Aggressively Truncated Taylor Series Method for Accurate Computation of Exponentials of Essentially Nonnegative Matrices

@article{Shao2014AggressivelyTT,
  title={Aggressively Truncated Taylor Series Method for Accurate Computation of Exponentials of Essentially Nonnegative Matrices},
  author={Meiyue Shao and Weiguo Gao and Jungong Xue},
  journal={SIAM J. Matrix Anal. Appl.},
  year={2014},
  volume={35},
  pages={317-338}
}
Small relative perturbations to the entries of an essentially nonnegative matrix introduce small relative errors to entries of its exponential. It is thus desirable to compute the exponential with high componentwise relative accuracy. Taylor series approximation coupled with scaling and squaring is used to compute the exponential of an essentially nonnegative matrix. An a priori componentwise relative error bound of truncation is established, from which one can choose the degree of Taylor… 

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References

SHOWING 1-10 OF 40 REFERENCES
Computing exponentials of essentially non-negative matrices entrywise to high relative accuracy
TLDR
Algorithm to compute exponentials of essentially non-negative matrices entrywise to high relative accuracy by determining the entries up to a condition number intrinsic to the exponential function.
A Schur-Parlett Algorithm for Computing Matrix Functions
An algorithm for computing matrix functions is presented. It employs a Schur decomposition with reordering and blocking followed by the block form of a recurrence of Parlett, with functions of the
Error analysis of two algorithms for the computation of the matrix exponential
TLDR
The reasons why the Taylor Series method can compete with Scaling and Squaring method if the norm of the original matrix is less than one are analyzed.
Numerical Computation of the Matrix Exponential with Accuracy Estimate
TLDR
An algorithm for computing the exponential of an arbitrary $n \times n$ matrix is presented and Diagonal Pade table approximations are used in conjunction with several techniques for reducing the norm of the matrix.
Dense and Structured Matrix Computations —the Parallel QR Algorithm and Matrix Exponentials
TLDR
This thesis proposes a method which efficiently computes all entries of the exponential of an essentially nonnegative matrix to high relative accuracy and proposes a repeated doubling strategy which works well even when a priori error estimates are pessimistic or not easy to compute.
Accurate computation of the smallest eigenvalue of a diagonally dominant M-matrix
TLDR
Rounding error analysis and numerical examples are presented to demonstrate the numerical behaviour of the algorithms that compute off-diagonal and diagonally dominant M-matrix quantities with relative errors in the magnitude of the machine precision.
On the Exponentiation of Interval Matrices
TLDR
The problem of computing a sharp enclosure of the interval matrix exponential is proved to be NP-hard and the scaling and squaring method is adapted to interval matrices and shown to drastically reduce the dependency loss w.r.t. the interval evaluation of the Taylor series.
Bounds for the Entries of Matrix Functions with Applications to Preconditioning
Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a well-known result of Demko, Moss and Smith on the decay of the inverse
The Padé method for computing the matrix exponential
...
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