Abstract
A new algorithm is developed for computing etA B, where A is an n × n matrix and B is n × n0 with n0>n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n × n 0 matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix etA B or a sequence etk B on an equally spaced grid of points tk. It uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential. It determines the amount of scaling and the Taylor degree using the recent analysis of AlMohy and Higham [SIAM J. Matrix Anal. Appl., 31 (2009), pp. 970989], which provides sharp truncation error bounds expressed in terms of the quantities ∥Ak ∥1/k for a few values of k, where the norms are estimated using a matrix norm estimator. Shifting and balancing are used as preprocessing steps to reduce the cost of the algorithm. Numerical experiments show that the algorithm performs in a numerically stable fashion across a wide range of problems, and analysis of rounding errors and of the conditioning of the problem provides theoretical support. Experimental comparisons with MATLAB codes based on Krylov subspace, Chebyshev polynomial, and Laguerre polynomial methods show the new algorithm to be sometimes much superior in terms of computational cost and accuracy. An important application of the algorithm is to exponential integrators for ordinary differential equations. It is shown that the sums of the form Σp k=0 φk(A)uk that arise in exponential integrators, where the φk are related to the exponential function, can be expressed in terms of a single exponential of a matrix of dimension n+p built by augmenting A with additional rows and columns, and the algorithm of this paper can therefore be employed. © 2011 Society for Industrial and Applied Mathematics.
Original language  English 

Pages (fromto)  488511 
Number of pages  23 
Journal  SIAM Journal on Scientific Computing 
Volume  33 
Issue number  2 
DOIs  
Publication status  Published  2011 
Keywords
 φ functions
 Backward error analysis
 Chebyshev polynomial
 Condition number
 Expm
 Exponential integrator
 Krylov method
 Laguerre polynomial
 MATLAB
 Matrix exponential
 ODE
 Ordinary differential equation
 Overscaling
 Taylor series
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