A general approach to transforming finite elements

The use of a reference element on which a finite element basis is constructed once and mapped to each cell in a mesh greatly expedites the structure and efficiency of finite element codes. However, many famous finite elements such as Hermite, Morley, Argyris, and Bell, do not possess the kind of equivalence needed to work with a reference element in the standard way. This paper gives a generalizated approach to mapping bases for such finite elements by means of studying relationships between the finite element nodes under push-forward.Comment: 28 page

Computing hypergeometric functions rigorously

We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions ${}_0F_1$, ${}_1F_1$, ${}_2F_1$ and ${}_2F_0$ (or the Kummer $U$-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function ${}_pF_q$ and computation of high-order parameter derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case E-G in section 8.5 (table 6, figure 2); adjusted paper siz

Hybrid Symbolic-Numeric Computing in Linear and Polynomial Algebra

In this thesis, we introduce hybrid symbolic-numeric methods for solving problems in linear and polynomial algebra. We mainly address the approximate GCD problem for polynomials, and problems related to parametric and polynomial matrices. For symbolic methods, our main concern is their complexity and for the numerical methods we are more concerned about their stability. The thesis consists of 5 articles which are presented in the following order: Chapter 1, deals with the fundamental notions of conditioning and backward error. Although our results are not novel, this chapter is a novel explication of conditioning and backward error that underpins the rest of the thesis. In Chapter 2, we adapt Victor Y. Pan\u27s root-based algorithm for finding approximate GCD to the case where the polynomials are expressed in Bernstein bases. We use the numerically stable companion pencil of G. F. Jónsson to compute the roots, and the Hopcroft-Karp bipartite matching method to find the degree of the approximate GCD. We offer some refinements to improve the process. In Chapter 3, we give an algorithm with similar idea to Chapter 2, which finds an approximate GCD for a pair of approximate polynomials given in a Lagrange basis. More precisely, we suppose that these polynomials are given by their approximate values at distinct known points. We first find each of their roots by using a Lagrange basis companion matrix for each polynomial. We introduce new clustering algorithms and use them to cluster the roots of each polynomial to identify multiple roots, and then marry the two polynomials using a Maximum Weight Matching (MWM) algorithm, to find their GCD. In Chapter 4, we define ``generalized standard triples\u27\u27 X, zC1 - C0, Y of regular matrix polynomials P(z) in order to use the representation X(zC1 - C0)-1 Y=P-1(z). This representation can be used in constructing algebraic linearizations; for example, for H(z) = z A(z)B(z) + C from linearizations for A(z) and B(z). This can be done even if A(z) and B(z) are expressed in differing polynomial bases. Our main theorem is that X can be expressed using the coefficients of 1 in terms of the relevant polynomial basis. For convenience we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases. We account for the possibility of common similarity transformations. We give explicit proofs for the less familiar bases. Chapter 5 is devoted to parametric linear systems (PLS) and related problems, from a symbolic computational point of view. PLS are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the coefficients of the system. We assume that the symbolic parameters appear polynomially in the coefficients and that the only variables to be solved for are those of the linear system. It is well-known that it is possible to specify a covering set of regimes, each of which is a semi-algebraic condition on the parameters together with a solution description valid under that condition.We provide a method of solution that requires time polynomial in the matrix dimension and the degrees of the polynomials when there are up to three parameters. Our approach exploits the Hermite and Smith normal forms that may be computed when the system coefficient domain is mapped to the univariate polynomial domain over suitably constructed fields. Our approach effectively identifies intrinsic singularities and ramification points where the algebraic and geometric structure of the matrix changes. Specially parametric eigenvalue problems can be addressed as well. Although we do not directly address the problem of computing the Jordan form, our approach allows the construction of the algebraic and geometric eigenvalue multiplicities revealed by the Frobenius form, which is a key step in the construction of the Jordan form of a matrix

Algebraic Companions and Linearizations

In this thesis, we look at a novel way of finding roots of a scalar polynomial using eigenvalue techniques. We extended this novel method to the polynomial eigenvalue problem (PEP). PEP have been used in many science and engineering applications such vibrations of structures, computer-aided geometric design, robotics, and machine learning. This thesis explains this idea in the order of which we discovered it. In Chapter 2, a new kind of companion matrix is introduced for scalar polynomials of the form $c(\lambda) = \lambda a(\lambda)b(\lambda)+c_0$, where upper Hessenberg companions are known for the polynomials $a(\lambda)$ and $b(\lambda)$. This construction can generate companion matrices with smaller entries than the Fiedler or Frobenius forms. This generalizes Piers Lawrence\u27s Mandelbrot companion matrix. The construction was motivated by use of Narayana-Mandelbrot polynomials. In Chapter 3, we define Euclid polynomials $E_{k+1}(\lambda) = E_{k} (\lambda) (E_{k} (\lambda) - 1) + 1$ where $E_{1}(\lambda) = \lambda + 1$ in analogy to Euclid numbers $e_k = E_{k} (1)$. We show how to construct companion matrices $E_{k}$, so $E_{k} (\lambda) = \det(\lambda I - E_{k} )$ is of height 1 (and thus of minimal height over all integer companion matrices for $E_{k}(\lambda)$). We prove various properties of these objects, and give experimental confirmation of some unproved properties. In Chapter 4, we show how to construct linearizations of matrix polynomials z\mat{a}(z)\mat{d}_0 + \mat{c}_0, \mat{a}(z)\mat{b}(z), \mat{a}(z) + \mat{b}(z) (when \deg(\mat{b}(z)) \u3c \deg(\mat{a}(z))), and z\mat{a}(z)\mat{d}_0\mat{b}(z) + \mat{c}_0 from linearizations of the component parts, matrix polynomials \mat{a}(z) and \mat{b}(z). This extends the new companion matrix construction introduced in Chapter 2 to matrix polynomials. In Chapter 5, we define ``generalized standard triples\u27\u27 which can be used in constructing algebraic linearizations; for example, for \H(z) = z \mat{a}(z)\mat{b}(z) + \mat{c}_0 from linearizations for \mat{a}(z) and \mat{b}(z). For convenience, we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases. We account for the possibility of common similarity transformations. We give explicit proofs for the less familiar bases. In Chapter 6, we investigate the numerical stability of algebraic linearization, which re-uses linearizations of matrix polynomials \mat{a}(\lambda) and \mat{b}(\lambda) to make a linearization for the matrix polynomial \mat{P}(\lambda) = \lambda \mat{a}(\lambda)\mat{b}(\lambda) + \mat{c}. Such a re-use \textsl{seems} more likely to produce a well-conditioned linearization, and thus the implied algorithm for finding the eigenvalues of \mat{P}(\lambda) seems likely to be more numerically stable than expanding out the product \mat{a}(\lambda)\mat{b}(\lambda) (in whatever polynomial basis one is using). We investigate this question experimentally by using pseudospectra

A gyrokinetic model for the plasma periphery of tokamak devices

A gyrokinetic model is presented that can properly describe strong flows, large and small amplitude electromagnetic fluctuations occurring on scale lengths ranging from the electron Larmor radius to the equilibrium perpendicular pressure gradient scale length, and large deviations from thermal equilibrium. The formulation of the gyrokinetic model is based on a second order description of the single charged particle dynamics, derived from Lie perturbation theory, where the fast particle gyromotion is decoupled from the slow drifts, assuming that the ratio of the ion sound Larmor radius to the perpendicular equilibrium pressure scale length is small. The collective behavior of the plasma is obtained by a gyrokinetic Boltzmann equation that describes the evolution of the gyroaveraged distribution function and includes a non-linear gyrokinetic Dougherty collision operator. The gyrokinetic model is then developed into a set of coupled fluid equations referred to as the gyrokinetic moment hierarchy. To obtain this hierarchy, the gyroaveraged distribution function is expanded onto a velocity-space Hermite-Laguerre polynomial basis and the gyrokinetic equation is projected onto the same basis, obtaining the spatial and temporal evolution of the Hermite-Laguerre expansion coefficients. The Hermite-Laguerre projection is performed accurately at arbitrary perpendicular wavenumber values. Finally, the self-consistent evolution of the electromagnetic fields is described by a set of gyrokinetic Maxwell's equations derived from a variational principle, with the velocity integrals of the gyroaveraged distribution function explicitly evaluated