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The construction of cubature formulae by continuation

Die Konstruktion von Kubaturformeln mit einer Fortsetzungsmethode

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Abstract

A cubature formulaQ is an approximation of ann-dimensional integralI. Q is exact for the space spanned by the polynomialsf 1, ...,f d if it verifies the system of equations:

$$Q[f_i ] = I[f_i ] i = 1,...,d.$$

The unknowns are knots and weights of the cubature formula. We suppose that there are as many unknowns as equations.

For searching solutions to this system, we construct a family of systems depending continuously on a parametert:

$$Q[f_i (t)] = I[f_i (t)] i = 1,...,d,$$

coinciding with the previous system fort=1 and whose solutions att=0 are easily computed. The solution curves originating from these solutions are followed numerically and may yield a solution fort=1.

Zusammenfassung

Eine KubaturformelQ ist eine Approximation für einn-dimensionales IntegralI. Q ist exakt für den von den Polynomenf 1, ...f d aufgespannten Vektorraum, wenn

$$Q[f_i ] = I[f_i ] i = 1,...,d.$$

Die Unbekannten sind die Knoten und Gewichte der Kubaturformel. Wir nehmen an, daß es soviele Unbekannte wie Gleichungen gibt.

Zur Bestimmung von Lösungen dieses Systems konstruieren wir eine Familie von Systemen, die stetig von einem Parametert abhängt:

$$Q[f_i (t)] = I[f_i (t)] i = 1,...,d,$$

die fürt=1 mit dem ersten System zusammenfällt und fürt=0 eine einfach zu bestimmende Lösungsmenge hat. Die Lösungszweige, die in jeder Lösung dieser Menge beginnen, werden numerisch verfolgt und können eine Lösung fürt=1 liefern.

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Verlinden, P., Haegemans, A. The construction of cubature formulae by continuation. Computing 45, 145–155 (1990). https://doi.org/10.1007/BF02247880

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  • DOI: https://doi.org/10.1007/BF02247880

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