Definition 2.5 (Arnoldi iteration). Let A e Rxn and be R". Arnoldi iteration computes basis {q1, .... qi} for K(A, b) in three steps b if i = 0 Step 1 qi+1 = Aqi otherwise Step 2 qi+1 = qi+1 -> qi+19kqk k=1 Step 3 qit1 = lqi+1/12 qi+1. Intuitively speaking, the numerical stability of Arnoldi iteration is due to orthogonalisation step that eliminates directions q1, .... qi-1 from qi. This prevents Aqi from turning to the direction of the largest eigenvector of A\fThe family of Krylov subspaces {K;(A, b) } associated to A c Roxn and b is defined as Ki(A, b) = span {b, Ab, . .., A'-lb]