[Rd] Why is matrix product slower when matrix has very small values?

[Serguei Sokol]

Le 20/11/2019 à 09:56, Hilmar Berger a écrit :
> Hi Florian,
>
> just a guess, but couldn't it be that the multiplication of very small 
> values leads to FP underflow exceptions which have to be handled by 
> BLAS in a less efficient way than "normal" multiplications handled by 
> SIMD instructions ?
Another guess is that you are caught by what is called "denormal 
numbers" https://en.wikipedia.org/wiki/Denormal_number.
Arithmetic operations on them are different and slower that those on 
"normal" numbers.

Best,
Serguei.

>
> Best regards,
> Hilmar
>
> On 19/11/2019 15:09, Florian Gerber wrote:
>> Hi,
>>
>> I experience surprisingly large timing differences for the
>> multiplication of matrices of the same dimension. An example is given
>> below. How can this be explained?
>> I posted the question on Stackoverflow:
>> https://stackoverflow.com/questions/58886111/r-why-is-matrix-product-slower-when-matrix-has-very-small-values 
>>
>> Somebody could reproduce the behavior but I did not get any useful
>> explanations yet.
>>
>> Many thanks for hints!
>> Florian
>>
>> ## disable openMP
>> library(RhpcBLASctl); blas_set_num_threads(1); omp_set_num_threads(1)
>>
>> A <- exp(-as.matrix(dist(expand.grid(1:60, 1:60))))
>> summary(c(A))
>> #     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
>> # 0.000000 0.000000 0.000000 0.001738 0.000000 1.000000
>>
>> B <- exp(-as.matrix(dist(expand.grid(1:60, 1:60)))*10)
>> summary(c(B))
>> #      Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
>> # 0.0000000 0.0000000 0.0000000 0.0002778 0.0000000 1.0000000
>>
>> identical(dim(A), dim(B))
>> ## [1] TRUE
>>
>> system.time(A %*% A)
>> #    user  system elapsed
>> #   2.387   0.001   2.389
>> system.time(B %*% B)
>> #    user  system elapsed
>> #  21.285   0.020  21.310
>>
>> sessionInfo()
>> # R version 3.6.1 (2019-07-05)
>> # Platform: x86_64-pc-linux-gnu (64-bit)
>> # Running under: Linux Mint 19.2
>>
>> # Matrix products: default
>> # BLAS:   /usr/lib/x86_64-linux-gnu/openblas/libblas.so.3
>> # LAPACK: /usr/lib/x86_64-linux-gnu/libopenblasp-r0.2.20.so
>>
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>