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c81f60b852
AVX-512ERI is available on all Knights Landing i.e. Intel Xeon Phi 7200 processors. These instructions may not be available on all Intel processors that support AVX-512F.
124 lines
4.4 KiB
Fortran
124 lines
4.4 KiB
Fortran
! In any Gaussian based QM code the performance of the 2-electron
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! integrals is likely to be critical (unless the wave function is
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! represented on a grid). Recent reviews on integral evaluations is
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! [1] and [2].
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! Wim Klopper suggested that the evaluation of the integrals could be
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! performed faster by approximating the natural exponent functions (EXP)
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! as a Taylor series [3]. To see to what extent using a Taylor series
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! may improve the performance this kernel was written. Given the order
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! of the Taylor series all other quantities are automatically
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! initialized. We also use the fact that we are interested only in
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! exp(x) for ln(eps) <= x <= 0 (where eps is the machine precision).
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! For x < ln(eps) we assume exp(x)=0.
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! If nord = 4 then the Taylor series approximation is 1.5 times faster
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! than the exp function on a Intel(R) Xeon(R) CPU E5-2670 0 @ 2.60GHz.
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! The interpolation table requires 1.4 MB of memory. The memory
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! requirements mean that the algorithm could be badly impacted by cache
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! misses. Knights Landing does not have such big caches. Also the Volta
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! GPU is unlikely going to have such big caches. In addition Knights
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! Landing has added AVX-512ER instructions to perform exponentiations
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! of the form 2^x. These might help boost the performance of the exp
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! function well beyond what a Taylor series brings. These instructions
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! are not available on all Intel processors though.
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!
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! [1] Simen Reine, Trygve Helgaker, Roland Lindh, "Multi-electron
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! integrals", WIREs Comput. Mol. Sci. 2012, 2:290-303,
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! doi:10.1002/wcms.78
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! [2] Justin T. Fermann, Edward F. Valeev, "Fundamentals of Molecular
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! Integrals Evaluation",
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! https://www.valeevgroup.chem.vt.edu/docs/ints.07032015.pdf
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! [3] Wim Klopper, "SCF methods, basis sets, and integrals:
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! Lecture IV: Integrals", ESQC 2011, Torre Normanna, 9/18 –
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! 10/01/2011,
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! http://esqc.ups-tlse.fr/11/lectures/KlopperLecture4-1x2.pdf
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! Slide: "Calculating the Boys function".
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!
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module taylor
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double precision, allocatable :: ts(:) ! table of exp(x_i)
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double precision :: h ! step size
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double precision :: xm ! x_min (exp(x) is needed only in the range
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! x_min <= x <= 0)
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double precision :: xmh ! = xm+h/2
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integer, parameter :: nord = 4 ! the maximum order
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double precision, parameter :: eps = 1.0d-16 ! machine precision
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contains
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subroutine t_init
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! Initialize the interpolation table
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implicit none
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integer :: npt ! number of points
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integer :: ipt ! current point
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double precision :: a, x
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xm = dlog(eps)
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h = 2.0d0*(eps**(1.0d0/dble(nord)))
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xmh = xm+0.5d0*h
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npt = (-xm/h)+1
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allocate(ts(npt))
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do ipt = 1, npt
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x=xmh+h*(ipt-1)
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ts(ipt) = dexp(x)
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enddo
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write(*,*)'xm ',xm
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write(*,*)'h ',h
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write(*,*)'npt',npt,' requires ',dble(8*npt)/1024,' KB'
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end subroutine t_init
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double precision function t_exp(x)
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! Use Taylor series to evaluate exp(x)
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implicit none
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double precision, intent(in) :: x
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double precision :: xi, xd
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double precision :: r
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integer ipt
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integer iord
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ipt = (x-xmh)/h
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xi = xmh+h*(ipt-1)
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xd = x-xi
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r = xd/dble(nord)+1.0d0
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do iord = nord-1, 1, -1
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r = r*xd/dble(iord)+1.0d0
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enddo
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t_exp = r*ts(ipt)
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end function t_exp
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end module taylor
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program performance
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use taylor
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implicit none
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#include "mpif.h"
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integer :: ierr, nmax, i
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double precision :: a, b
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double precision :: tt, te, t0
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nmax = 10000000
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a = -30.0d0
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call mpi_init(ierr)
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call t_init
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t0 = -mpi_wtime()
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b = 0.0d0
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do i = 1, nmax
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b = b + a/i
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enddo
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t0 = t0 + mpi_wtime()
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write(*,*)'Null time: ',t0,b
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tt = -mpi_wtime()
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b = 0.0d0
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do i = 1, nmax
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b = b + t_exp(a/i)
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enddo
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tt = tt + mpi_wtime()
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write(*,*)'t-exp time: ',tt-t0,b
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te = -mpi_wtime()
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b = 0.0d0
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do i = 1, nmax
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b = b + exp(a/i)
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enddo
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te = te + mpi_wtime()
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write(*,*)'Exp time: ',te-t0,b
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write(*,*)'Exp/t-exp: ',(te-t0)/(tt-t0)
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! do i = 1, 1000
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! write(*,*)a/i,exp(a/i),t_exp(a/i)
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! enddo
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call mpi_finalize(ierr)
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end
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