Complete pivoting strategy for the $IUL$ preconditioner obtained from Backward Factored APproximate INVerse process

‎In this paper‎, ‎we use a complete pivoting strategy to compute the IUL preconditioner obtained as the by-product of the Backward Factored APproximate INVerse process‎. ‎This pivoting is based on the complete pivoting strategy of the Backward IJK version of Gaussian Elimination process‎. ‎There is a parameter $alpha$ to control the complete pivoting process‎. ‎We have studied the effect of different values of $alpha$ on the quality of the IUL preconditioner‎. ‎For the numerical experiments section‎, ‎the IUL factorization which is coupled with the complete pivoting is compared to the ILUTP and to the left-looking version of RIF which is coupled with the complete pivoting strategy‎. ‎As the preprocessing‎, ‎we have applied the maximum weighted matching coupled the Reverse Cuthill-Mckee (RCM) and multilevel nested dissection reordering.