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

Table 8

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

Integrand	Error	Exact	Boole	Formula I	Formula II	Formula III	Formula IV
𝑥^5	Value	0.260400𝐸+4	0.260400𝐸+4	0.259848𝐸+4	0.260400𝐸+4	0.260400𝐸+4	0.260400𝐸+4
	%	0.0	0.0	0.212143𝐸+0	0.0	0.0	0.0
	Ratio	—	—	∞	—	—
𝑥^7	Value	0.488280𝐸+5	0.489560𝐸+5	0.482853𝐸+5	0.487880𝐸+5	0.489427𝐸+5	0.488261𝐸+5
	%	0	0.262145𝐸+0	0.111146𝐸+1	0.819202𝐸−1	0.234860𝐸+0	0.393217𝐸−2
	Ratio	—	1.0	4.2	0.31	0.90	0.015
Exp(𝑥)	Value	0.145695𝐸+3	0.145891𝐸+3	0.145043𝐸+3	0.145605𝐸+3	0.146011𝐸+3	0.145705𝐸+3
	%		0.134295𝐸+0	0.447095𝐸+0	0.619861𝐸−1	0.216874𝐸+0	0.666631𝐸−2
	Ratio	—	1.0	3.3	0.46	1.6	0.050