A Polynomial Preconditioner for the CMRH Algorithm

Algorithm 3

PCMRH(m).

Input: the dimension of Krylov subspace 𝑚; the prescribed stopping criterion 𝜀;

  the degree of the polynomial preconditioner 𝑘𝑘; the initial approximate solution 𝑥₀;

      the residual vector 𝑟₀=𝑏−𝐴𝑥₀.

1. The procedure of constructing polynomial preconditioner 𝑞_𝑘𝑘

 (1)  Let 𝑝(𝑖)=𝑖,𝑖=1,2,…,𝑛;

     determine 𝑖₀ so |(𝑟₀)|_𝑖0=‖𝑟₀‖_∞; 𝑙₁=𝑟₀(𝑟₀)_𝑖0; 𝑝(1)↔𝑝(𝑖₀),

     1𝐶(1,1)=(𝑟₀)_𝑖0

  (2)  for 𝑘=1,2,…,𝑘𝑘 do

    𝑢=𝐴𝑙_𝑘

        for𝑗=1,2,…,𝑘 do

   𝑐=𝑢_𝑝(𝑗),ℎ_𝑗,𝑘=𝑐,𝑢_𝑝(𝑗)=0

   for𝑙=𝑗+1,…,𝑛 do

      𝑢_𝑝(𝑙)=𝑢_𝑝(𝑙)−𝑐×(𝑙_𝑗)_𝑝(𝑙)

   end

     if 𝑘<𝑛

    compute 𝑖₀ so |(𝑢₀)|_𝑖0=‖𝑢₀‖_∞; 𝑝(𝑘+1)↔𝑝(𝑖₀)

    ℎ_𝑘+1,𝑘=𝑢_𝑖0; 𝑙_𝑘+1=𝑢(𝑢)_𝑖0

    let ℎ_𝑗=^ℎ1,𝑗⋮_ℎ𝑗,𝑗,𝐶_𝑗=^{𝑐11⋯𝑐1𝑗0⋱⋮}_{00𝑐𝑗𝑗}

    compute

    ^{𝑐1,𝑗+1𝑐2,𝑗+1}_{⋮𝑐𝑗+1,𝑗+1}=ℎ⁻¹_𝑗+1,𝑗^0𝑐1𝑗_{⋮𝑐𝑗𝑗}−ℎ⁻¹_𝑗+1,𝑗^{𝐶𝑗ℎ𝑗}₀

  end

 end

  (3) Form the approximate solution 𝑥_𝑘=𝑥₀+𝐿_𝑘𝑦_𝑘, where 𝑦_𝑘∈𝐶^𝑘 minimizes

            𝐹(𝑦)=‖𝐻_𝑘𝑦−(𝑟₀)_𝑖0𝑒₁‖.

(4) Form the polynomial preconditioner 𝑞_𝑘𝑘(𝑧)

            𝐶_𝑘𝑦_𝑘=(^{𝛼0⋯𝛼𝑘−1})^𝑇;

            𝑞_𝑘𝑘(𝑥)=𝛼₀+𝛼₁𝑥+⋯+𝛼_𝑘−1𝑥^𝑘−1.        

2. Using the CMRH(m) algorithm for 𝑞_𝑘𝑘(𝐴)𝐴𝑥=𝑞_𝑘𝑘(𝐴)𝑏.