Introduction to Computational Linear Algebra

130 Introduction to Computational Linear Algebra

5.4.4 Arnoldi Method for Computing an Eigenpair

Let us assume that we search an eigenpair (λ, x) of A where |λ| is maximum.

The dimension k of the Krylov basis must be large enough to get valid ap-

proximations of the eigenvalues of A but two obstacles imply limitations on

k:

• the storage V

k+1

is limited by the memory capacity. This matrix is dense

and therefore implies storage of n(k + 1) words. When n is huge, k must

be smaller than some value m.

• roundoff errors generate a loss of orthogonality. When k is too large the

computed system fl(V

k

) is not orthogonal and often is even not a full

column rank