I want to compute the summation more efficient. There are nested loops, which I want to avoid.
! If i+j <= I,then A_{j} = \sum_{m,n,i} C_{m, i+j} G_{ m, n, i, j} C_{n, i}
! else if i+j >= I, then A_{j} = \sum_{m,n,i} C_{m, i+j-I} G_{ m, n, i, j} C_{n, i}
program main
implicit none
real, allocatable :: A(:)
real, allocatable :: C(:,:), G(:,:,:,:)
integer :: i, j, m, n
integer, parameter :: N = 1500, I = 2000
allocate(A(J))
allocate(C(N,I))
allocate(G(N,N,I,I))
! If i+j <= I,then
! A_{j} = \sum_{m,n,i} C_{m, i+j} G_{ m, n, i, j} C_{n, i}
! else if i+j >= I, then
! A_{j} = \sum_{m,n,i} C_{m, i+j-I} G_{ m, n, i, j} C_{n, i}
do j = 1, I
do i = 1, I
if ( i + j <= I ) then
do n = 1, N
do m = 1, N
A(j) = A(j) + C(m,i+j) * G(m,n,i,j) * C(n,i)
end do
end do
else
do n = 1, N
do m = 1, N
A(j) = A(j) + C(m,i+j-I) * G(m,n,i,j) * C(n,i)
end do
end do
end if
end do
end do
end program main
Let's start by addressing @francescalus' comment, and rename I and N. Let's call them II and NN. No, this is not a good naming convention, but I don't know what these variables actually are.
Let's also assume you've initialised your variables, as per @lastchance's comment.
The first thing that leaps out is the lines
do j=1,II
do i=1,II
if (i+j <= II) then
...
else
...
this loop-with-an-if-inside can be replaced with a couple of loops with no if, which is usually a good idea. Something like:
do j=1,II
do i=1,II-j
...
end do
do i=II-j+1,II
...
end do
Now let's use matmul to clean up the loops over m and n, something like:
do j=1,II
do i=1,II-j
A(j) = A(j) + dot_product(matmul(C(:,i+j), G(:,:,i,j)), C(:,i))
end do
do i=II-j+1,II
A(j) = A(j) + dot_product(matmul(C(:,i+j-I), G(:,:,i,j)), C(:,i))
end do
end do
Note that this isn't doing any less work than your code (and indeed, since you're looping over all four indices of G, doing less work doesn't look possible), but it should at least do the work a little more efficiently.