MATH 372 -- Numerical Linear Algebra

Gauss-Seidel Iteration Example

April 13, 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


>> DPlusL=[4 0 0; 3 5 0; 1 1 3]

DPlusL =

     4     0     0
     3     5     0
     1     1     3

>> DLI=inv(DPlusL)

DLI =

    0.2500         0         0
   -0.1500    0.2000         0
   -0.0333   -0.0667    0.3333

>> x = [0; 0; 0]

x =

     0
     0
     0

>> for i = 1:7 
   end
>> U = A - DPlusL

U =

     0     1     2
     0     0     1
     0     0     0

>> for i=1:10
     x = -DLI*U*x + DLI*b
   end

x =

    0.2500
   -0.1500
   -0.0333


x =

    0.3042
   -0.1758
   -0.0428


x =

    0.3153
   -0.1807
   -0.0449


x =

    0.3176
   -0.1816
   -0.0453


x =

    0.3181
   -0.1818
   -0.0454


x =

    0.3182
   -0.1818
   -0.0454


x =

    0.3182
   -0.1818
   -0.0455

% MATLAB's solution:

>> A \ b

ans =

    0.3182
   -0.1818
   -0.0455

% Note: agreement to at least 4 decimal places after
% only 7 iterations versus more than 20 for Jacobi
% to achieve this level of accuracy.