Kombiniranje kvadraturnih formul v kvadraturne formule višjega reda : delo diplomskega seminarja

Deželak, Eva (author), Plestenjak, Bor (mentor)

Definiramo pozitivno in negativno pravilo ter spremljevalni pravili. Iz para spremljevalnih pravil sestavimo novo pravilo, ki ga imenujemo združeno pravilo in ima višjo stopnjo, kot začetni spremljevalni pravili. Posebej se osredotočimo na družine združenih pravil na dveh točkah in družine združenih pravil na treh točkah ter povemo nekaj lastnosti, ki zanje veljajo. V nadaljevanju s $\Theta$ označimo množico pravil stopnje $m$ in definiramo transformacijo $W: \Theta \times \Theta \to \Theta$. Nato lahko definiramo povprečno pravilo $W(g)$ in trdimo, da ima stopnjo vsaj $m+1$, pri čemer imata začetni pravili $A(g)$ in $B(g)$ stopnjo $m$. Na koncu se osredotočimo še na pravila poljubne stopnje z racionalnimi vozlišči, kjer kombiniramo pravila stopnje 1 in na ta način skonstruiramo pravilo višje stopnje. S tem si zagotovimo numerično stabilnost, saj zaradi racionalnosti vozlišč ne prihaja do vmesnih zaokroževanj.

• Language: Slovenian
• Keywords: pozitivno pravilo, negativno pravilo, spremljevalni pravili, združe-no pravilo, povprečno pravilo
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-119936
• UDC: 519.6
• COBISS.SI-ID: 58181891
• Publication date in RUL: 13.09.2020

DEŽELAK, Eva, 2020, Kombiniranje kvadraturnih formul v kvadraturne formule višjega reda : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 8 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=119936

Secondary language

• Language: English
• Title: Combining quadrature rules into higher-order quadrature rules

Abstract

We define the positive, negative and companion rules. From a given pair of the companion rules we construct a new rule, which we call the combined rule and it has a higher degree of precision. Then we focus separately on families of the companion two-point rules and families of the companion three-point rules and we state some facts which apply to them. In what follows, $\Theta$ denotes the set of rules with degree $m$ and we define the transformation $W: \Theta \times \Theta \to \Theta$. Then we can define the mean rule $W(g)$ and state that it has a degree of at least $m+1$, whereas beginning rules $A(g)$ and $B(g)$ have degree $m$. At the end we focus on the rules of arbitrary degree with rational nodes, where we combine rules of degree 1 and from them we construct a new rule of a higher degree. With that we provide the numerical stability; because of the rationality of the nodes it doesn't come to intermediate rounding.

• Keywords: positive rule, negative rule, companion rules, combined rule, mean rule