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Solution

The coefficient matrix

is diagonally dominant as

and the inequality is strictly greater than for at least one row. Hence the solution should converge using Gauss-Seidal method.

Rewriting the equations, we get

Assuming an initial guess of

Iteration 1:

=0.50000

=4.9000

=3.0923

The absolute relative approximate error at the end of first iteration is

=67.662%

=100.000%

=67.662%

The maximum absolute relative approximate error is 100.000%

Iteration 2:

=0.14679

=3.7153

=3.8118

At the end of second iteration, the absolute relative approximate error is

=240.62%

=31.887%

=18.876%.

The maximum absolute relative approximate error is 240.62%. This is greater than the value of 67.612% we obtained in the first iteration. Is the solution diverging? No, as you conduct more iterations, the solution converges as follows.

Iteration	a1		a2		a3
	0.50000 0.14679 0.74275 0.94675 0.99177 0.99919	67.662 240.62 80.23 21.547 4.5394 0.74260	4.900 3.7153 3.1644 3.0281 3.0034 3.0001	100.00 31.887 17.409 4.5012 0.82240 0.11000	3.0923 3.8118 3.9708 3.9971 4.0001 4.0001	67.662 18.876 4.0042 0.65798 0.07499 0.00000

This is close to the exact solution vectorof