import numpy as np

import scipy.sparse as sp

import matplotlib.pyplot as plt

import scipy.sparse.linalg as la

xmax = 10

ymax = 10

h = 0.1

Nx = int(xmax/h) #Nx is total number of points (including boundary) - 1

Ny = int(ymax/h)

def f(x,y):

return np.exp(-10*(x-5)**2 - 10*(y-5)**2)

reshaper0 = lambda u: np.reshape(u, (Nx-1)*(Ny-1), order = 'F')

reshaper1 = lambda u: np.reshape(u,(int(Ny)-1,int(Nx)-1))[::-1,:]

def grid_form(h, xmax, ymax):

return np.mgrid[h:xmax:h,h:ymax:h]

def L_form(h, Nx, Ny):

return -L

def sourcefn(h, Nx, Ny,grid, func):

return np.reshape(sourcefn, ((int(Nx)-1)*(int(Ny)-1)), order='F')

grid = grid_form(h, xmax, ymax)

u0 = reshaper0( f(grid[0],grid[1]) )

A = L_form(h, Nx, Ny)

source = sourcefn(h, Nx, Ny, grid, f)

residuals = []

def cback(rk):

residuals.append(rk)

xk, info = sp.linalg.gmres(A, source, tol = 1e-12, callback = cback, maxiter = 5000, restart = 5000)

save = residuals

fig = plt.figure()

fig.suptitle("GMRES", fontsize=10)

sour = plt.subplot(221)

sour.title.set_text('source function')

plt.imshow(reshaper1(source))

numba = 2

logresi = []

spsolvesol = []

eig = []

for i in [-40, 0, 40]:

key = 'key' + str(numba)

residuals = []

A += sp.eye(A.shape[0])*i

eigvalues = la.eigs(A)[0]

eig.append(eigvalues)

x = la.spsolve(A, source)

spsolvesol.append(x)

xk = sp.linalg.gmres(A, source, tol = 1e-12, callback = cback, maxiter = 5000, restart = 5000)[0]

lastrk = residuals[-1]

verify = abs(lastrk - np.linalg.norm(source - A*xk)/np.linalg.norm(source))

logresi.append(residuals)

print("Verification: Absolute difference = ", verify)

key = plt.subplot(220+numba)

key.title.set_text('$\gamma = %d$'%(i))

plt.imshow(reshaper1(x))

numba += 1

A -= sp.eye(A.shape[0])*i

plt.figure()

plt.semilogy(logresi[0], label = '$\gamma$ = -40')

plt.semilogy(logresi[1], label = '$\gamma$ = 0')

plt.semilogy(logresi[2], label = '$\gamma$ = +40')

plt.legend()

zeros = np.zeros(eig[0].shape)

fig3 = plt.figure()

fig3.suptitle("Eigenvalues", fontsize=10)

one = plt.subplot(311)

plt.scatter(eig[0],zeros)

one.title.set_text('$\gamma$ = -40')

two = plt.subplot(312)

plt.scatter(eig[1],zeros)

two.title.set_text('$\gamma$ = 0')

three = plt.subplot(313)

plt.scatter(eig[2],zeros)

three.title.set_text('$\gamma$ = 40')

plt.show()