Geodesic
Eigenvalue bounds for block preconditioned matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part X, Tome 219 (1994), pp. 5-41

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In the symmetric positive definite case we derive two-sided eigenvalue bounds for block Jacobi scaled matrices and upper eiqenvalue bounds for matrices preconditioned with an incomplete block factorization. A quantitative characterization of block matrix partitionings is also suggested that can be used when analyzing varions block preconditioning methods. Bibliography: 13 titles. 
@article{ZNSL_1994_219_a0,
     author = {L. Yu. Kolotilina},
     title = {Eigenvalue bounds for block preconditioned matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--41},
     publisher = {mathdoc},
     volume = {219},
     year = {1994},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1994_219_a0/}
}
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L. Yu. Kolotilina. Eigenvalue bounds for block preconditioned matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part X, Tome 219 (1994), pp. 5-41. http://geodesic.mathdoc.fr/item/ZNSL_1994_219_a0/