Topics in matrix analysis
ACM Transactions on Mathematical Software (TOMS)
A method for exponential propagation of large systems of stiff nonlinear differential equations
Journal of Scientific Computing
Analysis of some Krylov subspace approximations to the matrix exponential operator
SIAM Journal on Numerical Analysis
Efficient solution of parabolic equations by Krylov approximation methods
SIAM Journal on Scientific and Statistical Computing
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
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
The Scaling and Squaring Method for the Matrix Exponential Revisited
SIAM Journal on Matrix Analysis and Applications
An error analysis of the modified scaling and squaring method
Computers & Mathematics with Applications
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Implementation of exponential Rosenbrock-type integrators
Applied Numerical Mathematics
The scaling and modified squaring method for matrix functions related to the exponential
Applied Numerical Mathematics
Algorithm 894: On a block Schur--Parlett algorithm for ϕ-functions based on the sep-inverse estimate
ACM Transactions on Mathematical Software (TOMS)
Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators
SIAM Journal on Scientific Computing
A new class of exponential propagation iterative methods of Runge-Kutta type (EPIRK)
Journal of Computational Physics
Time-averaging and exponential integrators for non-homogeneous linear IVPs and BVPs
Applied Numerical Mathematics
Circuit simulation via matrix exponential method for stiffness handling and parallel processing
Proceedings of the International Conference on Computer-Aided Design
Locally exact modifications of numerical schemes
Computers & Mathematics with Applications
Improving the accuracy of the AVF method
Journal of Computational and Applied Mathematics
Journal of Computational Physics
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We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called ϕ-functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the matlab function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art.