A Family of Derivative-Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations

Table 3

Numerical results for function 𝑓₁₁(𝑥).

𝑥0=0.1	SM		STM (to𝑙𝑢=10^−11)		M7		M7mod
	iter	error	iter	error	iter	error	iter	error
𝛼=0	1	4.52𝑒−2	1	1.50𝑒−2	1	1.50𝑒−3	1	2.28𝑒−4
	2	4.60𝑒−3	2	2.32𝑒−4	2	3.49e-11	2	0
	⋮		3	5.38𝑒−8	3	0
	4	8.32𝑒−9	4	0
	5	0
|𝑥𝑘+1−𝑥𝑘|	8.32𝑒−9		5.38𝑒−8		3.49𝑒−11		2.28𝑒−4
|𝑓(𝑥𝑘+1)|	2.08𝑒−16		5.80𝑒−15		3.76𝑒−37		4.08𝑒−15
ACOC	2.0919		2.0000		4.2042		—
𝑥0=3	iter	error	iter	error	iter	error	iter	error
𝛼=1	1	0.29	1	0.80	1	3.45𝑒−2	1	6.22𝑒−2
	2	0.15	2	0.25	2	6.91𝑒−11	2	2.35𝑒−9
	⋮		⋮		3	NaN	3	0
	5	7.98𝑒−8	6	5.36𝑒−12
	6	1.0𝑒−14	7	0
𝑥𝑘+1−𝑥𝑘|	7.98𝑒−8		5.36𝑒−12		NaN		2.35𝑒−9
|𝑓(𝑥𝑘+1)|	1.27𝑒−14		0		NaN		0
ACOC	1.9997		2.0000		—		4.9686
𝑥0=−20	iter	error	iter	error	iter	error	iter	error
𝛼=−1	1	18.44	1	9.26	1	7.17	1	2.31
	2	17.88	2	4.39	2	2.38	2	8.82e-2
	⋮		⋮		⋮		3	2.29e-10
	30	5.65𝑒−5	8	1.98𝑒−6	4	2.72𝑒−8	4	0
	31	1.80𝑒−13	9	3.93𝑒−12	5	NaN
|𝑥𝑘+1−𝑥𝑘|	5.65𝑒−5		1.98𝑒−6		NaN		2.29𝑒−8
|𝑓(𝑥𝑘+1)|	1.80𝑒−13		3.93𝑒−12		NaN		0
ACOC	—		1.9991		—		—