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В011300-Биология

Rewriting the equations, we get

Assuming an initial guess of

the next six iterative values are given in the table below

Iteration	a1		a2		a3
	21.000 -196.15 -1995.0 -20149 2.0364x105 -2.0579x105	110.71 109.83 109.90 109.89 109.90 1.0990	0.80000 14.421 -116.02 1204.6 -12140 1.2272x105	100.00 94.453 112.43 109.63 109.92 109.89	5.0680 -462.30 4718.1 -47636 4.8144x105 -4.8653x106	98.027 110.96 109.80 109.90 109.89 109.89

You can see that this solution is not converging and the coefficient matrixis not diagonally dominant. The coefficient matrix

is not diagonally dominant as

Hence Gauss-Seidal method may or may not converge.

However, it is the same set of equations as the previous example and that converged. The only difference is that we exchanged first and the third equation with each other and that made the coefficient matrixnot diagonally dominant.

So it is possible that a system of equationscan be made diagonally dominant if one exchanges the equations with each other. But it is not possible for all cases. For example, the following set of equations.

can not be rewritten to make the coefficient matrixdiagonally dominant.

В011300-Биология