Uniform Approximations for Transcendental Functions

@inproceedings{Winitzki2003UniformAF,
  title={Uniform Approximations for Transcendental Functions},
  author={Sergei Winitzki},
  booktitle={ICCSA},
  year={2003}
}
A heuristic method to construct uniform approximations to analytic transcendental functions is developed as a generalization of the Hermite-Pade interpolation to infinite intervals. The resulting uniform approximants are built from elementary functions using known series and asymptotic expansions of the given transcendental function. In one case (Lambert's W function) we obtained a uniform approximation valid in the entire complex plane. Several examples of the application of this method to… 

Figures from this paper

Global approximation for some functions
Elementary Functions and Approximate Computing
  • J. Muller
  • Computer Science
    Proceedings of the IEEE
  • 2020
TLDR
This article reviews some of the classical methods used for quickly obtaining low-precision approximations to the elementary functions and examines what can be done for obtaining very fast estimates of a function, at the cost of a (controlled) loss in terms of accuracy.
Accurate Padé Global Approximations for the Mittag-Leffler Function, Its Inverse, and Its Partial Derivatives to Efficiently Compute Convergent Power Series
The fractional derivative operator provides a mathematical means to concisely describe a heterogeneous and relatively complex system that exhibits non-local, power-law behavior. Discretization of a
Supersymmetric features of the Error and Dawson's functions
Following a letter by Bassett, we show first that it is possible to find an analytical approximation to the error function in terms of a finite series of hyperbolic tangents from the supersymmetric
Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse
Abstract This paper proposes a global Padé approximation of the generalized Mittag-Leffler function Eα,β(−x) with x ∈ [0,+∞). This uniform approximation can account for both the Taylor series for
Rational Solutions for the Time-Fractional Diffusion Equation
A uniform rational approximation of the Mittag-Leffler function is derived which serves as a global approximant, accounting for both the Taylor series for small arguments and asymptotic series for ...
Series Expansion and Fourth-Order Global Padé Approximation for a Rough Heston Solution
The rough Heston model has recently been shown to be extremely consistent with the observed empirical data in the financial market. However, the shortcoming of the model is that the conventional
Discrete-space time-fractional processes
A time-fractional diffusion process defined in a discrete probability setting is studied. Working in continuous time, the infinitesimal generators of random processes are discretized and the
Fourth-order algorithms for solving the imaginary-time Gross-Pitaevskii equation in a rotating anisotropic trap.
  • S. Chin, E. Krotscheck
  • Computer Science
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2005
By implementing the exact density matrix for the rotating anisotropic harmonic trap, we derive a class of very fast and accurate fourth-order algorithms for evolving the Gross-Pitaevskii equation in
...
1
2
3
4
5
...

References

SHOWING 1-8 OF 8 REFERENCES
Modern Computer Algebra
TLDR
This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems.
I and i
TLDR
There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.
Canad. Math. Soc. Conf. Proc
  • Canad. Math. Soc. Conf. Proc
  • 2000
Abramowitz and I . Stegun , Handbook of special functions , National Bureau of Standards , 1964 . 3 . P . Borwein , Canad . Math
  • 1999
Abramowitz and I. Stegun, Handbook of special functions
  • Abramowitz and I. Stegun, Handbook of special functions
  • 1964
Anal. Ser. B J. L. Spouge, J. SIAM of Num. Anal
  • Anal. Ser. B J. L. Spouge, J. SIAM of Num. Anal
  • 1964
Handbook of special functions
  • 1964