Efficient computation of the exponential operator for large, sparse, symmetric matrices

In this paper we compare Krylov subspace methods with Chebyshev series expa nsion for approximating the matrix exponential operator on large, sparse, s ymmetric matrices. Experimental results upon negative-definite matrices wit h very large size, arising from (2D and 3D) FE and FD spatial discretizatio n of linear parabolic PDEs, demonstrate that the Chebyshev method can be an effective alternative to Krylov techniques, especially when memory bounds do not allow the storage of all Ritz vectors. We also discuss the sensitivi ty of Chebyshev convergence to extreme eigenvalue approximation, as well as the reliability of various a priori and a posteriori error estimates for b oth methods. Copyright (C) 2000 John Wiley & Sons, Ltd.