The Arnoldi method was first introduced as a direct algorithm for reducing a general matrix into upper Hessenberg form . 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.