Abstract

APPENDIX D. GMRES AND ARNOLDI ITERATIVE METHODS 154
To specify what the ’best’ approximate eigenvector means, the pair ( ˜ λ, ã) ∈ C × Kj
is the ’best’ approximate eigenpair if for all w ∈ Kj, wehave
Dã − ˜
λã, w =0.
This ’best’ approximate eigenpair is called the Ritz pair, where ˜ λ is the Ritz value and
ã is called the Ritz vector. Let Pj ∈ R M×j be the matrix whose columns correspond
to the orthonomal basis of Kj generated by the Arnoldi process. Then, as stated in
[25], ( ˜ λ, ã) is a Ritz pair if and only if for some γ ∈ R M ,wehave
ã = Pjγ and P T j DPjγ = ˜ λγ.
So computing a Ritz pair in the Krylov subspace Kj corresponds to finding the
coefficient vector γ ∈ R j such that γ is an eigenvector of the matrix P T j DPj ∈ R j×j
with eigenvalue ˜ λ. Therefore, to compute the j-th Arnoldi iteration, we can solve a
j-dimensional eigenvalue problem to compute the Ritz pair ( ˜ λ, ã). To see how well the
Ritz pair approximates the eigenpair (λ,a), a Rayleigh quotient residual is computed.
The Rayleigh quotient residual, denoted by ˆr, is
ˆr = ˜ λ − ãT Dã
ã T ã .