PDM research project

RIQI: Rational Interpolatory Quadrature formulas on the Interval (1-10-2009 to 30-09-2012)

Phase 1

Given a (possibly complex) weight function and a set of arbitrary nodes inside the interval [-1,1], we will first investigate the properties of convergence for RIQs on the interval. Although we will focus on developing a numerical procedure to compute the nodes and weights in Gauss-type RIQs, these nodes and a well-chosen transformation of the weights can be used in more general RIQs. We will therefore not restrict this investigation to the case of Gauss-type RIQs. As part of this research we will investigate whether the method and results in [1], for the case of polynomial interpolation, can be generalized to the case of RIQs on the interval.

Next, we will concentrate on the case in which the weight function is positive. From a numerical point of view the weights in the RIQ will then need to be positive too. Therefore, we will try to find a characterization for RIQs on the interval with positive weights, and we will investigate the asymptotical behavior for the weights. To do so, we will need to generalize the results in [2,3] for the case of polynomial interpolation to the case of rational interpolation.

Gauss-type RIQs are a special kind of RIQs, in such a way that they have maximal domain of validity. More specifically, Gaussian RIQs are exact in a space of rational functions of dimension 2n, while Gauss-Radau (1 node fixed in advance) and Gauss-Lobatto (2 nodes fixed in advance) RIQs generally are exact in a space of rational functions of dimension 2n-1 and 2n-2 respectively. The existence of the Gauss-Radau (Gauss-Lobatto) RIQs, however, depends on the choice of the node(s) to be fixed in advance (for certain choices, one of the nodes may lie outside the interval). Therefore, we will first investigate under which conditions the Gauss-Radau and Gauss-Lobatto RIQs do exist, and under which conditions the weights are positive.

The Joukowski transform gives a relation between a point x = cosθ on the interval [-1,1] and a point z = exp(iθ), with i² = -1, on the complex unit circle. Let there be given a weight function w(x) on the interval, and define the weight function u(θ) on the complex unit circle by u(θ) = w(cosθ)|sinθ|. Then it holds that ∫_[-1,1] f(x)w(x)dx = ½ ∫_[-π,π] f(cosθ)u(θ)dθ. Consequently, the Joukowski transform implies (under certain conditions) a relation between Gauss-type and Szegö RIQs (i.e., RIQs on the complex unit circle with maximal domain of validity). This relation has already been studied by Szegö [4, Chapter 11.5] and Geronimus [5, Chapter 9], and has later been generalized to the case of rational interpolatory functions with all real poles for Gaussian RIQs by Van gucht [6] and Van Deun [7, Chapter 4.4]. Our aim is to investigate under which conditions this relation holds for the case of rational interpolatory functions with arbitrary complex poles for each Gauss-type RIQ. By means of this relation we can then transfer known results (theoretical as well as numerical) for Szegö RIQs to the case of Gauss-type RIQs.

Finally, we will look for expressions for the error on the approximation of the integral in terms of the number of nodes n.

Phase 2

In the second phase we will investigate numerical procedures to compute the nodes and weights in Gauss-type RIQs and RIQs with positive weights. To do so, we will start from the theoretical analysis in the first phase and from (quasi-)orthogonal rational functions with arbitrary complex poles. These are a generalization of (quasi-)orthogonal polynomials in such a way that they are of increasing degree and the polynomial case results when all the poles are at infinity (see [8]).

The nodes in RIQs with positive weights are the zeros of a (quasi-)orthogonal rational function with respect to the inner product ⟨f,g&rang = ∫_[-1,1] f(x)g^c(x)w(x)dx. Here the superscript c denotes the complex conjugate. Further, orthogonal rational functions (ORFs) with arbitrary complex poles satisfy (under certain conditions) a three-term recurrence relation (see [9,10]). For a first possible numerical procedure we will start from this three-term recurrence relation (cfr. [11,12] for the case of ORFs with all real poles). Further, we will investigate the influence of a rational modification of the weight function w(x) on the recurrence coefficients (in terms of the recurrence coefficients with respect to the initial weight function).

For a second possible numerical procedure we will build on known numerical procedures to compute the nodes and weights in Szegö RIQs, and the relation with Gauss-type RIQs (see first phase).

Finally, we will consider the case of the Chebyshev weight functions w(x) = (1-x)^a(1+x)^b, where a and b belong to {±½}, because explicit expressions are known for the so-called Chebyshev ORFs on the interval and on the complex unit circle (see [13,14]). So, for a third numerical procedure we will start from these explicit expressions (cfr. [15] for the case of Gaussian RIQs).

Phase 3

Finally, our aim is to provide a set of professional software routines that offer engineers and scientists a user-friendly method to obtain a fast, accurate and reliable result. To do so, we will need to use the comparative study of the numerical procedures from the second phase to create an implementation that meets the requirements for contemporary professional numerical software. This will involve an examination of as many exceptional cases as possible, the clear documentation of the code in view of possible later adaptations by others, reports on and clarification of the mathematical techniques employed in publications, and informing the scientific community and target user group on conferences and symposia.

References

1. R. Cruz-Barroso, P. González-Vera, and F. Perdomo-Pío. “Orthogonality, interpolation and quadratures on the unit circle and the interval [−1,1]”. Submitted.
2. F. Peherstorfer. “On positive quadrature formulas”, volume 112 of International Series of Numerical Mathematics, pp. 297–313, Birkhaüser Verlag, Basel, 1993.
3. F. Peherstorfer. “Positive quadrature formulas III: Asymptotics of weights”, Math. Comp. 77(264):2261–2275, 2008.
4. G. Szegő. Orthogonal Polynomials, volume 33 of Amer. Math. Soc. Colloq. Publ., American Mathematical Society, Providence, Rhode Island, 1975. 4th ed.
5. Ya. L. Geronimus. Polynomials Orthogonal on a Circle and Interval, volume 18 of International Series of Monographs in Pure and Applied Mathematics, Pergamon Press, Oxford, 1960.
6. P. Van gucht and A. Bultheel. “A relation between orthogonal rational functions on the unit circle and the interval [−1,1]”, Communications in the Analytic Theory of Continued Fractions 8:170–182, 2000.
7. J. Van Deun. Orthogonal Rational Functions: Asymptotic Behaviour and Computational Aspects, PhD thesis, Department of Computer Science, K.U.Leuven, may 2005.
8. A. Bultheel, P. González-Vera, E. Hendriksen, and O. Njåstad. Orthogonal Rational Functions, volume 5 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 1999.
9. J. Van Deun and A. Bultheel. “Orthogonal rational functions on an interval”, Technical Report TW322, Department of Computer Science, K.U.Leuven, March 2001.
10. K. Deckers and A. Bultheel. “Recurrence and asymptotics for orthogonal rational functions on an interval”, IMA Journal of Numerical Analysis 29(1):1–23, 2009.
11. A. Bultheel, P. González-Vera, E. Hendriksen, and O. Njåstad. “Orthogonal rational functions and tridiagonal matrices”, J. Comput. Appl. Math. 153(1–2):89–97, 2003.
12. D. Fasino and L. Gemignani. “Structured eigenvalue problems for rational Gauss quadrature”. Submitted.
13. K. Deckers, J. Van Deun, and A. Bultheel. “Rational Gauss-Chebyshev quadrature formulas for complex poles outside [−1,1]”, Math. Comp. 77(262):967-983, 2008.
14. A. Bultheel, R. Cruz-Barroso, K. Deckers, and P. González-Vera. “Rational Szegő quadratures associated with Chebyshev weight functions”, Math. Comp., 2008. To appear.
15. J. Van Deun, K. Deckers, A. Bultheel, and J.A.C. Weideman. “Algorithm 882: Near best fixed pole rational interpolation with applications in spectral methods”, ACM Trans. Math. Software 35(2):14:1–14:21, 2008.