The following nonlinear eigenvalue problem is studied : Let $T(\lambda )$ be an $n \times n$ matrix, whose elements are analytical functions of the complex number $\lambda $. We seek $\lambda $ and vectors x and y, such that $T(\lambda )x = 0$, and $y^H T(\lambda ) = 0$.Several algorithms for the numerical solution of this problem are studied. These algorithms are extensions of algorithms for the linear eigenvalue problem such as inverse iteration and the $QR$ algorithm, and algorithms that reduce the nonlinear problem into a sequence of linear problems. It is found that this latter method can be extended into a global strategy, finding a complete basis of eigenvectors in the cases where it is proved that such a basis exists.Numerical tests, performed in order to compare the different algorithms, are reported, and a few numerical examples illustrating their behavior are given.