Geodesic
Solution of the eigenvalue problem for the regular bundle $D(\lambda)=\lambda A_0-A_1$ using deflated subspaces
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms, Tome 80 (1978), pp. 48-65

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We consider the solution of the generalized eigenvalue problem $$ (A_0\lambda-A_1)x=0, $$ in the case where one or both of the matrices $A_0$, $A_1$ are degenerate but the intersection of their null spaces is empty. Using orthogonal matrices $\mathscr P$ and which are independent of $\lambda$ the original problem is transformed to a simpler one in which the pencil is of smaller dimension. The construction of $P$ and $Q$ uses the normalization process. We include an Algol program and sample runs. 
@article{ZNSL_1978_80_a2,
     author = {T. Ya. Kon'kova},
     title = {Solution of the eigenvalue problem for the regular bundle $D(\lambda)=\lambda A_0-A_1$ using deflated subspaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {48--65},
     publisher = {mathdoc},
     volume = {80},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1978_80_a2/}
}
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T. Ya. Kon'kova. Solution of the eigenvalue problem for the regular bundle $D(\lambda)=\lambda A_0-A_1$ using deflated subspaces. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms, Tome 80 (1978), pp. 48-65. http://geodesic.mathdoc.fr/item/ZNSL_1978_80_a2/