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Efficient solution of parabolic equations by Krylov approximation methods
SIAM Journal on Scientific and Statistical Computing
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
Computing the logarithm of a symmetric positive definite matrix
Applied Numerical Mathematics
A Padé Approximation Method for Square Roots of Symmetric Positive Definite Matrices
SIAM Journal on Matrix Analysis and Applications
Exponential Integrators for Large Systems of Differential Equations
SIAM Journal on Scientific Computing
EXPINT---A MATLAB package for exponential integrators
ACM Transactions on Mathematical Software (TOMS)
An error analysis of the modified scaling and squaring method
Computers & Mathematics with Applications
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
Journal of Computational and Applied Mathematics
| Hi-index | 7.30 |
An efficient numerical method is developed for evaluating ϕ(A), where A is a symmetric matrix and ϕ is the function defined by ϕ(x)= (ex - 1)/x = 1 +x/2 + x2/6 +.... This matrix function is useful in the so-called exponential integrators for differential equations. In particular, it is related to the exact solution of the ODE system dy/dt =Ay + b, where A and b are t-independent. Our method avoids the eigenvalue decomposition of the matrix A and it requires about 10n3/3 operations for a general symmetric n × n matrix. When the matrix is tridiagonal, the required number of operations is only O(n2) and it can be further reduced to O(n) if only a column of the matrix function is needed. These efficient schemes for tridiagonal matrices are particularly useful when the Lanczos method is used to calculate the product of this matrix function (for a large symmetric matrix) with a given vector.