Geodesic
Solution of an eigenvalue problem for pencils of band matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part VII, Tome 139 (1984), pp. 41-50

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An iterative algorithm is proposed for solving the complete eigenvalue problem of a regular, linear pencil $A-\lambda B$ of matrices $A$ and $B$ of band structure which under certain conditions preserves the band structure of matrices of the pencil. It is modification of the algorithm $AB-1$ based on applying nonorthogonal transformations. A detailed description of the algorithm is presented in application to the pencils with tridiagonal and pentadiagonal matrices. 
@article{ZNSL_1984_139_a2,
     author = {T. V. Vashchenko},
     title = {Solution of an eigenvalue problem for pencils of band matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {41--50},
     publisher = {mathdoc},
     volume = {139},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_139_a2/}
}
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T. V. Vashchenko. Solution of an eigenvalue problem for pencils of band matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part VII, Tome 139 (1984), pp. 41-50. http://geodesic.mathdoc.fr/item/ZNSL_1984_139_a2/