Direct methods for sparse matrices
Direct methods for sparse matrices
On the rate of rational approximation of analytic functions
Proceedings of the international seminar on Approximation and optimization
Analysis of some Krylov subspace approximations to the matrix exponential operator
SIAM Journal on Numerical Analysis
Matrix computations (3rd ed.)
Expokit: a software package for computing matrix exponentials
ACM Transactions on Mathematical Software (TOMS)
Exponential Integrators for Large Systems of Differential Equations
SIAM Journal on Scientific Computing
A numerical study of large sparse matrix exponentials arising in Markov chains
Computational Statistics & Data Analysis
Scientific Computations on Mathematical Problems and Conjectures
Scientific Computations on Mathematical Problems and Conjectures
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
An estimator for the diagonal of a matrix
Applied Numerical Mathematics
SIAM Journal on Matrix Analysis and Applications
Domain-Decomposition-Type Methods for Computing the Diagonal of a Matrix Inverse
SIAM Journal on Scientific Computing
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We are interested in computing the Fermi---Dirac matrix function in which the matrix argument is the Hamiltonian matrix arising from density functional theory (DFT) applications. More precisely, we are really interested in the diagonal of this matrix function. We discuss rational approximation methods to the problem, specifically the rational Chebyshev approximation and the continued fraction representation. These schemes are further decomposed into their partial fraction expansions, leading ultimately to computing the diagonal of the inverse of a shifted matrix over a series of shifts. We describe Lanczos and sparse direct methods to address these systems. Each approach has advantages and disadvantages that are illustrated with experiments.