human
I am working on the file ".ipynb_checkpoints/Making_a_Preconditioner-checkpoint.ipynb".
The first task is:
Let us define the Poisson matrix also as a Linear Operator
Can you write Python program for it?
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])
    ut=np.zeros([nx,ny])
    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. Homogenous Dirichlet
    u[ 0,:] = -u[ 1,:]
    u[-1,:] = -u[-2,:]
    u[:, 0] = -u[:, 1]
    u[:,-1] = -u[:,-2]
  
    for i in range(1,nx+1):
      for j in range(1,ny+1):
        ut[i-1,j-1]=(Ax*(u[i+1,j]+u[i-1,j])+Ay*(u[i,j+1]+u[i,j-1]) - 2.0*(Ax+Ay)*u[i,j])
    return ut.reshape(v.shape)
  return mv