Numerical solution of large sparse linear systems
Preconditioned iterative solver (Krylov subspace method, e.g. GMRES)
Incomplete LU decomposition methods
RULE OF THUMB
Iterative solvers perform quite well if , where (strict for certain iterative solvers like GMRES)
PARTIAL INCOMPLETE LU DECOMPOSITION
has a specific nonzero pattern.
, drop tolerance.
INVERSE TRIANGULAR FACTORS
and may become very large pivoting recommended.
What kind of pivoting is appropriate?
the entries being dropped from and .
Lemma[B.,Saad '04]. Error bounds for the inverse error .
PRECONDITIONED ITERATIVE SOLVERS
Solve instead of , where
Preconditioned system requires or to be small.
, drop tolerance
Prescribe a bound and apply diagonal pivoting such that and .
Rows/columns that exceed the prescribed bound are pushed to the end