# Jacobi method

(Difference between revisions)
 Revision as of 20:47, 15 December 2005 (view source)Tsaad (Talk | contribs) (towards a uniform notation for linear systems : A*Phi = B)← Older edit Revision as of 20:48, 15 December 2005 (view source)Tsaad (Talk | contribs) Newer edit → Line 11: Line 11: === Algorithm === === Algorithm === ---- ---- - :    Chose an intital guess $X^{0}$ to the solution
+ :    Chose an intital guess $\Phi^{0}$ to the solution
:    for k := 1 step 1 untill convergence do
:    for k := 1 step 1 untill convergence do
::  for i := 1 step until n do
::  for i := 1 step until n do

## Revision as of 20:48, 15 December 2005

We seek the solution to set of linear equations:

$A \cdot \Phi = B$

In matrix terms, the definition of the Jacobi method can be expressed as :
$\phi^{(k)} = D^{ - 1} \left( {L + U} \right)\phi^{(k - 1)} + D^{ - 1} B$
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 $\Phi^{0}$ 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
$\sigma = \sigma + a_{ij} \phi_j^{(k-1)}$
end if
end (j-loop)
$\phi_i^{(k)} = {{\left( {b_i - \sigma } \right)} \over {a_{ii} }}$
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 $\Phi$ are used.