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I'm applying finite-difference method to a system of 3 coupled equations. Two of the equations are not coupled, however the third equation couples to both the other two. I noticed that by changing the order of equations, say from $(x, y, z)$ to $(x, z, y)$ that the coefficient matrix becomes symmetric.

Is there any advantage to doing this? For example, in terms of stability or efficiency/speed of solution. The matrices are highly sparse, if that is important, the non-zero terms are along central diagonals.

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  • $\begingroup$ Yes, it takes much less effort to solve a symmetric system than an unsymmetric one. If, in addition, you can show that your coefficient matrix is positive-definite, then you are in a good place. $\endgroup$ – J. M. May 9 '13 at 11:04
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Absolutely!

First off, some linear algebra systems are smart enough to only store half of the matrix, this could save you a bunch of memory. But even if this isn't the case, various algorithms in numerical linear algebra will exploit the symmetry.

For example, given a symmetric matrix, any eigensolver will immediately know that all eigenvalues are real-valued, and the solution method may use that fact.

A typical thing that many people will think of are Krylov subspace methods for the solution of equation systems $Ax=b$: If your problem is symmetric, you know that you don't need methods for nonsymmetric problem like GMRES, and can reside to something less memory-intensive like MINRES, or -- if your matrix is also positive-definite -- CG. The convergence behavior of Krylov methods is not influenced by permutations though, so you could even use symmetric methods for your unpermuted system.

Another example is the factorization of your matrix $A=LU$ in to an lower-triangular part $L$ and an upper triangular part $U$. If $A$ is symmetric, then $A=LL^T$, and you only have to store one factor (Cholesky decomposition).

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    $\begingroup$ "...and the solution method may use that fact by, e.g., cutting of round-off errors in the imaginary part during the computation." - more like the computing environment uses a method that exploits the symmetry and is guaranteed to give real results. $\endgroup$ – J. M. May 9 '13 at 12:30

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