Letter | Published:

On Approximate Integration

Nature volume 113, page 714 (17 May 1924) | Download Citation

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Abstract

IN vol. 105 of NATURE, 1920, several very interesting letters were published concerning approximate integration. It is to be noted that all the formulæ alluded to in these letters are founded on the analytical method which consists in replacing the integrand by a polynomial. It appears that one very good formula based on geometrical reasoning and discovered by Poncelet is not so well known in Great Britain as it is in France. It may interest readers of NATURE, therefore, to try this formula. The general form is the interval (a, b) being divided into 2n sub-intervals each equal to h, P being the sum of those of the ordinates y1, y2n+1 the indices of which are even, E1 being the sum of the extreme ordinates y1+y2n+1, and E2 being the sum of the ordinates next to them, y2+y2n. The most remarkable feature of this formula is that, when the graph of y is everywhere concave to the x-axis (or everywhere convex), it gives an upper bound of the error, extremely simple and not necessitating the knowledge of the derivatives of y, namely:1 . For the case where , Poncelet formula reduces to This seven-ordinate formula may be compared with Simpson's seven-ordinate rule. Taking the numerical examples of Mr. Dufton, the errors of these two seven-ordinate rules are shown in the following table: It will be noted that the abscissas involved in the seven-ordinate Simpson's rule are , and in the seven-ordinate Poncelet rule they are 0, 0·1, 0·3, 0·5, 0·7, 0·9, 1, so that the computation of the ordinates in Poncelet's rule will often be more easy.

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  1. 141a rue Principale, Oberhausbergen, par Strasbourg (France).

    • MAURICE FRÉCHET

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DOI

https://doi.org/10.1038/113714a0

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