# [petsc-users] PCLU diverges where PCILU converges on Dense Matrix

Ali Berk Kahraman aliberkkahraman at yahoo.com
Mon Mar 12 05:59:59 CDT 2018

```Thank you for your answers. Barry's answer allowed me to get up to 50x
faster code, so it was a huge help. I decided not to use ksp context
altogether for this small operation.

Thank you again,

Ali

On 11-03-2018 18:53, Matthew Knepley wrote:

> On Sun, Mar 11, 2018 at 1:14 AM, Smith, Barry F. <bsmith at mcs.anl.gov
> <mailto:bsmith at mcs.anl.gov>> wrote:
>
>
>       1) Run the problem with -ksp_view_mat and -ksp_view_rhs and mail
>     petsc-maint at mcs.anl.gov <mailto:petsc-maint at mcs.anl.gov>  the
>     resulting file produced called binaryoutput
>
>        2) By default PCLU does a reordering to reduce fill that could
>     introduce a zero pivoit, PCILU does not do a reordering by
>     default. You can use -pc_factor_mat_ordering_type none to force no
>     reordering (PCLU does not do numerical pivoting for stability so
>     can fail with zero pivots).
>
>        3) If you need to solve these tiny 7 by 7 systems many times
>     (presumably you are solving these to set up a large algebraic
>     system solved afterwards) then you probably don't want to use KSP
>     to solve them. You can use the low level kernel
>     PetscKernel_A_gets_inverse_A_7() that does do pivoting followed by
>     a multiply like
>
>     s1 = v[0]*x1 + v[6]*x2  + v[12]*x3 + v[18]*x4 + v[24]*x5 + v[30]*x6;
>           s2 = v[1]*x1 + v[7]*x2  + v[13]*x3 + v[19]*x4 + v[25]*x5 +
>     v[31]*x6;
>           s3 = v[2]*x1 + v[8]*x2  + v[14]*x3 + v[20]*x4 + v[26]*x5 +
>     v[32]*x6;
>           s4 = v[3]*x1 + v[9]*x2  + v[15]*x3 + v[21]*x4 + v[27]*x5 +
>     v[33]*x6;
>           s5 = v[4]*x1 + v[10]*x2 + v[16]*x3 + v[22]*x4 + v[28]*x5 +
>     v[34]*x6;
>           s6 = v[5]*x1 + v[11]*x2 + v[17]*x3 + v[23]*x4 + v[29]*x5 +
>     v[35]*x6;
>
>     where v is the dense 7 by 7 matrix (stored column oriented like
>     Fortran) an the x are the seven values of the right hand side.
>
>
> Note that PCPBJACOBI will do this automatically.
>
>    Matt
>
>        Barry
>
>
>
>
>
>     > On Mar 10, 2018, at 5:22 AM, Ali Berk Kahraman
>     <aliberkkahraman at yahoo.com <mailto:aliberkkahraman at yahoo.com>> wrote:
>     >
>     > Hello All,
>     >
>     > I am trying to get the finite difference coefficients for a
>     given irregular grid. For this, I follow the following webpage,
>     which tells me to solve a linear system.
>     >
>     > http://web.media.mit.edu/~crtaylor/calculator.html
>     <http://web.media.mit.edu/%7Ecrtaylor/calculator.html>
>     >
>     > I solve a 7 unknown linear system with a 7x7 dense matrix to get
>     the finite difference coefficients. Since I will call this code
>     many many many times in my overall project, I need it to be as
>     fast, yet as exact as possible. So I use PCLU. I make sure that
>     there are no zero diagonals on the matrix, I swap required rows
>     for it. However, PCLU still diverges with the output at the end of
>     this e-mail. It indicates "FACTOR_NUMERIC_ZEROPIVOT" , but as I
>     have written above I make sure there are no zero main diagonal
>     entries on the matrix. When I use PCILU instead, it converges
>     pretty well.
>     >
>     > So my question is, is PCILU the same thing mathematically as
>     PCLU when applied on a small dense matrix? I need to know if I get
>     the exact solution with PCILU, because my whole project will
>     depend on the accuracy of the finite differences.
>     >
>     > Best Regards,
>     >
>     > Ali Berk Kahraman
>     > M.Sc. Student, Mechanical Engineering Dept.
>     > Boğaziçi Uni., Istanbul, Turkey
>     >
>     > Linear solve did not converge due to DIVERGED_PCSETUP_FAILED
>     iterations 0
>     >                PCSETUP_FAILED due to FACTOR_NUMERIC_ZEROPIVOT
>     > KSP Object: 1 MPI processes
>     >   type: gmres
>     >     restart=30, using Classical (unmodified) Gram-Schmidt
>     Orthogonalization with no iterative refinement
>     >     happy breakdown tolerance 1e-30
>     >   maximum iterations=10000, initial guess is zero
>     >   tolerances:  relative=1e-05, absolute=1e-50, divergence=10000.
>     >   left preconditioning
>     >   using PRECONDITIONED norm type for convergence test
>     > PC Object: 1 MPI processes
>     >   type: lu
>     >     out-of-place factorization
>     >     tolerance for zero pivot 2.22045e-14
>     >     matrix ordering: nd
>     >     factor fill ratio given 5., needed 1.
>     >       Factored matrix follows:
>     >         Mat Object: 1 MPI processes
>     >           type: seqaij
>     >           rows=7, cols=7
>     >           package used to perform factorization: petsc
>     >           total: nonzeros=49, allocated nonzeros=49
>     >           total number of mallocs used during MatSetValues calls =0
>     >             using I-node routines: found 2 nodes, limit used is 5
>     >   linear system matrix = precond matrix:
>     >   Mat Object: 1 MPI processes
>     >     type: seqaij
>     >     rows=7, cols=7
>     >     total: nonzeros=49, allocated nonzeros=49
>     >     total number of mallocs used during MatSetValues calls =0
>     >       using I-node routines: found 2 nodes, limit used is 5
>     >
>     >
>
>
>
>
> --
> What most experimenters take for granted before they begin their
> experiments is infinitely more interesting than any results to which
> their experiments lead.
> -- Norbert Wiener
>
> https://www.cse.buffalo.edu/~knepley/ <http://www.caam.rice.edu/%7Emk51/>

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