Direct solvers are less sensitive to whether the matrix is diagonal dominate but in the extreme, since matrices that are not diagonally dominate are generally more ill-conditioned, direct solvers in that extreme region will produce less accurate answers. Direct solvers have the additional problem they do not, and likely cannot, scale to really large problems, 10's of millions to billions of unknowns while iterative solves can (assuming the matrix is suitable for direct solvers) can solve problems with billions of unknowns.
Barry > On Dec 16, 2016, at 8:26 PM, Massoud Rezavand <[email protected]> wrote: > > Thanks you very much. > > As far as I know, PETSc provides direct solvers, as well. How about direct > solvers and the performance for a diagonally dominant matrix and a random > matrix? > > Massoud > > On Sat, Dec 17, 2016 at 3:22 AM, Barry Smith <[email protected]> wrote: > > > On Dec 16, 2016, at 8:17 PM, Massoud Rezavand <[email protected]> wrote: > > > > Dear PETSc team, > > > > Sorry if my question is more related to math. > > Using PETSc, how important is the structure of the matrix A for > > performance? I mean mainly the diagonal and off-diagonal parts. > > > > For example, solving with a matrix which is dense in diagonal part and > > sparse in off-diagonal part is faster than with a matrix in which the > > non-zeros are distributed randomly? > > Yes, loosely speaking this is true. More technically there is a term > "diagonally dominate", or block diagonally dominate, for those matrices > generally iterative methods perform better. > > Barry > > > > > > Thanks in advance. > > > > Massoud > >
