Analysis of some Krylov subspace approximations to the matrix exponential operator
TLDR
A theoretical analysis of some Krylov subspace approximations to the matrix exponential operation $\exp (A)v$ is presented, and a priori and a posteriors error estimates are established.Abstract:
In this note a theoretical analysis of some Krylov subspace approximations to the matrix exponential operation $\exp (A)v$ is presented, and a priori and a posteriors error estimates are established. Several such approximations are considered. The main idea of these techniquesis to approximately project the exponential operator onto a small Krylov subspace and to carry out the resulting small exponential matrix computation accurately. This general approach, which has been used with success in several applications, provides a systematic way of defining high-order explicit-type schemes for solving systems of ordinary differential equations or time-dependent partial differential equations.read more
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Journal ArticleDOI
The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)
Journal ArticleDOI
Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later ⁄
Cleve B. Moler,Charles Van Loan +1 more
TL;DR: Methods involv- ing approximation theory, dierential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed, indicating that some of the methods are preferable to others, but that none are completely satisfactory.
Book
Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-dependent Problems
TL;DR: This book discusses infinite difference approximations, Iterative methods for sparse linear systems, and zero-stability and convergence for initial value problems for ordinary differential equations.
Journal ArticleDOI
Fourth-Order Time-Stepping for Stiff PDEs
TL;DR: A modification of the exponential time-differencing fourth-order Runge--Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators.
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Expokit: a software package for computing matrix exponentials
TL;DR: The toolkit provides a set of routines aimed at computing matrix exponentials that computes either a small matrix exponential in full, the action of a large sparse matrix exponential on an operand vector, or the solution of a system of linear OBEs with constant inhomogeneity.
References
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Perturbation theory for linear operators
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
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GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
Youcef Saad,Martin H. Schultz +1 more
TL;DR: An iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace.
Book
The Theory of Matrices
TL;DR: In this article, the Routh-Hurwitz problem of singular pencils of matrices has been studied in the context of systems of linear differential equations with variable coefficients, and its applications to the analysis of complex matrices have been discussed.
Related Papers (5)
On Krylov Subspace Approximations to the Matrix Exponential Operator
Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later ⁄
Cleve B. Moler,Charles Van Loan +1 more