Stationary Iterative Methods for Linear Systems

Copyright (C) 2020 Andreas Kloeckner

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In [2]:
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 [3]:
n = 50



mesh = np.linspace(0, 1, n)

h = mesh[1] - mesh[0]

Set up the system matrix A to carry out centered finite differences

$$ u''(x)\approx \frac{u(x+h) - 2u(x) + u(x-h)}{h^2}. $$

Use np.eye(n, k=...). What needs to be in the first and last row?

In [5]:
#clear

A = (np.eye(n, k=1) + -2*np.eye(n) + np.eye(n, k=-1))/h**2

A[0] = 0

A[-1] = 0

A[0,0] = 1

A[-1,-1] = 1

Next, fix the right hand side:

In [9]:
b = -30*mesh**2

b[0] = 1

b[-1] = -1

Compute a reference solution x_true to the linear system:

In [10]:
x_true = la.solve(A, b)

pt.plot(mesh, x_true)
Out[10]:
[<matplotlib.lines.Line2D at 0x7f1cd81932e8>]

Next, we'll try all the stationary iterative methods we have seen.

Jacobi

In [44]:
x = np.zeros(n)

Next, apply a Jacobi step:

In [42]:
x_new = np.empty(n)



for i in range(n):

    x_new[i] = b[i]

    for j in range(n):

        if i != j:

            x_new[i] -= A[i,j]*x[j]

        

    x_new[i] = x_new[i] / A[i,i]



x = x_new
In [43]:
pt.plot(mesh, x)

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

pt.legend()
Out[43]:
<matplotlib.legend.Legend at 0x7f1cd2f38ef0>

  • Ideas to accelerate this?

  • Multigrid

Gauss-Seidel

In [45]:
x = np.zeros(n)
In [64]:
x_new = np.empty(n)



for i in range(n):

    x_new[i] = b[i]

    for j in range(i):

        x_new[i] -= A[i,j]*x_new[j]

    for j in range(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[64]:
<matplotlib.legend.Legend at 0x7f1cd28169e8>

And now Successive Over-Relaxation ("SOR")

In [92]:
x = np.zeros(n)
In [106]:
x_new = np.empty(n)



for i in range(n):

    x_new[i] = b[i]

    for j in range(i):

        x_new[i] -= A[i,j]*x_new[j]

    for j in range(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[106]:
(-1.3, 1.3)
In [ ]: