GAIL---Guaranteed Automatic Integration Library in MATLAB: Documentation for Version 2.1

Choi, Sou‐Cheng T.; Ding, Yuhan; Hickernell, Fred J.; Jiang, Lan; Rugama, Lluís Antoni Jiménez; Tong, Xin; Zhang, Yizhi; Zhou, Xuan

doi:10.48550/arxiv.1503.06544

Cited by 1 publication

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“…Plaskota [19] has developed an adaptive Simpson's algorithm for approximating b a f (x) dx assuming that the fourth derivative f (4)…”

Section: Related Work On Adaptive Algorithmsmentioning

confidence: 99%

“…It implements our best adaptive algorithms and upholds the principles of reproducible and reliable computational science as elucidated in Choi et al [6,3]. We have adopted practices including input parsing, extensive testing, code comments, a user guide [4], and case studies. Algorithms A and M described here are implemented as GAIL functions funappx g and funmin g, respectively in GAIL version 2.2.…”

Section: Numerical Examplesmentioning

confidence: 99%

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Local adaption for approximation and minimization of univariate functions

Choi

Ding

Hickernell

et al. 2017

Journal of Complexity

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Most commonly used adaptive algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a user-specified absolute error tolerance for a cone, C, of non-spiky input functions in the Sobolev space W 2,∞ [a, b]. Our algorithms automatically determine where to sample the function-sampling more densely where the second derivative is larger. The computational cost of our algorithm for approximating a univariate function f on a bounded interval with L ∞ -error no greater than ε is O f 1 2 /ε as ε → 0. This is the same order as that of the best function approximation algorithm for functions in C. The computational cost of our global minimization algorithm is of the same order and the cost can be substantially less if f significantly exceeds its minimum over much of the domain. Our Guaranteed Automatic Integration Library (GAIL) contains these new algorithms. We provide numerical experiments to illustrate their superior performance.

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“…Plaskota [19] has developed an adaptive Simpson's algorithm for approximating b a f (x) dx assuming that the fourth derivative f (4)…”

Section: Related Work On Adaptive Algorithmsmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%