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

Algorithm 1

Biconjugate 𝐴-orthonormalization procedure.

(1) Choose 𝑣₁, 𝜔₁, such that 𝜔₁,𝐴𝑣₁=1

(2) Set 𝛽₁=𝛿₁≡0, 𝜔₀=𝑣₀≡𝟎∈ℂ^𝑁

(3) for  𝑗=1,2,…  do

(4)  𝛼_𝑗=𝜔_𝑗,𝐴(𝐴𝑣_𝑗)

(5)  𝑣_𝑗+1=𝐴𝑣_𝑗−𝛼_𝑗𝑣_𝑗−𝛽_𝑗𝑣_𝑗−1

(6)  𝜔_𝑗+1=𝐴^𝐻𝜔_𝑗−𝛼_𝑗𝜔_𝑗−𝛿_𝑗𝜔_𝑗−1

(7)  𝛿_𝑗+1=𝜔_𝑗+1,𝐴𝑣_𝑗+1^1/2

(8)  𝛽_𝑗+1=𝜔_𝑗+1,𝐴𝑣_𝑗+1/𝛿_𝑗+1

(9)  𝑣_𝑗+1=𝑣_𝑗+1/𝛿_𝑗+1

(10)   𝜔_𝑗+1=𝜔_𝑗+1/𝛽_𝑗+1

(11) end for