Comprehensive Interpretation of a Three-Point Gauss Quadrature with Variable Sampling Points and Its Application to Integration for Discrete Data

Table 1

Different types of 3-point Gauss integration of variable sampling points and the integration characteristics.

Type	Weight ratio φ (=𝑊2/ 𝑊1)	Weighting factor α	Integration weights	Sampling points	Integration characteristics
(a)	0	0.0	1.00000000; 0.00000000	±0.57735027; 0.00000000	Conventional two-point rule; (Legendre sampling points)
(b)	𝜀	1×10^−4	0.99995000; 0.00010000	±0.57736470; 0.00000000	Quasi two-point rule; (near-zero center-weight; three-point rule)
(c)	1	2/3	0.66666667; 0.66666667	±0.70710678; 0.00000000	Three-point rule; (even weight)
(d)	8/5	8/9	0.55555556; 0.88888889	±0.77459667; 0.00000000	Conventional three-point rule; (Legendre sampling points)
(e)	2	1.0	0.50000000; 1.00000000	±0.81649658; 0.00000000	Three-point rule; (double-center-weight)
(f)	4	4/3	0.33333333; 1.33333333	±1.00000000; 0.00000000	Three-point rule; (end-point rule and Simpson’s 1/3 rule)
(g)	22	22/12	0.08333333; 1.83333333	±2.00000000; 0.00000000	Three-point rule; (1st extended end-point)
(h)	52	52/27	0.03703704; 1.92592592	±3.00000000; 0.00000000	Three-point rule; (2nd extended end-point)
(i)	∞	2.0	0.00000000; 2.00000000	∞; 0.00000000	Conventional one-point rule; (Legendre sampling point)