# Jacobi method

(Difference between revisions)
 Revision as of 05:15, 15 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 18:04, 28 September 2005 (view source)Rparedes (Talk | contribs) Newer edit → Line 6: Line 6: In matrix terms, the definition of the Jacobi method can be expressed as :
In matrix terms, the definition of the Jacobi method can be expressed as :
$[itex] - x^{(k)} = D^{ - 1} \left( {L + U} \right)x^{(k - 1)} + D^{ - 1} q + x^{(k)} = D^{ - 1} \left( {L + U} \right)x^{(k - 1)} + D^{ - 1} Q$
[/itex]
Where '''D''','''L''' and '''U''' represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix '''A''' and k is iteration counter.
Where '''D''','''L''' and '''U''' represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix '''A''' and k is iteration counter.

## Revision as of 18:04, 28 September 2005

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 Jacobi method can be expressed as :
$x^{(k)} = D^{ - 1} \left( {L + U} \right)x^{(k - 1)} + D^{ - 1} Q$
Where D,L and U represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix A and k is iteration counter.

### 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 n do
if j != i then
$\sigma = \sigma + a_{ij} x_j^{(k-1)}$
end if
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)

Note: The major difference between the Gauss-Seidel method and Jacobi method lies in the fact that for Jacobi method the values of solution of previous iteration (here k) are used, where as in Gauss-Seidel method the latest available values of solution vector X are used.