Gauss-Seidel Iteration
Each iteration of Jacobi method updates the whole set of N variables at a time. However, so long as we
do not use a multiprocessor computer capable of parallel processing, each one of N variables is updated
sequentially one by one. Therefore, it is no wonder that we could speed up the convergence by using all the
most recent values of variables for updating each variable even in the same iteration as follows:
1
2
x1,k+1 = − x2,k + ,
3
3
1
1
x2,k+1 = − x1,k −
3
2
This scheme is called Gauss-Seidel iteration, which can be generalized for an N × N matrix-vector equation
as follows:
P
P
k+1
k
bm − m−1
− N
(k+1)
n=1 amn xn
n=m+1 amn xn
xm
=
amm
for m = 1, . . . , N and for each time stage k
This is implemented in the following MATLAB routine gauseid(), which we will use to solve the above
equation.
function X = gauseid(A,B,X0,kmax)
%This function finds x = A^-1 B by Gauss-Seidel iteration.
if nargin < 4, tol = 1e-6; kmax = 100;
elseif kmax < 1, tol = max(kmax,1e-16); kmax = 1000;
else tol = 1e-6;
end if nargin < 4, tol = 1e-6; kmax = 100; end
if nargin < 3, X0 = zeros(size(B)); end
NA = size(A,1); X = X0;
for k = 1: kmax
X(1,:) = (B(1,:)-A(1,2:NA)*X(2:NA,:))/A(1,1);
for m = 2:NA-1
tmp = B(m,:)-A(m,1:m-1)*X(1:m - 1,:)-A(m,m + 1:NA)*X(m + 1:NA,:);
X(m,:) = tmp/A(m,m); %Eq.(2.5.4)
end
X(NA,:) = (B(NA,:)-A(NA,1:NA - 1)*X(1:NA - 1,:))/A(NA,NA);
if nargout == 0, X, end %To see the intermediate results
if norm(X - X0)/(norm(X0) + eps)<tol, break; end
X0 = X;
end
>>A = [3 2;1 2]; b = [1 -1]’; %the coefficient matrix and RHS vector
>>x0 = [0 0]ß; %the initial value
>>gauseid(A,b,x0,10) %omit output argument to see intermediate results
X = 0.3333 0.7778 0.9259 0.9753 0.9918 ......
-0.6667 -0.8889 -0.9630 -0.9877 -0.9959 ......
As with the Jacobi iteration in the previous section, we can see this Gauss-Seidel iteration converging to
the true solution xO = [1 − 1]T and that with fewer iterations. But, if we use a multiprocessor computer
capable of parallel processing, the Jacobi iteration may be better in speed even with more iterations, since it
can exploit the advantage of simultaneous parallel computation.
Note that the Jacobi/Gauss-Seidel iterative scheme seems unattractive and even unreasonable if we are
given a standard form of linear equations as
Ax = b
because the computational overhead for converting it into the form of Jacobi iteration may be excessive.