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I am a computational physics postgrad student, working with libraries like ATLAS and MAGMA. I have a matrix which is upper-triangular, and is the result of a Cholesky decomposition. I need to convert the upper-triangular matrix in to a symmetric, square matrix where the elements of the lower triangle are

mat(j,i) = mat(i,j)

I have a naive routine in C++ that simply does the above in a loop, but it's incredibly slow. I believe it's so slow because the CPU cache is poorly utilised - element i,j is quite far from element j,i.

Is there a clever mathematical trick that I can use to optimise this loop? Or alternatively, does anyone know of any fuctions in BLAS/similar libraries that can do this kind of operation?

Many thanks in advance for any advice you can give.

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If the libraries have efficient routines for matrix transposition, maybe you could just try $S=U+U^T$? (Provided that $U$ is strictly upper triangular of course, otherwise you have to divide the diagonal by 2 afterwards.) – Hans Lundmark Jan 29 '11 at 15:36
Thanks Hans, funnily enough that approach occurred to me not long after posting this question! I am trying to find out whether such a tranposition method exists and how efficient it is. – user6383 Jan 29 '11 at 15:45
May I ask out of curiosity: what is a postgrad student? Someone who has completed a master's degree but hasn't started PhD studies? – Rasmus Jan 29 '11 at 15:57
"Postgrad" is a catch all term in the UK for anybody studying a postgraduate degree, whether that be a master's, Ph.D. or Other. I'm studying an M.Sc. by Research, which is a bastard child of Master's and Ph.D. – user6383 Jan 29 '11 at 20:08
@Rasmus: In other words, "Postgrad" in the UK = "Graduate Student" in the US. – Arturo Magidin Jan 29 '11 at 21:56

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