Abstract The dissertation consists of three articles in which irrationality measures for some values of certain special cases of the Gauss hypergeometric function are considered in both archimedean and non-archimedean metrics. The first presents a general result and a divisibility criterion for certain products of binomial coefficients upon which the sharpenings of the general result in special cases rely. The paper also provides an improvement concerning th e values of the logarithmic function. The second paper includes two other special cases, the first of which gives irrationality measures for some values of the arctan function, for example, and the second concerns values of the binomial function. All the results of the first two papers are effective, but no computation of the constants for explicit presentation is carried out. This task is fulfilled in the third article for logarithmic and binomial cases. The results of the latter case are applied to some Diophantine equations.

No / The key equations of BiCGStab are summarised to show its connections with Pade and vector-Pade approximation. These considerations lead naturally to stabilised vector-Pade approximation of a vector-valued function (VPAStab), and an algorithm for the acceleration of convergence of a linearly generated sequence of vectors. A generalisation of this algorithm for the acceleration of convergence of a nonlinearly generated system is proposed here, and comparative numerical results are given.

No / The four vector extrapolation methods, minimal polynomial extrapolation, reduced rank extrapolation, modified minimal polynomial extrapolation and the topological epsilon algorithm, when applied to linearly generated vector sequences are Krylov subspace methods and it is known that they are equivalent to some well-known conjugate gradient type methods. However, the vector -algorithm is an extrapolation method, older than the four extrapolation methods above, and no similar results are known for it. In this paper, a determinantal formula for the vector -algorithm is given. Then it is shown that, when applied to a linearly generated vector sequence, the algorithm is also a Krylov subspace method and for a class of matrices the method is equivalent to a preconditioned Lanczos method. A new determinantal formula for the CGS is given, and an algebraic comparison between the vector -algorithm for linear systems and CGS is also given.

Die Literatur von den Autoren Hairer, Wanner, Nørsett und Butcher, die der Arbeit als wichtige Quellen zugrunde lag, beschäftigt sich auch intensiv mit Mehrschrittverfahren. Hier wird jeweils nur ein kurzer Ausblick auf Ordnungssterne bzw. Ordnungspfeile bei Mehrschrittverfahren mit einem Beispiel gegeben. Auch auf eine Behandlung der Ordnungssterne im Gebiet der Approximationstheorie wird mit einem Beispiel kurz hingewiesen.

Abstract The focus of this thesis is on two functions: the exponential function and Euler’s factorial series. By constructing explicit Padé approximations, we are able to improve lower bounds for linear forms in the values of these functions. In particular, the dependence on the height of the coefficients of the linear form will be sharpened in the lower bound. The first chapter contains some necessary definitions and auxiliary results needed in later chapters.We give precise definitions for a transcendence measure and Padé approximations of the second type. Siegel’s lemma will be introduced as a fundamental tool in Diophantine approximation. A brief excursion to exterior algebras shows how they can be used to prove determinant expansion formulas. The reader will also be familiarised with valuations of number fields. In Chapter 2, a new transcendence measure for e is proved using type II Hermite-Padé approximations to the exponential function. An improvement to the previous transcendence measures is achieved by estimating the common factors of the coefficients of the auxiliary polynomials. The exponential function is the underlying topic of the third chapter as well. Now we study the common factors of the maximal minors of some large block matrices that appear when constructing Padé-type approximations to the exponential function. The factorisation of these minors is of interest both because of Bombieri and Vaaler’s improved version of Siegel’s lemma and because they are connected to finding explicit expressions for the approximation polynomials. In the beginning of Chapter 3, two general theorems concerning factors of Vandermonde-type block determinants are proved. In the final chapter, we concentrate on Euler’s factorial series which has a positive radius of convergence in p-adic fields. We establish some non-vanishing results for a linear form in the values of Euler’s series at algebraic integer points. A lower bound for this linear form is derived as well.

Abstract This dissertation consists of four articles on irrationality measures. In the first paper we derive explicit irrationality measures by using the simple continued fraction expansions in a completely new way. In the second and third articles we use Padé approximations to construct irrationality measures. In the second paper we obtain an explicit irrationality measure for the values of q-exponential series, for which the earlier corresponding results are not as explicit. Furthermore, we construct a restricted irrationality measure for the values of q-exponential series, which is an improvement on the earlier results in the restricted case. In the third article we derive the best possible asymptotic restricted irrationality exponent for the values of Jacobi's triple product. In the last paper we consider Cantor series. We generalize the earlier results by deriving Sondow's irrationality measure for some Cantor series. / Tiivistelmä Tämä väitöskirja koostuu neljästä artikkelista, jotka kaikki käsittelevät irrationaalisuusmittoja. Ensimmäisessä artikkelissa irrationaalisuusmittoja johdetaan uudella tavalla irrationaalilukujen yksinkertaisista ketjumurtolukuesityksistä. Toisessa ja kolmannessa artikkelissa irrationaalisuusmitat konstruoidaan Padé-approksimaatioiden avulla. Toisessa artikkelissa saadaan eksplisiittinen irrationaalisuusmitta q-eksponenttisarjan arvoille, joiden vastaavat aikaisemmat irrationaalisuusmitat eivät ole näin eksplisiittisiä. Lisäksi samassa artikkelissa konstruoidaan q-eksponenttisarjan arvoille rajoitettu eksplisiittinen irrationaalisuusmitta, mikä parantaa aikaisempia tuloksia rajoitetussa tapauksessa. Kolmannessa artikkelissa johdetaan paras mahdollinen asymptoottinen irrationaalisuuseksponentti Jacobin kolmitulon arvoille. Viimeisessä artikkelissa käsitellään Cantorin sarjoja. Siinä yleistetään aikaisempia tuloksia johtamalla Sondowin irrationaalisuusmitta tietylle joukolle Cantorin sarjoja.

Dans cette thèse, nous étudions des algorithmes pour un problème de recherche de relations à une ou plusieurs variables. Il généralise celui de calculer une solution à un système d’équations linéaires modulaires sur un anneau de polynômes, et inclut par exemple le calcul d’approximants de Hermite-Padé ou d’interpolants bivariés. Plutôt qu’une seule solution, nous nous attacherons à calculer un ensemble de générateurs possédant de bonnes propriétés. Précisément, l’entrée de notre problème consiste en un module de dimension finie spécifié par l’action des variables sur ses éléments, et en un certain nombre d’éléments de ce module ; il s’agit de calculer une base de Gröbner du modules des relations entre ces éléments. En termes d’algèbre linéaire, l’entrée décrit une matrice avec une structure de type Krylov, et il s’agit de calculer sous forme compacte une base du noyau de cette matrice. Nous proposons plusieurs algorithmes en fonction de la forme des matrices de multiplication qui représentent l’action des variables. Dans le cas d’une matrice de Jordan,nous accélérons le calcul d’interpolants multivariés sous certaines contraintes de degré ; nos résultats pour une forme de Frobenius permettent d’accélérer le calcul de formes normales de matrices polynomiales univariées. Enfin, dans le cas de plusieurs matrices denses, nous accélérons le changement d’ordre pour des bases de Gröbner d’idéaux multivariés zéro-dimensionnels. / In this thesis, we study algorithms for a problem of finding relations in one or several variables. It generalizes that of computing a solution to a system of linear modular equations over a polynomial ring, including in particular the computation of Hermite- Padéapproximants and bivariate interpolants. Rather than a single solution, we aim at computing generators of the solution set which have good properties. Precisely, the input of our problem consists of a finite-dimensional module given by the action of the variables on its elements, and of some elements of this module; the goal is to compute a Gröbner basis of the module of syzygies between these elements. In terms of linear algebra, the input describes a matrix with a type of Krylov structure, and the goal is to compute a compact representation of a basis of the nullspace of this matrix. We propose several algorithms in accordance with the structure of the multiplication matrices which specify the action of the variables. In the case of a Jordan matrix, we accelerate the computation of multivariate interpolants under degree constraints; our result for a Frobenius matrix leads to a faster algorithm for computing normal forms of univariate polynomial matrices. In the case of several dense matrices, we accelerate the change of monomial order for Gröbner bases of multivariate zero-dimensional ideals.