Matrix analysis
Analysis of some Krylov subspace approximations to the matrix exponential operator
SIAM Journal on Numerical Analysis
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
Exponential Integrators for Large Systems of Differential Equations
SIAM Journal on Scientific Computing
A new class of time discretization schemes for the solution of nonlinear PDEs
Journal of Computational Physics
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Exponential time differencing for stiff systems
Journal of Computational Physics
Computing a matrix function for exponential integrators
Journal of Computational and Applied Mathematics
Fourth-Order Time-Stepping for Stiff PDEs
SIAM Journal on Scientific Computing
Generalized integrating factor methods for stiff PDEs
Journal of Computational Physics
The Scaling and Squaring Method for the Matrix Exponential Revisited
SIAM Journal on Matrix Analysis and Applications
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Exponential Runge-Kutta methods for parabolic problems
Applied Numerical Mathematics - Tenth seminar on and differential-algebraic equations (NUMDIFF-10)
Journal of Computational Physics
Comparison of methods for evaluating functions of a matrix exponential
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)
ACM Transactions on Mathematical Software (TOMS)
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As a time discretization scheme for an ordinary differential equation with a stiff linear term, there is a class of methods that utilize the exponential or related functions of the coefficient matrix of the linear term. To implement these methods, we must compute a set of matrix functions called ''@f-function'', that includes the exponential itself, and it is important to compute these functions efficiently and accurately. In this paper, we consider the modified scaling and squaring method for the computation of @f-function. An algorithm based on Higham's method is defined, and the bounding parameter @q"m appropriate for @f-function is determined from an analysis of the truncation error under the assumption of the exact arithmetic. We also consider the propagation of the rounding error in the squaring process, and show that the error of @f-function is expected to be less than or roughly equal to that of the matrix exponential. Several evaluations are performed for famous test matrices, and the result shows that when the matrix exponential is computed accurately, the other @f-functions can also be obtained with the same level of accuracy.