Preconditioned Krylov Subspace Methods for Sixth Order Compact Approximations of the Helmholtz Equation

Algorithm 1

Simplified Krylov subspace (SKS) algorithm.

(1) Let 𝑌⁽⁰⁾=𝐴_𝑝⁻¹𝑈⁽⁰⁾=0 be the initial approximation, let the tolerance be 𝑡𝑜𝑙,

and let the maximum number of iterations be 𝑀

(2) 𝑟₀=𝐹, 𝑌=𝑟₀ and 𝛿₀=𝑟₀₂

(3) 𝑗=1

(4) while  𝑗<𝑀 and 𝜀≥𝑡𝑜𝑙  do

(5) 𝑈^(𝑗)=𝐴_𝑝⁻¹𝑌

(6)  𝑤=𝐴𝑈^(𝑗)

(7)  𝑟=𝑟₀−𝑤

(8)  𝛿=‖𝑟‖₂

(9)  𝛿𝜀=𝛿₀

(10)  𝑌=𝑌+𝑟

(11)  𝑗=𝑗+1

(12) end while

(13) 𝑈^(𝑗) is the iterative solution of 𝐴𝑈=𝐹.