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Jacobi method

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(fixed dot product notation)
(towards a uniform notation for linear systems : A*Phi = B)
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We seek the solution to set of linear equations: <br>
We seek the solution to set of linear equations: <br>
-
:<math> A \cdot X = Q </math> <br>
+
:<math> A \cdot \Phi = B </math> <br>
-
For the given matrix '''A''' and vectors '''X''' and '''Q'''. <br>
 
In matrix terms, the definition of the Jacobi method can be expressed as : <br>
In matrix terms, the definition of the Jacobi method can be expressed as : <br>
<math>  
<math>  
-
x^{(k)}  = D^{ - 1} \left( {L + U} \right)x^{(k - 1)}  + D^{ - 1} Q
+
\phi^{(k)}  = D^{ - 1} \left( {L + U} \right)\phi^{(k - 1)}  + D^{ - 1} B
</math><br>
</math><br>
Where '''D''','''L''' and '''U''' represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix '''A''' and k is iteration counter.<br>
Where '''D''','''L''' and '''U''' represent the diagonal, lower triangular and upper triangular matrices of coefficient matrix '''A''' and k is iteration counter.<br>
Line 18: Line 17:
:::  for j := 1 step until n do <br>
:::  for j := 1 step until n do <br>
::::  if j != i then
::::  if j != i then
-
:::::      <math> \sigma  = \sigma  + a_{ij} x_j^{(k-1)} </math>
+
:::::      <math> \sigma  = \sigma  + a_{ij} \phi_j^{(k-1)} </math>
::::  end if
::::  end if
:::    end (j-loop) <br>
:::    end (j-loop) <br>
-
:::    <math>  x_i^{(k)}  = {{\left( {q_i - \sigma } \right)} \over {a_{ii} }} </math>
+
:::    <math>  \phi_i^{(k)}  = {{\left( {b_i - \sigma } \right)} \over {a_{ii} }} </math>
::  end (i-loop)
::  end (i-loop)
::  check if convergence is reached
::  check if convergence is reached
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----
----
-
'''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. <br>
+
'''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 <math>\Phi</math> are used. <br>
----
----
<i> Return to [[Numerical methods | Numerical Methods]] </i>
<i> Return to [[Numerical methods | Numerical Methods]] </i>

Revision as of 20:47, 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 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} \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.


Return to Numerical Methods

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