The Arnoldi method was first introduced as a direct algorithm for reducing a general matrix into upper Hessenberg form [19]. It was later discovered that this algorithm leads to a good iterative technique for approximating eigenvalues of large sparse matrices.
The algorithm works for non-Hermitian matrices.
It is most useful for cases when the matrix is large
but matrix-vector products are relatively inexpensive to perform.
This is the situation, for example, when
is large and sparse.
We begin with a presentation of the basic algorithm and then describe
a number of variations.