The best replacement for gaussian elimination is lu decomposition.


The best alternatives to gaussian elimination are rated as following:
★★★★★  lu decomposition
★★★★  gauss jordan elimination
★★★  cramer's rule
★★  lu factorization
★  gauss-jordan elimination
★  gauss-seidel

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Speaking about gaussian elimination, one probably shouldn't forget about partial pivoting.

Other search terms related to gaussian elimination and their ratings are:

★★★★★ partial pivoting
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gaussian elimination
lu decomposition
gauss jordan elimination