MATH 372 -- Numerical Linear Algebra

Jacobi Iteration Example

April 11, 2007


% We wish to solve the linear system 
% Ax = b for coefficient matrix and
% right-hand side vector A, b as follows:

>> A = [4 1 2; 3 5 1; 1 1 3]

A =

     4     1     2
     3     5     1
     1     1     3

>> b = [1; 0; 0]

b =

     1
     0
     0

% The diagonal entries in A form the diagonal
% matrix D; the Rest is L + U 

>> D = [4 0 0; 0 5 0; 0 0 3]

D =

     4     0     0
     0     5     0
     0     0     3

>> Rest = A - D

Rest =

     0     1     2
     3     0     1
     1     1     0

% Jacoby iteration in matrix form is the 
% iteration scheme  x^(k+1) = D^(-1)(-(L+U)x^(k) + b)

ID = inv(D);

for i = 1:20
    x = -ID*Rest*x + ID*b
end

% after 5th iteration:

x =

    0.3303
   -0.1669
   -0.0364


% after 10th iteration

x =

    0.3156
   -0.1846
   -0.0479



% after 15th iteration

x =

    0.3187
   -0.1812
   -0.0449


% after 20th iteration

x =

    0.3181
   -0.1819
   -0.0456

% MATLAB's solution:

>> A \ b

ans =

    0.3182
   -0.1818
   -0.0455