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A finite dimensional analog of the Krein formula. (English) Zbl 1003.39016

The author presents a simple and useful formula for the resolvent of a small rank perturbation of large matrices. He gives applications of this formula for analytical and numerical resolution of difference boundary value problems. For the difference Laplacian the numerical efficiency of the corresponding algorithms is estimated.

MSC:

39A12 Discrete version of topics in analysis
65N06 Finite difference methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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[1] Akhiezer N I Glazman I N Theory of Linear Operators in Hilbert Space; Second revised and augmented edition, Nauka - Moscow, 1966 (in Russian); Third edition, corrected and augmented. Vishcha Shkola - Kharkov, Vol. I, 1977, Vol. II, 1978 (in Russian); Translated from the Russian and with a preface by Merlynd Nestell. Reprint of the 1961 and 1963 translations. Two volumes bound as one. Dover Publications, Inc. New York, 1993
[2] Gerasimenko N I, Teoret. Mat Fiz. 74 pp 345– (1988)
[3] Kostrykin V, J. Phys. A: Math. Gen. 32 pp 595– (1999) · Zbl 0928.34066 · doi:10.1088/0305-4470/32/4/006
[4] Albeverio S, Solvable Models in Quantum Mechanics, Texts and Monographs in Physics (1988) · doi:10.1007/978-3-642-88201-2
[5] Gordon E I Nonstandard Methods in Commutative Harmonic Analysis, Translations of Mathematical Monographs, Vol. 164, Providence, R.I., American Mathematical Society, 1997
[6] Albeverio S, Singular Perturbations of Differential Operators, London Mathematical Society Lecture Notes 271 (2000) · doi:10.1017/CBO9780511758904
[7] Pavlov B S, Uspekhi Matematicheskih Nauk (Russian Mathematical Survays) 42 (6) pp 99– (1987)
[8] Antonets M A, Russian Acad. Sci. Dokl. Math. 48 (2) pp 286– (1994)
[9] Bakhvalov N S, Numerical Methods (1987)
[10] Vysheslavtsev P P, Izvestija VUZ’ov, Radiofizika 40 pp 213– (1997)
[11] Nefedov I M, J. Nonlin. Math. Phys. 8 (3) pp 313– (2001) · Zbl 0996.65088 · doi:10.2991/jnmp.2001.8.3.1
[12] Okomelkova I A, Mat. Model. 7 (5) pp 89– (1995)
[13] Prasolov V V, Problems and Theorems in Linear Algebra, Translations of Mathematical Monographs 134 (1994) · Zbl 0803.15001
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