Table 1 Outline of the Krylov subspace projection method

Input: A, b and ${\widehat{\psi }}_0$

$r=b-A{\widehat{\psi }}_0$

ρ₁ = r^Hr

p= r

q= Ap

α = ρ₁/p^Hq

${\widehat{\psi }}_1={\widehat{\psi }}_0+\alpha p$

r= r- α q

for s = 2,..., S _k (or while ρ _s > ϵ)

ρ _s = r^Hr

β = ρ _s /ρ_s-1

p= r+ β p

q= Ap

α = ρ _s /p^Hq

${\widehat{\psi }}_s={\widehat{\psi }}_{s-1}+\alpha p$

r= r- α q

end

Output: ${\widehat{\psi }}_{S_k}$