Introduction to Krylov subspace methods - IMAGe

Projection methods5 KRYLOV SUBSPACE METHODS 33Generate Krylov subspace from scratch:Proof. (Homework)Arnoldi’s method : (1951) A → H unitaryArnoldi’s procedure builds on orthonormal basis for the Krylov subspace K m (A; v).(Exact arithmetic supposed)Pick v 1 s.t. ‖v 1 ‖ 2 = 1for j = 1, 2, · · · , m doendh ij = (Av j , v i ) for i = 1, 2, · · · , jw j = Av j − ∑ ji=1 h ijv ih j+1,j = ‖w j ‖ 2 (If h j+1,j = 0, then Stop!! ∗)v j+1 = w j /h j+1,jDef. The grade of a vector v w.r.t. A is the lowest degree monic polynomialp(≠ 0) s.t. p(A)v = 0. (Trsf ForA Ainto ∈ Rupper m×m , Heissenberg v ∈ R m , weform have using thatunitary the grade transformations) of v isalways equal or smaller than m.31