منابع مشابه
What Is the Complexity of Stieltjes Integration?
We study the complexity of approximating the Stieltjes integral R 1 0 f (x)dg(x) for functions f having r continuous derivatives and functions g whose sth derivative has bounded variation. Let r(n) denote the nth minimal error attainable by approximations using at most n evaluations of f and g, and let comp(ε) denote the ε-complexity (the minimal cost of computing an ε-approximation). We show t...
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The Stieltjes constants γn are the coefficients appearing in the Laurent series of the Riemann zeta function at s = 1. We give a simple and efficient method to compute a p-bit approximation of γn with rigorous error bounds. Starting from an integral representation due to Blagouchine, we shift the contour to eliminate cancellation. The integral is then evaluated numerically in ball arithmetic us...
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This paper presents Stieltjes-type integration for operator-valued functions with respect to spectral families. The relation between RiemannStieltjes integrals associated with some classes of spectral families including, in particular, those that arise in the context of the Stone representation theorem for Banach spaces is established. The developed approach is applied to the structural analysi...
متن کاملWhat is the complexity of Stieltjes integration ? Arthur
We study the complexity of approximating the Stieltjes integral R 1 0 f (x) dg(x) for functions f having r continuous derivatives and functions g whose sth derivative has bounded variation. Let r(n) denote the nth minimal error attainable by approximations using at most n evaluations of f and g, and let comp(") denote the "-complexity (the minimal cost of computing an "-approximation). We show ...
متن کاملNewton-Cotes integration for approximating Stieltjes (generalized Euler) constants
In the Laurent expansion ζ(s, a) = 1 s− 1 + ∞ ∑ k=0 (−1)γk(a) k! (s− 1) , 0 < a ≤ 1, of the Riemann-Hurwitz zeta function, the coefficients γk(a) are known as Stieltjes, or generalized Euler, constants. [When a = 1, ζ(s, 1) = ζ(s) (the Riemann zeta function), and γk(1) = γk.] We present a new approach to high-precision approximation of γk(a). Plots of our results reveal much structure in the gr...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1956
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-1956-0075280-9