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 1

ILUFF algorithm.
(1) 𝑀 1 = 𝑒 𝑇 1 , 𝑧 1 = 𝑒 1 , 𝑑 1 = π‘Ž 1 1 .
(2) for 𝑗 = 2 to 𝑛   do
(3)   𝑀 𝑗 = 𝑒 𝑇 𝑗 , 𝑧 𝑗 = 𝑒 𝑗 .
(4)  for   𝑖 = 1 to 𝑗 βˆ’ 1 do
(5)    𝐿 𝑗 𝑖 = 𝐴 𝑗 , ∢ 𝑧 𝑖 𝑑 𝑖 , π‘ˆ 𝑖 𝑗 = 𝑀 𝑖 𝐴 ∢ , 𝑗 𝑑 𝑖
(6)   apply a dropping rule to 𝐿 𝑗 𝑖 and to π‘ˆ 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 ≀ 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 ≀ 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 = ( 𝐿 𝑖 𝑗 ) and π‘ˆ = ( 𝑑 𝑖 π‘ˆ 𝑖 𝑗 )