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Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic
Author(s) -
Druskin Vladimir,
Knizhnerman Leonid
Publication year - 1995
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1680020303
Subject(s) - mathematics , krylov subspace , lanczos resampling , eigenvalues and eigenvectors , matrix function , generalized minimal residual method , superconvergence , lanczos algorithm , matrix (chemical analysis) , convergence (economics) , iterative method , symmetric matrix , algorithm , finite element method , physics , materials science , quantum mechanics , economics , composite material , thermodynamics , economic growth
Many researchers are now working on computing the product of a matrix function and a vector, using approximations in a Krylov subspace. We review our results on the analysis of one implementation of that approach for symmetric matrices, which we call the spectral lanczos decomposition method (SLDM). We have proved a general convergence estimate, relating SLDM error bounds to those obtained through approximation of the matrix function by a part of its Chebyshev series. Thus, we arrived at effective estimates for matrix functions arising when solving parabolic, hyperbolic and elliptic partial differential equations. We concentrate on the parabolic case, where we obtain estimates that indicate superconvergence of SLDM. For this case a combination of SLDM and splitting methods is also considered and some numerical results are presented. We implement our general estimates to obtain convergence bounds of Lanczos approximations to eigenvalues in the internal part of the spectrum. Unlike Kaniel‐Saad estimates, our estimates are independent of the set of eigenvalues between the required one and the nearest spectrum bound. We consider an extension of our general estimate to the case of the simple Lanczos method (without reorthogonalization) in finite computer arithmetic which shows that for a moderate dimension of the Krylov subspace the results, proved for the exact arithmetic, are stable up to roundoff.