Preconditioners, or accelerators are used to accelerate an iterative solution process. In some cases, their use can reduce the number of iterations dramatically and thus lead to better solver performance. Although the terms

preconditioner

and

accelerator

are synonyms, hereafter only

preconditioner

is used.

Intel® oneAPI Math Kernel Library

provides two preconditioners, ILU0 and ILUT, for sparse matrices presented in the format accepted in the

Intel® oneAPI Math Kernel Library

direct sparse solvers (three-array variation of the CSR storage format described inSparse Matrix Storage Format ). The algorithms used are described in [Saad03].

The ILU0 preconditioner is based on a well-known factorization of the original matrix into a product of two triangular matrices: lower and upper triangular matrices. Usually, such decomposition leads to some fill-in in the resulting matrix structure in comparison with the original matrix. The distinctive feature of the ILU0 preconditioner is that it preserves the structure of the original matrix in the result.

Unlike the ILU0 preconditioner, the ILUT preconditioner preserves some resulting fill-in in the preconditioner matrix structure. The distinctive feature of the ILUT algorithm is that it calculates each element of the preconditioner and saves each one if it satisfies two conditions simultaneously: its value is greater than the product of the given tolerance and matrix row norm, and its value is in the given bandwidth of the resulting preconditioner matrix.

Both ILU0 and ILUT preconditioners can apply to any non-degenerate matrix. They can be used alone or together with the

Intel® oneAPI Math Kernel Library

RCI FGMRES solver (seeSparse Solver Routines). Avoid using these preconditioners with MKL RCI CG solver because in general, they produce a non-symmetric resulting matrix even if the original matrix is symmetric. Usually, an inverse of the preconditioner is required in this case. To do this the

Intel® oneAPI Math Kernel Library

triangular solver routine

mkl_dcsrtrsv

must be applied twice: for the lower triangular part of the preconditioner, and then for its upper triangular part.