Explicit solutions of Genz test integrals

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 , where d ∈ Z + , 0 < w i < 1 , and c i ∈ R + for all i ∈ { 1 , … , d } .