Paweł Keller

Stanisław Lewanowicz, Paweł Keller, Paweł Woźny

We propose a novel approach to the problem of polynomial approximation of rational Bézier triangular patches with prescribed boundary control points. The method is very efficient thanks to using recursive properties of the bivariate dual Bernstein polynomials and applying a smart algorithm for evaluating a collection of two-dimensional integrals. Some illustrative examples are given.

Stanisław Lewanowicz, Paweł Keller, Paweł Woźny

Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining Bézier form of the $L^2$-solution of the problem of best polynomial approximation of Bézier curve or surface. In this connection, the Bézier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost. In this paper, a set of recurrence relations satisfied by the Bézier coefficients of dual bivariate Bernstein polynomials is derived and an efficient algorithm for evaluation of these coefficients is proposed. Applications of this result to some approximation problems of Computer Aided Geometric Design (CAGD) are discussed.

Stanisław Lewanowicz, Paweł Woźny, Paweł Keller

We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bézier curve by the Bézier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, associated with the Jacobi scalar product, and exploits their recursive properties. Examples are given, showing the effectiveness of the algorithm.

Paweł Keller, Iwona Wróbel

We investigate the possibility of fast, accurate and reliable computation of the Cauchy principal value integrals $\mathrm{P}\!\int_{a}^{b} f(x)(x-τ)^{-1} dx$ $(a < τ< b)$ using standard adaptive quadratures. In order to properly control the error tolerance for the adaptive quadrature and to obtain a~reliable estimation of the approximation error, we research the possible influence of round-off errors on the computed result. As the numerical experiments confirm, the proposed method can successfully compete with other algorithms for computing such type integrals. Moreover, the presented method is very easy to implement on any system equipped with a reliable adaptive integration subroutine.

Paweł Keller, Iwona Wróbel

If $A$ is a tridiagonal matrix, then the equations $AX=I$ and $XA=I$ defining the inverse $X$ of $A$ are in fact the second order recurrence relations for the elements in each row and column of $X$. Thus, the recursive algorithms should be a natural and commonly used way for inverting tridiagonal matrices -- but they are not. Even though a variety of such algorithms were proposed so far, none of them can be applied to numerically invert an arbitrary tridiagonal matrix. Moreover, some of the methods suffer a huge instability problem. In this paper, we investigate these problems very thoroughly. We locate and explain the different reasons the recursive algorithms for inverting such matrices fail to deliver satisfactory (or any) result, and then propose new formulae for the elements of $X=A^{-1}$ that allow to construct the asymptotically fastest possible algorithm for computing the inverse of an arbitrary tridiagonal matrix $A$, for which both residual errors, $\|AX-I\|$ and $\|XA-I\|$, are always very small.