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

Algorithm 4

Computation of the AINV preconditioner.

Input: Matrix 𝐴, the tolerance 𝜏

Output: Matrices 𝑊, 𝑍 and 𝐷 such that 𝑀⁻¹=𝑍𝐷⁻¹𝑊^𝑇

Set p  =  q  =  0, 𝑍=𝑊=𝐼

for   𝑘=1,…,𝑛  do

 𝑝_𝑘=𝐰^𝑇_𝑘𝐴𝐳_𝑘,𝑞_𝑘=𝐰^𝑇_𝑘𝐴𝐳_𝑘

 for  𝑖=𝑘+1,…,𝑛  do

  𝑝_𝑖=(𝐰^𝑇_𝑖𝐴𝐳_𝑘)/𝑝_𝑘,𝑞_𝑖=(𝐰^𝑇_𝑘𝐴𝐳_𝑖)/𝑞_𝑘

  𝐰_𝑖←𝐰_𝑖−𝑝_𝑖𝐰_𝑘,𝐳_𝑖←𝐳_𝑖−𝑞_𝑖𝐳_𝑘

  for  𝑗=1,…,𝑖  do

   Drop the 𝑗th component of 𝐰_𝑖 and 𝐳_𝑖 if smaller than 𝜏

   end

  end

end

for  𝑘=1,…,𝑛  do

   𝑑_𝑘=𝑝_𝑘

end