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

Table 7

Comparison of the integrals using four formulae of extrapolation methods (end-point Gauss quadrature) with Simpson’s 1/3 rule.

Integrand	Error	Exact	Simpson	Formula I	Formula II	Formula III	Formula IV
𝑥^3	Value	0.200000𝐸+2	0.200000𝐸+2	0.200000𝐸+2	0.200000𝐸+2	0.200000𝐸+2	0.200000𝐸+2
	%	0.0	0.0	0.0	0.0	0.0	0.0
	Ratio	—	—	—	—	—	—
𝑥^7	Value	0.820000𝐸+3	0.900000𝐸+3	0.742218𝐸+3	0.806667𝐸+3	0.858226𝐸+3	0.819360𝐸+3
	%	0	0.975610𝐸+1	0.948565𝐸+1	0.162602𝐸+1	0.466169𝐸+1	0.780488𝐸−1
	Ratio	—	1.0	0.97	0.17	0.48	0.0080
Exp(𝑥)	Value	0.173673𝐸+2	0.174533𝐸+2	0.172896𝐸+2	0.173665𝐸+2	0.174049𝐸+2	0.173684𝐸+2
	%		0.495718𝐸+0	0.447095𝐸+0	0.619861𝐸−1	0.216874𝐸+0	0.666631𝐸−2
	Ratio	—	1.0	0.90	0.13	0.44	0.013