[petsc-users] solve complex linear system: real or complex formulation

[Jack Poulson]

Xiangdong,

Nearly all of the time in a serial sparse-direct factorization goes into
performing many different dense "frontal" factorizations. Efficient
implementations of dense LU factorization spend almost all of their time
within dense matrix-matrix multiplication, and complex matrix-matrix
multiplication requires about 4 times as much work as real matrix-matrix
multiplication, as half the work is in adds and half the work is in
multiplies with the standard approach (complex addition and multiplication
are respectively require 2 and 6 real flops).

Overall, a complex sparse-direct solve takes about 4 times as many flops
(and twice as much memory) as a real sparse-direct solve, but due to having
a higher arithmetic intensity (roughly, more flops per piece of data), the
complex version will almost always take significantly less than 4 times as
long as the real version.

Jack

On Fri, Dec 9, 2011 at 11:54 PM, Xiangdong Liang <xdliang at gmail.com> wrote:

> Hello everyone,
>
> I am solving a complex linear system C x = d, where C= A+ iB (A, B are
> real), in sparse-direct solver. So far, I use the real formulation by
> solving the linear system [A, -B; B,A].  The reason we chose this
> approach is to use the property of that the imaginary part B in our
> problem is sparser than A. However, I just found that the spare-direct
> solver cannot benefit from this property.
>
> Now, I am thinking to re-implement it in to complex version by solving
> Cx=d in the complex version. Of course, complex formulation would
> perform better for iterative solvers since it only has half
> eigenvalues than the corresponding real formulations. I am wondering
> the comparison between real and complex formulation for sparse-direct
> solver. Can anyone share experiences on this? Thanks.
>
> Xiangdong
>
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