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

Algorithm 1

The Gl-CMRH(𝑚) method for 𝐴𝑋=𝐵 [18].

(1) Choose an initial guess 𝑋₀, the restarting frequency 𝑚 and the tolerance tol. Compute 𝑅₀=𝐵−𝐴𝑋₀.

Determine 𝑖₀,𝑗₀ such that (𝑅₀)_𝑖0,𝑗0=max_{1≤𝑖≤𝑛,1≤𝑗≤𝑠}{(𝑅₀)_𝑖0,𝑗0}. Set 𝛽=(𝑅₀)_𝑖0,𝑗0, 𝑉₁=𝑅₀/𝛽, 𝑝_1,1=𝑖₀ and 𝑝_1,2=𝑗₀.

(2) Construct the matrix basis 𝒱_𝑚=[𝑉₁,𝑉₂,…,𝑉_𝑚] and 𝐻_𝑚 by the global Hessenberg process [18].

(3) Solve 𝑦_𝑚 by (10) and update 𝑋_𝑚 by (5).

(4) Compute 𝑅_𝑚=𝐵−𝐴𝑋_𝑚. If 𝑅_𝑚𝐹/𝑅_0𝐹≤tol, then stop; otherwise set 𝑋₀=𝑋_𝑚, 𝑅₀=𝑅_𝑚.

 Choose 𝑖₀,𝑗₀ such that (𝑅₀)_𝑖0,𝑗0=max_{1≤𝑖≤𝑛,1≤𝑗≤𝑠}{(𝑅₀)_𝑖0,𝑗0}.

 Set 𝛽=(𝑅₀)_𝑖0,𝑗0, 𝑉₁=𝑅₀/𝛽, 𝑝_1,1=𝑖₀ and 𝑝_1,2=𝑗₀. Go to Step >2.