On Corrected Quadrature Rules and Optimal Error Bounds

Table 4

Results for the midpoint rule.

Midpoint rule
	Standard rule ()
	Computed	Absolute error	Relative error	Estimated order
	integral	ABS()	REL()
5	1.4527054409211020	0.0062641086729669
10	1.4479877945462096	0.0015464622980745		2.0
15	1.4471270587692653	0.0006857265211302		2.0
20	1.4468267417338132	0.0003854094856781		2.0
25	1.4466879021519083	0.0002465699037733		2.0
	Corrected rule ()
	Computed	Absolute error	Relative error	Estimated order
	integral	ABS()	REL()
5	1.4465452831301751	0.0001039508820400
10	1.4464477550984778	0.0000064228503427		4.0
15	1.4464425967924956	0.0000012645443606		4.0
20	1.4464417318718803	0.0000003996237452		4.0
25	1.4464414958402714	0.0000001635921363		4.0