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- W1966597140 abstract "A definition of a length scale is given for analytic functions. Quadrature error bounds based on this length scale can be used to compare different orders and types of quadrature rules. A familiar theorem about convergence of Taylor series states that an analytic function has a bound on its derivatives so that, if the nearest singularity is at a radius greater than r from the point x in the complex plane, then there exists some M such that, for all n, (1) If(nf(x)l If(x)I, and M 0, there exists an r greater than zero and less than the distance to the nearest singularity such that Eq. (1) holds for all n. The lemma is proved as follows: Define r0 as a radius less than the distance to the nearest singularity, and define Mo as an M which satisfies (1) for r = r0 according to the theorem. Then it is straightforward to show that, given an M, M _ If(x)l and M > 0, an r which satisfies the lemma is given by r = min(r0, Mro/M0). We now define two functionals of functions analytic in the neighborhood of a given interval. The first functional, M, is the maximum absolute value of the function on the interval under consideration. The second functional r is the largest value r which satisfies (1) for the given function for every n and for every value x in the interval when M is as defined above in this paragraph. The lemma assures that such an r exists. This value r will be smaller than (or equal to) the smallest radius of convergence of the function as measured from points along the interval. The functional r is valuable because it can be used as a scale of length for the variability of the function. A search for other useful definitions of scale might be valuable, but no other definition will be considered here. If a quadrature rule over this interval has a degree of precision (n 1), then the error for that rule can be written" @default.
- W1966597140 created "2016-06-24" @default.
- W1966597140 creator A5033862354 @default.
- W1966597140 date "1972-09-01" @default.
- W1966597140 modified "2023-09-26" @default.
- W1966597140 title "An error bound for quadratures" @default.
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- W1966597140 doi "https://doi.org/10.1090/s0025-5718-1972-0331732-2" @default.
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