Geodesic
Counting the eigenvalues surrounded by a closed curve
Sibirskij žurnal industrialʹnoj matematiki, Tome 4 (2001) no. 2, pp. 73-94 Cet article a éte moissonné depuis la source Math-Net.Ru

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A procedure for counting the number of eigenvalues of a given matrix which are surrounded by a curve is presented. It is based on the application of the Caushy formula. The quadrature can be performed by different formulae (series expension, trapezoidal rule, ODE solvers). The main point consists of selecting an appropriate step size. Numerical tests are reported for matrices ranging from medium to large orders. 
@article{SJIM_2001_4_2_a4,
     author = {O. Bertrand and B. Philippe},
     title = {Counting the eigenvalues surrounded by a~closed curve},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {73--94},
     year = {2001},
     volume = {4},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2001_4_2_a4/}
}
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O. Bertrand; B. Philippe. Counting the eigenvalues surrounded by a closed curve. Sibirskij žurnal industrialʹnoj matematiki, Tome 4 (2001) no. 2, pp. 73-94. http://geodesic.mathdoc.fr/item/SJIM_2001_4_2_a4/