Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Algorithm 7

Dynamic construction of the Block FSAI preconditioner.

Input: Matrix 𝐴, the initial pattern 𝒮⁰_𝐹

Output: Matrix 𝐹

for  𝑖=1,…,𝑛  do

 Extract 𝒥⁰ from 𝒮⁰_𝐹

 if  𝒥⁰≠∅  then

  Gather 𝐴[𝒥⁰,𝒥⁰] and 𝐴[𝒥⁰,𝑖] from 𝐴

  Solve 𝐴[𝒥⁰,𝒥⁰]𝐟_𝑖[𝒥⁰]=−𝐴[𝒥⁰,𝑖]

 end

 Set 𝑘=0 and 𝜖_𝑖>𝜖_𝐹

 while  (𝑘≤𝑘_max) and (𝜖_𝑖<𝜖_𝐹)  do

  𝑘←𝑘+1

  Compute 𝐠_𝑖=2(𝐴_𝑖𝑏𝐟_𝑖̃+𝐚_𝑖)

  Update 𝒥^𝑘−1 by adding the index of the largest component of 𝐠_𝑖

  Gather 𝐴[𝒥^𝑘,𝒥^𝑘] and 𝐴[𝒥^𝑘,𝑖] from 𝐴

  Solve 𝐴[𝒥^𝑘,𝒥^𝑘]𝐟_𝑖[𝒥^𝑘]=−𝐴[𝒥^𝑘,𝑖]

  Compute the diagonal term [𝐹𝐴𝐹^𝑇]^𝑘_𝑖𝑖 at the current step

  Compute 𝜖_𝑖=([𝐹𝐴𝐹^𝑇]^𝑘_𝑖𝑖−[𝐹𝐴𝐹^𝑇]^𝑘−1_𝑖𝑖)/[𝐹𝐴𝐹^𝑇]^𝑘−1_𝑖𝑖

 end

end