fortranopenmpblasintel-mkl

# Repeated single precison complex matrix vector multiplication (speed and accuracy improvement)

I've boiled a long running function down to a "simple" series of matrix vector multiplications. The matrix does not change, but there are a a lot of vectors. I have put together a test program with the current state of the algorithm.

I've chased a few options for performance, but what is below is the best I have and it seems to work pretty well.

``````module maths

contains
subroutine lots_of_MVM(Y,P,t1,t2,startRow)
implicit none
! args
complex, intent(in), contiguous    :: Y(:,:),P(:,:)
complex, intent(inout), contiguous :: t1(:,:),t2(:,:)
integer, intent(in)                :: startRow

! locals
integer :: ii,jj,zz,nrhs,n,pCol,tRow,yCol
! indexing
nrhs = size(P,2)/2
n    = size(Y,1)

! Do lots of maths
!\$OMP PARALLEL PRIVATE(jj,pCol,tRow,yCol,zz)
!\$OMP DO
do jj=1,nrhs
pCol = jj*2-1
tRow = startRow
do yCol=1,size(Y,2)
! This is faster than doing sum(P(:,pCol)*Y(:,yCol))
do zz=1,n
t1(tRow,jj) = t1(tRow,jj) + P(zz,pCol  )*Y(zz,yCol)
t2(tRow,jj) = t2(tRow,jj) + P(zz,pCol+1)*Y(zz,yCol)
end do
tRow = tRow + 1
end do
end do
!\$OMP END DO
!\$OMP END PARALLEL

end subroutine

end module

program test
use maths
use omp_lib
implicit none
! variables
complex, allocatable,dimension(:,:) :: Y,P,t1,t2
integer :: n,nrhs,nY,yStart,yStop,mult
double precision startTime

! setup (change mult to make problem larger)
! real problem size mult = 1000 to 2000
mult = 300
n = 10*mult
nY = 30*mult
nrhs = 20*mult
yStart = 5
yStop  = yStart + nrhs - 1

! allocate
allocate(Y(n,nY),P(n,nrhs*2))
allocate(t1(nrhs,nrhs),t2(nrhs,nrhs))

! make some data
call random_number(Y%re)
call random_number(Y%im)
call random_number(P%re)
call random_number(P%im)
t1 = 0
t2 = 0

! do maths
startTime = omp_get_wtime()
call lots_of_MVM(Y(:,yStart:yStop),P,t1,t2,1)
write(*,*) omp_get_wtime()-startTime
end program
``````

Things I tried for performance (maybe incorrectly)

• Alignment of the data to 64 byte boundaries (start of matrix and each columns) I used associated directive to tell the compiler this and it seemed to make no difference. This was implement by copying Y and P with extra padding. I would like to avoid this increase in memory anyways.
• MKL cgemv_batch_strided. The OMP nested do loops wins over MKL. MKL is probably not optimized for stride 0 on the A matrix.
• Swapping 2nd and 3rd loops so t1 and t2 fill full columns. Requires in place transpose at the end

In addition to performance I would like better accuracy. More accuracy for similar performance would be acceptable. I tried a few things for this, but ended up with significantly slower performance.

Restrictions

• I can't just throw more cores at it with OMP. Probably will use only 2-4 cores
• Memory consumption should not increase drastically

Other info

• I'm using intel fortran compiler on RHEL 8
• Compiler flags -O3 -xHost
• Set mult variable to 1000 or 2000 for more representative problem size
• This code runs on a dual socket intel system at the moment, but could go to AMD.
• I want to run as many instance of this code as possible at a time. I'm currently able to run 24 concurrently on a dual 28 core processor system with 768GB of RAM. I am basically RAM and core limited for that run (2 cores per run)
• This is part of a larger code. Much of the rest is single threaded and its not trivial to make it multi-threaded. I'm targeting this section because it is the most time consuming portion.
• I have implemented this using CUDA API batched cgemv on GPU (GV100). It is much faster there (6x), but the CPU wins on total throughput as a single run on the GPU saturates compute or memory bandwidth.

Solution

• Rewriting it using `cgemm`/`zgenn` increases the speed by a factor of about 4-10 using either ifort or gfortran with openblas. Here's the code I knocked together:

``````Module Precision_module

Use, Intrinsic :: iso_fortran_env, Only : wp => real64
!  Use, Intrinsic :: iso_fortran_env, Only : wp => real32

Implicit None

Private

Public :: wp

End Module Precision_module

Module maths

!  Use, Intrinsic :: iso_fortran_env, Only : wp => real64

Use Precision_module, Only : wp

Contains
Subroutine lots_of_MVM(Y,P,t1,t2,startRow)
Implicit None
! args
Complex( wp ), Intent(in), Contiguous    :: Y(:,:),P(:,:)
Complex( wp ), Intent(inout), Contiguous :: t1(:,:),t2(:,:)
Integer, Intent(in)                :: startRow

! locals
Integer :: jj,zz,nrhs,n,pCol,tRow,yCol
! indexing
nrhs = Size(P,2)/2
n    = Size(Y,1)

! Do lots of maths
!\$OMP PARALLEL PRIVATE(jj,pCol,tRow,yCol,zz)
!\$OMP DO
Do jj=1,nrhs
pCol = jj*2-1
tRow = startRow
Do yCol=1,Size(Y,2)
! This is faster than doing sum(P(:,pCol)*Y(:,yCol))
Do zz=1,n
t1(tRow,jj) = t1(tRow,jj) + P(zz,pCol  )*Y(zz,yCol)
t2(tRow,jj) = t2(tRow,jj) + P(zz,pCol+1)*Y(zz,yCol)
End Do
tRow = tRow + 1
End Do
End Do
!\$OMP END DO
!\$OMP END PARALLEL

End Subroutine lots_of_MVM

End Module maths

Program test

Use, Intrinsic :: iso_fortran_env, Only : numeric_storage_size, real32

Use Precision_module, Only : wp

Use maths
Use omp_lib, Only : omp_get_wtime, omp_get_max_threads
Implicit None

! variables
Complex( wp ), Allocatable,Dimension(:,:) :: Y,P,t1,t2
Integer :: n,nrhs,nY,yStart,yStop,mult
Real( wp ) :: startTime

Complex( wp ), Allocatable, Dimension( :, : ) :: t3, t4
Real( wp ) :: mem_reqd, mem_reqd_Gelements
Real( wp ) :: tloop, tblas

! setup (change mult to make problem larger)
! real problem size mult = 1000 to 2000
mult = 300
!mult = 50 ! for debug
n = 10*mult
nY = 30*mult
nrhs = 20*mult
yStart = 5
yStop  = yStart + nrhs - 1

! allocate
Allocate(Y(n,nY),P(n,nrhs*2))
Allocate(t1(nrhs,nrhs),t2(nrhs,nrhs))

mem_reqd = Size( Y ) + Size( P ) + Size( t1 ) + Size( t2 )
mem_reqd_Gelements = mem_reqd / ( 1024.0_wp * 1024.0_wp * 1024.0_wp )
Write( *, * ) 'Mem reqd: ', mem_reqd_Gelements, ' Gelements'

! make some data
Call random_Number(Y%re)
Call random_Number(Y%im)
Call random_Number(P%re)
Call random_Number(P%im)
t1 = 0
t2 = 0

! do maths
Write( *, * ) 'Using ', omp_get_max_threads(), ' threads'
Write( *, * ) 'Using ', Merge( 'single', 'double', Kind( y ) == real32 ), ' precision'
startTime = Real( omp_get_wtime(), wp )
Call lots_of_MVM(Y(:,yStart:yStop),P,t1,t2,1)
tloop = Real( omp_get_wtime(), wp ) - startTime
Write(*,*) 'TLoop: ', tloop

Allocate( t3, mold = t1 )
Allocate( t4, mold = t2 )

t3 = 0.0_wp
t4 = 0.0_wp

startTime = Real( omp_get_wtime(), wp )
Call zgemm( 'T', 'N', nrhs, nrhs, n, ( 1.0_wp, 0.0_wp ), Y ( 1, ystart ), n    , &
P ( 1, 1      ), 2 * n, &
( 1.0_wp, 0.0_wp ), t3             , nrhs )
Call zgemm( 'T', 'N', nrhs, nrhs, n, ( 1.0_wp, 0.0_wp ), Y ( 1, ystart ), n    , &
P ( 1, 2      ), 2 * n, &
( 1.0_wp, 0.0_wp ), t4             , nrhs )
tblas = Real( omp_get_wtime(), wp ) - startTime
Write(*,*) 'TBlas: ', tblas
Write( *, * ) 'Time ratio ', tloop / tblas, ' ( big means blas better )'
Write( *, * ) "Max diff in t1 ", Maxval( Abs( t3 - t1 ) )
Write( *, * ) "Max diff in t2 ", Maxval( Abs( t4 - t2 ) )

End Program test
``````

Note I have used double precision throughout as I know what to expect in terms of errors here. Results for ifort on 2 threads:

``````ijb@ijb-Latitude-5410:~/work/stack\$ ifort -O3 -qopenmp mm.f90 -lopenblas
ijb@ijb-Latitude-5410:~/work/stack\$ ./a.out
Mem reqd:   0.125728547573090       Gelements
Using            2  threads
Using double precision
TLoop:    71.4670290946960
TBlas:    17.8680319786072
Time ratio    3.99971464010481       ( big means blas better )
Max diff in t1   1.296029998720414E-011
Max diff in t2   1.273302296896508E-011
``````

Results for gfortran:

``````ijb@ijb-Latitude-5410:~/work/stack\$ gfortran-12 -fopenmp -O3 -Wall -Wextra -pedantic -Werror -std=f2018  mm.f90 -lopenblas
ijb@ijb-Latitude-5410:~/work/stack\$ ./a.out
Mem reqd:   0.12572854757308960       Gelements
Using            2  threads
Using double precision
TLoop:    185.08875890000490
TBlas:    16.093782140000258
Time ratio    11.500637779852656       ( big means blas better )
Max diff in t1    1.2732928769591198E-011
Max diff in t2    1.3642443193628513E-011
``````

Those differences are about what I would expect for double precision.

If the arguments to `zgemm` are confusing take a look at BLAS LDB using DGEMM

Running in single precision (and changing the call to be to cgemm) has a comparable story, obviously with bigger differences, around 10^-3 to 10^-4, at least for gfortran. As I don't use single precision in my own work I have less of a feel for what to expect here, but this doesn't seem unreasonable:

`````` ijb@ijb-Latitude-5410:~/work/stack\$ ./a.out
Mem reqd:   0.125728548      Gelements
Using            2  threads
Using single precision
TLoop:    147.786453
TBlas:    8.18331814
Time ratio    18.0594788      ( big means blas better )
Max diff in t1    7.32427742E-03
Max diff in t2    6.83614612E-03
``````

As for what precision you want and what you consider accurate, well you don't say so I can't really address that, save to say the simplest way is to move to double precision if you can take the memory hit - the use of zgemm will easily outweigh any performance hit you would take. But for performance it's the same story for any code - if you can rewrite it in terms of a matrix multiply then you will win.