R devel - Nov 2019 - Why is matrix product slower when matrix has very small values?

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

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 ?

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
>
> ______________________________________________
> R-devel at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-devel
-- 
Dr. Hilmar Berger, MD
Max Planck Institute for Infection Biology
Charit?platz 1
D-10117 Berlin
GERMANY

Phone:  + 49 30 28460 430
Fax:    + 49 30 28460 401
  
E-Mail: berger at mpiib-berlin.mpg.de
Web   : www.mpiib-berlin.mpg.de

Le 20/11/2019 ? 09:56, Hilmar Berger a ?crit?:
> 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
>>
>> ______________________________________________
>> R-devel at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-devel
>