```! Direct banded-matrix inversion of Poisson's equation
! ====================================================

USE rbmat   ! real banded matrix operations
USE gnuplot ! prepare for plotting

ny=20
nx=3∗ny
dvar=ny+1  ! band width
REAL base(0..nx,0..ny)=0,    ! 2D solution array
A(0..nx,0..ny,-dvar..dvar) ! banded matrix
A(**)=1  ! ∗∗ compounds the first two indices into 1; A is initialized to
! a 2D identity matrix in banded storage
LOOP FOR ix=1 TO nx-1 AND iy=1 TO ny-1
A(ix,iy,0)=4  ! center of the laplacian stencil
A(ix,iy,1)=-1 ! 1 step forward in the y direction
A(ix,iy,-1)=-1 ! 1 step backward in the y direction
A(ix,iy,dvar)=-1 ! 1 step forward in the x direction
A(ix,iy,-dvar)=-1 ! 1 step backward in the x direction
REPEAT
DO base(0,iy)=iy/ny FOR ALL iy ! initialize some Dirichlet boundary conditions
DO base(ix,ny)=1 FOR ix=1 TO ny
LOOP FOR ix=ny+1 TO nx-1 ! and Neumann boundary conditions
A(ix,ny,0)=4
A(ix,ny,-1)=-2
A(ix,ny,dvar)=-1
A(ix,ny,-dvar)=-1
REPEAT
LUdecomp A(**) ! banded LU decomposition
base(**)=A(**)\~ ! invert system
SPLOT base ! plot result
READ ! pause to see plot
```