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Next: The Gauss-Seidel Method
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Next Page: Convergence of the Jacobi method
The Jacobi method is easily derived by examining each of
the n equations in the linear system in isolation. If in
the
th equation
we solve for the value of while assuming the other entries
of
remain fixed, we obtain
This suggests an iterative method defined by
which is the Jacobi method. Note that the order in which the equations are examined is irrelevant, since the Jacobi method treats them independently. For this reason, the Jacobi method is also known as the method of simultaneous displacements, since the updates could in principle be done simultaneously.
In matrix terms, the definition of the Jacobi method
in () can be expressed as
where the matrices ,
and
represent the diagonal, the
strictly lower-triangular, and the strictly upper-triangular parts of
,
respectively.
The pseudocode for the Jacobi method is given in Figure .
Note that an auxiliary storage vector,
is used in the
algorithm. It is not possible to update the vector
in place,
since values from
are needed throughout the
computation of
.
