Advances in Numerical Analysis

Research Article

Different Versions of ILU and IUL Factorizations Obtained from Forward and Backward Factored Approximate Inverse Processesβ€”Part I

Algorithm 2

IULBF algorithm.
(1) 𝑀 𝑛 = 𝑒 𝑇 𝑛 , 𝑧 𝑛 = 𝑒 𝑛 , 𝑑 𝑛 = π‘Ž 𝑛 𝑛 .
(2) for   𝑗 = 𝑛 βˆ’ 1 to 1 do
(3)   𝑀 𝑗 = 𝑒 𝑇 𝑗 , 𝑧 𝑗 = 𝑒 𝑗 .
(4)  for 𝑖 = 𝑗 + 1 to 𝑛   do
(5)    π‘ˆ 𝑗 𝑖 = 𝐴 𝑗 , ∢ 𝑧 𝑖 𝑑 𝑖 , 𝐿 𝑖 𝑗 = 𝑀 𝑖 𝐴 ∢ , 𝑗 𝑑 𝑖
(6)   apply a dropping rule to π‘ˆ 𝑗 𝑖 and to 𝐿 𝑖 𝑗
(7)    𝑧 𝑗 = 𝑧 𝑗 βˆ’ ξ‚΅ 𝑀 𝑖 𝐴 ∢ , 𝑗 𝑑 𝑖 ξ‚Ά 𝑧 𝑖 , 𝑀 𝑗 = 𝑀 𝑗 βˆ’ ξ‚΅ 𝐴 𝑗 , ∢ 𝑧 𝑖 𝑑 𝑖 ξ‚Ά 𝑀 𝑖
(8)   for all 𝑙 β‰₯ 𝑖 apply a dropping rule to 𝑧 𝑙 𝑗 and to 𝑀 𝑗 𝑙
    (first format of dropping for π‘Š and 𝑍 )
(9)   end for
(10) for all 𝑙 β‰₯ 𝑗 apply a dropping rule to 𝑧 𝑙 𝑗 and to 𝑀 𝑗 𝑙
     (second format of dropping for π‘Š and 𝑍 )
(11)   𝑑 𝑗 = 𝑀 𝑗 𝐴 ∢ , 𝑗 (if 𝐴 is not positive definite)
(12)   𝑑 𝑗 = 𝑀 𝑗 𝐴 𝑀 𝑇 𝑗 (if 𝐴 is positive definite)
(13) end for
(14) Return 𝐿 = ( 𝑑 𝑗 𝐿 𝑖 𝑗 ) and π‘ˆ = ( π‘ˆ 𝑖 𝑗 )