Xiangdong,<div><br></div><div>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).</div>
<div><br></div><div>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.</div>
<div><br></div><div>Jack<br><br><div class="gmail_quote">On Fri, Dec 9, 2011 at 11:54 PM, Xiangdong Liang <span dir="ltr"><<a href="mailto:xdliang@gmail.com">xdliang@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
Hello everyone,<br>
<br>
I am solving a complex linear system C x = d, where C= A+ iB (A, B are<br>
real), in sparse-direct solver. So far, I use the real formulation by<br>
solving the linear system [A, -B; B,A]. The reason we chose this<br>
approach is to use the property of that the imaginary part B in our<br>
problem is sparser than A. However, I just found that the spare-direct<br>
solver cannot benefit from this property.<br>
<br>
Now, I am thinking to re-implement it in to complex version by solving<br>
Cx=d in the complex version. Of course, complex formulation would<br>
perform better for iterative solvers since it only has half<br>
eigenvalues than the corresponding real formulations. I am wondering<br>
the comparison between real and complex formulation for sparse-direct<br>
solver. Can anyone share experiences on this? Thanks.<br>
<span class="HOEnZb"><font color="#888888"><br>
Xiangdong<br>
</font></span></blockquote></div><br></div>