Explicit solutions of Genz test integrals

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED Applied Mathematics Letters Pub Date : 2025-04-01 Epub Date: 2024-12-26 DOI:10.1016/j.aml.2024.109444

Vesa Kaarnioja

引用次数: 0

Abstract

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

\int_{0}^{1} \dots \int_{0}^{1} cos (2 π w_{1} + \sum_{i = 1}^{d} c_{i} x_{i}) d x_{d} \dots d x_{1} = 2^{d} cos (2 π w_{1} + \frac{1}{2} \sum_{i = 1}^{d} c_{i}) \prod_{i = 1}^{d} \frac{sin (\frac{c_{i}}{2})}{c_{i}}, \int_{0}^{1} \dots \int_{0}^{1} \prod_{i = 1}^{d} \frac{1}{c_{i}^{- 2} + {(x_{i} - w_{i})}^{2}} d x_{d} \dots d x_{1} = \prod_{i = 1}^{d} c_{i} (arctan (c_{i} w_{i}) + arctan (c_{i} - c_{i} w_{i})), \int_{0}^{1} \dots \int_{0}^{1} {(1 + \sum_{i = 1}^{d} c_{i} x_{i})}^{- (d + 1)} d x_{d} \dots d x_{1} = \frac{1}{d! \prod_{i = 1}^{d} c_{i}} \sum_{u \subseteq {c_{1}, \dots, c_{d}}} \frac{{(- 1)}^{# u}}{1 + \sum_{i \in u} i}, \int_{0}^{1} \dots \int_{0}^{1} exp (- \sum_{i = 1}^{d} c_{i}^{2} {(x_{i} - w_{i})}^{2}) d x_{d} \dots d x_{1} = \frac{π^{d / 2}}{2^{d}} \prod_{i = 1}^{d} \frac{\erf (c_{i} w_{i}) + \erf (c_{i} - c_{i} w_{i})}{c_{i}}, \int_{0}^{1} \dots \int_{0}^{1} exp (- \sum_{i = 1}^{d} c_{i} | x_{i} - w_{i} |) d x_{d} \dots d x_{1} = \prod_{i = 1}^{d} \frac{exp (c_{i} w_{i} - c_{i}) - exp (- c_{i} w_{i})}{c_{i}}, \int_{0}^{w_{1}} \int_{0}^{w_{2}} \int_{0}^{1} \dots \int_{0}^{1} exp (\sum_{i = 1}^{d} c_{i} x_{i}) d x_{d} \dots d x_{3} d x_{2} d x_{1} = \prod_{i = 1}^{2} (exp (c_{i} w_{i}) - 1) \frac{\prod_{i = 3}^{d} (exp (c_{i}) - 1)}{\prod_{i = 1}^{d} c_{i}},

where

d \in Z_{+}

0 < w_{i} < 1

, and

c_{i} \in R_{+}

for all

i \in {1, \dots, d}

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Genz测试积分的显式解

Genz（1984）引入的一组测试积分至今仍很流行，用于评估高维数值积分算法的鲁棒性。然而，在现有的文献中，这些积分的显式解似乎并不容易得到：通常，测试积分的真实值只是简单地用“过度”数值解来近似。在本文中，分析解决方案提出了贞测试积分∫01⋯∫01 cos(2πw1 +∑i = 1 dcixi) dxd⋯dx1 = 2 dco(2πw1 + 12∑我dci = 1)∏i = 1 dsin ci2 ci,∫01⋯∫01∏i = 1 d1ci−2 +(ξ−wi) 2 dxd⋯dx1 dci =∏i = 1(反正切(ciwi) +反正切(ci−ciwi)), 01⋯∫∫01(1 +∑i = 1 dcixi)−(d + 1) dxd⋯dx1 = 1 d !∏i = 1 dci∑u⊆{c1,…,cd}(−1)# u1 +∑我∈ui, 01⋯∫∫01 exp(−∑i = 1 dci2(ξ−wi) 2) dxd⋯dx1 22 =πd / d∏i = 1 derf (ciwi) +小块土地(ci−ciwi) ci, 01⋯∫∫01 exp(−∑i = 1 dci | xi−wi |) dxd⋯dx1 =∏i = 1 dexp (ciwi−ci)−exp(−ciwi) ci, w1 0 w2∫∫∫0 01⋯∫01 exp(∑i = 1 dcixi) dxd⋯dx3dx2dx1 =∏i = 12 (exp (ciwi)−1)∏i = 3 d (exp (ci)−1)∏我dci = 1,d∈Z +, 0 & lt; wi< ci∈R + 1,我∈{1,…,d}。

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来源期刊

Applied Mathematics Letters 数学-应用数学

CiteScore

7.70

自引率

5.40%

发文量

347

审稿时长

10 days

期刊介绍： The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.