Applications of Number Theory to Numerical Analysis by S.K. Zaremba

By S.K. Zaremba

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By S.K. Zaremba

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For all α between 1 and I shall denote the minimum value obtained by and any minimizing α by α (Ν,s). The calculation is of practicable length for interesting values of Ν and s, and some results are presented in Tables IV-IX. Some econo- mizing was possible in the actual calculation. For example, noting that 1 cuts in half the number of a s for which Β is computed. In the tables below I have, together with tabulated and It appears from the data, particularly from Tables IV-VII, that while the first of these two quantities is increasing somewhat as Ν increases, the second is decreasing.

It should be noted that while the present calculations were done to investigate the asymptotic behavior of certain sequences of quadrature formulas, they have also produced a large number of actual quadrature formulas together with error bounds for them. The parameters defining them are listed in the tables below; those for s < 6, at 22 EXPERIMENTS ON COEFFICIENTS least, should be of practical value. It should also be noted that in applying the error bound (9), the quantity C used need not satisfy (4') for all m but only for those = 0 mod N.

L. KRYLOV, Approximate Calculation of Integrals, Macmillan, New York, 1962. 37 La Methode des "Bons Treillis" pour le Calcul des Integrales Multiples S. K. ZAREMBA 1. Introduction ! La méthode de Monte Carlo a suggéré qu'il n est pas nécessaire que la quantité de travail demandée par le calcul numérique d'une intégrale multiple avec une précision donnée croisse à peu près exponentiellement avec le nombre de dimensions. f Or c est ce qui a lieu quand on applique les méthodes traditionnelles, qui sont basées sur f l itération de l'intégration dans une dimension.

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