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Let us take Jacobi’s Method one step further. Where the true solution is *x* = (*x*_{1}, *x*_{2}, … , *x*_{n}), if *x*_{1}^{(k+1)} is a better approximation to the true value of *x*_{1} than *x*_{1}^{(k)} is, then it would make sense that once we have found the *new* value *x*_{1}^{(k+1)} to use it (rather than the old value *x*_{1}^{(k)}) in finding *x*_{2}^{(k+1)}, … , *x*_{n}^{(k+1)}. So *x*_{1}^{(k+1)} is found as in Jacobi's Method, but in finding *x*_{2}^{(k+1)}, instead of using the *old* value of *x*_{1}^{(k)} and the old values *x*_{3}^{(k)}, … , *x*_{n}^{(k)}, we now use the *new* value *x*_{1}^{(k+1)} and the old values *x*_{3}^{(k)}, … , *x*_{n}^{(k)} , and similarly for finding *x*_{3}^{(k+1)}, … , *x*_{n}^{(k+1)}. This technique is called the Gauss-Seidel Method -- even though, as noted by Gil Strang in his *Introduction to Applied Mathematics*, Gauss didn’t know about it and Seidel didn’t recommend it. It is described by
,
so that
#### Example 2

.
.
.

This can also be written

That is,

Let's apply the Gauss-Seidel Method to the system from

At each step, given the current values *x*_{1}^{(k)}, *x*_{2}^{(k)}, *x*_{3}^{(k)},*x*_{1}^{(k+1)}, *x*_{2}^{(k+1)}, *x*_{3}^{(k+1)}

To compare our results from the two methods, we again choose *x*^{(0)} = (0, 0, 0). We then find *x*^{(1)} = (*x*_{1}^{(1)}, *x*_{2}^{(1)}, *x*_{3}^{(1)})

Let us be clear about how we solve this system. We first solve for *x*_{1}^{(1)}

*x*_{1}^{(1)} = 3/4 = 0.750.

We then solve for *x*_{2}^{(1)}*x*_{1}^{(1)} = 0.750,

*x*_{2}^{(1)} = [9 + 2(0.750)] / 6 = 1.750.

Finally, we solve for *x*_{3}^{(1)}*x*_{1}^{(1)} = 0.750*x*_{2}^{(1)} = 1.750,

*x*_{3}^{(1)} = [-6 + 0.750 − 1.750] / 7 = − 1.000.

The result of this first iteration of the Gauss-Seidel Method is

*x*^{(1)} = (*x*_{1}^{(1)}, *x*_{2}^{(1)}, *x*_{3}^{(1)})

We iterate this process to generate a sequence of increasingly better approximations *x*^{(0)}, *x*^{(1)}, *x*^{(2)}, …

k |
x^{(k)} |
x^{(k)} − x^{(k-1)} |
e^{(k)} = − xx^{(k)} |
||e^{(k)}|| |
||||||
---|---|---|---|---|---|---|---|---|---|---|

0 | -0.000 | -0.000 | -0.000 | − | − | − | -0.000 | -0.000 | -1.000 | 2.449 |

1 | 0.750 | 1.750 | -1.000 | -0.000 | -0.000 | -1.000 | 0.250 | 0.250 | 0.000 | 0.354 |

2 | 0.938 | 1.979 | -1.006 | 0.188 | 0.229 | -0.006 | 0.063 | 0.021 | 0.006 | 0.066 |

3 | 0.993 | 1.999 | -1.001 | 0.056 | 0.020 | 0.005 | 0.007 | 0.001 | 0.001 | 0.007 |

4 | 0.999 | 2.000 | -1.000 | 0.006 | 0.001 | 0.001 | 0.001 | 0.000 | 0.000 | 0.001 |

5 | 1.000 | 2.000 | -1.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

As in Example 1, we stop iterating after **x**^{(k)} − **x**^{(k-1)},*e*^{(k)}*e*^{(k)}||

David M. Strong, "Iterative Methods for Solving [i]Ax[/i] = [i]b[/i] - Gauss-Seidel Method," *Convergence* (July 2005)

Journal of Online Mathematics and its Applications