A Polynomial Preconditioner for the CMRH Algorithm

Algorithm 1

Hessenberg process with pivoting [2].

set 𝑝=(1,…,𝑛)^𝑇.

Determine 𝑖₀ so |(𝑣)_𝑖0|=‖𝑣‖_∞; 𝑙₁=𝑣₀(𝑣)_𝑖0; 𝑝(1)↔𝑝(𝑖₀) (interchange 𝑝(1)  and 𝑝(𝑖₀));

iterate:

for  𝑘=1,2,…,𝑛

     𝑢=𝐴𝑙_𝑘

     for  𝑗=1,2,…,𝑘

  𝑐=𝑢_𝑝(𝑗); ℎ_𝑗,𝑘=𝑐; 𝑢_𝑝(𝑗)=0

  for  𝑙=𝑗+1,…,𝑛

   𝑢_𝑝(𝑙)=𝑢_𝑝(𝑙)−𝑐∗(𝑙_𝑗)_𝑝(𝑙)

  end

   end

   if  𝑘<𝑛

  determine 𝑖₀  so  |(𝑢)_𝑖0|=‖𝑢‖_∞; 𝑝(𝑘+1)↔𝑝(𝑖₀)

   ℎ_𝑘+1,𝑘=𝑢_𝑖0; 𝑙_𝑘+1=𝑢/(𝑢)_𝑖0

   end

end