NAGnews Issue 146
In this issue:
New Technical Report: Extending error function and related functions to complex arguments
The NAG Library Chapter 'Approximations of Special Functions' (Chapter S) contains routines for evaluating Bessel functions, error functions, elliptic integrals and Fresnel integrals among others. In this report we extend the error functions available in the NAG Library for real argument to complex argument. These extensions will allow us to evaluate more difficult special functions related to the error functions.
The error functions appear in several mathematical and physics applications; in mathematics the complementary error function is extensively used in the development of uniform asymptotic expansions for evaluating more involved special functions such as incomplete gamma functions and appear in several exponentially-improved asymptotic expansions, for example in the generalized exponential integral. Furthermore, it plays a key role in providing a smooth interpretation of the Stokes phenomenon. Applications in statistics and probability theory are well known, for example the use of the error function in the normal distribution function and in the asymptotic of arbitrary probability density functions. The Faddeeva function, Fresnel integrals, and other related functions are present in several physics applications, from analysis of the diffraction of light to atomic physics and astrophysics. Voigt functions appear in the analysis of light absorption, plasma diagnostics, neutron diffraction, laser spectroscopy among others. Other closely related functions with important applications in physics will be explored in Section 3 of the report.
The report is organized into 4 sections. In Section 2 we present the extensions of the Faddeeva function. In Section 3 we explore several applications of some related functions. Finally, Section 4 includes the necessary code to run the examples using the NAG Toolbox for MATLAB®.
You can read the report here.
The Role of Matrix Functions - register for our 'how to 'webinar
Thursday 9 March 2017, 16:00 GMT, 12:00 EDT, 09:00 PDT
Matrix Functions are playing an increasingly important role in science, finance and engineering. Dr Edvin Hopkins will deliver this 30-minute live webinar covering a variety of Matrix Functions related topics including:
- What are matrix functions and how are they defined?
- Examples of the many applications where they are used
- Algorithms for computing functions of matrices
- Using NAG routines to evaluate matrix functions
- Practical examples using Python and C* to call the NAG Library
- Q & A session
*Examples also available in MATLAB and C/C++
New Mathematical Optimization Collaboration with the University of Oxford
NAG has recently started an academic collaboration with the Centre for Doctoral Training in Industrially Focused Mathematical Modelling (InFoMM) at the University of Oxford. Lindon Roberts is the main researcher supervised by Coralia Cartis, Associate Professor in Numerical Optimization. NAG is a strong supporter of InFoMM, offering student projects, providing training courses and sitting on the Industrial Engagement Committee.
This project focuses on mathematical optimization where derivatives are not readily available, so called derivative-free optimization (DFO). It is not easy or even possible to evaluate derivatives of functions which appear in the optimization model and thus many well-established approaches in mathematical optimization might not be satisfactory. Moving to a derivative-free regime presents novel approaches for approximating the solution without computing or estimating derivatives. NAG added its first derivative-free solver to the NAG Library about five years ago. Since then this field has attracted significant academic attention, resulting in numerous advances.
NAG started the collaboration with Lindon, Coralia and the InFoMM CDT earlier this year when a mini-project was sponsored to investigate DFO for nonlinear least squares optimization, a problem which is very common in the calibration of models in finance and engineering. After successful completion, NAG received not only a review of state-of-the-art DFO software, but also a working solver which will be adopted into the Library in 2017. We believe it will be the first such commercial solver available to the public anywhere in the world.
The full Doctoral project will focus on several open problems in DFO such as performance for noisy problems and the curse of dimensionality for large-scale problems. NAG will assist throughout the project by providing technical expertise and guidance, and by collaborating closely with Lindon to integrate his research into the NAG Library, enabling smooth and timely commercialisation of his research.
Academic collaborations remain a core part of our business and help the adoption of cutting-edge research into the NAG Library. The optimization software in the Library is an important part of its value to our customers and Lindon's research into new techniques for DFO will enhance this further. We look forward to the continuation of this research.
Best of the Blog
Out & About with NAG
Come and see us at various conferences and events over the next few months.
Rice University Oil & Gas HPC Conference
15-16 March 2017
Fortran Modernization Workshop, University of Warwick
27-28 April 2017
The Trading Show Chicago 2017
17-18 May 2017