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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. Specifically, we are interested in the simplest type of factorization in which no ``fill'' is allowed, even if the matrix has a nonzero in the fill position. If we split the matrix as in diagonal, lower and upper triangular part, then the factorization will be of the form . In this way, we only need to store a diagonal matrix containing the pivots of the factorization, and a preconditioner solve code is needed for both The transpose preconditioner solve is for an iterative method involving a transpose matrix-vector product.