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

Algorithm 3

Left preconditioned CSBiCOR method.

(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). Set 𝑝₀=𝑟₀, 𝑝₀=𝑟₀̃, 𝑞₀=𝐴𝑝₀, 𝑞₀=𝐴^𝐻𝑝₀.

(3) Compute 𝜌₀=𝑟₀̃,𝐴𝑟₀.

(4) Begin LOOP (𝑛=0,1,2,…)

(5) 𝜎_𝑛=𝑞_𝑛,𝑞_𝑛

(6) 𝑠_𝑛+1=𝜎_𝑛𝑟_𝑛−𝜌_𝑛𝑞_𝑛

(7) 𝑠̃_𝑛+1=𝜎_𝑛𝑟_𝑛̃−𝜌_𝑛𝑞_𝑛

(8) 𝑦_𝑛+1=𝐴𝑠_𝑛+1

(9) 𝑦_𝑛+1=𝐴^𝐻𝑠̃_𝑛+1

(10) 𝜃_𝑛+1=𝑠̃_𝑛+1,𝑦_𝑛+1

(11) 𝜁_𝑛+1=𝑦_𝑛+1,𝑦_𝑛+1

(12) if 1 × 1 step then

(13)  𝛼_𝑛=𝜌_𝑛/𝜎_𝑛

(14)  𝜌_𝑛+1=𝜃_𝑛+1/𝜎²_𝑛

(15)  𝛽_𝑛+1=𝜌_𝑛+1/𝜌_𝑛

(16)  𝑥_𝑛+1=𝑥_𝑛+𝛼_𝑛𝑝_𝑛

(17)  𝑟_𝑛+1=𝑟_𝑛−𝛼_𝑛𝑞_𝑛

(18)  𝑟̃_𝑛+1=𝑟_𝑛̃−𝛼_𝑛𝑞_𝑛

(19)  𝑝_𝑛+1=𝑠_𝑛+1/𝜎_𝑛+𝛽_𝑛+1𝑝_𝑛

(20)  𝑞_𝑛+1=𝑦_𝑛+1/𝜎_𝑛+𝛽_𝑛+1𝑞_𝑛

(21)  𝑞_𝑛+1=𝑦_𝑛+1/𝜎_𝑛+𝛽_𝑛+1𝑞_𝑛

(22)  𝑛←𝑛+1

(23) else

(24)  𝛿_𝑛=𝜎_𝑛𝜁_𝑛+1𝜌²_𝑛−𝜃²_𝑛+1

(25)  𝛼_𝑛=𝜁_𝑛+1𝜌³_𝑛/𝛿_𝑛

(26)  𝛼_𝑛+1=𝜃_𝑛+1𝜌²_𝑛/𝛿_𝑛

(27)  𝑥_𝑛+2=𝑥_𝑛+𝛼_𝑛𝑝_𝑛+𝛼_𝑛+1𝑠_𝑛+1

(28)  𝑟_𝑛+2=𝑟_𝑛−𝛼_𝑛𝑞_𝑛−𝛼_𝑛+1𝑦_𝑛+1

(29)  𝑟̃_𝑛+2=𝑟_𝑛̃−𝛼_𝑛𝑞_𝑛−𝛼_𝑛+1𝑦_𝑛+1

(30)  solve  𝐵𝑧_𝑛+2=𝑟_𝑛+2

(31)  solve  𝐵^𝐻𝑧_𝑛+2=𝑟̃_𝑛+2

(32)  𝑧_𝑛+2=𝐴𝑧_𝑛+2

(33)  𝑧_𝑛+2=𝐴^𝐻𝑧_𝑛+2

(34)  𝜌_𝑛+2=𝑧_𝑛+2,𝑟_𝑛+2

(35)  𝛽_𝑛+1=𝜌_𝑛+2/𝜌_𝑛

(36)  𝛽_𝑛+2=𝜌_𝑛+2𝜎_𝑛/𝜃_𝑛+1

(37)  𝑝_𝑛+2=𝑧_𝑛+2+𝛽_𝑛+1𝑝_𝑛+𝛽_𝑛+2𝑠_𝑛+1

(38)  𝑞_𝑛+2=𝑧_𝑛+2+𝛽_𝑛+1𝑞_𝑛+𝛽_𝑛+2𝑦_𝑛+1

(39)  𝑞_𝑛+2=𝑧_𝑛+2+𝛽_𝑛+1𝑞_𝑛+𝛽_𝑛+2𝑦_𝑛+1

(40)  𝑛←𝑛+2

(41) end if

(42) Check convergence; continue if necessary

(43) End LOOP