TRn/nn - NAG Technical Report series number

TR1/20 Comparison of Wavelets for Adaptive Mesh Refinement
PDF Format
J Knipping & C Vuik, (Delft University of Technology) and J du Toit (NAG Ltd, Oxford)

In the field of Scientific Computing there is a big focus on solving time dependent Partial Differential Equations (PDEs) as efficiently as possible. Adaptive mesh refinement (AMR) can be used to construct a sparse mesh at every time step which maintains an accurate approximation to the solution. Interpolating wavelets are often used in AMR. We present a detailed comparison of two wavelets for AMR: Donoho's interpolating wavelet and a lifted version (also called second generation wavelets) of Donoho's interpolating wavelet. The wavelets are compared on PDE problems from computational finance and computational fluid dynamics.   We also examine different ways of handling the boundaries and the impact thereof.

Donoho's interpolating wavelet with lower order boundary stencil implementation appears to be the most accurate, whilst resulting in very high compression compared to the original mesh. For one data set Donoho's interpolating wavelet keeps fewer than 5% of the points whilst having an error smaller than 0.0001. In general, Donoho's interpolating wavelet produces sparse meshes while maintaining good accuracy, even for very irregular shapes. Lastly, an improvement on the inverse transform during the adaptive mesh refinement leads to promising results.

TR1/19 Using the NAG Library for Python with Kdb+ and PyQ
PDF Format
Christopher Brandt (NAG)

This paper provides detailed instructions on how to use the NAG Library for Python with kdb+ and PyQ. PyQ is an extension to kdb+ featuring zero-copy sharing of data between Python and the q programming language. The paper provides examples that illustrate how to access routines within the NAG Library for Python using data stored in kdb+. 

TR1/18 Using the NAG Library with Kdb+ in a Pure Q Environment
PDF Format
Christopher Brandt (NAG)

In the present technical report, we demonstrate how to integrate the NAG Library with kdb+ using the Foreign Function Interface (FFI) from Kx Systems. The procedure outlined herein leverages FFI to drastically simplify the development process for users. The enclosed three examples were carefully chosen to illustrate usage cases that extend to most of the 1700+ routines contained within the NAG Library.

TR4/16 Extending Error Function and related functions to Complex Arguments
PDF Format
Guillermo Navas-Palencia (NAG)

In this short communication several extensions of the Faddeeva function are implemented using functions currently available in the NAG Library. These extensions allow the evaluation of error and related functions with complex arguments. Finally, two relevant applications employing these extensions are presented.

TR2/16 Index-tracking Portfolio Optimization Model
PDF Format
Guillermo Navas-Palencia (NAG)

In the present tutorial report we examine the theory and computational aspects behind the index-tracking portfolio optimization model. This model is compared with Markowitz mean-variance model. This report is distributed with an example in C using the NAG C Library.
tr02_16.zip

TR2/15 Pricing Bermudan Swaptions on the LIBOR Market Model using the Stochastic Grid Bundling Method
PDF Format
Stef Maree (Delft University of Technology) and Jacques du Toit (NAG)

We examine using the Stochastic Grid Bundling Method (SGBM) to price a Bermudan swaption driven by a one-factor LIBOR Market Model (LMM). Using a well known approximation formula from the finance literature, we implement SGBM with one basis function and show that it is around six times faster than the equivalent Longstaff–Schwartz method. The two methods agree in price to one basis point, and the SGBM path estimator gives better (higher) prices than the Longstaff–Schwartz prices. A closer examination shows that inaccuracies in the approximation formula introduce a small bias into the SGBM direct estimator.

TR3/14 Adjoint Algorithmic Differentiation Tool Support for Typical Numerical Patterns in Computational Finance
PDF Format
Uwe Naumann (Aachen University) and Jacques du Toit (NAG)

We demonstrate the flexibility and ease of use of C++ algorithmic differentiation (AD) tools based on overloading to numerical patterns (kernels) arising in computational finance. While adjoint methods and AD have been known in the finance literature for some time, there are few tools capable of handling and integrating with the C++ codes found in production. Adjoint methods are also known to be very powerful but to potentially have infeasible memory requirements. We present several techniques for dealing with this problem and demonstrate them on numerical kernels which occur frequently in finance. We build the discussion around our own AD tool dco/c++ which is designed to handle arbitrary C++ codes and to be highly flexible, however the sketched concepts can certainly be transferred to other AD solutions including in-house tools. An archive of the source code for the numerical kernels as well as all the AD solutions discussed can be downloaded from our website. This includes documentation for the code and dco/c++. Trial licences for dco/c++ are available from NAG.

TR1/14 Generating Realisations of Stationary Gaussian Random Fields by Circulant Embedding
PDF Format
Catherine E. Powell, School of Mathematics, University of Manchester, UK

Random fields are families of random variables, indexed by a d-dimensional parameter x with d > 1.They are important in many applications and are used, for example, to model properties of biological tissue, velocity fields in turbulent flows and permeability coefficients of rocks. Mark 24 of the NAG Fortran Library includes new routines for generating realisations of stationary Gaussian random fields using the method of circulant embedding. This short note illustrates the main ideas behind circulant embedding and how to use the routines g05zr and g05zs in the NAG Toolbox for MATLAB. The routines g05zm, g05zn and g05zp can also be used to generate realisations of stationary Gaussian stochastic processes (the d = 1 case).1

TR3/12 Variable Selection in a Cox Proportional Hazards Model
PDF Format

In this article, and the associated example programs, we show how to use existing NAG library routines to perform automatic variable selection for a Cox proportional hazards model, a commonly type of model commonly used in the analysis of censored data. The three approaches described are; forward, backward and stepwise selection.
Accompanying materials:
tr3_12.zip

TR1/12 Solving partial differential equations using the NAG Library
PDF Format
Jeremy Walton, (NAG)

We describe the characteristics of partial differential equations (PDEs), including their uses, classification, subsidiary conditions and some of the ways in which they may be solved. In this context, we demonstrate how routines from the NAG Library can be used in their numerical solution. These routines come not only from the Library’s PDE chapter, but also from the chapters which deal with mesh generation and the solution of large linear systems. The combination of mesh generators and large linear solvers is applicable in the implementation of the so-called finite element method, which may be used in cases where the complexity of the geometry of the domain over which the PDE is to be solved prevents the application of the comparatively simple finite differencing method (as used, for example, in the Library’s PDE chapter). We illustrate the use of the NAG routines using a variety of example problems; the solutions are generated using the NAG Toolbox for MATLAB ® and plotted using tools in that environment.

TR2/11 Nonlinear Optimization Made Easier: A Tutorial for using the AMPL modelling language with NAG routines.
PDF Format
Jan Fiala, (NAG)

Optimization, or Operational Research in general, nowadays plays an important role in our lives. No matter if you are a respected finance house or a student of mathematics, you have probably used some sort of optimization routines. The field itself has changed rapidly since linear programming was introduced in the mid 1940s. More powerful computers allowed us to consider much more realistic and complex models using sophisticated algorithms. Whereas the input for linear programming problems is relatively simple, it is a much more delicate task in the case of general nonlinear programming. One way to tackle it is to introduce a specialised language for the problem description. In this tutorial we will focus on a particular one called AMPL which we have equipped with two of our NAG solvers, namely E04UFF and E04UGF.

Accompanying materials:
package_linux32.tgz
package_selfbuilt.zip
package_win32.zip

TR6/10 Solving an Optimization Problem using the NAG Library for .NET from F#
PDF Format
Sorin Serban, (NAG)

NAG has just released their latest numerical library; the NAG Library for .NET. This is the first release of the library and includes over 400 methods for key mathematical and statistical areas, including Wavelet Transforms, Integration, Interpolation and Approximation, Random Number Generators, Time Series Analysis, and Optimization. The Optimization chapter contains methods for solving LP-, QP-, LS- and NLP-problems without constraints or with constraints. A global optimizer is also included, solving problems without constraints but with bounds on the variables.

TR4/10 Flexible delivery of visualization software and services
PDF Format
Jason Wood, Jungwook Seo, David Duke Ken Brodlie (University of Leeds) and Jeremy Walton (NAG)

An important issue in the design of visualization systems is the allowance of flexibility in providing a range of interfaces to a single body of algorithmic software. In this paper we describe how the ADVISE architecture provides exactly this flexibility. The architecture is cleanly separated into three layers (user interface, web service middleware and visualization components) which gives us the flexibility to provide a range of different delivery options, but all making use of the same basic set of visualization components. These delivery options comprise a range of user interfaces (visual pipeline editor, tailored application, web page), coupled with installation choice between a stand-alone desktop application, or a distributed client-server application. This work was carried out within the ADVISE project.

TR2/10 Using the NAG Library to calculate financial option prices in Excel
PDF Format
Marcin Krzysztofik, Jeremy Walton, NAG

In finance, an option is a contract that conveys the right, but not the obligation, to buy or sell a specific asset. Options are widely traded on financial markets, and so some method of determining their value (or price) is required. Several option pricing models have been developed, which have then been implemented using a range of mathematical methods; some of these implementations have been made available in the latest release of the NAG Library. We have used these routines to calculate option prices in Microsoft Excel, and present some examples (which may be downloaded from the NAG website) that illustrate the way in which NAG routines can be called from within an Excel spreadsheet.

TR5/09 Calling NAG Library Routines from Scilab
HTML Format
Nathaniel Fenton, NAG Ltd, Oxford

This report gives detailed instructions on how to call routines in the NAG C and Fortran Libraries from the Scilab programming environment.

TR3/09 Fitting a Seasonal ARIMA Model using the NAG C Library
PDF Format
Martyn Byng, NAG Ltd, Oxford

This article gives a brief description of how to fit a seasonal ARIMA (autoregressive integrated moving average) model using the NAG C Library routine g13bec (/numeric/cl/nagdoc_cl26.2/html/g13/g13bec.html), and how to forecast from such a model using the NAG C Library routine g13bjc (/numeric/cl/nagdoc_cl26.2/html/g13/g13bjc.html). The article should be read in conjunction with the documentation for these two routines. A full set of example source code, data and expected results is available in the accompanying materials linked below.

Accompanying materials:
tr03_09.zip

TR1/09 A Web Services Architecture for Visualization
PDF Format
Jason Wood, Ken Brodlie, Jungwook Seo, David Duke (University of Leeds) and Jeremy Walton (NAG Ltd, Oxford)

Service-oriented architectures are increasingly being used in the creation of large distributed applications. This paper examines the provision of visualization as a service which can be made available for application designers to combine with other services. It describes a three-layer architecture which exploits the strengths of web service technologies in providing standardized access, and which also enables the efficient and flexible construction of visualization applications. A realization of the architecture is illustrated by re-visiting an early example of web-based visualization. This work was carried out within the ADVISE project.