Iwona Wróbel

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.

Alicja Smoktunowicz, Iwona Wróbel

It is known that Goertzel's algorithm is much less numerically accurate than the Fast Fourier Transform (FFT)(Cf. \cite{gen:69}). In order to improve accuracy we propose modifications of both Goertzel's and Horner's algorithms based on the divide-and-conquer techniques. The proof of the numerical stability of these two modified algorithms is given. The numerical tests in Matlab demonstrate the computational advantages of the proposed modifications. The appendix contains the proof of numerical stability of Goertzel's algorithm of polynomial evaluation.