The Jacobi Method for linear system of equations

Description

MATLAB and Python code for Jacobi method :

 

Given the linear system of equations:

From the above equation, follows that:

The Jacobi method is an iterative method, which starts from an initial guess for the solution

Then, the solution in iteration k is used to find an approximation for the system solution in iteration k + 1. This is done as follows:

Generally, the solution in iteration

can be written in the form:

 

The Jacobi iteration stops when,

for some arbitrary

 

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Example 2.1 Write the first three iterations of the Jacobi method, for the linear system:

starting from the zeros vector

 

Solution:

We write:

1. First iteration k = 0:

2. Second iteration k = 1:

3. Third iteration k = 2:

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Example 2.2 The Jacobi method will be applied for solving the linear system:

MATLAB code :

function x = JacobiSolve(A, b, Eps)
n = length(b) ;
x0 = zeros(3, 1) ;
x = ones(size(x0)) ;
while norm(x-x0, inf) >= Eps
x0 = x ;
for i = 1 : n
x(i) = b(i) ;
for j = 1 : n
if j ~= i
x(i) = x(i) - A(i, j)*x0(j) ;
end
end
x(i) = x(i) / A(i, i) ;
end
end

Python code :