human
I am working on the file ".ipynb_checkpoints/Making_a_Preconditioner-vectorized-checkpoint.ipynb".
The first task is:
Let us define the Poisson matrix also as a LinearOperator
Can you write Python program for it?

Give me a the Python program you requested?
assistant
#Here is the Python program you requested.

def Laplace(nx,ny):
  '''
  Action of the Laplace matrix on a vector v
  '''
  def mv(v):
    u =np.zeros([nx+2,ny+2])
  
    u[1:nx+1,1:ny+1]=v.reshape([nx,ny])
    dx=1.0/nx; dy=1.0/ny
    Ax=1.0/dx**2; Ay=1.0/dy**2
  
    #BCs. Needs to be generalized!
    u[ 0,:] = -u[ 1,:]
    u[-1,:] = -u[-2,:]
    u[:, 0] = -u[:, 1]
    u[:,-1] = -u[:,-2]

    ut = (Ax*(u[2:nx+2,1:ny+1]+u[0:nx,1:ny+1])
        + Ay*(u[1:nx+1,2:ny+2]+u[1:nx+1,0:ny])
        - 2.0*(Ax+Ay)*u[1:nx+1,1:ny+1])
    return ut.reshape(v.shape)
  A = LinearOperator((nx*ny,nx*ny), matvec=mv)
  return A