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I am preparing for an exam and I have an exercise type, without any example. Give the following system : $$ \begin{bmatrix} 2 & 1\\ -1 & 2\\ \end{bmatrix} *x = \begin{bmatrix} 3\\ 1\\ \end{bmatrix} $$ study the convergence of Gauss-Seidel method.I have no ideea what should I look for , so any tips wold be welcome. Thank you.

user137209
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  • BTW the question probably wants you to compute the spectral radius of $\begin{bmatrix}1 & 0\0& 1\end{bmatrix}-\begin{bmatrix}2 & 1\0& 2\end{bmatrix}^{-1}\begin{bmatrix}2 & 1\-1& 2\end{bmatrix}$, which gives you an idea about the convergence rate of the method (which is usually defined as the negative decimal logarithm of this spectral radius giving an idea of how many digits in the approximation of $x$ become valid per iteration). – Algebraic Pavel Jun 10 '14 at 11:50

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The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either:

A is symmetric positive-definite or A is strictly or irreducibly diagonally dominant.

[By http://en.wikipedia.org/wiki/Gauss–Seidel_method#Convergence ]

ABC
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  • Found examples on that wikipedia link.I should have find myself, but i didn't expect to find examples on wikipedia. Thank you. – user137209 Jun 10 '14 at 10:56