A Polynomial Preconditioned Global CMRH Method for Linear Systems with Multiple Right-Hand Sides

Algorithm 2

PGl-CMRH(𝑚,deg) method for 𝐴𝑋=𝐵.

(1) Input: 𝑚, deg and 𝑋₀.

(2) Phase I: Compute 𝑅₀=𝐵−𝐴𝑋₀. Determine 𝑖₀,𝑗₀ such that (𝑅₀)_𝑖0,𝑗0=max_{1≤𝑖≤𝑛,1≤𝑗≤𝑠}|(𝑅₀)_𝑖,𝑗|.

Set 𝛽=(𝑅₀)_𝑖0,𝑗0, 𝑉₁=𝑅₀/𝛽, 𝑝_1,1=𝑖₀ and 𝑝_1,2=𝑗₀. Let 𝑈_1,1=(𝑅₀)_𝑖0,𝑗0 (the upper triangular matrix

defined in (16)).

(3) for  𝑘=1∶deg

(4)  𝑀=𝐴𝑉_𝑘

(5)  for 𝑗=1∶𝑘

(6)   (𝐻)_𝑗,𝑘=(𝑀)_{𝑝𝑗,1,𝑝𝑗,2}

(7)   𝑀=𝑀−(𝐻)_𝑗,𝑘𝑉_𝑗

(8)  end for

(9)  Determine 𝑖₀,𝑗₀ such that (𝑀)_𝑖0,𝑗0=max_{1≤𝑖≤𝑛,1≤𝑗≤𝑠}(𝑀)_𝑖,𝑗.

(10)  Set (𝐻)_𝑘+1,𝑘=(𝑀)_𝑖0,𝑗0, 𝑉_𝑘+1=𝑀/(𝐻)_𝑘+1,𝑘, 𝑝_𝑘+1,1=𝑖₀ and 𝑝_𝑘+1,2=𝑗₀.

(11)  Compute (𝑈_𝑘+1)_.,𝑘+1 from (20).

(12) end for

(13) 𝑋_deg=𝑋₀+𝒱_deg∗𝑦_deg, where 𝑦_deg=argmin_{𝑦∈ℝdeg}(𝑅₀)_𝑖0,𝑗0𝑒₁^(deg+1)−𝐻_deg𝑦₂.

(14) Construct the polynomial preconditioner 𝑄_deg−1(𝐴) with its coefficients decided by entries of 𝑈_deg𝑦_deg.

(15)  Phase II: Solve 𝑄_deg−1(𝐴)𝐴𝑋=𝑄_deg−1(𝐴)𝐵 using Algorithm 1.