Geodesic
Solving a nonlinear spectral problem for a matrix
Zapiski Nauchnykh Seminarov POMI, Computational methods and automatic programming, Tome 58 (1976), pp. 54-66 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper examines the solving of the eigenvalue problem for a matrix $M(\lambda)$ with a nonlinear occurrence of the spectral parameter. Two methods are suggested for replacing the equation $\det M(\lambda)=0$ by a scalar equation $f(\lambda)=0$. Here the function $f(\lambda)$ is not written formally, but a rule for computing $f(\lambda)$ at a fixed point of the domain in which the desired roots lie is indicated. Mьller's method is used to solve the equation $f(\lambda)=0$. The eigenvalue found is refined by Newton's method based on the normalized expansion of matrix $M(\lambda)$ and the linearly independent vectors corresponding to it are computed. An ALGOL program and test examples are presented. 
@article{ZNSL_1976_58_a6,
     author = {T. Ya. Kon'kova and V. N. Kublanovskaya and L. T. Savinova},
     title = {Solving a nonlinear spectral problem for a matrix},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {54--66},
     year = {1976},
     volume = {58},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1976_58_a6/}
}
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T. Ya. Kon'kova; V. N. Kublanovskaya; L. T. Savinova. Solving a nonlinear spectral problem for a matrix. Zapiski Nauchnykh Seminarov POMI, Computational methods and automatic programming, Tome 58 (1976), pp. 54-66. http://geodesic.mathdoc.fr/item/ZNSL_1976_58_a6/