Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems

Algorithm 1

𝑘=0;

while (not convergent)

 𝑟^(𝑘)=𝑏−𝐴𝑥^(𝑘);

 approximately solve (𝛼𝑃₁+𝑊)𝑧^(𝑘)=𝑟^(𝑘) by employing CG method,

 such that the residual 𝑝^(𝑘)=𝑟^(𝑘)−(𝛼𝑃₁+𝑊)𝑧^(𝑘) of the iteration satisfies

 ‖𝑝^(𝑘)‖₂≤𝜂_𝑘‖𝑟^(𝑘)‖₂;

 𝑥^(𝑘+1/2)=𝑥^(𝑘)+𝑧^(𝑘);

 𝑟^(𝑘+1/2)=𝑏+𝑖𝐴𝑥^(𝑘+1/2);

 approximately solve (𝛽𝑃₂+𝑇)𝑧^(𝑘+1/2)=𝑟^(𝑘+1/2) by employing CG method,

 such that the residual 𝑞^(𝑘+1/2)=𝑟^(𝑘+1/2)−(𝛽𝑃₂+𝑇)𝑧^(𝑘+1/2) of the

 iteration satisfies ‖𝑞^(𝑘+1/2)‖₂≤𝜏_𝑘‖𝑟^(𝑘+1/2)‖₂;

 𝑥^(𝑘+1)=𝑥^(𝑘+1/2)+𝑧^(𝑘+1/2);

 𝑘=𝑘+1;

end