MatLUFactor for Complex Matrices - Returns inverse of the diagonal in LU??

Barry Smith bsmith at mcs.anl.gov
Wed May 27 19:59:14 CDT 2009

```    Paul,

In PETSc for the AIJ sparse matrix storage format we store the
inverse of the diagonal entries of U. This is because then when it
comes to the triangular solves they only involve floating point
multiplies and adds (no time consuming divisions). I think this is
pretty common for sparse matrix solvers. You could argue that the
MatView() should invert those values so what gets printed out is
clearer; I view the MatView() for factored matrices as only a useful
tool for debugging for developers so it just prints out the stored
values. Yes it is a bit confusing.

Barry

For BAIJ matrices it gets even more confusing. With BAIJ the
factorizations use the little blocks of the matrix as the basic
computational unit of the factorization. Here we actually compute the
exact inverse of the diagonal block and store that (a block version of
what we do for AIJ). Again this is to make the solves as efficient as
possible.

On May 27, 2009, at 6:20 PM, Paul Dostert wrote:

> I'm a beginner with PETSc, so please forgive me if this is obvious,
> but I
> couldn't seem to find any help in the archives.
>
> I'm trying to just the hang of the software, so I've been messing
> around
> with routines. I'm going to need complex matrices (Maxwell's with
> PML) so
> everything is configured for this. I'm messing around with some very
> simple
> test cases, and have a symmetric (but not Hermitian) complex matrix
> with
> 2-2i on the diagonal and -1+i on the upper and lower diagonals. I am
> this in from a petsc binary file (again, for testing purposes,
> eventually
> I'm going to be just reading in my matrix and RHS). I view the
> matrix, and
> it has been read in correctly. I perform LU factorization by doing the
> following (where ISCreateStride(PETSC_COMM_WORLD,m,0,1,&perm); has
> been
> called earlier):
>
> MatConvert(A,MATSAME,MAT_INITIAL_MATRIX,&LU);
> MatFactorInfoInitialize(&luinfo);
> MatLUFactor(LU,perm,perm,&luinfo);
>
> I get that LU is:
>
>   (1,1)      0.2500 + 0.2500i
>   (2,1)     -0.5000
>   (1,2)     -1.0000 + 1.0000i
>   (2,2)      0.3333 + 0.3333i
>   (3,2)     -0.6667
>   (2,3)     -1.0000 + 1.0000i
>   (3,3)      0.3750 + 0.3750i
>   (4,3)     -0.7500
>   (3,4)     -1.0000 + 1.0000i
>   (4,4)      0.4000 + 0.4000i
>   (5,4)     -0.8000
>   (4,5)     -1.0000 + 1.0000i
>   (5,5)      0.4167 + 0.4167i
>   (6,5)     -0.8333
>   (5,6)     -1.0000 + 1.0000i
>   (6,6)      0.4286 + 0.4286i
>
> Now, I am interpreting this as L being unit on the diagonal and the
> lower diagonal portion of this "LU" matrix, while U being the diagona
> + upper of this "LU" matrix. I can interpret this the other way around
> as well, and it doesn't matter.
>
> However, knowing the LU factorization, it is VERY clear the the proper
> LU decomposition would have the inverse of the diagonal elements
> presented here. So I believe I should have  LU as:
>
>   (1,1)      2.0000 - 2.0000i
>   (2,1)     -0.5000
>   (1,2)     -1.0000 + 1.0000i
>   (2,2)      1.5000 - 1.5000i
>   (3,2)     -0.6667
>   (2,3)     -1.0000 + 1.0000i
>   (3,3)      1.3333 - 1.3333i
>   (4,3)     -0.7500
>   (3,4)     -1.0000 + 1.0000i
>   (4,4)      1.2500 - 1.2500i
>   (5,4)     -0.8000
>   (4,5)     -1.0000 + 1.0000i
>   (5,5)      1.2000 - 1.2000i
>   (6,5)     -0.8333
>   (5,6)     -1.0000 + 1.0000i
>   (6,6)      1.1667 - 1.1667i
>
> Is there some reason this returns the inverse of the diagonal entries,
> or am I completely missing something? Is returning the inverse
> something standard??
>
> Since I'm new, I'm not quite sure where to look for actual source
> code. Is there a location where the LU factorization code is written
> and well commented?
>
> Thank you very much!

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