; i++)
> 	    if (ISNAN(x[i])) {have_na = TRUE; break;}
> 	if (!have_na)
> 	    for (R_xlen_t i = 0; i < NRY*ncy; i++)
> 		if (ISNAN(y[i])) {have_na = TRUE; break;}
> 
> I tried deleting the 'break'. By inspecting the asm code, I
> verified that the loop was not being vectorized before, but
> now is vectorized. Total time goes down:
> 
> system.time (C <- B %*% A)
> nancheck: wall time  1.898667s
>    dgemm: wall time 43.913621s
>  matprod: wall time 45.812468s
>     user   system  elapsed 
> 2727.877   20.723   45.859
> 
> The break accelerates the case when there is a NaN, at the
> expense of the much more common case when there isn't a
> NaN. If a NaN is detected, it doesn't call dgemm and calls its
> own matrix multiply, which makes the NaN check time
> insignificant so I doubt the early exit provides any benefit.
> 
> I was a little surprised that the O(n) NaN check is costly
> compared to the O(n**2) dgemm that follows. I think the reason
> is that nan check is single thread and not vectorized, and my
> machine can do 2048 floating point ops/cycle when you consider
> the cores/dual issue/8 way SIMD/muladd, and the constant
> factor will be significant for even large matrices.
> 
> Would you consider deleting the breaks? I can submit a patch
> if that will help. Thanks.
> 
> Robert

Thank you Robert for bringing the issue up ("again", possibly).
Within R core, some have seen somewhat similar timing on some
platforms (gcc) .. but much less dramatical differences e.g. on
macOS with clang.

As seen in the source code you cite above, the current
implementation was triggered by a nasty BLAS bug .. actually
also showing up only on some platforms, possibly depending on
runtime libraries in addition to the compilers used.

Do you have R code (including set.seed(.) if relevant) to show
on how to generate the large square matrices you've mentioned in
the beginning?  So we get to some reproducible benchmarks?

With best regards,
Martin Maechler