Exploiting the Composite Step Strategy to the Biconjugate 𝐴-Orthogonal Residual Method for Non-Hermitian Linear Systems

Algorithm 2

Algorithm BiCOR.

(1) Compute 𝑟₀=𝑏−𝐴𝑥₀ for some initial guess 𝑥₀.

(2) Choose 𝑟^∗₀=𝑃(𝐴)𝑟₀ such that 𝑟^∗₀,𝐴𝑟₀≠0, where 𝑃(𝑡) is a polynomial in 𝑡.

   (e.g., 𝑟^∗₀=𝐴𝑟₀).

(3) Set 𝑝₀=𝑟₀, 𝑝^∗₀=𝑟^∗₀, 𝑞₀=𝐴𝑝₀, 𝑞^∗₀=𝐴^𝐻𝑝^∗₀, 𝑟₀̂=𝐴𝑟₀, 𝜌₀=⟨𝑟^∗₀,𝑟₀̂⟩.

(4) for  𝑛=0,1,…  do

(5)  𝜎_𝑛=⟨𝑞^∗_𝑛,𝑞_𝑛⟩

(6)  𝛼_𝑛=𝜌_𝑛/𝜎_𝑛

(7)  𝑥_𝑛+1=𝑥_𝑛+𝛼_𝑛𝑝_𝑛

(8)  𝑟_𝑛+1=𝑟_𝑛−𝛼_𝑛𝑞_𝑛

(9)  𝑥^∗_𝑛+1=𝑥^∗_𝑛+𝛼_𝑛𝑝^∗_𝑛

(10)   𝑟^∗_𝑛+1=𝑟^∗_𝑛−𝛼_𝑛𝑞^∗_𝑛

(11)   𝑟̂_𝑛+1=𝐴𝑟_𝑛+1

(12)   𝜌_𝑛+1=𝑟^∗_𝑛+1,𝑟̂_𝑛+1

(13)   if  𝜌_𝑛+1=0, method fails

(14)   𝛽_𝑛+1=𝜌_𝑛+1/𝜌_𝑛

(15)   𝑝_𝑛+1=𝑟_𝑛+1+𝛽_𝑛+1𝑝_𝑛

(16)   𝑝^∗_𝑛+1=𝑟^∗_𝑛+1+𝛽_𝑛+1𝑝^∗_𝑛

(17)   𝑞_𝑛+1=𝑟̂_𝑛+1+𝛽_𝑛+1𝑞_𝑛

(18)   𝑞^∗_𝑛+1=𝐴^𝐻𝑝^∗_𝑛

(19)  check convergence; continue if necessary

(20) end for