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

Algorithm 6

Recursive application for 𝜇 times of the V-cycle AMG function MGV(𝐴,𝑅^𝐻_ℎ,𝑃^ℎ_𝐻,𝑆_ℎ,v,u).

Input: Matrix 𝐴^(𝑘), the known vector 𝐯^(𝑘), the restriction and prolongation

   operators 𝑅_ℎ^𝐻(𝑘) and 𝑃_𝐻^ℎ(𝑘), the smoothers 𝑆_ℎ^(𝑘), 𝑘, 𝜈₁, 𝜈₂ and 𝜇

Output: Vector 𝐮^(𝑘)=𝑀⁻¹𝐯^(𝑘)

if  𝑘=𝜇  then

 Solve 𝐴^(𝑘)𝐮^(𝑘)=𝐯^(𝑘)

else

 Apply 𝜈₁ iterations of the smoother 𝑆_ℎ^(𝑘) to 𝐴^(𝑘)𝐮^(𝑘)=𝐯^(𝑘)

 𝐫^(𝑘)=𝐯^(𝑘)−𝐴^(𝑘)𝐮^(𝑘)

 𝐫^(𝑘+1)=𝑅_ℎ^𝐻(𝑘)𝐫^(𝑘)

 Call MGV(𝐴^(𝑘+1),𝑅_ℎ^𝐻(𝑘+1),𝑃_𝐻^ℎ(𝑘+1),𝑆_ℎ^(𝑘+1),𝐫^(𝑘+1),𝐞^(𝑘+1))

 𝐞^(𝑘)=𝑃_𝐻^ℎ(𝑘)𝐞^(𝑘+1)

 𝐮^(𝑘)←𝐮^(𝑘)+𝐞^(𝑘)

 Apply 𝜈₂ iterations of the smoother 𝑆_ℎ^(𝑘) to 𝐴^(𝑘)𝐮^(𝑘)=𝐯^(𝑘)

end