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Preconditioning Lanczos approximations to the matrix exponential. (English) Zbl 1105.65051

The concept of preconditioning for linear systems is generalized to compute the matrix exponential \(\exp(-\tau A),\) for a symmetric and positive semidefinite \(A.\) The exponential function is a quickly decaying function. Hence the vector \(\exp(-\tau A)v\) is mostly determined by the smallest eigenvalues and their corresponding invariant subspaces.
In this paper, the Lanczos method is applied to the inverse of \((I+\gamma A)\) with \(\gamma > 0,\) to emphasize the first eigenvalues and their eigenvectors using an inner-outer iteration. The convergence speed is independent of the norm of the matrix. In addition, a strategy for choosing the tolerances for the errors in the solution of the shifted system is proposed. The numerical results indicate the effectiveness of the method.

MSC:

65F35 Numerical computation of matrix norms, conditioning, scaling
65F30 Other matrix algorithms (MSC2010)
65F10 Iterative numerical methods for linear systems
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