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Gauss Elimination


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By Gaussian elimination any LSE with invertible $ n\times n$ coefficient matrix $ A$ can be brought to upper triangular form in at most $ n-1$ steps. For this purpose all coefficients below the diagonal are successively nullified, that is, after $ \ell-1$ steps the LSE has the form

\begin{displaymath}\begin{array}{rrrrrrrrrrcccrrcl}
a_{1,1}&x_1&+&a_{1,2}&x_2&+...
...a_{n,\ell}&x_{\ell}&+&\hdots&+&a_{n,n}&x_n&=&b_n
\end{array}
\end{displaymath}

In detail the $ \ell$-th elimination step proceeds as follows:

Example:


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  automatically generated 5/17/2011