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According to the book Linear algebra and its applications by Strang, "(The) good method is Gaussian Elimination. This is the algorithm that is constantly used to solve large systems of equations".

Is it really constantly used to solve large systems of equations, or are there better algorithms for it? If yes, which algorithms are usually used to actually solve them?

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  • $\begingroup$ Maybe this can help you: cs.stackexchange.com/questions/76623/… $\endgroup$ – dcolazin Jun 17 at 17:18
  • $\begingroup$ There are also sub-cubic (randomized?) algorithms to compute the Frobenius normal form, which is a very sparse matrix. See Storjohann's work. That simplifies every linear algebraic problem, including the solution of systems of linear equations, the computation of the inverse matrix, the similarity problem (input: two matrices, question: are they similar), the computation of the rank, the computation of the determinant, etc. $\endgroup$ – A. Pongrácz Jun 17 at 17:22
  • $\begingroup$ Depends on the structure of the matrix and the purpose. $\endgroup$ – воитель Jun 17 at 17:27
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It depends on what you mean by large. On most regular computers you can solve dense systems up to 5k x 5k fairly easily. If the matrix is ill-conditioned you can use other algorithms and if it pretty large it is probably better to use iterative methods.

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