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An extension of the rayleigh‐ritz method for finding upper and lower bounds of eigenvalues
Author(s) -
Sack R. A.
Publication year - 2009
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560010656
Subject(s) - eigenvalues and eigenvectors , subspace topology , mathematics , hermitian matrix , rayleigh–ritz method , upper and lower bounds , ritz method , simple (philosophy) , matrix (chemical analysis) , extension (predicate logic) , mathematical analysis , pure mathematics , physics , quantum mechanics , chemistry , philosophy , epistemology , chromatography , boundary value problem , computer science , programming language
A simple variational method is described for the determination of upper and lower bounds for the eigenvalues of an Hermitian operator M . For a selected vector subspace of n dimensions it involves the diagonalization of a 2n × 2n matrix, the elements of which depend on the matrix elements of M and M 2 as well as on upper and lower bounds of M in the complementary subspace. The method is equivalent to one given by H. F. Weinberger; it includes the Rayleigh‐Ritz procedure and a modified form of Temple's formula as limiting cases.