Algorithm 919
Top Cited Papers
- 1 April 2012
- journal article
- research article
- Published by Association for Computing Machinery (ACM) in ACM Transactions on Mathematical Software
- Vol. 38 (3), 1-19
- https://doi.org/10.1145/2168773.2168781
Abstract
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.Keywords
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Funding Information
- Australian Research Council (DP0559083)
This publication has 25 references indexed in Scilit:
- Computing the Action of the Matrix Exponential, with an Application to Exponential IntegratorsSIAM Journal on Scientific Computing, 2011
- Implementation of exponential Rosenbrock-type integratorsApplied Numerical Mathematics, 2009
- Numerical Methods for Ordinary Differential EquationsPublished by Wiley ,2008
- The Scaling and Squaring Method for the Matrix Exponential RevisitedSIAM Journal on Matrix Analysis and Applications, 2005
- A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency OptionsThe Review of Financial Studies, 1993
- Efficient Solution of Parabolic Equations by Krylov Approximation MethodsSIAM Journal on Scientific and Statistical Computing, 1992
- A method for exponential propagation of large systems of stiff nonlinear differential equationsJournal of Scientific Computing, 1989
- Sparse matrix test problemsACM Transactions on Mathematical Software, 1989
- On the numerical solution of heat conduction problems in two and three space variablesTransactions of the American Mathematical Society, 1956
- A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction typeMathematical Proceedings of the Cambridge Philosophical Society, 1947