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I have a system of linear equations, $Ax=b$, with $N$ equations and $N$ unknowns ($N$ large) If I am interested in the solution for only one of the unknowns, what are the best approaches? for example, Assume N=50,000 and we want the solution for $x_1$ through $x_{100}$ only. is there any trick that does not require $O(n^{3})$ (or O(matrix inversion) ) for that ? |
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Unless your matrix is sparse or structured (e.g. Vandermonde, Hankel, or those other named matrix families that admit a fast solution method), there is not much hope of doing things better than $O(n^3)$ effort. Even if one were to restrict himself to solving for just one of the 50,000 variables, Cramer will demand computing two determinants for your answer, and the effort for computing a determinant is at least as much as decomposing/inverting a matrix to begin with. |
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[this should be a comment to @Juan Joder's answer or to the OP's question, but not enough repo] strang gilbert - linear algebra and its applications 3rd edition page 16 at the footer mentions that complexity had fallen already fallen below $O(Cn^{2.376})$ at the date of writing 1988, altough $C$ is so large that makes the algorithm impractical for most matrix sizes found in practice today. It does not mention the name of the algorithm though. personal guess: not sure about solving for single variables, but I don't think you can reduce complexity like that since the entire system is coupled. |
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