Introduction to Krylov subspace methods - IMAGe

α i−1 〈Ap i−1 , p i−1 〉 〈r i−1 , p i−1 〉 〈Ap i−1 , p i−1 〉∴ β i = 〈ri , r i 〉〈r i−1 , p i−1 〉Conjugate gradient(Hestenes and Stiefel, [3], Lanczos, [4])Given x 0 , r 0 = b − Ax 0 , p 0 = r 0 ,k = 0do ”until convergence”end doα k = 〈rk , p k 〉〈Ap k , p k 〉x k+1 = x k + α k p kr k+1 = r k − α k Ap kβ k+1 = 〈rk+1 , r k+1 〉〈r k , r k 〉p k+1 = r k+1 + β k+1 p kk = k + 1Stiefel (1952) and Lanczos (1950)1) Krylov Subspace Methods27