In a recent article, Bai and Perron (2003, Journal of Applied Econometrics) present a comprehensive discussion of computational aspects of multiple structural change models along with several empirical examples. Here, we report on the results of a replication study using the R statistical software package. We are able to verify most of their findings; however, some confidence intervals associated with breakpoints cannot be reproduced. These confidence intervals require computation of the quantiles of a nonstandard distribution, the distribution of the argmax functional of a certain stochastic process. Interestingly, the difficulties appear to be due to numerical problems in GAUSS, the software package used by Bai and Perron. / Series: Research Report Series / Department of Statistics and Mathematics

In a recent article, Bai and Perron (2003, Journal of Applied Econometrics) present a comprehensive discussion of computational aspects of multiple structural change models along with several empirical examples. Here, we report on the results of a replication study using the R statistical software package. We are able to verify most of their findings; however, some confidence intervals associated with breakpoints cannot be reproduced. These confidence intervals require computation of the quantiles of a nonstandard distribution, the distribution of the argmax functional of a certain stochastic process. Interestingly, the difficulties appear to be due to numerical problems in GAUSS, the software package used by Bai and Perron. / Series: Research Report Series / Department of Statistics and Mathematics

First-principle modelling of chemical processes and their unit operations has been of great interest in the chemical process, as well as the control and allied industries over the past decades. This is because it offers the opportunity to develop virtual representations (models) of real process systems, which can be used to describe and predict the dynamic behaviour of those systems. These models are based on the fundamentals of the transport phenomena of fluid dynamics (involving momentum transfer), mass transfer, and energy transfer of the systems they describe. A first-principle model of a semi-batch rhodium dissolution chemical process has been developed. It describes the dynamic behaviour of two exothermic reactions, occurring simultaneously in a semi-batch process. The dissolution of 29 kg of solid crude rhodium sponge (Rh) into 546 L of a solution of hydrochloric acid (HCl(aq)), to produce a solution of aqueous rhodium(III) chloride (RhCl3.H2O), as well as the reaction of chlorine (Cl2(aq)) with water (H2O(l)) to produce some more HCl(aq) in the reactor. The model was formulated as a system of explicit ordinary differential equations (ODEs), which demonstrated some good and stable qualitative tracking of the temperature and pressure data of the real reactor. The molar responses of all chemical species, as well as the heats of reactions, showed to be consistent with the description of the process, and no negative values of those variables were generated. Estimates of the key parameters of heat and mass transfer coefficients, arrhenius constants, and activation energies of reactions were assumed and tuned to satisfaction by trial-and-error, but not optimised. This is because during simulations, the numerical solver would often fail to integrate the equations, due to the appearance of large derivatives in some model equations whenever those parameters varied, thereby stopping simulations. Finally, the model was validated with a set of data from 45 batches. For all simulations done, the simulated temperature responses showed better prediction of data than the simulated pressure responses did, with an average percentage accuracy of 80% against 60 percent, respectively. / Dissertation (MSc)--University of Pretoria, 2017. / Anglo American Platinum / BluESP (Pty) Ltd / Chemical Engineering / MSc / Unrestricted

Pour réduire les coûts des systèmes embarqués, ces derniers sont livrés avec des micro-processeurs peu puissants. Ces processeurs sont dédiés à l'exécution de tâches calculatoires dont certaines, comme la transformée de Fourier rapide, peuvent s'avérer exigeantes en termes de ressources de calcul. Afin que les implémentations de ces algorithmes soient efficaces, les programmeurs utilisent l'arithmétique à virgule fixe qui est plus adaptée aux processeurs dépourvus d'unité flottante. Cependant, ils se retrouvent confrontés à deux difficultés: D'abord, coder en virgule fixe est fastidieux et exige que le programmeur gère tous les détails arithmétiques. Ensuite, et en raison de la faible dynamique des nombres à virgule fixe par rapport aux nombres flottants, les calculs en fixe sont souvent perçus comme intrinsèquement peu précis. La première partie de cette thèse propose une méthodologie pour dépasser ces deux limitations. Elle montre comment concevoir et mettre en œuvre des outils pour générer automatiquement des programmes en virgule fixe. Ensuite, afin de rassurer l'utilisateur quant à la qualité numérique des codes synthétisés, des certificats sont générés qui fournissent des bornes sur les erreurs d'arrondi. La deuxième partie de cette thèse est dédiée à l'étude des compromis lors de la génération de programmes en virgule fixe pour les briques d'algèbre linéaire. Des données expérimentales y sont fournies sur la synthèse de code pour la multiplication et l'inversion matricielles. / To be cost effective, embedded systems are shipped with low-end micro-processors. These processors are dedicated to one or few tasks that are highly demanding on computational resources. Examples of widely deployed tasks include the fast Fourier transform, convolutions, and digital filters. For these tasks to run efficiently, embedded systems programmers favor fixed-point arithmetic over the standardized and costly floating-point arithmetic. However, they are faced with two difficulties: First, writing fixed-point codes is tedious and requires that the programmer must be in charge of every arithmetical detail. Second, because of the low dynamic range of fixed-point numbers compared to floating-point numbers, there is a persistent belief that fixed-point computations are inherently inaccurate. The first part of this thesis addresses these two limitations as follows: It shows how to design and implement tools to automatically synthesize fixed-point programs. Next, to strengthen the user's confidence in the synthesized codes, analytic methods are suggested to generate certificates. These certificates can be checked using a formal verification tool, and assert that the rounding errors of the generated codes are indeed below a given threshold. The second part of the thesis is a study of the trade-offs involved when generating fixed-point code for linear algebra basic blocks. It gives experimental data on fixed-point synthesis for matrix multiplication and matrix inversion through Cholesky decomposition.

Les systèmes critiques basés sur l’arithmétique flottante exigent un processus rigoureux de vérification et de validation pour augmenter notre confiance en leur sureté et leur fiabilité. Malheureusement, les techniques existentes fournissent souvent une surestimation d’erreurs d’arrondi. Nous citons Arian 5 et le missile Patriot comme fameux exemples de désastres causés par les erreurs de calculs. Ces dernières années, plusieurs techniques concernant la transformation d’expressions arithmétiques pour améliorer la précision numérique ont été proposées. Dans ce travail, nous allons une étape plus loin en transformant automatiquement non seulement des expressions arithmétiques mais des programmes complets contenant des affectations, des structures de contrôle et des fonctions. Nous définissons un ensemble de règles de transformation permettant la génération, sous certaines conditions et en un temps polynômial, des expressions pluslarges en appliquant des calculs formels limités, au sein de plusieurs itérations d’une boucle. Par la suite, ces larges expressions sont re-parenthésées pour trouver la meilleure expression améliorant ainsi la précision numérique des calculs de programmes. Notre approche se base sur les techniques d’analyse statique par interprétation abstraite pour sur-rapprocher les erreurs d’arrondi dans les programmes et au moment de la transformation des expressions. Cette approche est implémenté dans notre outil et des résultats expérimentaux sur des algorithmes numériques classiques et des programmes venant du monde d’embarqués sont présentés. / Critical software based on floating-point arithmetic requires rigorous verification and validation process to improve our confidence in their reliability and their safety. Unfortunately available techniques for this task often provide overestimates of the round-off errors. We can cite Arian 5, Patriot rocket as well-known examples of disasters. These last years, several techniques have been proposed concerning the transformation of arithmetic expressions in order to improve their numerical accuracy and, in this work, we go one step further by automatically transforming larger pieces of code containing assignments, control structures and functions. We define a set of transformation rules allowing the generation, under certain conditions and in polynomial time, of larger expressions by performing limited formal computations, possibly among several iterations of a loop. These larger expressions are better suited to improve, by re-parsing, the numerical accuracy of the program results. We use abstract interpretation based static analysis techniques to over-approximate the round-off errors in programs and during the transformation of expressions. A tool has been implemented and experimental results are presented concerning classical numerical algorithms and algorithms for embedded systems.

Numerické metody inverzní Laplaceovy transformace (NILT) se staly důležitou částí numerické sady nástrojů praktikujících a výzkumných pracovníků v mnoha vědeckých a inženýrských oborech, zejména v aplikované elektrotechnice. Techniky NILT zejména pomáhají při získávání výsledků simulací v časové oblasti v různých aplikacích. Příkladem jsou řešení obyčejných diferenciálních rovnic, které se objevují např. při analýze obvodů se soustředěnými parametry, nebo řešení parciálních diferenciálních rovnic objevujících se v systémech s rozprostřenými parametry, např. při zkoumání problematiky integrity signálů. Obecně platí, že většina dostupných 1D NILT metod je velmi specifická, tj. funguje dobře na několika typech funkcí a tudíž na omezeném počtu aplikací; Cílem této práce je podrobně se věnovat těmto numerickým metodám, vývoji univerzálních metod NILT a jejich rozšíření na multidimenzionální NILT, které mohou pokrývat širokou oblast aplikací a mohly by poskytnout praktický mechanism pro efektivnější způsob analýzy a simulace v časové oblasti. Myšlenky výzkumu jsou prezentovány v rámci diskusí nad širokou škálou případových studií a aplikací; Například metody NILT se používají při řešení přenosových vedení, včetně vícevodičových, a dokonce i při řešení slabě nelinárních obvodů při použití NILT více proměnných. Pomocí metody NILT mohou být s výhodou uvažovány parametry prvků závislé na kmitočtu a prvky necelistvých řádů v jejich příslušných modelech mohou být zahrnuty velmi přesným a jednoduchým způsobem.