Barycentric coordinates are functions on a polygon, one for each vertex, whose values are coefficients that provide an expression of a point of the polygon as a convex combination of the vertices. Wachspress barycentric coordinates are barycentric coordinates that are defined by rational functions of minimal degree. We study the rational map on P2 defined by Wachspress barycentric coordinates, the Wachspress map, and we describe polynomials that set-theoretically cut out the closure of the image, the Wachspress variety. The map has base points at the intersection points of non-adjacent edges. The Wachspress map embeds the polygon into projective space of dimension one less than the number of vertices. Adjacent edges are mapped to lines meeting at the image of the vertex common to both edges, and base points are blown-up into lines. The deformed image of the polygon is such that its non-adjacent edges no longer intersect but both meet the exceptional line over the blown-up corresponding base point. We find an ideal that cuts out the Wachspress variety set-theoretically. The ideal is generated by quadratics and cubics with simple expressions along with other polynomials of higher degree. The quadratic generators are scalar products of vectors of linear forms and the cubics are determinants of 3 x 3 matrices of linear forms. Finally, we conjecture that the higher degree generators are not needed, thus the ideal is generated in degrees two and three.

Dynamical systems are widely used to model physical phenomena and, in many cases, these physical phenomena are parameter dependent. In this thesis we investigate three prominent problems related to the simulation of parametric dynamical systems and develop the analysis and computational framework to solve each of them. In many cases we have access to data resulting from simulations of a parametric dynamical system for which an explicit description may not be available. We introduce the parametric AAA (p-AAA) algorithm that builds a rational approximation of the underlying parametric dynamical system from its input/output measurements, in the form of transfer function evaluations. Our algorithm generalizes the AAA algorithm, a popular method for the rational approximation of nonparametric systems, to the parametric case. We develop p-AAA for both scalar and matrix-valued data and study the impact of parameter scaling. Even though we present p-AAA with parametric dynamical systems in mind, the ideas can be applied to parametric stationary problems as well, and we include such examples. The solution of a dynamical system can often be expressed in terms of an eigenvalue problem (EVP). In many cases, the resulting EVP is nonlinear and depends on a parameter. A common approach to solving (nonparametric) nonlinear EVPs is to approximate them with a rational EVP and then to linearize this approximation. An existing algorithm can then be applied to find the eigenvalues of this linearization. The AAA algorithm has been successfully applied to this scheme for the nonparametric case. We generalize this approach by using our p-AAA algorithm to find a rational approximation of parametric nonlinear EVPs. We define a corresponding linearization that fits the format of the compact rational Krylov (CORK) algorithm for the approximation of eigenvalues. The simulation of dynamical systems may be costly, since the need for accuracy may yield a system of very large dimension. This cost is magnified in the case of parametric dynamical systems, since one may be interested in simulations for many parameter values. Interpolatory model order reduction (MOR) tackles this problem by creating a surrogate model that interpolates the original, is of much smaller dimension, and captures the dynamics of the quantities of interest well. We generalize interpolatory projection MOR methods from parametric linear to parametric bilinear systems. We provide necessary subspace conditions to guarantee interpolation of the subsystems and their first and second derivatives, including the parameter gradients and Hessians. Throughout the dissertation, the analysis is illustrated via various benchmark numerical examples. / Doctor of Philosophy / Simulation of mathematical models plays an important role in the development of science. There is a wide range of models and approaches that depend on the information available and the goal of the problem. In this dissertation we focus on three problems whose solution depends on parameters and for which we have either data resulting from simulations of the model or a explicit structure that describes the model. First, for the case when only data are available, we develop an algorithm that builds a data-driven approximation that is then easy to reevaluate. Second, we embed our algorithm in an already developed framework for the solution of a specific kind of model structure: nonlinear eigenvalue problems. Third, given a model with a specific nonlinear structure, we develop a method to build a model with the same structure, smaller dimension (for faster computation), and that provides an accurate approximation of the original model.

O problema de reconstruir uma função f a partir de um número finito de valores conhecidos f(x0), f(x1), ..., f(xn) aparece com frequência em modelagem matemática. Em geral, não é possível determinar f completamente a partir de f(x0), f(x1), ..., f(xn), mas, em muitos casos de interesse, podemos encontrar aproximações razoáveis para f usando interpolação, que consiste em determinar uma função (um polinômio, ou uma função racional ou trigonométrica, etc) g que satisfaça g(xi) = f(xi); i = 0, 1, ..., n: Na prática, a função interpoladora g é avaliada em precisão finita e o valor final computado de g(x) pode diferir do valor exato g(x) devido a erros de arredondamento. Essa diferença pode, inclusive, ultrapassar o erro de interpolação E(x) = f(x) - g(x) em várias ordens de magnitude, comprometendo todo o processo de aproximação. A estabilidade numérica de um algoritmo reflete sua sensibilidade em relação a erros de arredondamento. Neste trabalho apresentamos uma análise detalhada da estabilidade numérica de alguns algoritmos utilizados no cálculo de interpoladores polinomiais ou racionais que podem ser postos na forma baricêntrica. Os principais resultados deste trabalho também estão disponíveis em língua inglesa nos artigos - Mascarenhas, W e Camargo, A. P., On the backward stability of the second barycentric formula for interpolation, Dolomites research notes on approximation v. 7 (2014) pp. 1-12. - Camargo, A. P., On the numerical stability of Floater-Hormann\'s rational interpolant, Numerical Algorithms, DOI 10.1007/s11075-015-0037-z. - Camargo, A. P., Erratum: On the numerical stability of Floater-Hormann\'s rational interpolant\", Numerical Algorithms, DOI 10.1007/s11075-015-0071-x. - Camargo, A. P. e Mascarenhas, W., The stability of extended Floater-Hormann interpolants, Numerische Mathematik, submetido. arXiv:1409.2808v5 / The problem of reconstructing a function f from a finite set of known values f(x0), f(x1), ..., f(xn) appears frequently in mathematical modeling. It is not possible, in general, to completely determine f from f(x0), f(x1), ..., f(xn) but, in several cases of interest, it is possible to find reasonable approximations for f by interpolation, which consists in finding a suitable function (a polynomial function, a rational or trigonometric function, etc.) g such that g(xi) = f(xi); i = 0, 1, ..., n: In practice, the interpolating function g is evaluated in finite precision and the final computed value of g(x) may differ from the exact value g(x) due to rounding. In fact, such difference can even exceed the interpolation error E(x) = f(x)-g(x) in several orders of magnitude, compromising the entire approximation process. The numerical stability of an algorithm reflect is sensibility with respect to rounding. In this work we present a detailed analysis of the numerical stability of some algorithms used to evaluate polynomial or rational interpolants which can be put in the barycentric format. The main results of this work are also available in english in the papers - Mascarenhas, W e Camargo, A. P., On the backward stability of the second barycentric formula for interpolation, Dolomites research notes on approximation v. 7 (2014) pp. 1-12. - Camargo, A. P., On the numerical stability of Floater-Hormann\'s rational interpolant, Numerical Algorithms, DOI 10.1007/s11075-015-0037-z. - Camargo, A. P., Erratum: On the numerical stability of Floater-Hormann\'s rational interpolant\", Numerical Algorithms, DOI 10.1007/s11075-015-0071-x. - Camargo, A. P. e Mascarenhas, W., The stability of extended Floater-Hormann interpolants, Numerische Mathematik, submetido. arXiv:1409.2808v5

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This thesis discusses several topics related to interpolation and how it is used in numerical analysis. It begins with an overview of the aspects of interpolation theory that are relevant to the discussion at hand before presenting three new contributions to the field. The first new result is a detailed error analysis of the barycentric formula for trigonometric interpolation in equally-spaced points. We show that, unlike the barycentric formula for polynomial interpolation in Chebyshev points (and contrary to the main view in the literature), this formula is not always stable. We demonstrate how to correct this instability via a rewriting of the formula and establish the forward stability of the resulting algorithm. Second, we consider the problem of trigonometric interpolation in grids that are perturbations of equally-spaced grids in which each point is allowed to move by at most a fixed fraction of the grid spacing. We prove that the Lebesgue constant for these grids grows at a rate that is at most algebraic in the number of points, thus answering questions put forth by Trefethen and Weideman about the robustness of numerical methods based on trigonometric interpolation in points that are uniformly distributed but not equally-spaced. We use this bound to derive theorems about the convergence rate of trigonometric interpolation in these grids and also discuss the related question of quadrature. Specifically, we prove that if a function has V ≥ 1 derivatives, the Vth of which is Hölder continuous (with a Hölder exponent that depends on the size of the maximum allowable perturbation), then the interpolants converge uniformly to the function at an algebraic rate; larger values of V lead to more rapid convergence. A similar statement holds for the corresponding quadrature rule. We also consider what analogue, if any, there is for trigonometric interpolation of the famous 1/4 theorem of Kadec from sampling theory that restricts the size of the perturbations one can make to the integers and still be guaranteed to have a set of stable sampling for the Paley-Wiener space. We present numerical evidence suggesting that in the discrete case, the 1/4 threshold takes the form of a threshold for the boundedness of a "2-norm Lebesgue constant" and does not appear to have much significance in practice. We believe that these are the first results regarding this problem to appear in the literature. While we do not believe the results we establish are the best possible quantitatively, they do (rigorously) capture the main features of trigonometric interpolation in perturbations of equally-spaced grids. We make several conjectures as to what the optimal results may be, backed by extensive numerical results. Finally, we consider a new application of interpolation to numerical linear algebra. We show that recently developed methods for computing the eigenvalues of a matrix by dis- cretizing contour integrals of its resolvent are equivalent to computing a rational interpolant to the resolvent and finding its poles. Using this observation as the foundation, we develop a method for computing the eigenvalues of real symmetric matrices that enjoys the same advantages as contour integral methods with respect to parallelism but employs only real arithmetic, thereby cutting the computational cost and storage requirements in half.

O problema de reconstruir uma função f a partir de um número finito de valores conhecidos f(x0), f(x1), ..., f(xn) aparece com frequência em modelagem matemática. Em geral, não é possível determinar f completamente a partir de f(x0), f(x1), ..., f(xn), mas, em muitos casos de interesse, podemos encontrar aproximações razoáveis para f usando interpolação, que consiste em determinar uma função (um polinômio, ou uma função racional ou trigonométrica, etc) g que satisfaça g(xi) = f(xi); i = 0, 1, ..., n: Na prática, a função interpoladora g é avaliada em precisão finita e o valor final computado de g(x) pode diferir do valor exato g(x) devido a erros de arredondamento. Essa diferença pode, inclusive, ultrapassar o erro de interpolação E(x) = f(x) - g(x) em várias ordens de magnitude, comprometendo todo o processo de aproximação. A estabilidade numérica de um algoritmo reflete sua sensibilidade em relação a erros de arredondamento. Neste trabalho apresentamos uma análise detalhada da estabilidade numérica de alguns algoritmos utilizados no cálculo de interpoladores polinomiais ou racionais que podem ser postos na forma baricêntrica. Os principais resultados deste trabalho também estão disponíveis em língua inglesa nos artigos - Mascarenhas, W e Camargo, A. P., On the backward stability of the second barycentric formula for interpolation, Dolomites research notes on approximation v. 7 (2014) pp. 1-12. - Camargo, A. P., On the numerical stability of Floater-Hormann\'s rational interpolant, Numerical Algorithms, DOI 10.1007/s11075-015-0037-z. - Camargo, A. P., Erratum: On the numerical stability of Floater-Hormann\'s rational interpolant\", Numerical Algorithms, DOI 10.1007/s11075-015-0071-x. - Camargo, A. P. e Mascarenhas, W., The stability of extended Floater-Hormann interpolants, Numerische Mathematik, submetido. arXiv:1409.2808v5 / The problem of reconstructing a function f from a finite set of known values f(x0), f(x1), ..., f(xn) appears frequently in mathematical modeling. It is not possible, in general, to completely determine f from f(x0), f(x1), ..., f(xn) but, in several cases of interest, it is possible to find reasonable approximations for f by interpolation, which consists in finding a suitable function (a polynomial function, a rational or trigonometric function, etc.) g such that g(xi) = f(xi); i = 0, 1, ..., n: In practice, the interpolating function g is evaluated in finite precision and the final computed value of g(x) may differ from the exact value g(x) due to rounding. In fact, such difference can even exceed the interpolation error E(x) = f(x)-g(x) in several orders of magnitude, compromising the entire approximation process. The numerical stability of an algorithm reflect is sensibility with respect to rounding. In this work we present a detailed analysis of the numerical stability of some algorithms used to evaluate polynomial or rational interpolants which can be put in the barycentric format. The main results of this work are also available in english in the papers - Mascarenhas, W e Camargo, A. P., On the backward stability of the second barycentric formula for interpolation, Dolomites research notes on approximation v. 7 (2014) pp. 1-12. - Camargo, A. P., On the numerical stability of Floater-Hormann\'s rational interpolant, Numerical Algorithms, DOI 10.1007/s11075-015-0037-z. - Camargo, A. P., Erratum: On the numerical stability of Floater-Hormann\'s rational interpolant\", Numerical Algorithms, DOI 10.1007/s11075-015-0071-x. - Camargo, A. P. e Mascarenhas, W., The stability of extended Floater-Hormann interpolants, Numerische Mathematik, submetido. arXiv:1409.2808v5

Interpolation error estimates in terms of geometric quality measures are established for harmonic coordinates on polytopes in two and three dimensions. First we derive interpolation error estimates over convex polygons that depend on the geometric quality of the triangles in the constrained Delaunay triangulation of the polygon. This characterization is sharp in the sense that families of polygons with poor quality triangles in their constrained Delaunay triangulations are shown to produce large error when interpolating a basic quadratic function. Non-convex polygons exhibit a similar limitation: large constrained Delaunay triangles caused by vertices approaching a non-adjacent edge also lead to large interpolation error. While this relationship is generalized to convex polyhedra in three dimensions, the possibility of sliver tetrahedra in the constrained Delaunay triangulation prevent the analogous estimate from sharply reflecting the actual interpolation error. Non-convex polyhedra are shown to be fundamentally different through an example of a family of polyhedra containing vertices which are arbitrarily close to non-adjacent faces yet the interpolation error remains bounded.

Complex number is a major mathematical discovery. It can be used in many scientific fields, including engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. The aim of this paper investigates the problems of algebra, trigonometry and geometry, which can be solved cleverly by the properties of complex numbers.

This work contains a few basic terms of analytical geometry. We mention some of ray-triangle intersection computation algorithms and present some use-case examples. We discuss capabilities of CUDA, optimization techniques of this architecture and implementation with focus on given issues. Algorithms of ray-triangle intersection are tested and results are discussed.

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