Gauss-type quadrature rules with respect to the external zeros of the integrand

In the present paper, we propose a Gauss-type quadrature rule into which the external zeros of the integrand (the zeros of the integrand outside the integration interval) are incorporated. This new formula with $n$ nodes, denoted by $\mathcal G_n$, proves to be exact for certain polynomials of degrees greater than $2n-1$ (while the Gauss quadrature formula with the same number of nodes is exact for all polynomials of degrees less than or equal to $2n-1$). It turns out that $\mathcal G_n$ has several good properties: all its nodes belong to the interior of the integration interval, all its weights are positive, it converges, and it is applicable both when the external zeros of the integrand are known exactly and when they are known approximately. In order to economically estimate the error of $\mathcal G_n$, we construct its extensions that inherit the $n$ nodes of $\mathcal G_n$, and that are analogous to the Gauss-Kronrod, averaged Gauss and generalized averaged Gauss quadrature rules. Further, we show that $\mathcal G_n$ with respect to the pairwise distinct external zeros of the integrand represents a special case of the (slightly modified) Gauss quadrature formula with preassigned nodes. The accuracy of $\mathcal G_n$ and its extensions is confirmed by numerical experiments

Averaged quadrature formulas and vatiants with applications

Numeriqka integracija prouqava kako se moe izraqunati brojevna vrednost integrala. Formule numeriqke integracije nazivaju se kvadraturama. Jedinstvena optimalna interpolaciona kvadratura sa n qvorova jeste Gausova formula Gn, koja ima algebarski stepen taqnosti 2n−1. Vano pitanje u praktiqnim izraqunavanjima je kako (ekonomiqno) proceniti grexku Gausove formule. U te svrhe moe se koristiti odgovarajua Gaus-Kronrodova formula K2n+1 sa 2n+1 qvorova i algebarskim stepenom taqnosti 3n+1. U situacijam kada Gaus-Kronrodova formula ne postoji, treba nai adekvatnu alternativu i ta alternativa moe biti uopxtena usrednjena Gausova formula Gb2n+1 sa 2n + 1 qvorova i algebarskim stepenom taqnosti 2n + 2. Prednosti Gb2n+1 su to xto uvek postoji i to xto je njena numeriqka konstrukcija jednostavnija od konstrukcije K2n+1. Glavna tema ove doktorske disertacije je uopxtena usrednjena Gausova formula Gb2n+1. Uopxtene usrednjene Gausove formule mogu imati qvorove van intervala integracije. Kvadrature sa qvorovima van intervala integracije ne mogu se koristiti za aproksimaciju integrala kod kojih je integrand definisan samo na intervalu integracije. U ovoj disertaciji ispitano je kada uopxtene usrednjene Gausove formule i njihova skraenja sa Bernxtajn-Segeovim teinskim funkcijama imaju sve qvorove unutar intervala integracije. Neki integrali po m-dimenzionalnim oblastima mogu se aproksimirati formulama Gm n konstruisanim uzastopnom primenom Gausovih kvadratura Gn. Koristei odgovarajue Gaus-Kronrodove kvadrature K2n+1 ili odgovarajue uopxtene usrednjene Gausove kvadrature Gb2n+1 umesto Gn, u ovoj disertaciji konstruixemo formule K2mn+1 i Gbm 2n+1. Kako bismo procenili grexku jIm − Gm n j koristimo razlike jK2mn+1 − Gm n j i jGbm 2n+1 − Gm n j. Razmatramo integrale po m-dimenzionalnoj kocki, simpleksu, sferi i lopti.Numerical integration is the study of how numerical value of an integral can be calculated. Formulas for numerical integration are called quadrature rules. The unique optimal interpolatory quadrature rule with n nodes is Gauss formula Gn, which has algebraic degree os exactness 2n − 1. An important task in practical calculations is how to (economically) estimate the error of Gauss formula. For this purpose corresponding Gauss-Kronrod formula K2n+1 with 2n + 1 nodes and algebraic degree of exactness 3n + 1 can be used. In the situations when GaussKronrod formula doesn’t exist, it is of interest to find adequate alternative and this alternative can be corresponding generalized averaged Gauss formula Gb2n+1 with 2n + 1 nodes and algebraic degree of exactness 2n + 2. The adventages of Gb2n+1 are that it always exists, and that it’s numerical construction is simpler than the construction of K2n+1. The principal topic of this doctoral dissertation is generalized averaged Gauss formula Gb2n+1. Generalized averaged Gauss formulas may have nodes outside the interval of integration. Quadrature rules with nodes outside the interval of integration cannot be applied to approximate integrals with an integrand that is defined on the interval of integration only. This thesis investigates when generalized averaged Gauss formulas and their truncations for Bernstein-Szeg˝o weight functions have all nodes in the interval of integration. Some integrals Im over m-dimensional regions can be approximated by cubature formulas Gm n constructed by the product of Gauss quadrature rules Gn. Using corresponding Gauss-Kronrod rules K2n+1 or corresponding generalized averaged Gauss rules Gb2n+1 instead of Gn, in this thesis we construct cubature formulas Km 2n+1 and Gbm 2n+1. In order to estimate the error jIm − Gm n j we use the differences jK2mn+1 − Gm n j and jGbm 2n+1 − Gm n j. We consider integrals over m-dimensional cube, simplex, sphere and ball

Incorporating the external zeros of the integrand into certain quadrature rules

Quadrature formulas are often constructed to be exact on the space of functions that are easily integrated and that are in some sense similar to the integrand. This motivates us to examine how the known properties of the integrand, such as its external zeros (zeros outside the (closed) interval of integration), can be used in order to improve the accuracy of certain quadrature formulas. In particular, we consider Gauss-type quadrature rules into which the external zeros of the integrand are incorporated

Rational Averaged Gauss Quadrature Rules

It is important to be able to estimate the quadrature error in Gauss rules. Several approaches have been developed, including the evaluation of associated Gauss-Kronrod rules (if they exist), or the associated averaged Gauss and generalized averaged Gauss rules. Integrals with certain integrands can be approximated more accurately by rational Gauss rules than by Gauss rules. This paper introduces associated rational averaged Gauss rules and rational generalized averaged Gauss rules, which can be used to estimate the error in rational Gauss rules. Also rational Gauss-Kronrod rules are discussed. Computed examples illustrate the accuracy of the error estimates determined by these quadrature rules

Rational Averaged Gauss Quadrature Rules

It is important to be able to estimate the quadrature error in Gauss rules. Several approaches have been developed, including the evaluation of associated Gauss-Kronrod rules (if they exist), or the associated averaged Gauss and generalized averaged Gauss rules. Integrals with certain integrands can be approximated more accurately by rational Gauss rules than by Gauss rules. This paper introduces associated rational averaged Gauss rules and rational generalized averaged Gauss rules, which can be used to estimate the error in rational Gauss rules. Also rational Gauss-Kronrod rules are discussed. Computed examples illustrate the accuracy of the error estimates determined by these quadrature rules

Error Estimates for Certain Cubature Formulae

We estimate the errors of selected cubature formulae constructed by the product of Gauss quadrature rules. The cases of multiple and (hyper-)surface integrals over n-dimensional cube, simplex, sphere and ball are considered. The error estimates are obtained as the absolute value of the difference between cubature formula constructed by the product of Gauss quadrature rules and cubature formula constructed by the product of corresponding Gauss-Kronrod or corresponding generalized averaged Gaussian quadrature rules. Generalized averaged Gaussian quadrature rule (G) over cap (2l+1) is (2l + 1)-point quadrature formula. It has 2l + 1 nodes and the nodes of the corresponding Gauss rule G(l) with l nodes form a subset, similar to the situation for the (2l + 1)-point Gauss-Kronrod rule H2l+1 associated with G(l). The advantages of (G) over cap (2l+1) are that it exists also when H2l+1 does not, and that the numerical construction of (G) over cap (2l+1), based on recently proposed effective numerical procedure, is simpler than the construction of H2l+1

Signature of weather conditions in the canine babesiosis spring peak in Belgrade, Serbia

Background: Canine babesiosis, a tick-borne disease caused by Babesia canis, shows a seasonality whose relationship with local weather conditions has not been fully investigated. Objectives: Meteorological conditions can favour the tick-vector activity, and thus lead to an increased number of cases of canine babesiosis. Hence, our study looks into the link between the number of recorded cases, on the one hand, and temperature and relative humidity on the other with an aim to quantify their correlations. Material and Methods: Over 2013–2016, the data were collected in Belgrade, the capital of the Republic of Serbia. The meteorological parameters were obtained from the Republic Hydrometeorological Service of Serbia. The analysis includes correlations with a time lag, given in number of weeks, which shifts corresponding correlation pairs and shows a delayed effect of weather conditions. The time lag ranges between 0 and 52. Results: Canine babesiosis occurrence shows a pronounced maximum in the spring and a less marked one in the autumn. For the spring period, statistically significant correlation coefficients imply that over one year prior to the disease spring peak, temperature is more strongly linked with the number of cases than relative humidity. Conclusion: Temperature and relative humidity, through their influence on population of infected ticks, seem to be important meteorological drivers of the spring maximum of canine babesios in Belgrade. Further understanding of this interplay can help better contain the disease, and project its possible spread to other regions prompted by climate change