*
*

Numerical solution of large sparse linear systems

*
*

Preconditioned iterative solver (Krylov subspace method, e.g. GMRES)

*
*

Incomplete *LU* decomposition methods

*
*

Iterative solvers perform quite well if , where (strict for certain iterative solvers like GMRES)

*
*

*
*

has a specific nonzero pattern.

*
*

, drop tolerance.

*
*

and may become very large pivoting recommended.

**What kind of pivoting is appropriate?**

*
*

-->

Denote by
the entries being dropped from and .

**L**emma[B.,Saad '04]. Error bounds for the inverse error
.

- Standard case, e.g. [Meijerink & Van der Vorst '77, Munksgaard '80, Saad '94]

- There exists an ILU (``Tismenetsky approach'') such that

- Standard case
- Tismenetsky case

*
*

- and contribute to the inverse error
- For the standard case even their PRODUCT !

*
*

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

m.bollhoefer@tu-bs.de

Last modified: October 11, 2011