Efficient preconditioners for iterative methods can be found by performing an incomplete factorization of the coefficient matrix. In this section, we discuss the incomplete factorization of an matrix stored in the CRS format, and routines to solve a system with such a factorization. At first we only consider a factorization of the - type, that is, the simplest type of factorization in which no ``fill'' is allowed, even if the matrix has a nonzero in the fill position (see section ). Later we will consider factorizations that allow higher levels of fill. Such factorizations considerably more complicated to code, but they are essential for complicated differential equations. The solution routines are applicable in both cases.
For iterative methods, such as , that involve a transpose matrix vector product we need to consider solving a system with the transpose of as factorization as well.