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There are several methods known for solving a linear equation system Ax = b (like Jacobi or Gauss-Seidel) by iterating

$x_{n+1}=Mx_n+c$ with a matrix M, for which some norm is smaller than 1. But what can be done if we have an arbitary matrix A with no special properties (not symmetric and not diagonaldominant) ? Must the numerical gauss-algorithm be used, or is there a fix-point-iteration converging in any case ?

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    $\begingroup$ I don't know if it may interest you, but take a look at en.m.wikipedia.org/wiki/Biconjugate_gradient_method $\endgroup$ – Ant Jul 23 '14 at 16:21
  • $\begingroup$ I did not quite get it. Is this an iterative method ? $\endgroup$ – Peter Jul 23 '14 at 16:26
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    $\begingroup$ It is a generalization of the conjgate gradient method, which technically is a direct method because it converges to the solution in at most $n$ steps (where $n$ is the dimension of the matrix). On the other hand is often implemented as an iterative method. (It also feel like one, because you are iterating a simple procedure $n$ times) $\endgroup$ – Ant Jul 23 '14 at 16:29
  • $\begingroup$ Krylov methods like GMRES, FOM, BiCG, CGS, BiCGStab, QMR, TFQMR, IDR, CGNE, CGNE, LSQR, etc. etc. etc. are the usual methods of choice for such systems. $\endgroup$ – Algebraic Pavel Jul 23 '14 at 18:07
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A linear equation system $Ax=b$ with arbritray $A$ isn't solvable; just imagine $A_{i,j}=0$ everywhere; you will at least have to assume that the system is solvable aka that $A$ has full rank.

In this case, amongst all iterative methods, GMRES makes the least assumptions. The system doesn't have to be definite or symmetric.

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