Example: jacobi method in python
import numpy as np
from numpy.linalg import *
def jacobi(A, b, x0, tol, maxiter=200):
"""
Performs Jacobi iterations to solve the line system of
equations, Ax=b, starting from an initial guess, ``x0``.
Terminates when the change in x is less than ``tol``, or
if ``maxiter`` [default=200] iterations have been exceeded.
Returns 3 variables:
1. x, the estimated solution
2. rel_diff, the relative difference between last 2
iterations for x
3. k, the number of iterations used. If k=maxiter,
then the required tolerance was not met.
"""
n = A.shape[0]
x = x0.copy()
x_prev = x0.copy()
k = 0
rel_diff = tol * 2
while (rel_diff > tol) and (k < maxiter):
for i in range(0, n):
subs = 0.0
for j in range(0, n):
if i != j:
subs += A[i,j] * x_prev[j]
x[i] = (b[i] - subs ) / A[i,i]
k += 1
rel_diff = norm(x - x_prev) / norm(x)
print(x, rel_diff)
x_prev = x.copy()
return x, rel_diff, k
GL = 1.6
d = 0.8
A = np.array([
[1.0, 0, 0, 0, 0],
[ GL, -(d+1), 1.0, 0, 0],
[ 0, d, -(d+1), 1.0, 0],
[ 0, 0, d, -(d+1), 1.0],
[ 0, 0, 0, 0, 1.0]])
b = [0.5, 0, 0, 0, 0.1]
x0 = np.zeros(5);
tol = 1E-9
maxiter = 200
x, rel_diff, k = jacobi(A, b, x0, tol, maxiter)
if k == maxiter:
print(('WARNING: the Jacobi iterations did not '
'converge within the required tolerance.'))
print(('The solution is %s; within a tolerance of %g, '
'using %d iterations.' % (x, rel_diff, k)))
print('Solution error = norm(Ax-b) = %g' % \
norm(np.dot(A,x)-b))
print('Condition number of A = %0.5f' % cond(A))
print('Solution from built-in functions = %s' % solve(A, b))