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Gauss-Seidel method

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Revision as of 05:07, 15 September 2005 by Zxaar (Talk | contribs)
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We seek the solution to set of linear equations:

 A \bullet X = Q

For the given matrix A and vectors X and Q.
In matrix terms, the definition of the Gauss-Seidel method can be expressed as :
x^{(k)}  = \left( {D - L} \right)^{ - 1} \left( {Ux^{(k - 1)}  + Q} \right)
Where D,L and U represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix A and k is iteration counter.

The pseudocode for the Gauss-Seidel algorithm:


Chose an intital guess X^{0} to the solution
for k := 1 step 1 untill convergence do
for i := 1 step until n do
 \sigma = 0
for j := 1 step until i-1 do
 \sigma  = \sigma  + a_{ij} x_j^{(k)}
end (j-loop)
for j := i+1 step until n do
 \sigma  = \sigma  + a_{ij} x_j^{(k-1)}
end (j-loop)
  x_i^{(k)}  = {{\left( {q_i  - \sigma } \right)} \over {a_{ii} }}
end (i-loop)
check if convergence is reached
end (k-loop)

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