Iterative techniques are suited for large matrices. A simple iterative technique for solving linear equations is the Jacobi iteration, which is suited for matrices that have nonzero diagonal elements. Assume we are given the system of linear equations

(20.47) c20e047

we can express the ith row of the above system explicitly:

(20.48) c20e048

where we have isolated the term involving xi. We can “solve” for xi from the above equation as

(20.49) c20e049

Of course, we need to iterate several times before we converge to the correct solution. At iteration k, we can estimate c20ue006 as

(20.50) c20e050

Gauss–Seidel iteration differs from the Jacobi iteration in that it uses the most recently found values of c20ue007 in the iterations:

(20.51) c20e051

The order of evaluation of ...

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