In [30]:

from __future__ import division
import numpy as np
import scipy.linalg as la

import matplotlib.pyplot as pt

Let's solve \(u''=-30x^2\) with \(u(0)=1\) and \(u(1)=-1\).

In [61]:

n = 50

mesh = np.linspace(0, 1, n)
h = mesh[1] - mesh[0]

In [136]:

A = ...

(edit cell for solution)

In [145]:

b = -30*mesh**2
b[0] = 1
b[-1] = -1

In [138]:

x_true = la.solve(A, b)
pt.plot(mesh, x_true)

Out[138]:

[<matplotlib.lines.Line2D at 0xc1d4710>]

And now with Jacobi

In [440]:

x = np.zeros(n)

In [444]:

x_new = np.empty(n)

for i in xrange(n):
    x_new[i] = b[i]
    for j in xrange(i):
        x_new[i] -= ...
    for j in xrange(i+1, n):
        x_new[i] -= ...
        
    x_new[i] = ...

x = x_new
pt.plot(mesh, x)
pt.plot(mesh, x_true, label="true")
pt.legend()

Out[444]:

<matplotlib.legend.Legend at 0x204e4a10>

(Edit cell for solution)

Ideas to accelerate this?
Multigrid

And now Gauss-Seidel

In [148]:

x = np.zeros(n)

In [425]:

x_new = np.empty(n)

for i in xrange(n):
    x_new[i] = b[i]
    for j in xrange(i):
        x_new[i] -= A[i,j]*x_new[j]
    for j in xrange(i+1, n):
        x_new[i] -= A[i,j]*x[j]
        
    x_new[i] = x_new[i] / A[i,i]

x = x_new
pt.plot(mesh, x)
pt.plot(mesh, x_true, label="true")
pt.legend()

Out[425]:

<matplotlib.legend.Legend at 0x1f603ad0>

And now Successive Over-Relaxation ("SOR")

In [426]:

x = np.zeros(n)

In [439]:

x_new = np.empty(n)

for i in xrange(n):
    x_new[i] = b[i]
    for j in xrange(i):
        x_new[i] -= A[i,j]*x_new[j]
    for j in xrange(i+1, n):
        x_new[i] -= A[i,j]*x[j]
        
    x_new[i] = x_new[i] / A[i,i]

direction = x_new - x
omega = 1.5
x = x + omega*direction

pt.plot(mesh, x)
pt.plot(mesh, x_true, label="true")
pt.legend()
pt.ylim([-1.3, 1.3])

Out[439]:

(-1.3, 1.3)

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