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Jacobi method

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Revision as of 01:22, 19 December 2005 by Jasond (Talk | contribs)
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We seek the solution to set of linear equations:

 A \phi = b

In matrix terms, the definition of the Jacobi method can be expressed as :

\phi^{(k+1)}  = D^{ - 1} \left[\left( {L + U} \right)\phi^{(k)}  +  b\right]

where D, L, and U represent the diagonal, lower triangular, and upper triangular parts of the coefficient matrix A and k is the iteration count. This matrix expression is mainly of academic interest, and is not used to program the method. Rather, an element-based approach is used:

\phi^{(k+1)}_i  = \frac{1}{a_{ii}} \left(b_i -\sum_{j\ne i}a_{ij}\phi^{(k)}_j\right),\, i=1,2,\ldots,n.

Note that the computation of \phi^{(k+1)}_i requires each element in \phi^{(k)} except itself. Then, unlike in the Gauss-Seidel method, we can't overwrite \phi^{(k)}_i with \phi^{(k+1)}_i, as that value will be needed by the rest of the computation. This is the most meaningful difference between the Jacobi and Gauss-Seidel methods. The minimum ammount of storage is two vectors of size n, and explicit copying will need to take place.


Chose an initial 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)
My wiki