Usrednjene kvadraturne formule sa varijantama i primene

Abstract

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.