The Arnoldi method for standard eigenvalue problems possesses several
attractive properties making it robust, reliable and efficient for
many problems. We will present here a new algorithm equivalent to the
Arnoldi method, but designed for nonlinear eigenvalue problems
corresponding to the problem associated with a matrix depending on a
parameter in a nonlinear but analytic way. As a first result we show
that the reciprocal eigenvalues of an infinite dimensional operator.
We consider the Arnoldi method for this and show that with a
particular choice of starting function and a particular choice of
scalar product, the structure of the operator can be exploited in a
very effective way. The structure of the operator is such that when
the Arnoldi method is started with a constant function, the iterates
will be polynomials. For a large class of NEPs, we show that we can
carry out the infinite dimensional Arnoldi algorithm for the operator
in arithmetic based on standard linear algebra operations on vectors
and matrices of finite size. This is achieved by representing the
polynomials by vector coefficients. The resulting algorithm is by
construction such that it is completely equivalent to the standard
Arnoldi method and also inherits many of its attractive properties,
which are illustrated with examples.