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Explicit solutions of Genz test integrals
Applied Mathematics Letters ( IF 2.9 ) Pub Date : 2024-12-26 , DOI: 10.1016/j.aml.2024.109444 Vesa Kaarnioja
Applied Mathematics Letters ( IF 2.9 ) Pub Date : 2024-12-26 , DOI: 10.1016/j.aml.2024.109444 Vesa Kaarnioja
A collection of test integrals introduced by Genz (1984) has remained popular to this day for assessing the robustness of high-dimensional numerical integration algorithms. However, the explicit solutions to these integrals do not appear to be readily available in the existing literature: typically the true values of the test integrals are simply approximated using “overkill” numerical solutions. In this paper, analytic solutions are presented for the Genz test integrals ∫ 0 1 ⋯ ∫ 0 1 cos ( 2 π w 1 + ∑ i = 1 d c i x i ) d x d ⋯ d x 1 = 2 d cos ( 2 π w 1 + 1 2 ∑ i = 1 d c i ) ∏ i = 1 d sin ( c i 2 ) c i , ∫ 0 1 ⋯ ∫ 0 1 ∏ i = 1 d 1 c i − 2 + ( x i − w i ) 2 d x d ⋯ d x 1 = ∏ i = 1 d c i ( arctan ( c i w i ) + arctan ( c i − c i w i ) ) , ∫ 0 1 ⋯ ∫ 0 1 ( 1 + ∑ i = 1 d c i x i ) − ( d + 1 ) d x d ⋯ d x 1 = 1 d ! ∏ i = 1 d c i ∑ u ⊆ { c 1 , … , c d } ( − 1 ) # u 1 + ∑ i ∈ u i , ∫ 0 1 ⋯ ∫ 0 1 exp ( − ∑ i = 1 d c i 2 ( x i − w i ) 2 ) d x d ⋯ d x 1 = π d / 2 2 d ∏ i = 1 d erf ( c i w i ) + erf ( c i − c i w i ) c i , ∫ 0 1 ⋯ ∫ 0 1 exp ( − ∑ i = 1 d c i | x i − w i | ) d x d ⋯ d x 1 = ∏ i = 1 d exp ( c i w i − c i ) − exp ( − c i w i ) c i , ∫ 0 w 1 ∫ 0 w 2 ∫ 0 1 ⋯ ∫ 0 1 exp ( ∑ i = 1 d c i x i ) d x d ⋯ d x 3 d x 2 d x 1 = ∏ i = 1 2 ( exp ( c i w i ) − 1 ) ∏ i = 3 d ( exp ( c i ) − 1 ) ∏ i = 1 d c i , d ∈ Z + 0 < w i < 1 c i ∈ R + i ∈ { 1 , … , d }
更新日期：2024-12-26
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