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

Algorithm 3

Computation of the SPAI preconditioner.

Input: Matrix 𝐴, the initial nonzero pattern 𝒮 and the tolerance 𝜀

Output: Matrix 𝑀⁻¹

for  𝑗=1,…,𝑛  do

 Compute 𝒥={𝑖∶(𝑖,𝑗)∈𝒮}

 Compute ℐ={𝑖∶𝑎_𝑖𝑘≠0,𝑘∈𝒥}

 Gather 𝐴[ℐ,𝒥] from 𝐴

 Solve the least square problem ‖𝐞_𝑗[ℐ]−𝐴[ℐ,𝒥]𝐦_𝑗[𝒥]‖₂→min

 𝐫_𝑗=𝐞_𝑗[ℐ]−𝐴[ℐ,𝒥]𝐦_𝑗[𝒥]

 while  ‖𝐫_𝑗‖₂>𝜀  do

  Enlarge 𝒥 and update ℐ

  Gather 𝐴[ℐ,𝒥] from 𝐴

  Solve the least square problem ‖𝐞_𝑗[ℐ]−𝐴[ℐ,𝒥]𝐦_𝑗[𝒥]‖₂→min

  𝐫_𝑗=𝐞_𝑗[ℐ]−𝐴[ℐ,𝒥]𝐦_𝑗[𝒥]

 end

end