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Numerical evaluation of two and three parameter Mittag-Leffler functions
Roberto Garrappa
TL;DR
Numerical evaluation of Mittag–Leffler functions is difficult because series truncation is inefficient or unstable beyond small arguments, while few software methods exist. The paper uses Laplace-transform inversion with an optimal parabolic contour selected from singularity information, achieving high-accuracy, computationally efficient evaluation and extending the method to the three-parameter case.
Problem
Series evaluation is generally confined to small arguments, and few programming languages provide built-in Mittag–Leffler functions or broadly effective numerical methods.
Method
The method numerically inverts the Laplace transform, selecting among parabolic contours and analytic regions using singularity location and strength to reduce nodes and round-off propagation.
Results
The OPC method achieves accuracy close to or below the requested tolerance of 10^-15 on tested negative real-axis points and behaves robustly on tested imaginary-axis values.
Takeaways & Limitations
The approach evaluates the two-parameter ML function with high accuracy and computational efficiency and extends to the three-parameter ML function.
Takeaways & Limitations
Many singularities or large imaginary parts can make it nearly impossible to find a single contour that both encloses them and provides rapid exponential decay.
Abstract
from arXiv · showhide
The Mittag-Leffler (ML) function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation. In this work we present a method for the efficient computation of the ML function based on the numerical inversion of its Laplace transform (LT): an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the LT, with the aim of minimizing the computational effort and reduce the propagation of errors. Numerical experiments are presented to show accuracy and efficiency of the proposed approach. The application to the three parameter ML (also known as Prabhakar) function is also presented.
1. Introduction.
The paper motivates new numerical methods for Mittag–Leffler functions because existing series-based evaluation is limited for moderate or large arguments, while applications increasingly require many accurate evaluations. It proposes Laplace-transform inversion with an optimally selected parabolic contour and extends the approach to the three-parameter function.
- Mittag–Leffler functions are important in fractional calculus and model anomalous processes with hereditary effects.
- Most programming languages lack built-in Mittag–Leffler functions, leaving few numerical evaluation methods available.
- Series truncation is generally confined to small arguments because convergence slows and intermediate terms can cause overflow or numerical cancellation.
- Recent fractional differential-equation methods motivate accurate and fast evaluation across the complex plane, including matrix arguments.
- The proposed approach numerically inverts the Laplace transform on a parabolic contour whose region and parameters minimize computational effort and errors.
2. Evaluation of the ML function by LT inversion.
The paper formulates Mittag–Leffler evaluation as numerical Laplace-transform inversion and analyzes how analyticity, singularities, contour geometry, and quadrature parameters govern accuracy. It uses parabolic contours, residue subtraction, and error estimates to support prescribed-accuracy computation.
- Laplace-transform inversion provides a simple, fast computational route with accurate error estimates and virtually prescribed accuracy.
- The LT of the original two- and three-parameter functions is not directly available, so a generalized ML function is introduced to express the transform.
- The inverse-Laplace representation uses a Bromwich integral and deforms its contour into the left half-plane to damp the exponential factor and reduce oscillations.
- Contour selection must account for poles and the branch point at the origin, whose number and locations depend on α and λ.
- Many singularities or large imaginary parts can make a single contour unsuitable, motivating residue subtraction and separate analytic regions.
- Trapezoidal-rule error combines discretization and truncation effects, while narrow analyticity strips can make error prefactors nonnegligible.
3. Parabolic contours and the OPC algorithm.
The OPC algorithm partitions the complex plane into analyticity regions and selects parabolic contours and quadrature parameters to meet a prescribed accuracy with minimal computational effort and controlled round-off error.
- Parabolic contours: Parabolic contours are preferred because their simple representation eases handling multiple singularities despite slightly slower convergence than cotangent and hyperbolic contours.The cited convergence rates are fast for all three contour families, while parabolas are easier to handle under tightened constraints.
- OPC algorithm: In each region, the OPC procedure estimates error bounds and determines contour geometry, step size, and quadrature-node count for tolerance ε.The parameters are selected after estimating discretization-error contributions and matching them to the prescribed accuracy.
- OPC algorithm: The algorithm selects the region requiring the fewest quadrature nodes, then adds residues for singularities excluded by the selected contour.Residue subtraction enlarges contour-selection freedom, while the selected contour performs the actual Laplace-transform inversion.
- Computational selection: Contour-selection overhead is negligible relative to Laplace-transform inversion, enabling the expensive computation to use the smallest possible number of floating-point operations.The method also addresses round-off propagation when choosing among admissible contours.
- Region partitioning: The complex plane is partitioned into regions bounded by parabolas determined from the ordered singularities of the Laplace-transform integrand.The function ϕ orders singularities and defines parabolas that delimit disjoint regions of analyticity.
- Extension: The general strategy extends to the three-parameter Mittag-Leffler function because it uses the location and strength of Laplace-transform singularities.The paper describes this extension as straightforward.
4. Numerical experiments.
Numerical experiments evaluate OPC accuracy and efficiency for two- and three-parameter Mittag-Leffler functions. Across tested settings, OPC achieves errors near the requested 10^-15 tolerance, while its computation time remains comparatively stable.
- With ε = 10^-15, OPC targets accuracy close to machine precision in the numerical experiments.The tolerance represents the absolute integral error, while residue calculations can introduce relative round-off error smaller than 10^-15.
- OPC achieves accuracy close to or below 10^-15 for Eα,β(z) with α = 0.7, β = 1.0 on the real negative axis.The few gaps in the error plot occur when approximated and reference values are exactly equal.
- OPC CPU time remains nearly constant, whereas mlf is some orders of magnitude slower for moderate |z|.The two methods require similar or slightly different time for very small and large |z|, but mlf becomes substantially more expensive at intermediate magnitudes because it uses Romberg integration.
- OPC remains robust within the requested tolerance for complex values on the imaginary axis with α = 0.5 and β = 1.0.This setting is identified as one where Matlab's mlf code does not provide accurate results.
- For Eγ α,β(z) with α = 0.6, β = 0.9, γ = 1.2 and arg(z) = 3π, OPC produces errors very close to the target tolerance.Most computation and CPU time are the same as in the two-parameter case; the three-parameter extension mainly changes coefficients in the error estimates.
5. Concluding remarks.
The OPC method evaluates the two-parameter Mittag–Leffler function with high accuracy and computational efficiency, and extends to the three-parameter function.
- The OPC method evaluates the two-parameter Mittag–Leffler function with high accuracy.
- Numerical experiments demonstrate the computational efficiency of the OPC method.
- The method is generalized to the three-parameter Mittag–Leffler function and tested.
- The corresponding Matlab code is made freely available.