35,668 research outputs found

    Fast and accurate evaluation of a generalized incomplete gamma function

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    We present a computational procedure to evaluate the integral ∫xy sp-1 e-μs ds, for 0 ≤ x 0, which generalizes the lower (x=0) and upper (y=+∞) incomplete gamma functions. To allow for large values of x, y, and p while avoiding under/overflow issues in the standard double precision floating point arithmetic, we use an explicit normalization that is much more efficient than the classical ratio with the complete gamma function. The generalized incomplete gamma function is estimated with continued fractions, integrations by parts, or, when x ≈ y, with the Romberg numerical integration algorithm. We show that the accuracy reached by our algorithm improves a recent state-of-the-art method by two orders of magnitude, and is essentially optimal considering the limitations imposed by the floating point arithmetic. Moreover, the admissible parameter range of our algorithm (0 ≤ p,x,y ≤ 1015) is much larger than competing algorithms and its robustness is assessed through massive usage in an image processing application

    On the use of Hadamard expansions in hyperasymptotic evaluation: differential equations of hypergeometric type

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    We describe how a modification of a common technique for developing asymptotic expansions of solutions of linear differential equations can be used to derive Hadamard expansions of solutions of differential equations. Hadamard expansions are convergent series that share some of the features of hyperasymptotic expansions, particularly that of having exponentially small remainders when truncated, and, as a consequence, provide a useful computational tool for evaluating special functions. The methods we discuss can be applied to linear differential equations of hypergeometric type and may have wider applicability
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