• Corpus ID: 223956835

# A residual concept for Krylov subspace evaluation of the $\varphi$ matrix function

@inproceedings{Botchev2020ARC,
title={A residual concept for Krylov subspace evaluation of the \$\varphi\$ matrix function},
author={Mike A. Botchev and Leonid A. Knizhnerman and Eugene E. Tyrtyshnikov},
year={2020}
}
• Published 16 October 2020
• Computer Science
An efficient Krylov subspace algorithm for computing actions of the φmatrix function for large matrices is proposed. This matrix function is widely used in exponential time integration, Markov chains and network analysis and many other applications. Our algorithm is based on a reliable residual based stopping criterion and a new efficient restarting procedure. For matrices with numerical range in the stable complex half plane, we analyze residual convergence and prove that the restarted method…

## References

SHOWING 1-10 OF 73 REFERENCES

### On Krylov Subspace Approximations to the Matrix Exponential Operator

• Computer Science, Mathematics
• 1997
A new class of time integration methods for large systems of nonlinear differential equations which use Krylov approximations to the exponential function of the Jacobian instead of solving linear or nonlinear systems of equations in every time step is proposed.

### Analysis of some Krylov subspace approximations to the matrix exponential operator

• Computer Science, Mathematics
• 1992
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.

### Deflated Restarting for Matrix Functions

• Computer Science
SIAM J. Matrix Anal. Appl.
• 2011
We investigate an acceleration technique for restarted Krylov subspace methods for computing the action of a function of a large sparse matrix on a vector. Its effect is to ultimately deflate a

### Algorithm 919: A Krylov Subspace Algorithm for Evaluating the ϕ-Functions Appearing in Exponential Integrators

• Computer Science
TOMS
• 2012
An algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity using Arnoldi or Lanczos iteration and projecting the function on this subspace using time-stepping to prevent the Krylov subspace from growing too large.

### A Restarted Krylov Subspace Method for the Evaluation of Matrix Functions

• Computer Science
SIAM J. Numer. Anal.
• 2006
The Arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted Krylov subspace methods for solving linear systems of equations and inherits the superlinear convergence property of its unrestarted counterpart for entire functions.

### Residual, Restarting, and Richardson Iteration for the Matrix Exponential

• Computer Science, Mathematics
SIAM J. Sci. Comput.
• 2013
It is shown how the residual can be computed efficiently within several iterative methods for the matrix exponential, and how this completely resolves the question of reliable stopping criteria for these methods.

### Expokit: a software package for computing matrix exponentials

Expokit provides a set of routines aimed at computing matrix exponentials. More precisely, it computes either a small matrix exponential in full, the action of a large sparse matrix exponential on an