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.