id	sid	tid	token	lemma	pos
ma-53	1	1	2022	2022	NUM
ma-53	1	2	ada	ada	PROPN
ma-53	1	3	academica	academica	PROPN
ma-53	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-53	1	5	.	.	PUNCT
ma-53	2	1	j.	j.	PROPN
ma-53	2	2	math	math	PROPN
ma-53	2	3	.	.	PUNCT
ma-53	3	1	anal	anal	ADJ
ma-53	3	2	.	.	PUNCT
ma-53	3	3	2	2	NUM
ma-53	3	4	(	(	PUNCT
ma-53	3	5	2022	2022	NUM
ma-53	3	6	)	)	PUNCT
ma-53	4	1	3doi	3doi	NUM
ma-53	4	2	:	:	PUNCT
ma-53	4	3	10.28924	10.28924	NUM
ma-53	4	4	/	/	SYM
ma-53	4	5	ada	ada	PROPN
ma-53	4	6	/	/	SYM
ma-53	4	7	ma.2.3	ma.2.3	PROPN
ma-53	4	8	on	on	ADP
ma-53	4	9	the	the	DET
ma-53	4	10	ostrowski	ostrowski	ADJ
ma-53	4	11	method	method	NOUN
ma-53	4	12	for	for	ADP
ma-53	4	13	solving	solve	VERB
ma-53	4	14	equations	equation	NOUN
ma-53	4	15	ioannis	ioannis	PROPN
ma-53	4	16	k.	k.	PROPN
ma-53	4	17	argyros1,∗	argyros1,∗	PROPN
ma-53	4	18	,	,	PUNCT
ma-53	4	19	santhosh	santhosh	PROPN
ma-53	4	20	george2	george2	PROPN
ma-53	4	21	,	,	PUNCT
ma-53	4	22	christopher	christopher	PROPN
ma-53	4	23	i.	i.	PROPN
ma-53	4	24	argyros3	argyros3	PROPN
ma-53	5	1	1department	1department	NUM
ma-53	5	2	of	of	ADP
ma-53	5	3	mathematical	mathematical	ADJ
ma-53	5	4	sciences	sciences	PROPN
ma-53	5	5	,	,	PUNCT
ma-53	5	6	cameron	cameron	PROPN
ma-53	5	7	university	university	PROPN
ma-53	5	8	,	,	PUNCT
ma-53	5	9	lawton	lawton	PROPN
ma-53	5	10	,	,	PUNCT
ma-53	5	11	ok	ok	PROPN
ma-53	5	12	73505	73505	NUM
ma-53	5	13	,	,	PUNCT
ma-53	5	14	usa	usa	PROPN
ma-53	5	15	iargyros@cameron.edu	iargyros@cameron.edu	PROPN
ma-53	5	16	2	2	NUM
ma-53	5	17	department	department	NOUN
ma-53	5	18	of	of	ADP
ma-53	5	19	mathematical	mathematical	ADJ
ma-53	5	20	and	and	CCONJ
ma-53	5	21	computational	computational	ADJ
ma-53	5	22	sciences	science	NOUN
ma-53	5	23	,	,	PUNCT
ma-53	5	24	national	national	PROPN
ma-53	5	25	institute	institute	PROPN
ma-53	5	26	of	of	ADP
ma-53	5	27	technology	technology	PROPN
ma-53	5	28	karnataka	karnataka	PROPN
ma-53	5	29	,	,	PUNCT
ma-53	5	30	india-575	india-575	ADJ
ma-53	5	31	025	025	NUM
ma-53	5	32	sgeorge@nitk.edu.in	sgeorge@nitk.edu.in	NOUN
ma-53	5	33	3department	3department	NUM
ma-53	5	34	of	of	ADP
ma-53	5	35	computing	computing	NOUN
ma-53	5	36	and	and	CCONJ
ma-53	5	37	technology	technology	NOUN
ma-53	5	38	,	,	PUNCT
ma-53	5	39	cameron	cameron	PROPN
ma-53	5	40	university	university	PROPN
ma-53	5	41	,	,	PUNCT
ma-53	5	42	lawton	lawton	PROPN
ma-53	5	43	,	,	PUNCT
ma-53	5	44	ok	ok	PROPN
ma-53	5	45	73505	73505	NUM
ma-53	5	46	,	,	PUNCT
ma-53	5	47	usa	usa	PROPN
ma-53	5	48	christopher.argyros@cameron.edu	christopher.argyros@cameron.edu	PROPN
ma-53	5	49	∗correspondence	∗correspondence	NOUN
ma-53	5	50	:	:	PUNCT
ma-53	5	51	iargyros@cameron.edu	iargyros@cameron.edu	X
ma-53	6	1	abstract	abstract	ADJ
ma-53	6	2	.	.	PUNCT
ma-53	7	1	in	in	ADP
ma-53	7	2	this	this	DET
ma-53	7	3	paper	paper	NOUN
ma-53	7	4	,	,	PUNCT
ma-53	7	5	we	we	PRON
ma-53	7	6	revisited	revisit	VERB
ma-53	7	7	the	the	DET
ma-53	7	8	ostrowski	ostrowski	NOUN
ma-53	7	9	’s	’s	PART
ma-53	7	10	method	method	NOUN
ma-53	7	11	for	for	ADP
ma-53	7	12	solving	solve	VERB
ma-53	7	13	banach	banach	NOUN
ma-53	7	14	space	space	NOUN
ma-53	7	15	valued	value	VERB
ma-53	7	16	equa	equa	NOUN
ma-53	7	17	-	-	PUNCT
ma-53	7	18	tions	tion	NOUN
ma-53	7	19	.	.	PUNCT
ma-53	8	1	we	we	PRON
ma-53	8	2	developed	develop	VERB
ma-53	8	3	a	a	DET
ma-53	8	4	technique	technique	NOUN
ma-53	8	5	to	to	PART
ma-53	8	6	determine	determine	VERB
ma-53	8	7	a	a	DET
ma-53	8	8	subset	subset	NOUN
ma-53	8	9	of	of	ADP
ma-53	8	10	the	the	DET
ma-53	8	11	original	original	ADJ
ma-53	8	12	convergence	convergence	NOUN
ma-53	8	13	domain	domain	NOUN
ma-53	8	14	and	and	CCONJ
ma-53	8	15	usingthis	usingthi	VERB
ma-53	8	16	new	new	ADJ
ma-53	8	17	lipschitz	lipschitz	NOUN
ma-53	8	18	constants	constant	NOUN
ma-53	8	19	derived	derive	VERB
ma-53	8	20	.	.	PUNCT
ma-53	9	1	these	these	DET
ma-53	9	2	constants	constant	NOUN
ma-53	9	3	are	be	AUX
ma-53	9	4	at	at	ADV
ma-53	9	5	least	least	ADJ
ma-53	9	6	as	as	ADV
ma-53	9	7	tight	tight	ADJ
ma-53	9	8	as	as	SCONJ
ma-53	9	9	the	the	DET
ma-53	9	10	earlier	early	ADJ
ma-53	9	11	ones	one	NOUN
ma-53	9	12	leadingto	leadingto	VERB
ma-53	9	13	a	a	DET
ma-53	9	14	finer	fine	ADJ
ma-53	9	15	convergence	convergence	NOUN
ma-53	9	16	analysis	analysis	NOUN
ma-53	9	17	in	in	ADP
ma-53	9	18	both	both	CCONJ
ma-53	9	19	the	the	DET
ma-53	9	20	semi	semi	ADJ
ma-53	9	21	-	-	ADJ
ma-53	9	22	local	local	ADJ
ma-53	9	23	and	and	CCONJ
ma-53	9	24	the	the	DET
ma-53	9	25	local	local	ADJ
ma-53	9	26	convergence	convergence	NOUN
ma-53	9	27	case	case	NOUN
ma-53	9	28	.	.	PUNCT
ma-53	10	1	these	these	DET
ma-53	10	2	tech	tech	NOUN
ma-53	10	3	-	-	PUNCT
ma-53	10	4	niques	nique	NOUN
ma-53	10	5	are	be	AUX
ma-53	10	6	very	very	ADV
ma-53	10	7	general	general	ADJ
ma-53	10	8	,	,	PUNCT
ma-53	10	9	so	so	SCONJ
ma-53	10	10	they	they	PRON
ma-53	10	11	can	can	AUX
ma-53	10	12	be	be	AUX
ma-53	10	13	used	use	VERB
ma-53	10	14	to	to	PART
ma-53	10	15	extend	extend	VERB
ma-53	10	16	the	the	DET
ma-53	10	17	applicability	applicability	NOUN
ma-53	10	18	of	of	ADP
ma-53	10	19	other	other	ADJ
ma-53	10	20	methods	method	NOUN
ma-53	10	21	withoutadditional	withoutadditional	ADJ
ma-53	10	22	hypotheses	hypothesis	NOUN
ma-53	10	23	.	.	PUNCT
ma-53	11	1	numerical	numerical	ADJ
ma-53	11	2	experiments	experiment	NOUN
ma-53	11	3	complete	complete	VERB
ma-53	11	4	this	this	DET
ma-53	11	5	study	study	NOUN
ma-53	11	6	.	.	PUNCT
ma-53	12	1	1	1	X
ma-53	12	2	.	.	X
ma-53	12	3	introduction	introduction	NOUN
ma-53	12	4	one	one	NUM
ma-53	12	5	of	of	ADP
ma-53	12	6	the	the	DET
ma-53	12	7	most	most	ADV
ma-53	12	8	challenging	challenging	ADJ
ma-53	12	9	tasks	task	NOUN
ma-53	12	10	in	in	ADP
ma-53	12	11	computational	computational	ADJ
ma-53	12	12	mathematics	mathematic	NOUN
ma-53	12	13	is	be	AUX
ma-53	12	14	the	the	DET
ma-53	12	15	problem	problem	NOUN
ma-53	12	16	of	of	ADP
ma-53	12	17	determininga	determininga	NOUN
ma-53	12	18	solution	solution	NOUN
ma-53	12	19	x∗	x∗	PROPN
ma-53	12	20	of	of	ADP
ma-53	12	21	equation	equation	NOUN
ma-53	12	22	f	f	X
ma-53	12	23	(	(	PUNCT
ma-53	12	24	x	x	X
ma-53	12	25	)	)	PUNCT
ma-53	12	26	=	=	SYM
ma-53	12	27	0	0	NUM
ma-53	12	28	,	,	PUNCT
ma-53	12	29	(	(	PUNCT
ma-53	12	30	1.1	1.1	NUM
ma-53	12	31	)	)	PUNCT
ma-53	12	32	where	where	SCONJ
ma-53	12	33	f	f	X
ma-53	12	34	:	:	PUNCT
ma-53	12	35	ω	ω	NUM
ma-53	12	36	⊂	⊂	PROPN
ma-53	13	1	b	b	X
ma-53	13	2	−→	−→	ADJ
ma-53	13	3	b1	b1	NOUN
ma-53	13	4	is	be	AUX
ma-53	13	5	an	an	DET
ma-53	13	6	operator	operator	NOUN
ma-53	13	7	acting	act	VERB
ma-53	13	8	between	between	ADP
ma-53	13	9	banach	banach	NOUN
ma-53	13	10	spaces	space	NOUN
ma-53	13	11	b	b	NOUN
ma-53	13	12	and	and	CCONJ
ma-53	13	13	b1	b1	VERB
ma-53	13	14	with	with	ADP
ma-53	13	15	ω	ω	PROPN
ma-53	13	16	6=	6=	ADP
ma-53	13	17	∅.	∅.	ADP
ma-53	13	18	theclosed	theclose	VERB
ma-53	13	19	form	form	NOUN
ma-53	13	20	derivation	derivation	NOUN
ma-53	13	21	of	of	ADP
ma-53	13	22	x∗	x∗	PROPN
ma-53	13	23	is	be	AUX
ma-53	13	24	possible	possible	ADJ
ma-53	13	25	only	only	ADV
ma-53	13	26	in	in	ADP
ma-53	13	27	rare	rare	ADJ
ma-53	13	28	cases	case	NOUN
ma-53	13	29	.	.	PUNCT
ma-53	14	1	this	this	PRON
ma-53	14	2	leads	lead	VERB
ma-53	14	3	practitioners	practitioner	NOUN
ma-53	14	4	and	and	CCONJ
ma-53	14	5	researchersin	researchersin	VERB
ma-53	14	6	developing	develop	VERB
ma-53	14	7	solution	solution	NOUN
ma-53	14	8	methods	method	NOUN
ma-53	14	9	that	that	PRON
ma-53	14	10	are	be	AUX
ma-53	14	11	iterative.in	iterative.in	PRON
ma-53	14	12	this	this	DET
ma-53	14	13	work	work	NOUN
ma-53	14	14	,	,	PUNCT
ma-53	14	15	we	we	PRON
ma-53	14	16	consider	consider	VERB
ma-53	14	17	ostrowski	ostrowski	PROPN
ma-53	14	18	’s	’s	PART
ma-53	14	19	method	method	NOUN
ma-53	14	20	defined	define	VERB
ma-53	14	21	for	for	ADP
ma-53	14	22	x0	x0	PROPN
ma-53	14	23	∈	∈	PROPN
ma-53	14	24	ω	ω	PROPN
ma-53	14	25	and	and	CCONJ
ma-53	14	26	each	each	DET
ma-53	14	27	n	n	NOUN
ma-53	14	28	=	=	SYM
ma-53	14	29	0	0	NUM
ma-53	14	30	,	,	PUNCT
ma-53	14	31	1	1	NUM
ma-53	14	32	,	,	PUNCT
ma-53	14	33	2	2	NUM
ma-53	14	34	,	,	PUNCT
ma-53	14	35	.	.	PUNCT
ma-53	14	36	.	.	PUNCT
ma-53	14	37	.	.	PUNCT
ma-53	15	1	by	by	ADP
ma-53	15	2	yn	yn	PROPN
ma-53	16	1	=	=	PUNCT
ma-53	16	2	xn	xn	PROPN
ma-53	17	1	−	−	PROPN
ma-53	17	2	f	f	PROPN
ma-53	17	3	′(xn)−1f	′(xn)−1f	PROPN
ma-53	17	4	(	(	PUNCT
ma-53	17	5	xn	xn	PROPN
ma-53	17	6	)	)	PUNCT
ma-53	17	7	xk+1	xk+1	PUNCT
ma-53	18	1	=	=	SYM
ma-53	18	2	yn	yn	PRON
ma-53	19	1	−	−	PROPN
ma-53	20	1	a−1n	a−1n	NOUN
ma-53	21	1	f	f	PROPN
ma-53	22	1	(	(	PUNCT
ma-53	22	2	yn	yn	PROPN
ma-53	22	3	)	)	PUNCT
ma-53	22	4	,	,	PUNCT
ma-53	22	5	(	(	PUNCT
ma-53	22	6	1.2	1.2	NUM
ma-53	22	7	)	)	PUNCT
ma-53	22	8	received	receive	VERB
ma-53	22	9	:	:	PUNCT
ma-53	22	10	7	7	NUM
ma-53	22	11	nov	nov	PROPN
ma-53	22	12	2021	2021	NUM
ma-53	22	13	.	.	PUNCT
ma-53	23	1	key	key	ADJ
ma-53	23	2	words	word	NOUN
ma-53	23	3	and	and	CCONJ
ma-53	23	4	phrases	phrase	NOUN
ma-53	23	5	.	.	PUNCT
ma-53	24	1	ostrowski	ostrowski	PROPN
ma-53	24	2	’s	’s	PART
ma-53	24	3	method	method	NOUN
ma-53	24	4	;	;	PUNCT
ma-53	24	5	banach	banach	NOUN
ma-53	24	6	space	space	NOUN
ma-53	24	7	;	;	PUNCT
ma-53	24	8	convergence	convergence	NOUN
ma-53	24	9	criterion.1	criterion.1	X
ma-53	24	10	https://adac.ee	https://adac.ee	PROPN
ma-53	24	11	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	24	12	eur	eur	NOUN
ma-53	24	13	.	.	PUNCT
ma-53	25	1	j.	j.	PROPN
ma-53	25	2	math	math	PROPN
ma-53	25	3	.	.	PUNCT
ma-53	26	1	anal	anal	PROPN
ma-53	26	2	.	.	PUNCT
ma-53	27	1	10.28924	10.28924	NUM
ma-53	27	2	/	/	SYM
ma-53	27	3	ada	ada	PROPN
ma-53	27	4	/	/	SYM
ma-53	27	5	ma.2.3	ma.2.3	PROPN
ma-53	27	6	2where	2where	NUM
ma-53	27	7	an	an	DET
ma-53	27	8	=	=	SYM
ma-53	27	9	2[yn	2[yn	NUM
ma-53	27	10	,	,	PUNCT
ma-53	27	11	xn;f	xn;f	PUNCT
ma-53	28	1	]	]	PUNCT
ma-53	28	2	−	−	PROPN
ma-53	28	3	f	f	PROPN
ma-53	28	4	′(xn	′(xn	PROPN
ma-53	28	5	)	)	PUNCT
ma-53	28	6	.	.	PUNCT
ma-53	29	1	the	the	DET
ma-53	29	2	convergence	convergence	NOUN
ma-53	29	3	order	order	NOUN
ma-53	29	4	is	be	AUX
ma-53	29	5	four	four	NUM
ma-53	29	6	obtained	obtain	VERB
ma-53	29	7	under	under	ADP
ma-53	29	8	certain	certain	ADJ
ma-53	29	9	conditionson	conditionson	NOUN
ma-53	29	10	the	the	DET
ma-53	29	11	initial	initial	ADJ
ma-53	29	12	data	datum	NOUN
ma-53	29	13	(	(	PUNCT
ma-53	29	14	ω	ω	PROPN
ma-53	29	15	,	,	PUNCT
ma-53	29	16	f	f	PROPN
ma-53	29	17	,	,	PUNCT
ma-53	29	18	f	f	PROPN
ma-53	29	19	′	′	NOUN
ma-53	29	20	,	,	PUNCT
ma-53	29	21	x0	x0	PROPN
ma-53	29	22	)	)	PUNCT
ma-53	29	23	and	and	CCONJ
ma-53	29	24	taylor	taylor	PROPN
ma-53	29	25	expansion	expansion	NOUN
ma-53	29	26	[	[	X
ma-53	29	27	17	17	NUM
ma-53	29	28	,	,	PUNCT
ma-53	29	29	25	25	NUM
ma-53	29	30	]	]	PUNCT
ma-53	29	31	.	.	PUNCT
ma-53	30	1	so	so	ADV
ma-53	30	2	,	,	PUNCT
ma-53	30	3	the	the	DET
ma-53	30	4	assumptions	assumption	NOUN
ma-53	30	5	on	on	ADP
ma-53	30	6	the	the	DET
ma-53	30	7	fourthderivative	fourthderivative	NOUN
ma-53	30	8	reduce	reduce	VERB
ma-53	30	9	the	the	DET
ma-53	30	10	applicability	applicability	NOUN
ma-53	30	11	of	of	ADP
ma-53	30	12	these	these	PRON
ma-53	30	13	schemes.for	schemes.for	ADP
ma-53	30	14	example	example	NOUN
ma-53	30	15	:	:	PUNCT
ma-53	30	16	let	let	VERB
ma-53	30	17	b	b	NOUN
ma-53	30	18	=	=	SYM
ma-53	30	19	b1	b1	NOUN
ma-53	30	20	=	=	SYM
ma-53	30	21	r	r	PROPN
ma-53	30	22	,	,	PUNCT
ma-53	30	23	ω	ω	NOUN
ma-53	30	24	=	=	PUNCT
ma-53	31	1	[	[	X
ma-53	31	2	−0.5	−0.5	PROPN
ma-53	31	3	,	,	PUNCT
ma-53	31	4	1.5	1.5	NUM
ma-53	31	5	]	]	PUNCT
ma-53	31	6	.	.	PUNCT
ma-53	32	1	define	define	VERB
ma-53	32	2	λ	λ	PROPN
ma-53	32	3	on	on	ADP
ma-53	32	4	ω	ω	NUM
ma-53	32	5	by	by	ADP
ma-53	32	6	λ(t	λ(t	NOUN
ma-53	32	7	)	)	PUNCT
ma-53	32	8	=	=	PRON
ma-53	32	9	{	{	PUNCT
ma-53	32	10	t3	t3	PROPN
ma-53	32	11	log	log	NOUN
ma-53	32	12	t2	t2	PROPN
ma-53	32	13	+	+	CCONJ
ma-53	32	14	t5	t5	PROPN
ma-53	32	15	−	−	PROPN
ma-53	33	1	t4	t4	PROPN
ma-53	34	1	i	i	PRON
ma-53	34	2	f	f	PROPN
ma-53	34	3	t	t	PROPN
ma-53	34	4	6=	6=	PROPN
ma-53	34	5	0	0	NUM
ma-53	34	6	0	0	NUM
ma-53	35	1	i	i	PRON
ma-53	35	2	f	f	NOUN
ma-53	35	3	t	t	NOUN
ma-53	35	4	=	=	SYM
ma-53	35	5	0	0	X
ma-53	35	6	.	.	PUNCT
ma-53	36	1	then	then	ADV
ma-53	36	2	,	,	PUNCT
ma-53	36	3	we	we	PRON
ma-53	36	4	get	get	VERB
ma-53	36	5	t∗	t∗	NOUN
ma-53	36	6	=	=	SYM
ma-53	36	7	1	1	NUM
ma-53	36	8	,	,	PUNCT
ma-53	36	9	and	and	CCONJ
ma-53	36	10	λ′′′(t	λ′′′(t	PROPN
ma-53	36	11	)	)	PUNCT
ma-53	36	12	=	=	SYM
ma-53	36	13	6	6	NUM
ma-53	36	14	log	log	NOUN
ma-53	36	15	t2	t2	NOUN
ma-53	36	16	+	+	CCONJ
ma-53	37	1	60t2	60t2	NUM
ma-53	37	2	−	−	NUM
ma-53	37	3	24	24	NUM
ma-53	37	4	t	t	NOUN
ma-53	37	5	+	+	NOUN
ma-53	37	6	22	22	NUM
ma-53	37	7	.	.	PUNCT
ma-53	38	1	obviously	obviously	ADV
ma-53	38	2	λ′′′(t	λ′′′(t	VERB
ma-53	38	3	)	)	PUNCT
ma-53	38	4	is	be	AUX
ma-53	38	5	not	not	PART
ma-53	38	6	bounded	bound	VERB
ma-53	38	7	on	on	ADP
ma-53	38	8	ω	ω	PROPN
ma-53	38	9	.	.	PUNCT
ma-53	39	1	so	so	ADV
ma-53	39	2	,	,	PUNCT
ma-53	39	3	the	the	DET
ma-53	39	4	convergence	convergence	NOUN
ma-53	39	5	of	of	ADP
ma-53	39	6	scheme	scheme	NOUN
ma-53	39	7	(	(	PUNCT
ma-53	39	8	1.2	1.2	NUM
ma-53	39	9	)	)	PUNCT
ma-53	39	10	is	be	AUX
ma-53	39	11	not	not	PART
ma-53	39	12	guaranteed	guarantee	VERB
ma-53	39	13	bythe	bythe	ADP
ma-53	39	14	analyses	analysis	NOUN
ma-53	39	15	in	in	ADP
ma-53	39	16	[	[	X
ma-53	39	17	17,24].we	17,24].we	NUM
ma-53	39	18	study	study	VERB
ma-53	39	19	two	two	NUM
ma-53	39	20	types	type	NOUN
ma-53	39	21	of	of	ADP
ma-53	39	22	convergence	convergence	NOUN
ma-53	39	23	called	call	VERB
ma-53	39	24	local	local	ADJ
ma-53	39	25	and	and	CCONJ
ma-53	39	26	semi	semi	ADJ
ma-53	39	27	-	-	ADJ
ma-53	39	28	local	local	ADJ
ma-53	39	29	.	.	PUNCT
ma-53	40	1	in	in	ADP
ma-53	40	2	the	the	DET
ma-53	40	3	first	first	ADJ
ma-53	40	4	one	one	NUM
ma-53	40	5	based	base	VERB
ma-53	40	6	on	on	ADP
ma-53	40	7	thesolution	thesolution	NOUN
ma-53	40	8	x∗	x∗	PROPN
ma-53	40	9	we	we	PRON
ma-53	40	10	find	find	VERB
ma-53	40	11	the	the	DET
ma-53	40	12	radii	radius	NOUN
ma-53	40	13	of	of	ADP
ma-53	40	14	the	the	DET
ma-53	40	15	convergence	convergence	NOUN
ma-53	40	16	balls	ball	NOUN
ma-53	40	17	.	.	PUNCT
ma-53	41	1	but	but	CCONJ
ma-53	41	2	in	in	ADP
ma-53	41	3	the	the	DET
ma-53	41	4	second	second	ADJ
ma-53	41	5	one	one	NUM
ma-53	41	6	based	base	VERB
ma-53	41	7	on	on	ADP
ma-53	41	8	the	the	DET
ma-53	41	9	starter	starter	NOUN
ma-53	41	10	x0	x0	PROPN
ma-53	41	11	we	we	PRON
ma-53	41	12	develop	develop	VERB
ma-53	41	13	criteria	criterion	NOUN
ma-53	41	14	that	that	PRON
ma-53	41	15	guarantee	guarantee	VERB
ma-53	41	16	convergence	convergence	NOUN
ma-53	41	17	of	of	ADP
ma-53	41	18	sequence	sequence	NOUN
ma-53	41	19	{	{	PUNCT
ma-53	41	20	xn	xn	NUM
ma-53	41	21	}	}	PUNCT
ma-53	41	22	.	.	PUNCT
ma-53	42	1	there	there	PRON
ma-53	42	2	is	be	VERB
ma-53	42	3	a	a	DET
ma-53	42	4	plethora	plethora	NOUN
ma-53	42	5	of	of	ADP
ma-53	42	6	thistypes	thistype	NOUN
ma-53	42	7	of	of	ADP
ma-53	42	8	results	result	NOUN
ma-53	42	9	[	[	X
ma-53	42	10	10,15,16,22,28,38	10,15,16,22,28,38	NUM
ma-53	42	11	]	]	PUNCT
ma-53	42	12	.	.	PUNCT
ma-53	43	1	but	but	CCONJ
ma-53	43	2	what	what	PRON
ma-53	43	3	all	all	DET
ma-53	43	4	these	these	DET
ma-53	43	5	results	result	NOUN
ma-53	43	6	have	have	AUX
ma-53	43	7	in	in	ADP
ma-53	43	8	common	common	ADJ
ma-53	43	9	is	be	AUX
ma-53	43	10	that	that	SCONJ
ma-53	43	11	the	the	DET
ma-53	43	12	regionof	regionof	PROPN
ma-53	43	13	accessibility	accessibility	NOUN
ma-53	43	14	(	(	PUNCT
ma-53	43	15	or	or	CCONJ
ma-53	43	16	convergence	convergence	NOUN
ma-53	43	17	region	region	NOUN
ma-53	43	18	)	)	PUNCT
ma-53	43	19	is	be	AUX
ma-53	43	20	limited	limit	VERB
ma-53	43	21	in	in	ADP
ma-53	43	22	general	general	ADJ
ma-53	43	23	reducing	reduce	VERB
ma-53	43	24	the	the	DET
ma-53	43	25	applicability	applicability	NOUN
ma-53	43	26	of	of	ADP
ma-53	43	27	newton’sand	newton’sand	PRON
ma-53	43	28	other	other	ADJ
ma-53	43	29	methods	method	NOUN
ma-53	43	30	[	[	X
ma-53	43	31	8,20,26,28,31	8,20,26,28,31	NUM
ma-53	43	32	]	]	PUNCT
ma-53	43	33	.	.	PUNCT
ma-53	44	1	moreover	moreover	ADV
ma-53	44	2	,	,	PUNCT
ma-53	44	3	the	the	DET
ma-53	44	4	error	error	NOUN
ma-53	44	5	bounds	bound	VERB
ma-53	44	6	on	on	ADP
ma-53	44	7	distances	distance	NOUN
ma-53	44	8	‖xk+1−xk‖	‖xk+1−xk‖	NUM
ma-53	44	9	or	or	CCONJ
ma-53	44	10	‖xk−x∗‖are	‖xk−x∗‖are	VERB
ma-53	44	11	pessimistic	pessimistic	ADJ
ma-53	44	12	.	.	PUNCT
ma-53	45	1	the	the	DET
ma-53	45	2	same	same	ADJ
ma-53	45	3	is	be	AUX
ma-53	45	4	true	true	ADJ
ma-53	45	5	for	for	ADP
ma-53	45	6	the	the	DET
ma-53	45	7	uniqueness	uniqueness	NOUN
ma-53	45	8	ball	ball	NOUN
ma-53	45	9	of	of	ADP
ma-53	45	10	these	these	DET
ma-53	45	11	methods	method	NOUN
ma-53	45	12	.	.	PUNCT
ma-53	46	1	these	these	DET
ma-53	46	2	problems	problem	NOUN
ma-53	46	3	becomemore	becomemore	ADP
ma-53	46	4	difficult	difficult	ADJ
ma-53	46	5	when	when	SCONJ
ma-53	46	6	studying	study	VERB
ma-53	46	7	methods	method	NOUN
ma-53	46	8	of	of	ADP
ma-53	46	9	convergence	convergence	NOUN
ma-53	46	10	order	order	NOUN
ma-53	46	11	three	three	NUM
ma-53	46	12	or	or	CCONJ
ma-53	46	13	higher	high	ADJ
ma-53	46	14	[	[	X
ma-53	46	15	8	8	NUM
ma-53	46	16	,	,	PUNCT
ma-53	46	17	17	17	NUM
ma-53	46	18	,	,	PUNCT
ma-53	46	19	19	19	NUM
ma-53	46	20	,	,	PUNCT
ma-53	46	21	31–33	31–33	NUM
ma-53	46	22	]	]	PUNCT
ma-53	46	23	.	.	PUNCT
ma-53	47	1	wehave	wehave	PROPN
ma-53	47	2	developed	develop	VERB
ma-53	47	3	different	different	ADJ
ma-53	47	4	techniques	technique	NOUN
ma-53	47	5	to	to	PART
ma-53	47	6	addres	addre	NOUN
ma-53	47	7	these	these	DET
ma-53	47	8	problems.in	problems.in	X
ma-53	47	9	technique	technique	NOUN
ma-53	47	10	1	1	NUM
ma-53	47	11	,	,	PUNCT
ma-53	47	12	we	we	PRON
ma-53	47	13	determine	determine	VERB
ma-53	47	14	a	a	DET
ma-53	47	15	subset	subset	NOUN
ma-53	47	16	ω	ω	PROPN
ma-53	47	17	of	of	ADP
ma-53	47	18	ω	ω	PROPN
ma-53	47	19	also	also	ADV
ma-53	47	20	containing	contain	VERB
ma-53	47	21	the	the	DET
ma-53	47	22	iterates	iterate	NOUN
ma-53	47	23	.	.	PUNCT
ma-53	48	1	but	but	CCONJ
ma-53	48	2	in	in	ADP
ma-53	48	3	this	this	DET
ma-53	48	4	set	set	NOUN
ma-53	48	5	ω	ω	NUM
ma-53	48	6	thelipschitz	thelipschitz	ADJ
ma-53	48	7	-	-	PUNCT
ma-53	48	8	like	like	ADJ
ma-53	48	9	parameters	parameter	NOUN
ma-53	48	10	(	(	PUNCT
ma-53	48	11	or	or	CCONJ
ma-53	48	12	functions	function	NOUN
ma-53	48	13	)	)	PUNCT
ma-53	48	14	are	be	AUX
ma-53	48	15	at	at	ADV
ma-53	48	16	least	least	ADJ
ma-53	48	17	as	as	ADV
ma-53	48	18	tight	tight	ADJ
ma-53	48	19	as	as	ADP
ma-53	48	20	the	the	DET
ma-53	48	21	original	original	ADJ
ma-53	48	22	ones	one	NOUN
ma-53	48	23	,	,	PUNCT
ma-53	48	24	so	so	SCONJ
ma-53	48	25	the	the	DET
ma-53	48	26	resultingconvergence	resultingconvergence	NOUN
ma-53	48	27	is	be	AUX
ma-53	48	28	finer	fine	ADJ
ma-53	48	29	.	.	PUNCT
ma-53	49	1	this	this	DET
ma-53	49	2	technique	technique	NOUN
ma-53	49	3	does	do	AUX
ma-53	49	4	not	not	PART
ma-53	49	5	depend	depend	VERB
ma-53	49	6	on	on	ADP
ma-53	49	7	the	the	DET
ma-53	49	8	convergence	convergence	NOUN
ma-53	49	9	order	order	NOUN
ma-53	49	10	of	of	ADP
ma-53	49	11	the	the	DET
ma-53	49	12	method	method	NOUN
ma-53	49	13	.	.	PUNCT
ma-53	50	1	butwe	butwe	NOUN
ma-53	50	2	shall	shall	AUX
ma-53	50	3	demonstrate	demonstrate	VERB
ma-53	50	4	it	it	PRON
ma-53	50	5	in	in	ADP
ma-53	50	6	case	case	NOUN
ma-53	50	7	of	of	ADP
ma-53	50	8	fourth	fourth	ADJ
ma-53	50	9	order	order	NOUN
ma-53	50	10	methods	method	NOUN
ma-53	50	11	.	.	PUNCT
ma-53	51	1	these	these	DET
ma-53	51	2	methods	method	NOUN
ma-53	51	3	require	require	VERB
ma-53	51	4	the	the	DET
ma-53	51	5	evaluation	evaluation	NOUN
ma-53	51	6	ofthe	ofthe	PRON
ma-53	51	7	second	second	ADJ
ma-53	51	8	order	order	NOUN
ma-53	51	9	fréchet	fréchet	VERB
ma-53	51	10	derivative	derivative	NOUN
ma-53	51	11	of	of	ADP
ma-53	51	12	operator	operator	NOUN
ma-53	51	13	f.	f.	PROPN
ma-53	51	14	notice	notice	VERB
ma-53	51	15	that	that	SCONJ
ma-53	51	16	for	for	ADP
ma-53	51	17	a	a	DET
ma-53	51	18	system	system	NOUN
ma-53	51	19	(	(	PUNCT
ma-53	51	20	nonlinear	nonlinear	NOUN
ma-53	51	21	)	)	PUNCT
ma-53	51	22	of	of	ADP
ma-53	51	23	i	i	PRON
ma-53	51	24	equationswith	equationswith	VERB
ma-53	51	25	i	i	PRON
ma-53	51	26	unknowns	unknown	VERB
ma-53	51	27	,	,	PUNCT
ma-53	51	28	the	the	DET
ma-53	51	29	first	first	ADJ
ma-53	51	30	derivative	derivative	NOUN
ma-53	51	31	is	be	AUX
ma-53	51	32	a	a	DET
ma-53	51	33	matrix	matrix	NOUN
ma-53	51	34	with	with	ADP
ma-53	51	35	i2	i2	PROPN
ma-53	51	36	entries	entry	NOUN
ma-53	51	37	(	(	PUNCT
ma-53	51	38	values	value	NOUN
ma-53	51	39	)	)	PUNCT
ma-53	51	40	,	,	PUNCT
ma-53	51	41	whereas	whereas	SCONJ
ma-53	51	42	the	the	DET
ma-53	51	43	second	second	ADJ
ma-53	51	44	fréchetderivative	fréchetderivative	NOUN
ma-53	51	45	has	have	VERB
ma-53	51	46	i3	i3	NOUN
ma-53	51	47	entries	entry	NOUN
ma-53	51	48	.	.	PUNCT
ma-53	52	1	that	that	PRON
ma-53	52	2	is	be	AUX
ma-53	52	3	why	why	SCONJ
ma-53	52	4	there	there	PRON
ma-53	52	5	is	be	VERB
ma-53	52	6	a	a	DET
ma-53	52	7	need	need	NOUN
ma-53	52	8	for	for	ADP
ma-53	52	9	avoiding	avoid	VERB
ma-53	52	10	f	f	PROPN
ma-53	52	11	′′.the	′′.the	DET
ma-53	52	12	rest	rest	NOUN
ma-53	52	13	of	of	ADP
ma-53	52	14	the	the	DET
ma-53	52	15	paper	paper	NOUN
ma-53	52	16	is	be	AUX
ma-53	52	17	organized	organize	VERB
ma-53	52	18	as	as	SCONJ
ma-53	52	19	follows	follow	VERB
ma-53	52	20	:	:	PUNCT
ma-53	52	21	in	in	ADP
ma-53	52	22	section	section	NOUN
ma-53	52	23	2	2	NUM
ma-53	52	24	we	we	PRON
ma-53	52	25	develop	develop	VERB
ma-53	52	26	the	the	DET
ma-53	52	27	second	second	ADJ
ma-53	52	28	technique	technique	NOUN
ma-53	52	29	basedon	basedon	NOUN
ma-53	52	30	majorizing	majorize	VERB
ma-53	52	31	sequences	sequence	NOUN
ma-53	52	32	.	.	PUNCT
ma-53	53	1	the	the	DET
ma-53	53	2	local	local	ADJ
ma-53	53	3	convergence	convergence	NOUN
ma-53	53	4	analysis	analysis	NOUN
ma-53	53	5	results	result	NOUN
ma-53	53	6	appear	appear	VERB
ma-53	53	7	in	in	ADP
ma-53	53	8	section	section	NOUN
ma-53	53	9	3	3	NUM
ma-53	53	10	.	.	PUNCT
ma-53	54	1	numericalexamples	numericalexample	NOUN
ma-53	54	2	can	can	AUX
ma-53	54	3	be	be	AUX
ma-53	54	4	found	find	VERB
ma-53	54	5	in	in	ADP
ma-53	54	6	section	section	NOUN
ma-53	54	7	4	4	NUM
ma-53	54	8	.	.	PUNCT
ma-53	55	1	the	the	DET
ma-53	55	2	paper	paper	NOUN
ma-53	55	3	ends	end	VERB
ma-53	55	4	with	with	ADP
ma-53	55	5	some	some	DET
ma-53	55	6	concluding	concluding	NOUN
ma-53	55	7	remarks	remark	NOUN
ma-53	55	8	.	.	PUNCT
ma-53	56	1	2	2	X
ma-53	56	2	.	.	X
ma-53	56	3	semi	semi	ADJ
ma-53	56	4	-	-	ADJ
ma-53	56	5	local	local	ADJ
ma-53	56	6	convergence	convergence	NOUN
ma-53	56	7	we	we	PRON
ma-53	56	8	base	base	VERB
ma-53	56	9	our	our	PRON
ma-53	56	10	semi	semi	ADJ
ma-53	56	11	-	-	ADJ
ma-53	56	12	local	local	ADJ
ma-53	56	13	convergence	convergence	NOUN
ma-53	56	14	analysis	analysis	NOUN
ma-53	56	15	on	on	ADP
ma-53	56	16	scalar	scalar	ADJ
ma-53	56	17	parameters	parameter	NOUN
ma-53	56	18	and	and	CCONJ
ma-53	56	19	functions	function	NOUN
ma-53	56	20	.	.	PUNCT
ma-53	57	1	let	let	VERB
ma-53	57	2	η	η	PROPN
ma-53	57	3	≥	≥	X
ma-53	57	4	0	0	NUM
ma-53	57	5	,	,	PUNCT
ma-53	57	6	k0	k0	PROPN
ma-53	57	7	>	>	X
ma-53	57	8	0	0	PROPN
ma-53	57	9	,	,	PUNCT
ma-53	57	10	k	k	PROPN
ma-53	57	11	>	>	X
ma-53	57	12	0	0	PROPN
ma-53	57	13	,	,	PUNCT
ma-53	57	14	k1	k1	X
ma-53	57	15	>	>	X
ma-53	57	16	0	0	PROPN
ma-53	57	17	,	,	PUNCT
ma-53	57	18	k2	k2	X
ma-53	57	19	>	>	X
ma-53	57	20	0	0	PROPN
ma-53	57	21	,	,	PUNCT
ma-53	57	22	k3	k3	VERB
ma-53	57	23	>	>	X
ma-53	57	24	0	0	PROPN
ma-53	57	25	,	,	PUNCT
ma-53	57	26	l0	l0	PROPN
ma-53	57	27	>	>	X
ma-53	57	28	0	0	PUNCT
ma-53	57	29	with	with	ADP
ma-53	57	30	k0	k0	PROPN
ma-53	57	31	≤	≤	PROPN
ma-53	57	32	k	k	PROPN
ma-53	57	33	,	,	PUNCT
ma-53	57	34	l0	l0	PROPN
ma-53	57	35	≤	≤	NOUN
ma-53	57	36	2k1	2k1	NUM
ma-53	57	37	and	and	CCONJ
ma-53	57	38	k4	k4	NOUN
ma-53	57	39	=	=	PROPN
ma-53	57	40	k2	k2	PROPN
ma-53	57	41	+	+	CCONJ
ma-53	57	42	k3.define	k3.define	ADJ
ma-53	57	43	polynomials	polynomial	NOUN
ma-53	57	44	g1	g1	PROPN
ma-53	57	45	and	and	CCONJ
ma-53	57	46	g2	g2	PROPN
ma-53	57	47	on	on	ADP
ma-53	57	48	the	the	DET
ma-53	57	49	interval	interval	NOUN
ma-53	57	50	[	[	X
ma-53	57	51	0	0	NUM
ma-53	57	52	,	,	PUNCT
ma-53	57	53	1	1	NUM
ma-53	57	54	)	)	PUNCT
ma-53	57	55	by	by	ADP
ma-53	57	56	g1(t	g1(t	NOUN
ma-53	57	57	)	)	PUNCT
ma-53	57	58	=	=	SYM
ma-53	57	59	k1	k1	PROPN
ma-53	57	60	t	t	PROPN
ma-53	57	61	5	5	NUM
ma-53	57	62	+	+	CCONJ
ma-53	57	63	(	(	PUNCT
ma-53	57	64	2k1	2k1	NUM
ma-53	57	65	+	+	ADJ
ma-53	57	66	k3)t	k3)t	ADJ
ma-53	57	67	4	4	NUM
ma-53	57	68	+	+	NOUN
ma-53	57	69	k1	k1	NOUN
ma-53	57	70	t	t	NOUN
ma-53	57	71	3	3	NUM
ma-53	58	1	+	+	CCONJ
ma-53	58	2	(	(	PUNCT
ma-53	58	3	k	k	PROPN
ma-53	58	4	2	2	NUM
ma-53	58	5	−k3)t2	−k3)t2	PROPN
ma-53	58	6	−	−	PROPN
ma-53	58	7	k	k	PROPN
ma-53	58	8	2	2	NUM
ma-53	58	9	(	(	PUNCT
ma-53	58	10	2.1	2.1	NUM
ma-53	58	11	)	)	PUNCT
ma-53	58	12	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	58	13	eur	eur	NOUN
ma-53	58	14	.	.	PUNCT
ma-53	59	1	j.	j.	PROPN
ma-53	59	2	math	math	PROPN
ma-53	59	3	.	.	PUNCT
ma-53	60	1	anal	anal	PROPN
ma-53	60	2	.	.	PUNCT
ma-53	61	1	10.28924	10.28924	NUM
ma-53	61	2	/	/	SYM
ma-53	61	3	ada	ada	PROPN
ma-53	61	4	/	/	SYM
ma-53	61	5	ma.2.3	ma.2.3	PROPN
ma-53	61	6	3and	3and	NUM
ma-53	61	7	g2(t	g2(t	PROPN
ma-53	61	8	)	)	PUNCT
ma-53	61	9	=	=	SYM
ma-53	61	10	l0	l0	PROPN
ma-53	61	11	t	t	NOUN
ma-53	61	12	4	4	NUM
ma-53	61	13	+	+	CCONJ
ma-53	61	14	(	(	PUNCT
ma-53	61	15	l0	l0	INTJ
ma-53	61	16	+	+	NOUN
ma-53	61	17	k3)t	k3)t	ADJ
ma-53	61	18	3	3	NUM
ma-53	61	19	+	+	NOUN
ma-53	61	20	k4	k4	PROPN
ma-53	61	21	t	t	PROPN
ma-53	61	22	2	2	NUM
ma-53	61	23	−k3	−k3	PROPN
ma-53	61	24	t	t	NOUN
ma-53	61	25	−k4	−k4	NUM
ma-53	61	26	.	.	PUNCT
ma-53	62	1	(	(	PUNCT
ma-53	62	2	2.2	2.2	NUM
ma-53	62	3	)	)	PUNCT
ma-53	62	4	we	we	PRON
ma-53	62	5	have	have	VERB
ma-53	62	6	g1(0	g1(0	NOUN
ma-53	62	7	)	)	PUNCT
ma-53	63	1	=	=	PUNCT
ma-53	64	1	−k2	−k2	X
ma-53	64	2	<	<	X
ma-53	64	3	0	0	NUM
ma-53	64	4	,	,	PUNCT
ma-53	64	5	g1(1	g1(1	NOUN
ma-53	64	6	)	)	PUNCT
ma-53	64	7	=	=	NUM
ma-53	64	8	4k1	4k1	NUM
ma-53	64	9	>	>	SYM
ma-53	64	10	0	0	NUM
ma-53	64	11	,	,	PUNCT
ma-53	64	12	g2(0	g2(0	NOUN
ma-53	64	13	)	)	PUNCT
ma-53	64	14	=	=	PUNCT
ma-53	65	1	−k4	−k4	X
ma-53	65	2	<	<	X
ma-53	65	3	0	0	PUNCT
ma-53	65	4	and	and	CCONJ
ma-53	65	5	g2(1	g2(1	NOUN
ma-53	65	6	)	)	PUNCT
ma-53	65	7	=	=	PUNCT
ma-53	65	8	2l0	2l0	NUM
ma-53	65	9	>	>	X
ma-53	65	10	0	0	X
ma-53	65	11	.	.	PUNCT
ma-53	66	1	it	it	PRON
ma-53	66	2	thenfollows	thenfollow	VERB
ma-53	66	3	from	from	ADP
ma-53	66	4	the	the	DET
ma-53	66	5	intermediate	intermediate	ADJ
ma-53	66	6	value	value	NOUN
ma-53	66	7	theorem	theorem	NOUN
ma-53	66	8	that	that	PRON
ma-53	66	9	polynomials	polynomial	VERB
ma-53	66	10	g1	g1	NOUN
ma-53	66	11	and	and	CCONJ
ma-53	66	12	g2	g2	PROPN
ma-53	66	13	have	have	VERB
ma-53	66	14	at	at	ADV
ma-53	66	15	least	least	ADV
ma-53	66	16	one	one	NUM
ma-53	66	17	root	root	NOUN
ma-53	66	18	in	in	ADP
ma-53	66	19	(	(	PUNCT
ma-53	66	20	0	0	NUM
ma-53	66	21	,	,	PUNCT
ma-53	66	22	1	1	NUM
ma-53	66	23	)	)	PUNCT
ma-53	66	24	.	.	PUNCT
ma-53	67	1	denote	denote	VERB
ma-53	67	2	by	by	ADP
ma-53	67	3	δ1	δ1	NOUN
ma-53	67	4	and	and	CCONJ
ma-53	67	5	δ2	δ2	VERB
ma-53	67	6	the	the	DET
ma-53	67	7	least	least	ADJ
ma-53	67	8	such	such	ADJ
ma-53	67	9	roots	root	NOUN
ma-53	67	10	,	,	PUNCT
ma-53	67	11	respectively	respectively	ADV
ma-53	67	12	.	.	PUNCT
ma-53	68	1	moreover	moreover	ADV
ma-53	68	2	,	,	PUNCT
ma-53	68	3	it	it	PRON
ma-53	68	4	is	be	AUX
ma-53	68	5	convenient	convenient	ADJ
ma-53	68	6	to	to	PART
ma-53	68	7	definescalar	definescalar	VERB
ma-53	68	8	sequences	sequence	NOUN
ma-53	68	9	and	and	CCONJ
ma-53	68	10	parameters	parameter	NOUN
ma-53	68	11	t0	t0	PROPN
ma-53	68	12	=	=	SYM
ma-53	68	13	0	0	NUM
ma-53	68	14	,	,	PUNCT
ma-53	68	15	s0	s0	PROPN
ma-53	68	16	=	=	SYM
ma-53	68	17	η	η	PROPN
ma-53	68	18	,	,	PUNCT
ma-53	68	19	t1	t1	NOUN
ma-53	68	20	=	=	SYM
ma-53	68	21	s0	s0	PROPN
ma-53	68	22	+	+	CCONJ
ma-53	68	23	k0	k0	PROPN
ma-53	68	24	2	2	NUM
ma-53	68	25	(	(	PUNCT
ma-53	68	26	s0	s0	PROPN
ma-53	68	27	−	−	PROPN
ma-53	68	28	t0)2	t0)2	NUM
ma-53	68	29	1−	1−	NUM
ma-53	68	30	2k1s0	2k1s0	NUM
ma-53	68	31	,	,	PUNCT
ma-53	68	32	sn+1	sn+1	X
ma-53	68	33	=	=	SYM
ma-53	68	34	tn+1	tn+1	PROPN
ma-53	68	35	+	+	CCONJ
ma-53	68	36	(	(	PUNCT
ma-53	68	37	k3(tn+1	k3(tn+1	PROPN
ma-53	68	38	−	−	PROPN
ma-53	68	39	sn	sn	NOUN
ma-53	68	40	)	)	PUNCT
ma-53	69	1	+	+	ADJ
ma-53	69	2	k4(sn	k4(sn	PROPN
ma-53	69	3	−	−	PROPN
ma-53	69	4	tn))(tn+1	tn))(tn+1	NUM
ma-53	69	5	−	−	PROPN
ma-53	69	6	sn	sn	PROPN
ma-53	69	7	)	)	PUNCT
ma-53	69	8	1−	1−	NUM
ma-53	69	9	l0tn+1	l0tn+1	NOUN
ma-53	69	10	tn+2	tn+2	ADV
ma-53	69	11	=	=	SYM
ma-53	69	12	sn+1	sn+1	PROPN
ma-53	69	13	+	+	SYM
ma-53	69	14	k(sn+1	k(sn+1	PROPN
ma-53	69	15	−	−	PROPN
ma-53	69	16	tn+1)2	tn+1)2	NUM
ma-53	69	17	2(1−	2(1−	NUM
ma-53	69	18	(	(	PUNCT
ma-53	69	19	k1(sn+1	k1(sn+1	X
ma-53	69	20	+	+	CCONJ
ma-53	69	21	tn+1	tn+1	NOUN
ma-53	69	22	)	)	PUNCT
ma-53	69	23	+	+	NOUN
ma-53	69	24	k3(sn+1	k3(sn+1	NOUN
ma-53	69	25	−	−	NOUN
ma-53	69	26	tn+1	tn+1	NOUN
ma-53	69	27	)	)	PUNCT
ma-53	69	28	)	)	PUNCT
ma-53	69	29	,	,	PUNCT
ma-53	69	30	(	(	PUNCT
ma-53	69	31	2.3	2.3	NUM
ma-53	69	32	)	)	PUNCT
ma-53	69	33	αn	αn	NOUN
ma-53	69	34	=	=	PUNCT
ma-53	69	35	k(sn	k(sn	PROPN
ma-53	69	36	−	−	PROPN
ma-53	69	37	tn	tn	PROPN
ma-53	69	38	)	)	PUNCT
ma-53	69	39	2(1−	2(1−	NUM
ma-53	69	40	(	(	PUNCT
ma-53	69	41	k1(sn+1	k1(sn+1	X
ma-53	69	42	+	+	CCONJ
ma-53	69	43	tn+1	tn+1	NOUN
ma-53	69	44	)	)	PUNCT
ma-53	69	45	+	+	NOUN
ma-53	69	46	k3(sn+1	k3(sn+1	NOUN
ma-53	69	47	−	−	NOUN
ma-53	69	48	tn+1	tn+1	NOUN
ma-53	69	49	)	)	PUNCT
ma-53	69	50	)	)	PUNCT
ma-53	69	51	,	,	PUNCT
ma-53	69	52	γn	γn	X
ma-53	69	53	=	=	PUNCT
ma-53	69	54	k3(tn	k3(tn	PROPN
ma-53	69	55	−	−	PROPN
ma-53	69	56	sn	sn	PROPN
ma-53	69	57	)	)	PUNCT
ma-53	70	1	+	+	ADJ
ma-53	70	2	k4(sn	k4(sn	PROPN
ma-53	70	3	−	−	PROPN
ma-53	70	4	tn	tn	PROPN
ma-53	70	5	)	)	PUNCT
ma-53	70	6	1−	1−	NUM
ma-53	70	7	l0tn+1	l0tn+1	NOUN
ma-53	70	8	,	,	PUNCT
ma-53	70	9	for	for	ADP
ma-53	70	10	all	all	DET
ma-53	70	11	n	n	NOUN
ma-53	70	12	=	=	SYM
ma-53	70	13	0	0	NUM
ma-53	70	14	,	,	PUNCT
ma-53	70	15	1	1	NUM
ma-53	70	16	,	,	PUNCT
ma-53	70	17	2	2	NUM
ma-53	70	18	,	,	PUNCT
ma-53	70	19	.	.	PUNCT
ma-53	70	20	.	.	PUNCT
ma-53	70	21	.	.	PUNCT
ma-53	71	1	,	,	PUNCT
ma-53	71	2	δn	δn	PROPN
ma-53	71	3	=	=	SYM
ma-53	71	4	max{αn	max{αn	NOUN
ma-53	71	5	,	,	PUNCT
ma-53	71	6	γn	γn	ADP
ma-53	71	7	}	}	PUNCT
ma-53	71	8	,	,	PUNCT
ma-53	71	9	λ	λ	X
ma-53	71	10	=	=	SYM
ma-53	71	11	min{δ1	min{δ1	NOUN
ma-53	71	12	,	,	PUNCT
ma-53	71	13	δ2	δ2	ADJ
ma-53	71	14	}	}	PUNCT
ma-53	71	15	and	and	CCONJ
ma-53	71	16	µ	µ	X
ma-53	71	17	=	=	SYM
ma-53	71	18	max{δ1	max{δ1	NOUN
ma-53	71	19	,	,	PUNCT
ma-53	71	20	δ2	δ2	PROPN
ma-53	71	21	}	}	PUNCT
ma-53	71	22	.	.	PUNCT
ma-53	72	1	next	next	ADV
ma-53	72	2	,	,	PUNCT
ma-53	72	3	we	we	PRON
ma-53	72	4	present	present	VERB
ma-53	72	5	a	a	DET
ma-53	72	6	convergence	convergence	NOUN
ma-53	72	7	result	result	NOUN
ma-53	72	8	for	for	ADP
ma-53	72	9	sequences	sequence	NOUN
ma-53	72	10	{	{	PUNCT
ma-53	72	11	tn	tn	NOUN
ma-53	72	12	}	}	PUNCT
ma-53	72	13	and	and	CCONJ
ma-53	72	14	{	{	PUNCT
ma-53	72	15	sn	sn	NOUN
ma-53	72	16	}	}	PUNCT
ma-53	72	17	.	.	PUNCT
ma-53	73	1	lemma	lemma	PROPN
ma-53	73	2	2.1	2.1	NUM
ma-53	73	3	.	.	PUNCT
ma-53	74	1	suppose	suppose	VERB
ma-53	74	2	:	:	PUNCT
ma-53	74	3	there	there	PRON
ma-53	74	4	exists	exist	VERB
ma-53	74	5	δ	δ	PROPN
ma-53	74	6	satisfying	satisfy	VERB
ma-53	74	7	0	0	NUM
ma-53	74	8	≤	≤	NUM
ma-53	74	9	δ0	δ0	NOUN
ma-53	74	10	≤	≤	NUM
ma-53	74	11	λ	λ	PROPN
ma-53	74	12	≤	≤	NUM
ma-53	74	13	δ	δ	PROPN
ma-53	74	14	≤	≤	PROPN
ma-53	74	15	µ	µ	X
ma-53	74	16	<	<	X
ma-53	74	17	1−k1η	1−k1η	NUM
ma-53	74	18	.	.	PUNCT
ma-53	75	1	(	(	PUNCT
ma-53	75	2	2.4	2.4	NUM
ma-53	75	3	)	)	PUNCT
ma-53	75	4	then	then	ADV
ma-53	75	5	,	,	PUNCT
ma-53	75	6	sequences	sequence	NOUN
ma-53	75	7	{	{	PUNCT
ma-53	75	8	tn	tn	NOUN
ma-53	75	9	}	}	PUNCT
ma-53	75	10	,	,	PUNCT
ma-53	75	11	{	{	PUNCT
ma-53	75	12	sn	sn	NOUN
ma-53	75	13	}	}	PUNCT
ma-53	75	14	are	be	AUX
ma-53	75	15	well	well	ADV
ma-53	75	16	defined	define	VERB
ma-53	75	17	nondecreasing	nondecrease	VERB
ma-53	75	18	,	,	PUNCT
ma-53	75	19	bounded	bound	VERB
ma-53	75	20	from	from	ADP
ma-53	75	21	above	above	ADP
ma-53	75	22	bt	bt	NOUN
ma-53	75	23	s∗∗	s∗∗	PUNCT
ma-53	75	24	=	=	SYM
ma-53	75	25	η	η	PROPN
ma-53	75	26	1−δ	1−δ	PROPN
ma-53	75	27	and	and	CCONJ
ma-53	75	28	as	as	ADP
ma-53	75	29	such	such	ADJ
ma-53	75	30	they	they	PRON
ma-53	75	31	converge	converge	VERB
ma-53	75	32	to	to	ADP
ma-53	75	33	their	their	PRON
ma-53	75	34	unique	unique	ADJ
ma-53	75	35	least	least	ADV
ma-53	75	36	upper	upper	ADJ
ma-53	75	37	bound	bind	VERB
ma-53	75	38	s∗	s∗	PROPN
ma-53	75	39	∈	∈	PROPN
ma-53	76	1	[	[	X
ma-53	76	2	η	η	X
ma-53	76	3	,	,	PUNCT
ma-53	76	4	s∗∗	s∗∗	ADJ
ma-53	76	5	]	]	X
ma-53	76	6	.	.	PUNCT
ma-53	77	1	moreover	moreover	ADV
ma-53	77	2	,	,	PUNCT
ma-53	77	3	the	the	DET
ma-53	77	4	following	follow	VERB
ma-53	77	5	error	error	NOUN
ma-53	77	6	estimates	estimate	NOUN
ma-53	77	7	hold	hold	VERB
ma-53	77	8	for	for	ADP
ma-53	77	9	all	all	DET
ma-53	77	10	n	n	NOUN
ma-53	77	11	=	=	SYM
ma-53	77	12	1	1	NUM
ma-53	77	13	,	,	PUNCT
ma-53	77	14	2	2	NUM
ma-53	77	15	,	,	PUNCT
ma-53	77	16	.	.	PUNCT
ma-53	77	17	.	.	PUNCT
ma-53	78	1	.	.	PUNCT
ma-53	79	1	0	0	NUM
ma-53	80	1	≤	≤	NUM
ma-53	80	2	tn+1	tn+1	PROPN
ma-53	80	3	−	−	PROPN
ma-53	80	4	sn	sn	PROPN
ma-53	80	5	≤	≤	PROPN
ma-53	80	6	δ(sn	δ(sn	VERB
ma-53	80	7	−	−	PROPN
ma-53	80	8	tn	tn	NOUN
ma-53	80	9	)	)	PUNCT
ma-53	80	10	≤	≤	NUM
ma-53	80	11	δ2n+1η	δ2n+1η	NOUN
ma-53	80	12	,	,	PUNCT
ma-53	80	13	(	(	PUNCT
ma-53	80	14	2.5	2.5	NUM
ma-53	80	15	)	)	PUNCT
ma-53	80	16	0	0	NUM
ma-53	81	1	≤	≤	NUM
ma-53	82	1	sn	sn	PROPN
ma-53	83	1	−	−	PROPN
ma-53	84	1	tn	tn	NOUN
ma-53	84	2	≤	≤	PUNCT
ma-53	84	3	δ(tn	δ(tn	PROPN
ma-53	84	4	−	−	NOUN
ma-53	84	5	sn−1	sn−1	PROPN
ma-53	84	6	)	)	PUNCT
ma-53	84	7	≤	≤	NOUN
ma-53	84	8	δ2nη	δ2nη	PUNCT
ma-53	84	9	(	(	PUNCT
ma-53	84	10	2.6	2.6	NUM
ma-53	84	11	)	)	PUNCT
ma-53	84	12	and	and	CCONJ
ma-53	84	13	tn	tn	NOUN
ma-53	84	14	≤	≤	PROPN
ma-53	84	15	sn	sn	PROPN
ma-53	84	16	≤	≤	PROPN
ma-53	84	17	tn+1	tn+1	NOUN
ma-53	84	18	.	.	PUNCT
ma-53	85	1	(	(	PUNCT
ma-53	85	2	2.7	2.7	NUM
ma-53	85	3	)	)	PUNCT
ma-53	85	4	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	85	5	eur	eur	NOUN
ma-53	85	6	.	.	PUNCT
ma-53	86	1	j.	j.	PROPN
ma-53	86	2	math	math	PROPN
ma-53	86	3	.	.	PUNCT
ma-53	87	1	anal	anal	PROPN
ma-53	87	2	.	.	PUNCT
ma-53	88	1	10.28924	10.28924	NUM
ma-53	88	2	/	/	SYM
ma-53	88	3	ada	ada	PROPN
ma-53	88	4	/	/	SYM
ma-53	88	5	ma.2.3	ma.2.3	PROPN
ma-53	88	6	4	4	NUM
ma-53	88	7	proof	proof	NOUN
ma-53	88	8	.	.	PUNCT
ma-53	89	1	items	item	NOUN
ma-53	89	2	(	(	PUNCT
ma-53	89	3	2.5)-(2.7	2.5)-(2.7	X
ma-53	89	4	)	)	PUNCT
ma-53	89	5	hold	hold	VERB
ma-53	89	6	if	if	SCONJ
ma-53	89	7	0	0	NUM
ma-53	89	8	≤	≤	NUM
ma-53	89	9	αm	αm	NOUN
ma-53	89	10	≤	≤	PROPN
ma-53	89	11	δ	δ	PROPN
ma-53	89	12	,	,	PUNCT
ma-53	89	13	(	(	PUNCT
ma-53	89	14	2.8	2.8	NUM
ma-53	89	15	)	)	PUNCT
ma-53	89	16	0	0	NUM
ma-53	90	1	≤	≤	NUM
ma-53	90	2	γm	γm	PRON
ma-53	90	3	≤	≤	NUM
ma-53	90	4	δ	δ	PROPN
ma-53	90	5	(	(	PUNCT
ma-53	90	6	2.9	2.9	NUM
ma-53	90	7	)	)	PUNCT
ma-53	90	8	and	and	CCONJ
ma-53	90	9	tm	tm	PROPN
ma-53	90	10	≤	≤	PROPN
ma-53	90	11	sm	sm	VERB
ma-53	90	12	≤	≤	NUM
ma-53	90	13	tm+1	tm+1	PROPN
ma-53	90	14	(	(	PUNCT
ma-53	90	15	2.10	2.10	NUM
ma-53	90	16	)	)	PUNCT
ma-53	90	17	are	be	AUX
ma-53	90	18	true	true	ADJ
ma-53	90	19	for	for	ADP
ma-53	90	20	all	all	DET
ma-53	90	21	m	m	NOUN
ma-53	90	22	=	=	NOUN
ma-53	90	23	0	0	NUM
ma-53	90	24	,	,	PUNCT
ma-53	90	25	1	1	NUM
ma-53	90	26	,	,	PUNCT
ma-53	90	27	2	2	NUM
ma-53	90	28	,	,	PUNCT
ma-53	90	29	.	.	PUNCT
ma-53	90	30	.	.	PUNCT
ma-53	90	31	.	.	PUNCT
ma-53	90	32	.	.	PUNCT
ma-53	91	1	notice	notice	VERB
ma-53	91	2	that	that	SCONJ
ma-53	91	3	by	by	ADP
ma-53	91	4	the	the	DET
ma-53	91	5	definition	definition	NOUN
ma-53	91	6	of	of	ADP
ma-53	91	7	s0	s0	PROPN
ma-53	91	8	,	,	PUNCT
ma-53	91	9	t1	t1	NOUN
ma-53	91	10	and	and	CCONJ
ma-53	91	11	(	(	PUNCT
ma-53	91	12	2.4	2.4	NUM
ma-53	91	13	)	)	PUNCT
ma-53	91	14	,	,	PUNCT
ma-53	91	15	t1	t1	PROPN
ma-53	91	16	≥	≥	NUM
ma-53	91	17	0	0	NUM
ma-53	91	18	.	.	PUNCT
ma-53	92	1	we	we	PRON
ma-53	92	2	alsohave	alsohave	VERB
ma-53	92	3	(	(	PUNCT
ma-53	92	4	2.8	2.8	NUM
ma-53	92	5	)	)	PUNCT
ma-53	92	6	and	and	CCONJ
ma-53	92	7	(	(	PUNCT
ma-53	92	8	2.9	2.9	NUM
ma-53	92	9	)	)	PUNCT
ma-53	92	10	hold	hold	VERB
ma-53	92	11	for	for	ADP
ma-53	92	12	m	m	PROPN
ma-53	92	13	=	=	SYM
ma-53	92	14	0	0	X
ma-53	92	15	.	.	PUNCT
ma-53	93	1	suppose	suppose	VERB
ma-53	93	2	(	(	PUNCT
ma-53	93	3	2.8)-(2.10	2.8)-(2.10	NUM
ma-53	93	4	)	)	PUNCT
ma-53	93	5	hold	hold	NOUN
ma-53	93	6	for	for	ADP
ma-53	93	7	m	m	PROPN
ma-53	93	8	=	=	SYM
ma-53	93	9	1	1	NUM
ma-53	93	10	,	,	PUNCT
ma-53	93	11	2	2	NUM
ma-53	93	12	,	,	PUNCT
ma-53	93	13	.	.	PUNCT
ma-53	93	14	.	.	PUNCT
ma-53	94	1	.	.	PUNCT
ma-53	95	1	,	,	PUNCT
ma-53	95	2	n.	n.	PROPN
ma-53	95	3	then	then	ADV
ma-53	95	4	,	,	PUNCT
ma-53	95	5	we	we	PRON
ma-53	95	6	canobtain	canobtain	VERB
ma-53	95	7	in	in	ADP
ma-53	95	8	turn	turn	NOUN
ma-53	95	9	that	that	PRON
ma-53	95	10	sm	sm	VERB
ma-53	95	11	≤	≤	PUNCT
ma-53	95	12	tm	tm	PROPN
ma-53	95	13	+	+	NOUN
ma-53	95	14	δ2mη	δ2mη	PUNCT
ma-53	95	15	≤	≤	ADJ
ma-53	95	16	sm−1	sm−1	NOUN
ma-53	95	17	+	+	CCONJ
ma-53	95	18	δ2m−1η	δ2m−1η	VERB
ma-53	95	19	+	+	CCONJ
ma-53	95	20	δ2mη	δ2mη	VERB
ma-53	95	21	≤	≤	PROPN
ma-53	95	22	η	η	PROPN
ma-53	95	23	+	+	PROPN
ma-53	95	24	.	.	PUNCT
ma-53	95	25	.	.	PUNCT
ma-53	96	1	.+	.+	NOUN
ma-53	96	2	δη	δη	PROPN
ma-53	97	1	+	+	CCONJ
ma-53	97	2	.	.	PUNCT
ma-53	97	3	.	.	PUNCT
ma-53	98	1	.+	.+	NOUN
ma-53	98	2	δ2mη	δ2mη	PUNCT
ma-53	99	1	=	=	SYM
ma-53	99	2	1−	1−	NUM
ma-53	99	3	δ2m+1	δ2m+1	PROPN
ma-53	99	4	1−	1−	NUM
ma-53	99	5	δ	δ	PROPN
ma-53	99	6	η	η	PROPN
ma-53	99	7	≤	≤	PROPN
ma-53	99	8	η	η	PROPN
ma-53	99	9	1−	1−	PROPN
ma-53	99	10	δ	δ	PROPN
ma-53	99	11	=	=	PUNCT
ma-53	99	12	s∗∗	s∗∗	PROPN
ma-53	99	13	,	,	PUNCT
ma-53	99	14	and	and	CCONJ
ma-53	99	15	tm+1	tm+1	PRON
ma-53	99	16	≤	≤	NOUN
ma-53	99	17	sm	sm	VERB
ma-53	100	1	+	+	CCONJ
ma-53	100	2	δ2m+1η	δ2m+1η	NOUN
ma-53	100	3	≤	≤	ADJ
ma-53	100	4	tm	tm	PROPN
ma-53	100	5	+	+	X
ma-53	100	6	δ2mη	δ2mη	X
ma-53	101	1	+	+	CCONJ
ma-53	101	2	δ2m+1η	δ2m+1η	PROPN
ma-53	101	3	≤	≤	PROPN
ma-53	101	4	η	η	PROPN
ma-53	101	5	+	+	PROPN
ma-53	101	6	δη	δη	PROPN
ma-53	101	7	+	+	X
ma-53	101	8	.	.	PUNCT
ma-53	101	9	.	.	PUNCT
ma-53	102	1	.+	.+	NOUN
ma-53	102	2	δ2m+1η	δ2m+1η	PRON
ma-53	102	3	=	=	SYM
ma-53	102	4	1−	1−	NUM
ma-53	102	5	δ2m+2	δ2m+2	NOUN
ma-53	102	6	1−	1−	NUM
ma-53	102	7	δ	δ	PROPN
ma-53	102	8	η	η	PROPN
ma-53	102	9	≤	≤	PROPN
ma-53	102	10	η	η	PROPN
ma-53	102	11	1−	1−	PROPN
ma-53	102	12	δ	δ	PROPN
ma-53	102	13	.	.	PUNCT
ma-53	103	1	hence	hence	ADV
ma-53	103	2	,	,	PUNCT
ma-53	103	3	by	by	ADP
ma-53	103	4	(	(	PUNCT
ma-53	103	5	2.7	2.7	NUM
ma-53	103	6	)	)	PUNCT
ma-53	103	7	and	and	CCONJ
ma-53	103	8	the	the	DET
ma-53	103	9	induction	induction	NOUN
ma-53	103	10	hypotheses	hypothese	VERB
ma-53	103	11	,	,	PUNCT
ma-53	103	12	we	we	PRON
ma-53	103	13	deduce	deduce	VERB
ma-53	103	14	that	that	SCONJ
ma-53	103	15	sequences	sequence	NOUN
ma-53	103	16	{	{	PUNCT
ma-53	103	17	tm	tm	NOUN
ma-53	103	18	}	}	PUNCT
ma-53	103	19	and	and	CCONJ
ma-53	103	20	{	{	PUNCT
ma-53	103	21	sm	sm	INTJ
ma-53	103	22	}	}	PUNCT
ma-53	103	23	arenondecreasing	arenondecrease	VERB
ma-53	103	24	.	.	PUNCT
ma-53	104	1	evidently	evidently	ADV
ma-53	104	2	,	,	PUNCT
ma-53	104	3	(	(	PUNCT
ma-53	104	4	2.8	2.8	NUM
ma-53	104	5	)	)	PUNCT
ma-53	104	6	holds	hold	VERB
ma-53	104	7	if	if	SCONJ
ma-53	104	8	k	k	PROPN
ma-53	104	9	2	2	NUM
ma-53	104	10	δ2nη	δ2nη	X
ma-53	104	11	+	+	CCONJ
ma-53	104	12	δk1	δk1	X
ma-53	104	13	(	(	PUNCT
ma-53	104	14	1−	1−	NUM
ma-53	104	15	δ2n+3	δ2n+3	NOUN
ma-53	104	16	1−	1−	NUM
ma-53	104	17	δ	δ	PROPN
ma-53	104	18	η	η	PROPN
ma-53	104	19	)	)	PUNCT
ma-53	105	1	+	+	NOUN
ma-53	105	2	δk1	δk1	NOUN
ma-53	105	3	1−	1−	NUM
ma-53	106	1	δ2n+2	δ2n+2	NOUN
ma-53	106	2	1−	1−	NUM
ma-53	107	1	δ	δ	PROPN
ma-53	107	2	η	η	PROPN
ma-53	107	3	+	+	PROPN
ma-53	107	4	k3δ	k3δ	PROPN
ma-53	107	5	2(n+1)η	2(n+1)η	NUM
ma-53	107	6	−	−	PROPN
ma-53	107	7	δ	δ	PROPN
ma-53	107	8	≤	≤	PROPN
ma-53	107	9	0	0	NUM
ma-53	107	10	.	.	PUNCT
ma-53	108	1	(	(	PUNCT
ma-53	108	2	2.11	2.11	NUM
ma-53	108	3	)	)	PUNCT
ma-53	108	4	estimate	estimate	NOUN
ma-53	108	5	(	(	PUNCT
ma-53	108	6	2.11	2.11	NUM
ma-53	108	7	)	)	PUNCT
ma-53	108	8	motivates	motivate	VERB
ma-53	108	9	us	we	PRON
ma-53	108	10	to	to	PART
ma-53	108	11	introduce	introduce	VERB
ma-53	108	12	recurrent	recurrent	ADJ
ma-53	108	13	functions	function	NOUN
ma-53	108	14	h(1)n	h(1)n	X
ma-53	108	15	(	(	PUNCT
ma-53	108	16	t	t	NOUN
ma-53	108	17	)	)	PUNCT
ma-53	108	18	on	on	ADP
ma-53	108	19	the	the	DET
ma-53	108	20	interval	interval	NOUN
ma-53	108	21	[	[	X
ma-53	108	22	0	0	NUM
ma-53	108	23	,	,	PUNCT
ma-53	108	24	1	1	NUM
ma-53	108	25	)	)	PUNCT
ma-53	108	26	by	by	ADP
ma-53	108	27	h	h	PROPN
ma-53	108	28	(	(	PUNCT
ma-53	108	29	1	1	NUM
ma-53	108	30	)	)	PUNCT
ma-53	108	31	n	n	CCONJ
ma-53	108	32	)	)	PUNCT
ma-53	108	33	t	t	PROPN
ma-53	108	34	)	)	PUNCT
ma-53	109	1	=	=	SYM
ma-53	109	2	k	k	PROPN
ma-53	109	3	2	2	NUM
ma-53	109	4	t2n−1η	t2n−1η	INTJ
ma-53	110	1	+	+	NOUN
ma-53	110	2	k1(1	k1(1	NOUN
ma-53	110	3	+	+	X
ma-53	110	4	t	t	NOUN
ma-53	110	5	+	+	X
ma-53	110	6	.	.	PUNCT
ma-53	110	7	.	.	PUNCT
ma-53	111	1	.+	.+	NOUN
ma-53	111	2	t2n+2)η	t2n+2)η	ADP
ma-53	111	3	+	+	NOUN
ma-53	111	4	k1(1	k1(1	NOUN
ma-53	111	5	+	+	X
ma-53	111	6	t	t	NOUN
ma-53	111	7	+	+	X
ma-53	111	8	.	.	PUNCT
ma-53	111	9	.	.	PUNCT
ma-53	112	1	.+	.+	NOUN
ma-53	112	2	t2n+1)η	t2n+1)η	VERB
ma-53	113	1	+	+	SYM
ma-53	113	2	k3	k3	X
ma-53	113	3	t	t	NOUN
ma-53	113	4	2n+3η	2n+3η	NUM
ma-53	113	5	−	−	NOUN
ma-53	113	6	1	1	NUM
ma-53	113	7	.	.	PUNCT
ma-53	114	1	(	(	PUNCT
ma-53	114	2	2.12	2.12	NUM
ma-53	114	3	)	)	PUNCT
ma-53	114	4	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	114	5	eur	eur	NOUN
ma-53	114	6	.	.	PUNCT
ma-53	115	1	j.	j.	PROPN
ma-53	115	2	math	math	PROPN
ma-53	115	3	.	.	PUNCT
ma-53	116	1	anal	anal	PROPN
ma-53	116	2	.	.	PUNCT
ma-53	117	1	10.28924	10.28924	NUM
ma-53	117	2	/	/	SYM
ma-53	117	3	ada	ada	PROPN
ma-53	117	4	/	/	SYM
ma-53	117	5	ma.2.3	ma.2.3	PROPN
ma-53	117	6	5	5	NUM
ma-53	117	7	we	we	PRON
ma-53	117	8	need	need	VERB
ma-53	117	9	a	a	DET
ma-53	117	10	relationship	relationship	NOUN
ma-53	117	11	between	between	ADP
ma-53	117	12	two	two	NUM
ma-53	117	13	consecutive	consecutive	ADJ
ma-53	117	14	functions	function	NOUN
ma-53	117	15	f	f	X
ma-53	117	16	(	(	PUNCT
ma-53	117	17	1)n	1)n	X
ma-53	117	18	(	(	PUNCT
ma-53	117	19	t	t	NOUN
ma-53	117	20	)	)	PUNCT
ma-53	117	21	.	.	PUNCT
ma-53	118	1	by	by	ADP
ma-53	118	2	this	this	DET
ma-53	118	3	definition	definition	NOUN
ma-53	118	4	,	,	PUNCT
ma-53	118	5	we	we	PRON
ma-53	118	6	have	have	VERB
ma-53	118	7	inturn	inturn	NOUN
ma-53	119	1	that	that	SCONJ
ma-53	120	1	h	h	NOUN
ma-53	120	2	(	(	PUNCT
ma-53	120	3	1	1	X
ma-53	120	4	)	)	PUNCT
ma-53	120	5	n+1(t	n+1(t	NUM
ma-53	120	6	)	)	PUNCT
ma-53	121	1	=	=	SYM
ma-53	121	2	k	k	X
ma-53	121	3	2	2	NUM
ma-53	121	4	t2n+1η	t2n+1η	NOUN
ma-53	121	5	+	+	NOUN
ma-53	121	6	k1(1	k1(1	ADJ
ma-53	121	7	+	+	X
ma-53	121	8	t	t	NOUN
ma-53	121	9	+	+	X
ma-53	121	10	.	.	PUNCT
ma-53	121	11	.	.	PUNCT
ma-53	122	1	.+	.+	NOUN
ma-53	122	2	t2n+4)η	t2n+4)η	VERB
ma-53	122	3	+	+	NOUN
ma-53	122	4	k1(1	k1(1	PROPN
ma-53	122	5	+	+	X
ma-53	122	6	t	t	NOUN
ma-53	122	7	+	+	X
ma-53	122	8	.	.	PUNCT
ma-53	122	9	.	.	PUNCT
ma-53	123	1	.+	.+	NOUN
ma-53	123	2	t2n+3)η	t2n+3)η	ADP
ma-53	124	1	+	+	NOUN
ma-53	124	2	k3	k3	VERB
ma-53	124	3	t	t	NOUN
ma-53	125	1	2n+3η	2n+3η	NUM
ma-53	125	2	−	−	NOUN
ma-53	125	3	1	1	NUM
ma-53	125	4	−	−	NOUN
ma-53	125	5	k	k	NOUN
ma-53	125	6	2	2	NUM
ma-53	125	7	t2n−1η	t2n−1η	NUM
ma-53	125	8	−k1(1	−k1(1	NUM
ma-53	125	9	+	+	NUM
ma-53	125	10	t	t	NOUN
ma-53	125	11	+	+	X
ma-53	125	12	.	.	PUNCT
ma-53	125	13	.	.	PUNCT
ma-53	126	1	.+	.+	NOUN
ma-53	126	2	t2n+2)η	t2n+2)η	PRON
ma-53	126	3	−k1(1	−k1(1	NUM
ma-53	126	4	+	+	NUM
ma-53	126	5	t	t	NOUN
ma-53	126	6	+	+	X
ma-53	126	7	.	.	PUNCT
ma-53	126	8	.	.	PUNCT
ma-53	127	1	.+	.+	NOUN
ma-53	127	2	t2n+1)η	t2n+1)η	VERB
ma-53	127	3	−k3t2n+1η	−k3t2n+1η	VERB
ma-53	128	1	+	+	X
ma-53	128	2	1	1	NUM
ma-53	129	1	+	+	NUM
ma-53	129	2	h	h	NOUN
ma-53	129	3	(	(	PUNCT
ma-53	129	4	1	1	NUM
ma-53	129	5	)	)	PUNCT
ma-53	129	6	n	n	PROPN
ma-53	129	7	(	(	PUNCT
ma-53	129	8	t	t	NOUN
ma-53	129	9	)	)	PUNCT
ma-53	129	10	=	=	SYM
ma-53	130	1	h	h	NOUN
ma-53	130	2	(	(	PUNCT
ma-53	130	3	1	1	NUM
ma-53	130	4	)	)	PUNCT
ma-53	130	5	n	n	PROPN
ma-53	130	6	(	(	PUNCT
ma-53	130	7	t	t	PROPN
ma-53	130	8	)	)	PUNCT
ma-53	131	1	+	+	CCONJ
ma-53	131	2	k	k	PROPN
ma-53	131	3	2	2	NUM
ma-53	131	4	t2n+1η	t2n+1η	NOUN
ma-53	131	5	−	−	PROPN
ma-53	131	6	k	k	PROPN
ma-53	131	7	2	2	NUM
ma-53	131	8	t2n−1η	t2n−1η	INTJ
ma-53	131	9	+	+	NOUN
ma-53	131	10	k1(t	k1(t	X
ma-53	132	1	2n+3	2n+3	NOUN
ma-53	132	2	+	+	CCONJ
ma-53	132	3	t2n+4)η	t2n+4)η	VERB
ma-53	133	1	+	+	NOUN
ma-53	133	2	k1(t	k1(t	X
ma-53	133	3	2n+2	2n+2	NOUN
ma-53	134	1	+	+	CCONJ
ma-53	134	2	t2n+3)η	t2n+3)η	NOUN
ma-53	134	3	+	+	NOUN
ma-53	134	4	k3	k3	ADJ
ma-53	134	5	t	t	PROPN
ma-53	134	6	2n+3η	2n+3η	NUM
ma-53	134	7	−k3t2n+1η	−k3t2n+1η	PUNCT
ma-53	135	1	=	=	X
ma-53	135	2	h	h	NOUN
ma-53	135	3	(	(	PUNCT
ma-53	135	4	1	1	NUM
ma-53	135	5	)	)	PUNCT
ma-53	135	6	n	n	PROPN
ma-53	135	7	(	(	PUNCT
ma-53	135	8	t	t	PROPN
ma-53	135	9	)	)	PUNCT
ma-53	135	10	+	+	CCONJ
ma-53	135	11	g1(t)t	g1(t)t	NOUN
ma-53	135	12	2n−1η	2n−1η	NUM
ma-53	135	13	.	.	PUNCT
ma-53	136	1	(	(	PUNCT
ma-53	136	2	2.13	2.13	NUM
ma-53	136	3	)	)	PUNCT
ma-53	136	4	notice	notice	VERB
ma-53	136	5	that	that	SCONJ
ma-53	136	6	by	by	ADP
ma-53	136	7	the	the	DET
ma-53	136	8	definition	definition	NOUN
ma-53	136	9	of	of	ADP
ma-53	136	10	δ1	δ1	NOUN
ma-53	136	11	h	h	PROPN
ma-53	136	12	(	(	PUNCT
ma-53	136	13	1	1	NUM
ma-53	136	14	)	)	PUNCT
ma-53	136	15	n+1(δ1	n+1(δ1	NOUN
ma-53	136	16	)	)	PUNCT
ma-53	136	17	=	=	SYM
ma-53	136	18	h	h	NOUN
ma-53	136	19	(	(	PUNCT
ma-53	136	20	1	1	NUM
ma-53	136	21	)	)	PUNCT
ma-53	136	22	n	n	CCONJ
ma-53	136	23	(	(	PUNCT
ma-53	136	24	δ1	δ1	NOUN
ma-53	136	25	)	)	PUNCT
ma-53	136	26	.	.	PUNCT
ma-53	137	1	by	by	ADP
ma-53	137	2	(	(	PUNCT
ma-53	137	3	2.11)-(2.13	2.11)-(2.13	NUM
ma-53	137	4	)	)	PUNCT
ma-53	137	5	,	,	PUNCT
ma-53	137	6	estimate	estimate	INTJ
ma-53	137	7	(	(	PUNCT
ma-53	137	8	2.11	2.11	NUM
ma-53	137	9	)	)	PUNCT
ma-53	137	10	shall	shall	AUX
ma-53	137	11	be	be	AUX
ma-53	137	12	true	true	ADJ
ma-53	137	13	if	if	SCONJ
ma-53	137	14	for	for	ADP
ma-53	137	15	4k1	4k1	NUM
ma-53	137	16	<	<	X
ma-53	137	17	k	k	PROPN
ma-53	137	18	h	h	PROPN
ma-53	137	19	(	(	PUNCT
ma-53	137	20	1	1	NUM
ma-53	137	21	)	)	PUNCT
ma-53	137	22	n	n	CCONJ
ma-53	137	23	(	(	PUNCT
ma-53	137	24	δ1	δ1	NOUN
ma-53	137	25	)	)	PUNCT
ma-53	137	26	≤	≤	NOUN
ma-53	137	27	0	0	NUM
ma-53	137	28	.	.	PUNCT
ma-53	138	1	(	(	PUNCT
ma-53	138	2	2.14	2.14	NUM
ma-53	138	3	)	)	PUNCT
ma-53	138	4	let	let	VERB
ma-53	138	5	h(1)∞	h(1)∞	PROPN
ma-53	138	6	(	(	PUNCT
ma-53	138	7	t	t	NOUN
ma-53	138	8	)	)	PUNCT
ma-53	138	9	=	=	PROPN
ma-53	138	10	lim	lim	PROPN
ma-53	138	11	n−→∞	n−→∞	PROPN
ma-53	138	12	h	h	PROPN
ma-53	138	13	(	(	PUNCT
ma-53	138	14	1	1	NUM
ma-53	138	15	)	)	PUNCT
ma-53	138	16	n	n	PROPN
ma-53	138	17	(	(	PUNCT
ma-53	138	18	t	t	PROPN
ma-53	138	19	)	)	PUNCT
ma-53	138	20	.	.	PUNCT
ma-53	139	1	(	(	PUNCT
ma-53	139	2	2.15	2.15	NUM
ma-53	139	3	)	)	PUNCT
ma-53	139	4	but	but	CCONJ
ma-53	139	5	then	then	ADV
ma-53	139	6	h(1)∞	h(1)∞	X
ma-53	139	7	(	(	PUNCT
ma-53	139	8	δ	δ	PROPN
ma-53	139	9	)	)	PUNCT
ma-53	139	10	=	=	SYM
ma-53	139	11	2k1η	2k1η	NUM
ma-53	139	12	1−	1−	NUM
ma-53	139	13	δ	δ	NOUN
ma-53	139	14	−	−	PROPN
ma-53	139	15	1	1	NUM
ma-53	139	16	.	.	PUNCT
ma-53	139	17	(	(	PUNCT
ma-53	139	18	2.16	2.16	NUM
ma-53	139	19	)	)	PUNCT
ma-53	139	20	hence	hence	ADV
ma-53	139	21	,	,	PUNCT
ma-53	139	22	instead	instead	ADV
ma-53	139	23	of	of	ADP
ma-53	139	24	(	(	PUNCT
ma-53	139	25	2.13	2.13	NUM
ma-53	139	26	)	)	PUNCT
ma-53	139	27	we	we	PRON
ma-53	139	28	can	can	AUX
ma-53	139	29	show	show	VERB
ma-53	139	30	h(1)∞	h(1)∞	PROPN
ma-53	139	31	(	(	PUNCT
ma-53	139	32	δ	δ	NOUN
ma-53	139	33	)	)	PUNCT
ma-53	139	34	≤	≤	NOUN
ma-53	139	35	0	0	NUM
ma-53	139	36	,	,	PUNCT
ma-53	139	37	(	(	PUNCT
ma-53	139	38	2.17	2.17	NUM
ma-53	139	39	)	)	PUNCT
ma-53	139	40	which	which	PRON
ma-53	139	41	is	be	AUX
ma-53	139	42	true	true	ADJ
ma-53	139	43	by	by	ADP
ma-53	139	44	(	(	PUNCT
ma-53	139	45	2.4	2.4	NUM
ma-53	139	46	)	)	PUNCT
ma-53	139	47	.	.	PUNCT
ma-53	140	1	if	if	SCONJ
ma-53	140	2	4k1	4k1	NUM
ma-53	140	3	≥	≥	NOUN
ma-53	140	4	k	k	NOUN
ma-53	140	5	then	then	ADV
ma-53	140	6	f∞(t	f∞(t	PROPN
ma-53	140	7	)	)	PUNCT
ma-53	140	8	≥	≥	NOUN
ma-53	140	9	fn(t	fn(t	NUM
ma-53	140	10	)	)	PUNCT
ma-53	140	11	,	,	PUNCT
ma-53	140	12	so	so	ADV
ma-53	140	13	again	again	ADV
ma-53	140	14	f∞(δ	f∞(δ	NOUN
ma-53	140	15	)	)	PUNCT
ma-53	140	16	≤	≤	NOUN
ma-53	140	17	0	0	NUM
ma-53	140	18	holds	hold	NOUN
ma-53	140	19	.	.	PUNCT
ma-53	141	1	similarly	similarly	ADV
ma-53	141	2	,	,	PUNCT
ma-53	141	3	(	(	PUNCT
ma-53	141	4	2.9)holds	2.9)holds	NUM
ma-53	141	5	if	if	SCONJ
ma-53	141	6	k3δ	k3δ	PROPN
ma-53	141	7	2n+1η	2n+1η	NUM
ma-53	141	8	+	+	ADJ
ma-53	141	9	k4δ	k4δ	X
ma-53	141	10	2nη	2nη	ADJ
ma-53	141	11	+	+	CCONJ
ma-53	141	12	δl0	δl0	NOUN
ma-53	141	13	1−	1−	NUM
ma-53	142	1	δ2n+2	δ2n+2	PROPN
ma-53	142	2	1−	1−	NUM
ma-53	142	3	δ	δ	PROPN
ma-53	142	4	η	η	PROPN
ma-53	142	5	−	−	PROPN
ma-53	142	6	δ	δ	PROPN
ma-53	142	7	≤	≤	X
ma-53	142	8	0	0	NUM
ma-53	142	9	(	(	PUNCT
ma-53	142	10	2.18	2.18	NUM
ma-53	142	11	)	)	PUNCT
ma-53	142	12	or	or	CCONJ
ma-53	142	13	h	h	NOUN
ma-53	142	14	(	(	PUNCT
ma-53	142	15	2	2	NUM
ma-53	142	16	)	)	PUNCT
ma-53	142	17	n	n	CCONJ
ma-53	142	18	(	(	PUNCT
ma-53	142	19	δ	δ	NOUN
ma-53	142	20	)	)	PUNCT
ma-53	142	21	≤	≤	NOUN
ma-53	142	22	0	0	NUM
ma-53	142	23	,	,	PUNCT
ma-53	142	24	(	(	PUNCT
ma-53	142	25	2.19	2.19	NUM
ma-53	142	26	)	)	PUNCT
ma-53	142	27	where	where	SCONJ
ma-53	142	28	h	h	NOUN
ma-53	142	29	(	(	PUNCT
ma-53	142	30	2	2	NUM
ma-53	142	31	)	)	PUNCT
ma-53	142	32	n	n	PROPN
ma-53	142	33	(	(	PUNCT
ma-53	142	34	t	t	NOUN
ma-53	142	35	)	)	PUNCT
ma-53	142	36	=	=	PUNCT
ma-53	142	37	k3	k3	VERB
ma-53	142	38	t	t	PROPN
ma-53	142	39	2nη	2nη	NOUN
ma-53	143	1	+	+	PROPN
ma-53	143	2	k4	k4	PROPN
ma-53	143	3	t	t	PROPN
ma-53	143	4	2n−1η	2n−1η	PROPN
ma-53	144	1	+	+	CCONJ
ma-53	144	2	l0(1	l0(1	PROPN
ma-53	144	3	+	+	CCONJ
ma-53	144	4	t	t	NOUN
ma-53	144	5	+	+	X
ma-53	144	6	.	.	PUNCT
ma-53	144	7	.	.	PUNCT
ma-53	145	1	.+	.+	NOUN
ma-53	145	2	t2n+1)η	t2n+1)η	VERB
ma-53	145	3	−	−	NUM
ma-53	146	1	1	1	NUM
ma-53	146	2	.	.	PUNCT
ma-53	146	3	(	(	PUNCT
ma-53	146	4	2.20	2.20	NUM
ma-53	146	5	)	)	PUNCT
ma-53	146	6	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	146	7	eur	eur	NOUN
ma-53	146	8	.	.	PUNCT
ma-53	147	1	j.	j.	PROPN
ma-53	147	2	math	math	PROPN
ma-53	147	3	.	.	PUNCT
ma-53	148	1	anal	anal	PROPN
ma-53	148	2	.	.	PUNCT
ma-53	149	1	10.28924	10.28924	NUM
ma-53	149	2	/	/	SYM
ma-53	149	3	ada	ada	PROPN
ma-53	149	4	/	/	SYM
ma-53	149	5	ma.2.3	ma.2.3	PROPN
ma-53	149	6	6this	6this	NUM
ma-53	149	7	time	time	NOUN
ma-53	149	8	we	we	PRON
ma-53	149	9	have	have	VERB
ma-53	149	10	h	h	NOUN
ma-53	149	11	(	(	PUNCT
ma-53	149	12	2	2	NUM
ma-53	149	13	)	)	PUNCT
ma-53	149	14	n+1(t	n+1(t	NUM
ma-53	149	15	)	)	PUNCT
ma-53	150	1	=	=	PUNCT
ma-53	150	2	k3	k3	VERB
ma-53	150	3	t	t	PROPN
ma-53	150	4	2n+2η	2n+2η	NUM
ma-53	151	1	+	+	PROPN
ma-53	151	2	k4	k4	PROPN
ma-53	151	3	t	t	PROPN
ma-53	151	4	2n+1η	2n+1η	NUM
ma-53	151	5	+	+	CCONJ
ma-53	151	6	l0(1	l0(1	PROPN
ma-53	151	7	+	+	CCONJ
ma-53	151	8	t	t	NOUN
ma-53	151	9	+	+	X
ma-53	151	10	.	.	PUNCT
ma-53	151	11	.	.	PUNCT
ma-53	152	1	.+	.+	NOUN
ma-53	152	2	t2n+3)η	t2n+3)η	NOUN
ma-53	152	3	−1−k3t2nη	−1−k3t2nη	X
ma-53	152	4	−k4t2n−1η	−k4t2n−1η	PROPN
ma-53	152	5	−	−	PROPN
ma-53	152	6	l0(1	l0(1	PROPN
ma-53	152	7	+	+	CCONJ
ma-53	152	8	t	t	NOUN
ma-53	152	9	+	+	X
ma-53	152	10	.	.	PUNCT
ma-53	152	11	.	.	PUNCT
ma-53	153	1	.+	.+	NOUN
ma-53	153	2	t2n+1)η	t2n+1)η	VERB
ma-53	154	1	+	+	CCONJ
ma-53	154	2	1	1	NUM
ma-53	155	1	+	+	NUM
ma-53	155	2	h	h	NOUN
ma-53	155	3	(	(	PUNCT
ma-53	155	4	2	2	NUM
ma-53	155	5	)	)	PUNCT
ma-53	155	6	n	n	PROPN
ma-53	155	7	(	(	PUNCT
ma-53	155	8	t	t	NOUN
ma-53	155	9	)	)	PUNCT
ma-53	155	10	=	=	SYM
ma-53	156	1	h	h	NOUN
ma-53	156	2	(	(	PUNCT
ma-53	156	3	2	2	NUM
ma-53	156	4	)	)	PUNCT
ma-53	156	5	n	n	PROPN
ma-53	156	6	(	(	PUNCT
ma-53	156	7	t	t	PROPN
ma-53	156	8	)	)	PUNCT
ma-53	157	1	+	+	NOUN
ma-53	157	2	k3	k3	PROPN
ma-53	157	3	t	t	PROPN
ma-53	157	4	2n+2η	2n+2η	NUM
ma-53	157	5	−k3t2nη	−k3t2nη	PROPN
ma-53	158	1	+	+	PROPN
ma-53	158	2	k4	k4	PROPN
ma-53	158	3	t	t	PROPN
ma-53	158	4	2n+1	2n+1	PROPN
ma-53	158	5	−k4t2n+1η	−k4t2n+1η	PUNCT
ma-53	159	1	+	+	ADV
ma-53	159	2	l0(t	l0(t	X
ma-53	159	3	2n+2	2n+2	NOUN
ma-53	159	4	+	+	CCONJ
ma-53	159	5	t2n+3)η	t2n+3)η	NOUN
ma-53	159	6	=	=	SYM
ma-53	159	7	h	h	NOUN
ma-53	159	8	(	(	PUNCT
ma-53	159	9	2	2	NUM
ma-53	159	10	)	)	PUNCT
ma-53	159	11	n	n	PROPN
ma-53	159	12	(	(	PUNCT
ma-53	159	13	t	t	PROPN
ma-53	159	14	)	)	PUNCT
ma-53	159	15	+	+	CCONJ
ma-53	160	1	[	[	X
ma-53	160	2	k3	k3	X
ma-53	160	3	t	t	PROPN
ma-53	160	4	3	3	NUM
ma-53	160	5	−k3	−k3	PROPN
ma-53	160	6	t	t	PROPN
ma-53	160	7	+	+	PROPN
ma-53	160	8	k4	k4	PROPN
ma-53	160	9	t	t	NOUN
ma-53	160	10	2	2	NUM
ma-53	160	11	−k4	−k4	NOUN
ma-53	160	12	+	+	CCONJ
ma-53	160	13	l0	l0	PROPN
ma-53	160	14	t	t	NOUN
ma-53	160	15	2	2	NUM
ma-53	160	16	+	+	CCONJ
ma-53	160	17	l−	l−	NOUN
ma-53	160	18	0t4]t2n−1δ	0t4]t2n−1δ	NUM
ma-53	160	19	=	=	SYM
ma-53	160	20	h	h	NOUN
ma-53	160	21	(	(	PUNCT
ma-53	160	22	2	2	NUM
ma-53	160	23	)	)	PUNCT
ma-53	160	24	n	n	PROPN
ma-53	160	25	(	(	PUNCT
ma-53	160	26	t	t	PROPN
ma-53	160	27	)	)	PUNCT
ma-53	160	28	+	+	CCONJ
ma-53	160	29	g2(t)t	g2(t)t	NOUN
ma-53	160	30	2n−1η	2n−1η	NUM
ma-53	160	31	.	.	PUNCT
ma-53	160	32	(	(	PUNCT
ma-53	160	33	2.21	2.21	NUM
ma-53	160	34	)	)	PUNCT
ma-53	160	35	by	by	ADP
ma-53	160	36	the	the	DET
ma-53	160	37	definition	definition	NOUN
ma-53	160	38	of	of	ADP
ma-53	160	39	δ2	δ2	ADJ
ma-53	160	40	h	h	NOUN
ma-53	160	41	(	(	PUNCT
ma-53	160	42	2	2	NUM
ma-53	160	43	)	)	PUNCT
ma-53	160	44	n+1(δ2	n+1(δ2	PROPN
ma-53	160	45	)	)	PUNCT
ma-53	161	1	=	=	SYM
ma-53	161	2	h	h	NOUN
ma-53	161	3	(	(	PUNCT
ma-53	161	4	2	2	NUM
ma-53	161	5	)	)	PUNCT
ma-53	161	6	n	n	CCONJ
ma-53	161	7	(	(	PUNCT
ma-53	161	8	δ).let	δ).let	NOUN
ma-53	161	9	h(2)∞	h(2)∞	PROPN
ma-53	161	10	(	(	PUNCT
ma-53	161	11	t	t	NOUN
ma-53	161	12	)	)	PUNCT
ma-53	161	13	=	=	PUNCT
ma-53	161	14	limn−→∞	limn−→∞	NOUN
ma-53	161	15	h	h	NOUN
ma-53	161	16	(	(	PUNCT
ma-53	161	17	2	2	NUM
ma-53	161	18	)	)	PUNCT
ma-53	161	19	n	n	PROPN
ma-53	161	20	(	(	PUNCT
ma-53	161	21	t	t	PROPN
ma-53	161	22	)	)	PUNCT
ma-53	161	23	.	.	PUNCT
ma-53	162	1	then	then	ADV
ma-53	162	2	,	,	PUNCT
ma-53	162	3	we	we	PRON
ma-53	162	4	get	get	VERB
ma-53	162	5	h(2)∞	h(2)∞	NOUN
ma-53	162	6	(	(	PUNCT
ma-53	162	7	δ	δ	NOUN
ma-53	162	8	)	)	PUNCT
ma-53	162	9	=	=	SYM
ma-53	163	1	l0η	l0η	PROPN
ma-53	163	2	1−	1−	NUM
ma-53	163	3	δ	δ	NOUN
ma-53	163	4	−	−	PROPN
ma-53	164	1	1	1	NUM
ma-53	164	2	.	.	PUNCT
ma-53	165	1	hence	hence	ADV
ma-53	165	2	,	,	PUNCT
ma-53	165	3	instead	instead	ADV
ma-53	165	4	of	of	ADP
ma-53	165	5	(	(	PUNCT
ma-53	165	6	2.19	2.19	NUM
ma-53	165	7	)	)	PUNCT
ma-53	165	8	,	,	PUNCT
ma-53	165	9	we	we	PRON
ma-53	165	10	can	can	AUX
ma-53	165	11	show	show	VERB
ma-53	165	12	h(2)∞	h(2)∞	PROPN
ma-53	165	13	(	(	PUNCT
ma-53	165	14	δ	δ	PROPN
ma-53	165	15	)	)	PUNCT
ma-53	165	16	≤	≤	NOUN
ma-53	165	17	0,which	0,which	PRON
ma-53	165	18	is	be	AUX
ma-53	165	19	true	true	ADJ
ma-53	165	20	by	by	ADP
ma-53	165	21	(	(	PUNCT
ma-53	165	22	2.4	2.4	NUM
ma-53	165	23	)	)	PUNCT
ma-53	165	24	.	.	PUNCT
ma-53	166	1	the	the	DET
ma-53	166	2	induction	induction	NOUN
ma-53	166	3	for	for	ADP
ma-53	166	4	(	(	PUNCT
ma-53	166	5	2.8)-(2.10	2.8)-(2.10	NUM
ma-53	166	6	)	)	PUNCT
ma-53	166	7	is	be	AUX
ma-53	166	8	completed	complete	VERB
ma-53	166	9	.	.	PUNCT
ma-53	167	1	therefore	therefore	ADV
ma-53	167	2	,	,	PUNCT
ma-53	167	3	sequences	sequence	NOUN
ma-53	167	4	{	{	PUNCT
ma-53	167	5	tn	tn	NOUN
ma-53	167	6	}	}	PUNCT
ma-53	167	7	,	,	PUNCT
ma-53	167	8	{	{	PUNCT
ma-53	167	9	sn}are	sn}are	PROPN
ma-53	167	10	nondecreasing	nondecreasing	PROPN
ma-53	167	11	,	,	PUNCT
ma-53	167	12	bounded	bound	VERB
ma-53	167	13	from	from	ADP
ma-53	167	14	above	above	ADV
ma-53	167	15	by	by	ADP
ma-53	167	16	s∗∗	s∗∗	ADJ
ma-53	167	17	and	and	CCONJ
ma-53	167	18	as	as	ADP
ma-53	167	19	such	such	ADJ
ma-53	167	20	they	they	PRON
ma-53	167	21	converge	converge	VERB
ma-53	167	22	to	to	ADP
ma-53	167	23	s∗.	s∗.	PROPN
ma-53	167	24	�	�	PROPN
ma-53	167	25	the	the	DET
ma-53	167	26	semi	semi	ADJ
ma-53	167	27	-	-	ADJ
ma-53	167	28	local	local	ADJ
ma-53	167	29	convergence	convergence	NOUN
ma-53	167	30	analysis	analysis	NOUN
ma-53	167	31	shall	shall	AUX
ma-53	167	32	be	be	AUX
ma-53	167	33	based	base	VERB
ma-53	167	34	on	on	ADP
ma-53	167	35	conditions	condition	NOUN
ma-53	167	36	(	(	PUNCT
ma-53	167	37	a).suppose:(a1	a).suppose:(a1	NUM
ma-53	167	38	)	)	PUNCT
ma-53	167	39	there	there	PRON
ma-53	167	40	exists	exist	VERB
ma-53	167	41	x0	x0	PROPN
ma-53	167	42	∈	∈	PROPN
ma-53	167	43	ω	ω	PROPN
ma-53	167	44	,	,	PUNCT
ma-53	167	45	η	η	PROPN
ma-53	167	46	≥	≥	X
ma-53	167	47	0	0	NUM
ma-53	167	48	such	such	ADJ
ma-53	167	49	that	that	SCONJ
ma-53	167	50	f	f	PROPN
ma-53	167	51	′(x0)−1	′(x0)−1	PROPN
ma-53	167	52	∈	∈	PROPN
ma-53	167	53	l(e1	l(e1	NOUN
ma-53	167	54	,	,	PUNCT
ma-53	167	55	e	e	NOUN
ma-53	167	56	)	)	PUNCT
ma-53	167	57	and	and	CCONJ
ma-53	167	58	‖f	‖f	ADJ
ma-53	167	59	′(x0)−1f	′(x0)−1f	NOUN
ma-53	167	60	(	(	PUNCT
ma-53	167	61	x0)‖	x0)‖	PROPN
ma-53	167	62	≤	≤	PROPN
ma-53	167	63	η	η	PROPN
ma-53	167	64	.	.	PROPN
ma-53	167	65	(	(	PUNCT
ma-53	167	66	a2	a2	PROPN
ma-53	167	67	)	)	PUNCT
ma-53	167	68	for	for	ADP
ma-53	167	69	each	each	DET
ma-53	167	70	x	x	SYM
ma-53	167	71	∈	∈	PROPN
ma-53	167	72	ω	ω	NOUN
ma-53	167	73	‖f	‖f	PRON
ma-53	167	74	′(x0)−1(f	′(x0)−1(f	PROPN
ma-53	167	75	′(x)−	′(x)−	PROPN
ma-53	167	76	f	f	PROPN
ma-53	167	77	′(x0))‖	′(x0))‖	NUM
ma-53	167	78	≤	≤	NUM
ma-53	168	1	l0‖x	l0‖x	NUM
ma-53	168	2	−	−	PROPN
ma-53	168	3	x0‖.	x0‖.	PROPN
ma-53	168	4	set	set	VERB
ma-53	168	5	ω0	ω0	ADV
ma-53	168	6	=	=	SYM
ma-53	168	7	u[x0	u[x0	NOUN
ma-53	168	8	,	,	PUNCT
ma-53	168	9	1	1	NUM
ma-53	168	10	l0	l0	NOUN
ma-53	168	11	]	]	PUNCT
ma-53	168	12	∩ω.(a3	∩ω.(a3	PROPN
ma-53	168	13	)	)	PUNCT
ma-53	168	14	for	for	ADP
ma-53	168	15	each	each	DET
ma-53	168	16	x	x	NOUN
ma-53	168	17	,	,	PUNCT
ma-53	168	18	y	y	PROPN
ma-53	168	19	∈	∈	PROPN
ma-53	168	20	ω0	ω0	NOUN
ma-53	168	21	‖f	‖f	PUNCT
ma-53	168	22	′(x0)−1(f	′(x0)−1(f	PROPN
ma-53	168	23	′(y)−	′(y)−	VERB
ma-53	168	24	f	f	PROPN
ma-53	168	25	′(x))‖	′(x))‖	PROPN
ma-53	168	26	≤	≤	NOUN
ma-53	168	27	k‖y	k‖y	VERB
ma-53	168	28	−	−	PROPN
ma-53	168	29	x‖	x‖	NOUN
ma-53	168	30	,	,	PUNCT
ma-53	168	31	‖f	‖f	ADP
ma-53	168	32	′(x0)−1([y	′(x0)−1([y	PUNCT
ma-53	168	33	,	,	PUNCT
ma-53	168	34	x	x	PROPN
ma-53	168	35	;	;	PUNCT
ma-53	168	36	f	f	X
ma-53	169	1	]	]	X
ma-53	169	2	−	−	PROPN
ma-53	169	3	f	f	PROPN
ma-53	169	4	′(x0))‖	′(x0))‖	NUM
ma-53	169	5	≤	≤	NUM
ma-53	169	6	k1(‖y	k1(‖y	NOUN
ma-53	169	7	−	−	PROPN
ma-53	169	8	x0‖+	x0‖+	PUNCT
ma-53	169	9	‖x	‖x	NOUN
ma-53	170	1	−	−	PROPN
ma-53	170	2	x0‖	x0‖	PROPN
ma-53	170	3	)	)	PUNCT
ma-53	170	4	,	,	PUNCT
ma-53	170	5	‖f	‖f	ADP
ma-53	170	6	′(x0)−1([z	′(x0)−1([z	NOUN
ma-53	170	7	,	,	PUNCT
ma-53	170	8	y	y	PROPN
ma-53	170	9	;	;	PUNCT
ma-53	170	10	f	f	X
ma-53	171	1	]	]	X
ma-53	171	2	−	−	PROPN
ma-53	172	1	[	[	X
ma-53	172	2	y	y	PROPN
ma-53	172	3	,	,	PUNCT
ma-53	172	4	x	x	PROPN
ma-53	172	5	;	;	PUNCT
ma-53	172	6	f	f	PROPN
ma-53	172	7	]	]	X
ma-53	172	8	)	)	PUNCT
ma-53	172	9	‖	‖	PROPN
ma-53	172	10	≤	≤	PROPN
ma-53	173	1	k2(‖z	k2(‖z	NOUN
ma-53	173	2	−	−	PROPN
ma-53	174	1	y‖+	y‖+	PROPN
ma-53	174	2	‖y	‖y	PUNCT
ma-53	175	1	−	−	PROPN
ma-53	175	2	x‖)and	x‖)and	PROPN
ma-53	175	3	‖f	‖f	PRON
ma-53	175	4	′(x0)−1([z	′(x0)−1([z	PROPN
ma-53	175	5	,	,	PUNCT
ma-53	175	6	y	y	PROPN
ma-53	175	7	;	;	PUNCT
ma-53	175	8	f	f	X
ma-53	175	9	]	]	X
ma-53	175	10	−	−	X
ma-53	175	11	f	f	PROPN
ma-53	175	12	′(y))‖	′(y))‖	PROPN
ma-53	175	13	≤	≤	PUNCT
ma-53	176	1	k3‖z	k3‖z	PROPN
ma-53	176	2	−	−	PROPN
ma-53	176	3	y‖.(a4	y‖.(a4	PROPN
ma-53	176	4	)	)	PUNCT
ma-53	176	5	u[x0	u[x0	NOUN
ma-53	176	6	,	,	PUNCT
ma-53	176	7	s∗	s∗	PROPN
ma-53	176	8	]	]	PUNCT
ma-53	177	1	⊂	⊂	PROPN
ma-53	177	2	ω	ω	PROPN
ma-53	177	3	and(a5	and(a5	ADJ
ma-53	177	4	)	)	PUNCT
ma-53	177	5	conditions	condition	NOUN
ma-53	177	6	of	of	ADP
ma-53	177	7	lemma	lemma	PROPN
ma-53	177	8	2.1	2.1	NUM
ma-53	177	9	hold	hold	NOUN
ma-53	177	10	.	.	PUNCT
ma-53	178	1	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	NOUN
ma-53	178	2	eur	eur	PROPN
ma-53	178	3	.	.	PUNCT
ma-53	179	1	j.	j.	PROPN
ma-53	179	2	math	math	PROPN
ma-53	179	3	.	.	PUNCT
ma-53	180	1	anal	anal	PROPN
ma-53	180	2	.	.	PUNCT
ma-53	181	1	10.28924	10.28924	NUM
ma-53	181	2	/	/	SYM
ma-53	181	3	ada	ada	PROPN
ma-53	181	4	/	/	SYM
ma-53	181	5	ma.2.3	ma.2.3	PROPN
ma-53	181	6	7next	7next	NUM
ma-53	181	7	,	,	PUNCT
ma-53	181	8	we	we	PRON
ma-53	181	9	present	present	VERB
ma-53	181	10	the	the	DET
ma-53	181	11	semi	semi	ADJ
ma-53	181	12	-	-	ADJ
ma-53	181	13	local	local	ADJ
ma-53	181	14	convergence	convergence	NOUN
ma-53	181	15	of	of	ADP
ma-53	181	16	method	method	NOUN
ma-53	181	17	(	(	PUNCT
ma-53	181	18	1.2	1.2	NUM
ma-53	181	19	)	)	PUNCT
ma-53	181	20	.	.	PUNCT
ma-53	182	1	theorem	theorem	VERB
ma-53	182	2	2.2	2.2	NUM
ma-53	182	3	.	.	PUNCT
ma-53	183	1	suppose	suppose	VERB
ma-53	183	2	that	that	SCONJ
ma-53	183	3	conditions	condition	NOUN
ma-53	183	4	(	(	PUNCT
ma-53	183	5	a	a	X
ma-53	183	6	)	)	PUNCT
ma-53	183	7	hold	hold	NOUN
ma-53	183	8	.	.	PUNCT
ma-53	184	1	then	then	ADV
ma-53	184	2	,	,	PUNCT
ma-53	184	3	sequences	sequence	NOUN
ma-53	184	4	{	{	PUNCT
ma-53	184	5	yn	yn	PROPN
ma-53	184	6	}	}	PUNCT
ma-53	184	7	,	,	PUNCT
ma-53	184	8	{	{	PUNCT
ma-53	184	9	xn	xn	X
ma-53	184	10	}	}	PUNCT
ma-53	184	11	generated	generate	VERB
ma-53	184	12	by	by	ADP
ma-53	184	13	method	method	NOUN
ma-53	184	14	(	(	PUNCT
ma-53	184	15	1.2	1.2	NUM
ma-53	184	16	)	)	PUNCT
ma-53	184	17	are	be	AUX
ma-53	184	18	well	well	ADV
ma-53	184	19	defined	define	VERB
ma-53	184	20	in	in	ADP
ma-53	184	21	u[x0	u[x0	NOUN
ma-53	184	22	,	,	PUNCT
ma-53	184	23	s∗	s∗	PROPN
ma-53	184	24	]	]	PUNCT
ma-53	184	25	,	,	PUNCT
ma-53	184	26	remain	remain	VERB
ma-53	184	27	in	in	ADP
ma-53	184	28	u[x0	u[x0	NOUN
ma-53	184	29	,	,	PUNCT
ma-53	184	30	s∗	s∗	PROPN
ma-53	184	31	]	]	PUNCT
ma-53	184	32	for	for	ADP
ma-53	184	33	each	each	DET
ma-53	184	34	n	n	NOUN
ma-53	184	35	=	=	SYM
ma-53	184	36	0	0	NUM
ma-53	184	37	,	,	PUNCT
ma-53	184	38	1	1	NUM
ma-53	184	39	,	,	PUNCT
ma-53	184	40	2	2	NUM
ma-53	184	41	,	,	PUNCT
ma-53	184	42	.	.	PUNCT
ma-53	184	43	.	.	PUNCT
ma-53	184	44	.	.	PUNCT
ma-53	185	1	and	and	CCONJ
ma-53	185	2	converge	converge	VERB
ma-53	185	3	to	to	ADP
ma-53	185	4	a	a	DET
ma-53	185	5	solution	solution	NOUN
ma-53	185	6	x∗	x∗	PROPN
ma-53	185	7	∈	∈	PROPN
ma-53	185	8	u[x0	u[x0	NOUN
ma-53	185	9	,	,	PUNCT
ma-53	185	10	s∗	s∗	PROPN
ma-53	185	11	]	]	PUNCT
ma-53	185	12	of	of	ADP
ma-53	185	13	equation	equation	NOUN
ma-53	185	14	f	f	X
ma-53	185	15	(	(	PUNCT
ma-53	185	16	x	x	X
ma-53	185	17	)	)	PUNCT
ma-53	185	18	=	=	SYM
ma-53	186	1	0	0	X
ma-53	186	2	.	.	PUNCT
ma-53	187	1	moreover	moreover	ADV
ma-53	187	2	,	,	PUNCT
ma-53	187	3	the	the	DET
ma-53	187	4	following	follow	VERB
ma-53	187	5	assertion	assertion	NOUN
ma-53	187	6	holds	hold	VERB
ma-53	187	7	‖xn	‖xn	PROPN
ma-53	187	8	−	−	PROPN
ma-53	187	9	x∗‖	x∗‖	PROPN
ma-53	187	10	≤	≤	PROPN
ma-53	187	11	s∗	s∗	VERB
ma-53	187	12	−	−	PROPN
ma-53	187	13	tn	tn	PROPN
ma-53	187	14	.	.	PUNCT
ma-53	188	1	(	(	PUNCT
ma-53	188	2	2.22	2.22	NUM
ma-53	188	3	)	)	PUNCT
ma-53	188	4	proof	proof	NOUN
ma-53	188	5	.	.	PUNCT
ma-53	189	1	mathematical	mathematical	ADJ
ma-53	189	2	induction	induction	NOUN
ma-53	189	3	on	on	ADP
ma-53	189	4	m	m	PROPN
ma-53	189	5	shall	shall	AUX
ma-53	189	6	be	be	AUX
ma-53	189	7	used	use	VERB
ma-53	189	8	to	to	PART
ma-53	189	9	show(im	show(im	VERB
ma-53	189	10	)	)	PUNCT
ma-53	189	11	‖ym	‖ym	NUM
ma-53	189	12	−	−	PROPN
ma-53	189	13	xm‖	xm‖	PROPN
ma-53	189	14	≤	≤	PROPN
ma-53	190	1	sm	sm	VERB
ma-53	190	2	−	−	PROPN
ma-53	190	3	tmand(iim	tmand(iim	NOUN
ma-53	190	4	)	)	PUNCT
ma-53	190	5	‖xm+1	‖xm+1	VERB
ma-53	191	1	−	−	PROPN
ma-53	191	2	ym‖	ym‖	PROPN
ma-53	191	3	≤	≤	NUM
ma-53	191	4	tm+1	tm+1	PROPN
ma-53	191	5	−	−	PROPN
ma-53	191	6	sm.by	sm.by	PROPN
ma-53	191	7	the	the	DET
ma-53	191	8	first	first	ADJ
ma-53	191	9	substep	substep	NOUN
ma-53	191	10	of	of	ADP
ma-53	191	11	method	method	NOUN
ma-53	191	12	(	(	PUNCT
ma-53	191	13	1.2	1.2	NUM
ma-53	191	14	)	)	PUNCT
ma-53	191	15	we	we	PRON
ma-53	191	16	have	have	AUX
ma-53	191	17	‖y0	‖y0	VERB
ma-53	192	1	−	−	PROPN
ma-53	192	2	x0‖	x0‖	PROPN
ma-53	192	3	=	=	PUNCT
ma-53	192	4	‖f	‖f	DET
ma-53	192	5	′(x0)−1f	′(x0)−1f	NOUN
ma-53	192	6	(	(	PUNCT
ma-53	192	7	x0)‖	x0)‖	PROPN
ma-53	192	8	≤	≤	NUM
ma-53	192	9	η	η	PROPN
ma-53	192	10	=	=	PROPN
ma-53	192	11	s0	s0	PROPN
ma-53	192	12	−	−	PROPN
ma-53	192	13	t0	t0	PROPN
ma-53	192	14	=	=	PROPN
ma-53	192	15	s0	s0	PROPN
ma-53	192	16	≤	≤	NUM
ma-53	192	17	s∗.	s∗.	ADJ
ma-53	192	18	so	so	ADV
ma-53	192	19	(	(	PUNCT
ma-53	192	20	i0	i0	PROPN
ma-53	192	21	)	)	PUNCT
ma-53	192	22	holds	hold	VERB
ma-53	192	23	and	and	CCONJ
ma-53	192	24	y0	y0	PROPN
ma-53	192	25	∈	∈	NOUN
ma-53	192	26	u[x0	u[x0	NOUN
ma-53	192	27	,	,	PUNCT
ma-53	192	28	s∗	s∗	PROPN
ma-53	192	29	]	]	PUNCT
ma-53	192	30	.	.	PUNCT
ma-53	193	1	by	by	ADP
ma-53	193	2	the	the	DET
ma-53	193	3	first	first	ADJ
ma-53	193	4	substep	substep	NOUN
ma-53	193	5	of	of	ADP
ma-53	193	6	method	method	NOUN
ma-53	193	7	(	(	PUNCT
ma-53	193	8	1.2	1.2	NUM
ma-53	193	9	)	)	PUNCT
ma-53	193	10	we	we	PRON
ma-53	193	11	can	can	AUX
ma-53	193	12	write	write	VERB
ma-53	193	13	f	f	PROPN
ma-53	193	14	(	(	PUNCT
ma-53	193	15	y0	y0	NOUN
ma-53	193	16	)	)	PUNCT
ma-53	194	1	=	=	SYM
ma-53	194	2	f	f	X
ma-53	194	3	(	(	PUNCT
ma-53	194	4	y0)−	y0)−	PROPN
ma-53	194	5	f	f	PROPN
ma-53	194	6	(	(	PUNCT
ma-53	194	7	x0)−	x0)−	PROPN
ma-53	194	8	f	f	PROPN
ma-53	194	9	′(x0)(y0	′(x0)(y0	NOUN
ma-53	194	10	−	−	PROPN
ma-53	194	11	x0	x0	PROPN
ma-53	194	12	)	)	PUNCT
ma-53	194	13	.	.	PUNCT
ma-53	195	1	(	(	PUNCT
ma-53	195	2	2.23	2.23	NUM
ma-53	195	3	)	)	PUNCT
ma-53	195	4	using	use	VERB
ma-53	195	5	(	(	PUNCT
ma-53	195	6	a2	a2	PROPN
ma-53	195	7	)	)	PUNCT
ma-53	195	8	and	and	CCONJ
ma-53	195	9	(	(	PUNCT
ma-53	195	10	2.23	2.23	NUM
ma-53	195	11	)	)	PUNCT
ma-53	195	12	,	,	PUNCT
ma-53	195	13	we	we	PRON
ma-53	195	14	have	have	VERB
ma-53	195	15	‖f	‖f	ADJ
ma-53	195	16	′(x0)−1f	′(x0)−1f	NOUN
ma-53	195	17	(	(	PUNCT
ma-53	195	18	y0)‖	y0)‖	PROPN
ma-53	195	19	≤	≤	PROPN
ma-53	195	20	k0	k0	PROPN
ma-53	195	21	2	2	NUM
ma-53	195	22	‖y0	‖y0	PROPN
ma-53	195	23	−	−	PROPN
ma-53	195	24	x0‖2	x0‖2	PROPN
ma-53	195	25	≤	≤	PROPN
ma-53	195	26	k0	k0	PROPN
ma-53	195	27	2	2	NUM
ma-53	195	28	(	(	PUNCT
ma-53	195	29	s0	s0	PROPN
ma-53	195	30	−	−	PROPN
ma-53	195	31	t0)2	t0)2	NUM
ma-53	195	32	.	.	PUNCT
ma-53	196	1	(	(	PUNCT
ma-53	196	2	2.24	2.24	NUM
ma-53	196	3	)	)	PUNCT
ma-53	196	4	we	we	PRON
ma-53	196	5	need	need	VERB
ma-53	196	6	to	to	PART
ma-53	196	7	show	show	VERB
ma-53	196	8	the	the	DET
ma-53	196	9	invertability	invertability	NOUN
ma-53	196	10	of	of	ADP
ma-53	196	11	linear	linear	ADJ
ma-53	196	12	operator	operator	NOUN
ma-53	196	13	a.	a.	NOUN
ma-53	196	14	we	we	PRON
ma-53	196	15	have	have	VERB
ma-53	196	16	by	by	ADP
ma-53	196	17	(	(	PUNCT
ma-53	196	18	a3	a3	NOUN
ma-53	196	19	)	)	PUNCT
ma-53	196	20	‖f	‖f	ADP
ma-53	196	21	′(x0)−1(a0	′(x0)−1(a0	VERB
ma-53	197	1	−	−	ADP
ma-53	197	2	f	f	PROPN
ma-53	197	3	′(x0))‖	′(x0))‖	NUM
ma-53	197	4	≤	≤	NUM
ma-53	197	5	2‖f	2‖f	NUM
ma-53	197	6	′(x0)−1([y0	′(x0)−1([y0	NOUN
ma-53	197	7	,	,	PUNCT
ma-53	197	8	x0;f	x0;f	PROPN
ma-53	197	9	]	]	X
ma-53	197	10	−	−	PROPN
ma-53	197	11	f	f	PROPN
ma-53	197	12	′(x0))‖	′(x0))‖	NUM
ma-53	197	13	≤	≤	NUM
ma-53	197	14	2k1(‖y0	2k1(‖y0	NUM
ma-53	197	15	−	−	PROPN
ma-53	197	16	x0‖+	x0‖+	SYM
ma-53	198	1	‖x0	‖x0	ADJ
ma-53	198	2	−	−	PROPN
ma-53	198	3	x0‖	x0‖	PROPN
ma-53	198	4	)	)	PUNCT
ma-53	198	5	≤	≤	NUM
ma-53	198	6	2k1(s0	2k1(s0	NUM
ma-53	199	1	+	+	NUM
ma-53	199	2	t0	t0	NOUN
ma-53	199	3	)	)	PUNCT
ma-53	200	1	<	<	X
ma-53	200	2	1	1	NUM
ma-53	200	3	,	,	PUNCT
ma-53	200	4	so	so	ADV
ma-53	200	5	‖a−10	‖a−10	ADP
ma-53	200	6	f	f	PROPN
ma-53	201	1	′(x0)‖	′(x0)‖	NOUN
ma-53	201	2	≤	≤	NUM
ma-53	201	3	1	1	NUM
ma-53	201	4	1−	1−	NUM
ma-53	201	5	2k1(s0	2k1(s0	NUM
ma-53	201	6	+	+	NUM
ma-53	201	7	t0	t0	PROPN
ma-53	201	8	)	)	PUNCT
ma-53	201	9	,	,	PUNCT
ma-53	201	10	(	(	PUNCT
ma-53	201	11	2.25	2.25	NUM
ma-53	201	12	)	)	PUNCT
ma-53	201	13	by	by	ADP
ma-53	201	14	the	the	DET
ma-53	201	15	banach	banach	ADV
ma-53	201	16	lemma	lemma	PROPN
ma-53	201	17	on	on	ADP
ma-53	201	18	linear	linear	ADJ
ma-53	201	19	invertible	invertible	ADJ
ma-53	201	20	operators	operator	NOUN
ma-53	201	21	[	[	X
ma-53	201	22	19	19	NUM
ma-53	201	23	,	,	PUNCT
ma-53	201	24	25	25	NUM
ma-53	201	25	]	]	PUNCT
ma-53	201	26	.	.	PUNCT
ma-53	202	1	then	then	ADV
ma-53	202	2	,	,	PUNCT
ma-53	202	3	iterate	iterate	NOUN
ma-53	202	4	x1	x1	PRON
ma-53	202	5	exists	exist	VERB
ma-53	202	6	by	by	ADP
ma-53	202	7	the	the	DET
ma-53	202	8	secondsubstep	secondsubstep	NOUN
ma-53	202	9	of	of	ADP
ma-53	202	10	method	method	NOUN
ma-53	202	11	(	(	PUNCT
ma-53	202	12	1.2	1.2	NUM
ma-53	202	13	)	)	PUNCT
ma-53	202	14	,	,	PUNCT
ma-53	202	15	and	and	CCONJ
ma-53	202	16	we	we	PRON
ma-53	202	17	can	can	AUX
ma-53	202	18	write	write	VERB
ma-53	202	19	x1	x1	PRON
ma-53	203	1	−	−	PUNCT
ma-53	203	2	y0	y0	NOUN
ma-53	203	3	=	=	SYM
ma-53	203	4	(	(	PUNCT
ma-53	203	5	a−10	a−10	PROPN
ma-53	203	6	f	f	PROPN
ma-53	203	7	′(x0))(f	′(x0))(f	PROPN
ma-53	203	8	′(x0	′(x0	NOUN
ma-53	203	9	)	)	PUNCT
ma-53	203	10	−1f	−1f	PROPN
ma-53	203	11	(	(	PUNCT
ma-53	203	12	y0	y0	NOUN
ma-53	203	13	)	)	PUNCT
ma-53	203	14	)	)	PUNCT
ma-53	203	15	.	.	PUNCT
ma-53	204	1	(	(	PUNCT
ma-53	204	2	2.26	2.26	NUM
ma-53	204	3	)	)	PUNCT
ma-53	204	4	by	by	ADP
ma-53	204	5	(	(	PUNCT
ma-53	204	6	2.24)-(2.26	2.24)-(2.26	NUM
ma-53	204	7	)	)	PUNCT
ma-53	204	8	,	,	PUNCT
ma-53	204	9	we	we	PRON
ma-53	204	10	get	get	VERB
ma-53	204	11	‖x1	‖x1	NOUN
ma-53	204	12	−	−	PROPN
ma-53	204	13	y0‖	y0‖	NOUN
ma-53	204	14	≤	≤	PROPN
ma-53	204	15	‖a−10	‖a−10	ADP
ma-53	204	16	f	f	PROPN
ma-53	204	17	′(x0)‖‖f	′(x0)‖‖f	X
ma-53	204	18	′(x0)−1f	′(x0)−1f	NOUN
ma-53	204	19	(	(	PUNCT
ma-53	204	20	y0)‖	y0)‖	PROPN
ma-53	204	21	≤	≤	PROPN
ma-53	204	22	k0(s0	k0(s0	ADV
ma-53	204	23	−	−	PROPN
ma-53	204	24	t0)2	t0)2	ADP
ma-53	204	25	2(1−	2(1−	NUM
ma-53	204	26	2k1(s0	2k1(s0	NUM
ma-53	204	27	+	+	NUM
ma-53	204	28	t0	t0	NOUN
ma-53	204	29	)	)	PUNCT
ma-53	204	30	)	)	PUNCT
ma-53	205	1	=	=	PUNCT
ma-53	205	2	t1	t1	NOUN
ma-53	205	3	−	−	PROPN
ma-53	205	4	s0	s0	PROPN
ma-53	205	5	,	,	PUNCT
ma-53	205	6	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	205	7	eur	eur	NOUN
ma-53	205	8	.	.	PUNCT
ma-53	206	1	j.	j.	PROPN
ma-53	206	2	math	math	PROPN
ma-53	206	3	.	.	PUNCT
ma-53	207	1	anal	anal	PROPN
ma-53	207	2	.	.	PUNCT
ma-53	208	1	10.28924	10.28924	NUM
ma-53	208	2	/	/	SYM
ma-53	208	3	ada	ada	PROPN
ma-53	208	4	/	/	SYM
ma-53	208	5	ma.2.3	ma.2.3	PROPN
ma-53	208	6	8showing	8showing	NUM
ma-53	208	7	(	(	PUNCT
ma-53	208	8	ii0	ii0	NOUN
ma-53	208	9	)	)	PUNCT
ma-53	208	10	.	.	PUNCT
ma-53	209	1	moreover	moreover	ADV
ma-53	209	2	,	,	PUNCT
ma-53	209	3	we	we	PRON
ma-53	209	4	have	have	VERB
ma-53	209	5	‖x1	‖x1	NOUN
ma-53	209	6	−	−	PROPN
ma-53	209	7	x0‖	x0‖	PROPN
ma-53	209	8	≤	≤	PROPN
ma-53	210	1	‖x1	‖x1	NOUN
ma-53	210	2	−	−	PROPN
ma-53	210	3	y0‖+	y0‖+	PUNCT
ma-53	211	1	‖y0	‖y0	NOUN
ma-53	212	1	−	−	PROPN
ma-53	212	2	x0‖	x0‖	PROPN
ma-53	212	3	≤	≤	PROPN
ma-53	212	4	t1	t1	NOUN
ma-53	212	5	−	−	PROPN
ma-53	212	6	s0	s0	PROPN
ma-53	212	7	+	+	CCONJ
ma-53	212	8	s0	s0	PROPN
ma-53	212	9	−	−	PROPN
ma-53	212	10	t0	t0	PROPN
ma-53	212	11	=	=	PUNCT
ma-53	213	1	t1	t1	PROPN
ma-53	213	2	≤	≤	NUM
ma-53	213	3	s∗	s∗	PROPN
ma-53	213	4	,	,	PUNCT
ma-53	213	5	so	so	ADV
ma-53	213	6	x1	x1	PROPN
ma-53	213	7	∈	∈	PROPN
ma-53	213	8	u[x0	u[x0	NOUN
ma-53	213	9	,	,	PUNCT
ma-53	213	10	s∗	s∗	PROPN
ma-53	213	11	]	]	PUNCT
ma-53	213	12	.	.	PUNCT
ma-53	213	13	suppose	suppose	VERB
ma-53	213	14	that	that	SCONJ
ma-53	213	15	(	(	PUNCT
ma-53	213	16	i	i	NOUN
ma-53	213	17	m	m	PROPN
ma-53	213	18	)	)	PUNCT
ma-53	213	19	and	and	CCONJ
ma-53	213	20	(	(	PUNCT
ma-53	213	21	iim	iim	NOUN
ma-53	213	22	)	)	PUNCT
ma-53	213	23	hold	hold	NOUN
ma-53	213	24	,	,	PUNCT
ma-53	213	25	ym	ym	PROPN
ma-53	213	26	,	,	PUNCT
ma-53	213	27	xm+1	xm+1	PROPN
ma-53	213	28	∈	∈	PROPN
ma-53	213	29	u[x0	u[x0	NOUN
ma-53	213	30	,	,	PUNCT
ma-53	213	31	s∗	s∗	PROPN
ma-53	213	32	]	]	PUNCT
ma-53	213	33	and	and	CCONJ
ma-53	213	34	f	f	PROPN
ma-53	213	35	′(xm)−1	′(xm)−1	NOUN
ma-53	213	36	,	,	PUNCT
ma-53	213	37	a−1	a−1	PROPN
ma-53	213	38	m	m	PROPN
ma-53	213	39	existfor	existfor	ADP
ma-53	213	40	each	each	PRON
ma-53	213	41	m	m	NOUN
ma-53	213	42	=	=	NOUN
ma-53	213	43	1	1	NUM
ma-53	213	44	,	,	PUNCT
ma-53	213	45	2	2	NUM
ma-53	213	46	,	,	PUNCT
ma-53	213	47	.	.	PUNCT
ma-53	213	48	.	.	PUNCT
ma-53	214	1	.	.	PUNCT
ma-53	215	1	,	,	PUNCT
ma-53	215	2	n.	n.	INTJ
ma-53	215	3	we	we	PRON
ma-53	215	4	shall	shall	AUX
ma-53	215	5	prove	prove	VERB
ma-53	215	6	they	they	PRON
ma-53	215	7	hold	hold	VERB
ma-53	215	8	for	for	ADP
ma-53	215	9	m	m	PROPN
ma-53	215	10	=	=	SYM
ma-53	215	11	n	n	PROPN
ma-53	215	12	+	+	NOUN
ma-53	215	13	1	1	NUM
ma-53	215	14	.	.	PUNCT
ma-53	215	15	using	use	VERB
ma-53	215	16	the	the	DET
ma-53	215	17	second	second	ADJ
ma-53	215	18	substep	substep	NOUN
ma-53	215	19	ofmethod	ofmethod	NOUN
ma-53	215	20	(	(	PUNCT
ma-53	215	21	1.2	1.2	NUM
ma-53	215	22	)	)	PUNCT
ma-53	215	23	,	,	PUNCT
ma-53	215	24	we	we	PRON
ma-53	215	25	get	get	VERB
ma-53	215	26	‖f	‖f	PRON
ma-53	215	27	(	(	PUNCT
ma-53	216	1	x0	x0	PROPN
ma-53	216	2	)	)	PUNCT
ma-53	216	3	−1f	−1f	PROPN
ma-53	216	4	(	(	PUNCT
ma-53	216	5	xn+1)‖	xn+1)‖	PROPN
ma-53	216	6	=	=	PROPN
ma-53	216	7	‖f	‖f	PROPN
ma-53	216	8	′(x0)−1(f	′(x0)−1(f	PROPN
ma-53	216	9	(	(	PUNCT
ma-53	216	10	xn+1)−	xn+1)−	PROPN
ma-53	216	11	f	f	PROPN
ma-53	216	12	(	(	PUNCT
ma-53	216	13	yn))−	yn))−	PROPN
ma-53	216	14	an(xn+1	an(xn+1	PROPN
ma-53	216	15	−	−	PROPN
ma-53	216	16	yn))‖	yn))‖	PROPN
ma-53	216	17	=	=	SYM
ma-53	216	18	‖f	‖f	PUNCT
ma-53	216	19	′(x0)−1([xn+1	′(x0)−1([xn+1	NOUN
ma-53	216	20	,	,	PUNCT
ma-53	216	21	yn;f	yn;f	NOUN
ma-53	216	22	]	]	PUNCT
ma-53	216	23	−	−	PROPN
ma-53	216	24	an)(xn+1	an)(xn+1	CCONJ
ma-53	216	25	−	−	PUNCT
ma-53	216	26	yn)‖	yn)‖	NOUN
ma-53	216	27	≤	≤	PROPN
ma-53	216	28	f	f	PROPN
ma-53	216	29	′(x0	′(x0	NOUN
ma-53	216	30	)	)	PUNCT
ma-53	216	31	−1([xn+1	−1([xn+1	PUNCT
ma-53	216	32	,	,	PUNCT
ma-53	216	33	yn;f	yn;f	NOUN
ma-53	216	34	]	]	PUNCT
ma-53	216	35	−	−	PROPN
ma-53	217	1	[	[	X
ma-53	217	2	yn	yn	X
ma-53	217	3	,	,	PUNCT
ma-53	217	4	xn;f	xn;f	PUNCT
ma-53	217	5	]	]	PUNCT
ma-53	217	6	)	)	PUNCT
ma-53	217	7	‖	‖	PROPN
ma-53	218	1	+	+	PROPN
ma-53	218	2	‖f	‖f	ADP
ma-53	218	3	′(x0)−1([yn	′(x0)−1([yn	PROPN
ma-53	218	4	,	,	PUNCT
ma-53	218	5	xn;f	xn;f	PUNCT
ma-53	218	6	]	]	PUNCT
ma-53	219	1	−	−	X
ma-53	219	2	f	f	PROPN
ma-53	219	3	′(xn))‖	′(xn))‖	NOUN
ma-53	219	4	≤	≤	NOUN
ma-53	219	5	(	(	PUNCT
ma-53	219	6	k2(‖xn+1	k2(‖xn+1	PROPN
ma-53	219	7	−	−	PROPN
ma-53	219	8	yn‖+	yn‖+	PROPN
ma-53	219	9	‖yn	‖yn	PROPN
ma-53	219	10	−	−	PROPN
ma-53	219	11	xn‖	xn‖	PROPN
ma-53	219	12	)	)	PUNCT
ma-53	220	1	+	+	ADJ
ma-53	220	2	k3‖yn	k3‖yn	NOUN
ma-53	220	3	−	−	PROPN
ma-53	220	4	xn‖)‖xn+1	xn‖)‖xn+1	ADP
ma-53	220	5	−	−	PROPN
ma-53	220	6	yn‖	yn‖	NOUN
ma-53	220	7	(	(	PUNCT
ma-53	220	8	2.27	2.27	NUM
ma-53	220	9	)	)	PUNCT
ma-53	220	10	we	we	PRON
ma-53	220	11	need	need	VERB
ma-53	220	12	to	to	PART
ma-53	220	13	show	show	VERB
ma-53	220	14	f	f	PROPN
ma-53	220	15	′(xn+1	′(xn+1	PROPN
ma-53	220	16	)	)	PUNCT
ma-53	220	17	is	be	AUX
ma-53	220	18	invertible	invertible	ADJ
ma-53	220	19	.	.	PUNCT
ma-53	221	1	by	by	ADP
ma-53	221	2	(	(	PUNCT
ma-53	221	3	a2	a2	PROPN
ma-53	221	4	)	)	PUNCT
ma-53	221	5	and	and	CCONJ
ma-53	221	6	the	the	DET
ma-53	221	7	induction	induction	NOUN
ma-53	221	8	hypotheses	hypothese	VERB
ma-53	221	9	we	we	PRON
ma-53	221	10	obtain	obtain	VERB
ma-53	221	11	‖f	‖f	ADP
ma-53	221	12	′(x0)−1(f	′(x0)−1(f	PROPN
ma-53	222	1	′(xn+1)−	′(xn+1)−	NOUN
ma-53	222	2	f	f	X
ma-53	222	3	′(x0))‖	′(x0))‖	NUM
ma-53	222	4	≤	≤	NUM
ma-53	222	5	l0‖xn+1	l0‖xn+1	NOUN
ma-53	222	6	−	−	PROPN
ma-53	222	7	x0‖	x0‖	PROPN
ma-53	222	8	≤	≤	PROPN
ma-53	222	9	l0(tn+1	l0(tn+1	VERB
ma-53	222	10	−	−	PROPN
ma-53	222	11	t0	t0	PROPN
ma-53	222	12	)	)	PUNCT
ma-53	223	1	=	=	VERB
ma-53	224	1	l0tn=1	l0tn=1	X
ma-53	224	2	<	<	X
ma-53	224	3	1	1	NUM
ma-53	224	4	,	,	PUNCT
ma-53	224	5	so	so	SCONJ
ma-53	224	6	‖f	‖f	ADP
ma-53	224	7	′(xn+1)−1f	′(xn+1)−1f	NOUN
ma-53	224	8	′(x0)‖	′(x0)‖	ADJ
ma-53	224	9	≤	≤	NUM
ma-53	224	10	1	1	NUM
ma-53	224	11	1−	1−	NUM
ma-53	224	12	l0tn+1	l0tn+1	NOUN
ma-53	224	13	.	.	PUNCT
ma-53	225	1	(	(	PUNCT
ma-53	225	2	2.28	2.28	NUM
ma-53	225	3	)	)	PUNCT
ma-53	225	4	hence	hence	ADV
ma-53	225	5	,	,	PUNCT
ma-53	225	6	we	we	PRON
ma-53	225	7	get	get	VERB
ma-53	225	8	by	by	ADP
ma-53	225	9	(	(	PUNCT
ma-53	225	10	2.27	2.27	NUM
ma-53	225	11	)	)	PUNCT
ma-53	225	12	,	,	PUNCT
ma-53	225	13	(	(	PUNCT
ma-53	225	14	2.28	2.28	NUM
ma-53	225	15	)	)	PUNCT
ma-53	225	16	and	and	CCONJ
ma-53	225	17	the	the	DET
ma-53	225	18	first	first	ADJ
ma-53	225	19	substep	substep	NOUN
ma-53	225	20	of	of	ADP
ma-53	225	21	method	method	NOUN
ma-53	225	22	(	(	PUNCT
ma-53	225	23	1.20	1.20	NUM
ma-53	225	24	that	that	PRON
ma-53	225	25	‖yn+1	‖yn+1	PUNCT
ma-53	225	26	−	−	PROPN
ma-53	225	27	xn+1‖	xn+1‖	PROPN
ma-53	225	28	=	=	PUNCT
ma-53	226	1	‖f	‖f	ADP
ma-53	226	2	′(xn+1)−1f	′(xn+1)−1f	NOUN
ma-53	226	3	(	(	PUNCT
ma-53	226	4	x0)‖‖f	x0)‖‖f	X
ma-53	226	5	′(x0)−1f	′(x0)−1f	PROPN
ma-53	226	6	(	(	PUNCT
ma-53	226	7	xn+1)‖	xn+1)‖	PROPN
ma-53	226	8	≤	≤	PROPN
ma-53	226	9	(	(	PUNCT
ma-53	226	10	k2(tn+1	k2(tn+1	VERB
ma-53	226	11	−	−	PROPN
ma-53	226	12	sn	sn	NOUN
ma-53	226	13	)	)	PUNCT
ma-53	226	14	+	+	CCONJ
ma-53	226	15	(	(	PUNCT
ma-53	226	16	sn	sn	INTJ
ma-53	226	17	−	−	PROPN
ma-53	226	18	tn	tn	PROPN
ma-53	226	19	)	)	PUNCT
ma-53	226	20	)	)	PUNCT
ma-53	227	1	+	+	ADP
ma-53	227	2	k3(sn	k3(sn	PROPN
ma-53	227	3	−	−	PROPN
ma-53	227	4	tn))(tn+1	tn))(tn+1	NUM
ma-53	227	5	−	−	PROPN
ma-53	227	6	sn	sn	PROPN
ma-53	227	7	)	)	PUNCT
ma-53	227	8	1−	1−	NUM
ma-53	227	9	l0tn+1	l0tn+1	NOUN
ma-53	227	10	=	=	SYM
ma-53	227	11	sn+1	sn+1	VERB
ma-53	227	12	−	−	PROPN
ma-53	227	13	tn+1	tn+1	PROPN
ma-53	227	14	,	,	PUNCT
ma-53	227	15	(	(	PUNCT
ma-53	227	16	2.29	2.29	NUM
ma-53	227	17	)	)	PUNCT
ma-53	227	18	since	since	SCONJ
ma-53	227	19	k4	k4	NOUN
ma-53	227	20	=	=	SYM
ma-53	227	21	k2	k2	PROPN
ma-53	227	22	+	+	X
ma-53	227	23	k3	k3	ADJ
ma-53	227	24	,	,	PUNCT
ma-53	227	25	showing	show	VERB
ma-53	227	26	(	(	PUNCT
ma-53	227	27	i	i	NOUN
ma-53	227	28	m	m	PROPN
ma-53	227	29	)	)	PUNCT
ma-53	227	30	for	for	ADP
ma-53	227	31	m	m	PROPN
ma-53	227	32	=	=	SYM
ma-53	227	33	n	n	PROPN
ma-53	227	34	+	+	NOUN
ma-53	227	35	1	1	NUM
ma-53	227	36	.	.	PUNCT
ma-53	228	1	then	then	ADV
ma-53	228	2	,	,	PUNCT
ma-53	228	3	we	we	PRON
ma-53	228	4	also	also	ADV
ma-53	228	5	have	have	VERB
ma-53	228	6	‖yn+1	‖yn+1	VERB
ma-53	228	7	−	−	DET
ma-53	228	8	x0‖	x0‖	PROPN
ma-53	228	9	≤	≤	PROPN
ma-53	228	10	‖yn+1	‖yn+1	PUNCT
ma-53	228	11	−	−	NOUN
ma-53	228	12	xn+1‖+	xn+1‖+	PUNCT
ma-53	229	1	‖xn+1	‖xn+1	NUM
ma-53	230	1	−	−	PROPN
ma-53	230	2	x0‖	x0‖	PROPN
ma-53	230	3	≤	≤	PROPN
ma-53	230	4	sn+1	sn+1	VERB
ma-53	230	5	−	−	PROPN
ma-53	230	6	tn+1	tn+1	NOUN
ma-53	230	7	+	+	CCONJ
ma-53	230	8	tn+1	tn+1	NOUN
ma-53	230	9	−	−	NOUN
ma-53	230	10	s0	s0	NOUN
ma-53	230	11	=	=	SYM
ma-53	230	12	sn+1	sn+1	PROPN
ma-53	230	13	≤	≤	PROPN
ma-53	230	14	s∗	s∗	PROPN
ma-53	230	15	,	,	PUNCT
ma-53	230	16	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	230	17	eur	eur	NOUN
ma-53	230	18	.	.	PUNCT
ma-53	231	1	j.	j.	PROPN
ma-53	231	2	math	math	PROPN
ma-53	231	3	.	.	PUNCT
ma-53	232	1	anal	anal	PROPN
ma-53	232	2	.	.	PUNCT
ma-53	233	1	10.28924	10.28924	NUM
ma-53	233	2	/	/	SYM
ma-53	233	3	ada	ada	PROPN
ma-53	233	4	/	/	PROPN
ma-53	233	5	ma.2.3	ma.2.3	PROPN
ma-53	233	6	9so	9so	NOUN
ma-53	233	7	yn+1	yn+1	PROPN
ma-53	233	8	∈	∈	PROPN
ma-53	233	9	u[x0	u[x0	NOUN
ma-53	233	10	,	,	PUNCT
ma-53	233	11	s∗	s∗	PROPN
ma-53	233	12	]	]	PUNCT
ma-53	233	13	.	.	PUNCT
ma-53	234	1	operator	operator	NOUN
ma-53	234	2	a−1n+1	a−1n+1	PROPN
ma-53	234	3	shall	shall	AUX
ma-53	234	4	be	be	AUX
ma-53	234	5	shown	show	VERB
ma-53	234	6	to	to	PART
ma-53	234	7	exist	exist	VERB
ma-53	234	8	‖f	‖f	PUNCT
ma-53	234	9	′(x0)−1(an+1	′(x0)−1(an+1	PROPN
ma-53	235	1	−	−	PROPN
ma-53	236	1	f	f	PROPN
ma-53	236	2	′(x0))‖	′(x0))‖	NUM
ma-53	236	3	≤	≤	NOUN
ma-53	236	4	‖f	‖f	ADP
ma-53	236	5	′(x0)−1([yn+1	′(x0)−1([yn+1	NOUN
ma-53	236	6	,	,	PUNCT
ma-53	236	7	xn+1;f	xn+1;f	PROPN
ma-53	236	8	]	]	PUNCT
ma-53	236	9	−	−	PROPN
ma-53	236	10	f	f	X
ma-53	236	11	′(x0))‖	′(x0))‖	NUM
ma-53	236	12	+	+	PROPN
ma-53	236	13	‖f	‖f	PUNCT
ma-53	236	14	′(x0)−1([yn+1	′(x0)−1([yn+1	NOUN
ma-53	236	15	;	;	PUNCT
ma-53	236	16	xn+1;f	xn+1;f	PROPN
ma-53	236	17	]	]	PUNCT
ma-53	236	18	−	−	PROPN
ma-53	236	19	f	f	X
ma-53	236	20	′(xn+1))‖	′(xn+1))‖	X
ma-53	236	21	≤	≤	X
ma-53	236	22	k1(‖yn+1	k1(‖yn+1	ADJ
ma-53	236	23	−	−	PROPN
ma-53	237	1	x0‖+	x0‖+	PUNCT
ma-53	237	2	‖xn+1	‖xn+1	X
ma-53	237	3	−	−	PROPN
ma-53	237	4	x0‖	x0‖	PROPN
ma-53	237	5	)	)	PUNCT
ma-53	238	1	+	+	PUNCT
ma-53	238	2	k3‖yn+1	k3‖yn+1	ADJ
ma-53	238	3	−	−	PROPN
ma-53	238	4	xn+1‖	xn+1‖	PROPN
ma-53	238	5	≤	≤	PROPN
ma-53	238	6	k1(sn+1	k1(sn+1	NOUN
ma-53	238	7	+	+	CCONJ
ma-53	238	8	tn+1	tn+1	NUM
ma-53	238	9	)	)	PUNCT
ma-53	238	10	+	+	NOUN
ma-53	238	11	k3(sn+1	k3(sn+1	NOUN
ma-53	238	12	−	−	NOUN
ma-53	238	13	tn+1	tn+1	NOUN
ma-53	238	14	)	)	PUNCT
ma-53	238	15	<	<	X
ma-53	238	16	1	1	NUM
ma-53	238	17	,	,	PUNCT
ma-53	238	18	so	so	ADV
ma-53	238	19	‖a−1n+1f	‖a−1n+1f	PUNCT
ma-53	238	20	′(x0)‖	′(x0)‖	NOUN
ma-53	238	21	≤	≤	NUM
ma-53	238	22	1	1	NUM
ma-53	238	23	1−	1−	NUM
ma-53	238	24	(	(	PUNCT
ma-53	238	25	k1(sn+1	k1(sn+1	X
ma-53	238	26	+	+	CCONJ
ma-53	238	27	tn+1	tn+1	NOUN
ma-53	238	28	)	)	PUNCT
ma-53	238	29	+	+	NOUN
ma-53	238	30	k3(sn+1	k3(sn+1	NOUN
ma-53	238	31	−	−	NOUN
ma-53	238	32	tn+1	tn+1	NOUN
ma-53	238	33	)	)	PUNCT
ma-53	238	34	)	)	PUNCT
ma-53	238	35	.	.	PUNCT
ma-53	239	1	(	(	PUNCT
ma-53	239	2	2.30	2.30	NUM
ma-53	239	3	)	)	PUNCT
ma-53	239	4	by	by	ADP
ma-53	239	5	the	the	DET
ma-53	239	6	first	first	ADJ
ma-53	239	7	substep	substep	NOUN
ma-53	239	8	of	of	ADP
ma-53	239	9	method	method	NOUN
ma-53	239	10	(	(	PUNCT
ma-53	239	11	1.2	1.2	NUM
ma-53	239	12	)	)	PUNCT
ma-53	239	13	,	,	PUNCT
ma-53	239	14	we	we	PRON
ma-53	239	15	can	can	AUX
ma-53	239	16	write	write	VERB
ma-53	239	17	f	f	PROPN
ma-53	239	18	(	(	PUNCT
ma-53	239	19	yn+1	yn+1	PROPN
ma-53	239	20	)	)	PUNCT
ma-53	239	21	=	=	SYM
ma-53	239	22	f	f	PROPN
ma-53	239	23	(	(	PUNCT
ma-53	239	24	yn+1)−	yn+1)−	NOUN
ma-53	239	25	f	f	PROPN
ma-53	239	26	(	(	PUNCT
ma-53	239	27	xn+1)−	xn+1)−	PROPN
ma-53	239	28	f	f	PROPN
ma-53	239	29	′(xn+1)(yn+1	′(xn+1)(yn+1	PROPN
ma-53	239	30	−	−	PROPN
ma-53	239	31	xn+1	xn+1	NUM
ma-53	239	32	)	)	PUNCT
ma-53	239	33	,	,	PUNCT
ma-53	239	34	so	so	SCONJ
ma-53	240	1	‖f	‖f	ADJ
ma-53	240	2	′(x0)−1f	′(x0)−1f	NOUN
ma-53	240	3	(	(	PUNCT
ma-53	240	4	yn+1)‖	yn+1)‖	NOUN
ma-53	240	5	≤	≤	X
ma-53	241	1	k	k	PROPN
ma-53	241	2	2	2	NUM
ma-53	241	3	‖yn+1	‖yn+1	NOUN
ma-53	241	4	−	−	PROPN
ma-53	241	5	xn+1‖2	xn+1‖2	PROPN
ma-53	241	6	≤	≤	NUM
ma-53	241	7	k	k	X
ma-53	241	8	2	2	NUM
ma-53	241	9	(	(	PUNCT
ma-53	241	10	sn+1	sn+1	VERB
ma-53	241	11	−	−	PROPN
ma-53	241	12	tn+1)2,so	tn+1)2,so	PROPN
ma-53	241	13	‖xn+2	‖xn+2	ADV
ma-53	241	14	−	−	PROPN
ma-53	241	15	yn+1‖	yn+1‖	PROPN
ma-53	241	16	≤	≤	NUM
ma-53	241	17	‖a−1n+1f	‖a−1n+1f	NOUN
ma-53	241	18	′(x0)‖‖f	′(x0)‖‖f	NOUN
ma-53	241	19	′(x0)−1f	′(x0)−1f	NOUN
ma-53	241	20	(	(	PUNCT
ma-53	241	21	yn+1)‖	yn+1)‖	PROPN
ma-53	241	22	≤	≤	PROPN
ma-53	241	23	k(sn+1	k(sn+1	PROPN
ma-53	241	24	−	−	PROPN
ma-53	241	25	tn+1)2	tn+1)2	NUM
ma-53	241	26	2(1−	2(1−	NUM
ma-53	241	27	(	(	PUNCT
ma-53	241	28	k1(sn+1	k1(sn+1	X
ma-53	241	29	+	+	CCONJ
ma-53	241	30	tn+1	tn+1	NOUN
ma-53	241	31	)	)	PUNCT
ma-53	241	32	+	+	NOUN
ma-53	241	33	k3(sn+1	k3(sn+1	NOUN
ma-53	241	34	−	−	NOUN
ma-53	241	35	tn+1	tn+1	NOUN
ma-53	241	36	)	)	PUNCT
ma-53	241	37	)	)	PUNCT
ma-53	241	38	)	)	PUNCT
ma-53	242	1	=	=	SYM
ma-53	242	2	tn+2	tn+2	INTJ
ma-53	242	3	−	−	NOUN
ma-53	242	4	sn+1	sn+1	PROPN
ma-53	242	5	,	,	PUNCT
ma-53	242	6	showing	show	VERB
ma-53	242	7	(	(	PUNCT
ma-53	242	8	iim	iim	NOUN
ma-53	242	9	)	)	PUNCT
ma-53	242	10	for	for	ADP
ma-53	242	11	m	m	PROPN
ma-53	242	12	=	=	SYM
ma-53	242	13	n	n	PROPN
ma-53	242	14	+	+	NOUN
ma-53	242	15	1	1	X
ma-53	242	16	.	.	X
ma-53	243	1	we	we	PRON
ma-53	243	2	can	can	AUX
ma-53	243	3	get	get	VERB
ma-53	243	4	‖xn+2	‖xn+2	ADV
ma-53	243	5	−	−	PROPN
ma-53	243	6	x0‖	x0‖	PROPN
ma-53	243	7	≤	≤	PROPN
ma-53	243	8	‖xn+2	‖xn+2	ADV
ma-53	243	9	−	−	PROPN
ma-53	243	10	yn+1‖+	yn+1‖+	SYM
ma-53	243	11	‖yn+1	‖yn+1	PROPN
ma-53	243	12	−	−	PROPN
ma-53	243	13	x0‖	x0‖	PROPN
ma-53	243	14	≤	≤	PROPN
ma-53	244	1	tn+2	tn+2	PRON
ma-53	244	2	−	−	NOUN
ma-53	244	3	sn+1	sn+1	PROPN
ma-53	244	4	+	+	SYM
ma-53	244	5	sn+1	sn+1	VERB
ma-53	244	6	−	−	PROPN
ma-53	244	7	t0	t0	NOUN
ma-53	244	8	=	=	PUNCT
ma-53	244	9	tn+2	tn+2	X
ma-53	244	10	≤	≤	NUM
ma-53	244	11	s∗	s∗	PROPN
ma-53	244	12	,	,	PUNCT
ma-53	244	13	so	so	ADV
ma-53	244	14	xn+2	xn+2	NUM
ma-53	244	15	∈	∈	PROPN
ma-53	244	16	u[x0	u[x0	NOUN
ma-53	244	17	,	,	PUNCT
ma-53	244	18	s∗	s∗	PROPN
ma-53	244	19	]	]	PUNCT
ma-53	244	20	.	.	PUNCT
ma-53	245	1	furthermore	furthermore	ADV
ma-53	245	2	,	,	PUNCT
ma-53	245	3	we	we	PRON
ma-53	245	4	obtain	obtain	VERB
ma-53	245	5	‖xn+1	‖xn+1	PUNCT
ma-53	245	6	−	−	NOUN
ma-53	245	7	xn‖	xn‖	PROPN
ma-53	245	8	≤	≤	X
ma-53	245	9	‖xn+1	‖xn+1	PUNCT
ma-53	245	10	−	−	PROPN
ma-53	245	11	yn‖+	yn‖+	PROPN
ma-53	246	1	‖yn	‖yn	PROPN
ma-53	246	2	−	−	NOUN
ma-53	246	3	xn‖	xn‖	PROPN
ma-53	246	4	=	=	SYM
ma-53	246	5	tn+1	tn+1	PROPN
ma-53	246	6	−	−	PROPN
ma-53	247	1	sn	sn	PROPN
ma-53	248	1	+	+	CCONJ
ma-53	249	1	sn	sn	PROPN
ma-53	249	2	−	−	PROPN
ma-53	249	3	tn	tn	NOUN
ma-53	249	4	=	=	SYM
ma-53	249	5	tn+1	tn+1	PROPN
ma-53	249	6	−	−	PROPN
ma-53	249	7	tn	tn	PROPN
ma-53	249	8	,	,	PUNCT
ma-53	249	9	so	so	ADV
ma-53	249	10	sequence	sequence	NOUN
ma-53	249	11	{	{	PUNCT
ma-53	249	12	xn	xn	NOUN
ma-53	249	13	}	}	PUNCT
ma-53	249	14	is	be	AUX
ma-53	249	15	fundamental	fundamental	ADJ
ma-53	249	16	in	in	ADP
ma-53	249	17	a	a	DET
ma-53	249	18	banach	banach	NOUN
ma-53	249	19	space	space	NOUN
ma-53	249	20	b	b	NOUN
ma-53	249	21	,	,	PUNCT
ma-53	249	22	so	so	SCONJ
ma-53	249	23	it	it	PRON
ma-53	249	24	converges	converge	VERB
ma-53	249	25	to	to	ADP
ma-53	249	26	some	some	DET
ma-53	249	27	x∗	x∗	PROPN
ma-53	249	28	∈	∈	PROPN
ma-53	249	29	u[x0	u[x0	NOUN
ma-53	249	30	,	,	PUNCT
ma-53	249	31	s∗	s∗	PROPN
ma-53	249	32	]	]	PUNCT
ma-53	249	33	.	.	PUNCT
ma-53	250	1	byletting	bylette	VERB
ma-53	250	2	n	n	ADP
ma-53	250	3	−→∞	−→∞	PROPN
ma-53	250	4	in	in	ADP
ma-53	250	5	(	(	PUNCT
ma-53	250	6	2.27	2.27	NUM
ma-53	250	7	)	)	PUNCT
ma-53	250	8	,	,	PUNCT
ma-53	250	9	we	we	PRON
ma-53	250	10	obtain	obtain	VERB
ma-53	250	11	‖f	‖f	DET
ma-53	250	12	′(x0)−1f	′(x0)−1f	NOUN
ma-53	250	13	(	(	PUNCT
ma-53	250	14	xk+1)‖	xk+1)‖	NOUN
ma-53	250	15	≤	≤	PROPN
ma-53	250	16	(	(	PUNCT
ma-53	250	17	k2(tn+1	k2(tn+1	VERB
ma-53	250	18	−	−	PROPN
ma-53	250	19	sn	sn	NOUN
ma-53	250	20	)	)	PUNCT
ma-53	251	1	+	+	ADJ
ma-53	251	2	k4(sn	k4(sn	PROPN
ma-53	251	3	−	−	PROPN
ma-53	251	4	tn))(sn+1	tn))(sn+1	NUM
ma-53	251	5	−	−	PROPN
ma-53	251	6	sn	sn	NOUN
ma-53	251	7	)	)	PUNCT
ma-53	251	8	−→	−→	NOUN
ma-53	251	9	0	0	NUM
ma-53	251	10	,	,	PUNCT
ma-53	251	11	so	so	SCONJ
ma-53	251	12	f	f	X
ma-53	251	13	(	(	PUNCT
ma-53	251	14	x∗	x∗	PROPN
ma-53	251	15	)	)	PUNCT
ma-53	251	16	=	=	SYM
ma-53	251	17	0	0	NUM
ma-53	251	18	by	by	ADP
ma-53	251	19	the	the	DET
ma-53	251	20	continuity	continuity	NOUN
ma-53	251	21	of	of	ADP
ma-53	251	22	f.	f.	PROPN
ma-53	251	23	�	�	PROPN
ma-53	251	24	a	a	DET
ma-53	251	25	uniqueness	uniqueness	NOUN
ma-53	251	26	of	of	ADP
ma-53	251	27	the	the	DET
ma-53	251	28	solution	solution	NOUN
ma-53	251	29	result	result	NOUN
ma-53	251	30	is	be	AUX
ma-53	251	31	given	give	VERB
ma-53	251	32	next	next	ADV
ma-53	251	33	.	.	PUNCT
ma-53	252	1	proposition	proposition	NOUN
ma-53	252	2	2.3	2.3	NUM
ma-53	252	3	.	.	PUNCT
ma-53	253	1	suppose	suppose	VERB
ma-53	253	2	:	:	PUNCT
ma-53	253	3	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	253	4	eur	eur	PROPN
ma-53	253	5	.	.	PUNCT
ma-53	254	1	j.	j.	PROPN
ma-53	254	2	math	math	PROPN
ma-53	254	3	.	.	PUNCT
ma-53	255	1	anal	anal	PROPN
ma-53	255	2	.	.	PUNCT
ma-53	256	1	10.28924	10.28924	NUM
ma-53	256	2	/	/	SYM
ma-53	256	3	ada	ada	PROPN
ma-53	256	4	/	/	SYM
ma-53	256	5	ma.2.3	ma.2.3	PROPN
ma-53	256	6	10(i	10(i	NUM
ma-53	256	7	)	)	PUNCT
ma-53	256	8	there	there	PRON
ma-53	256	9	exists	exist	VERB
ma-53	256	10	a	a	DET
ma-53	256	11	simple	simple	ADJ
ma-53	256	12	solution	solution	NOUN
ma-53	256	13	x∗	x∗	PROPN
ma-53	256	14	of	of	ADP
ma-53	256	15	equation	equation	NOUN
ma-53	256	16	f	f	X
ma-53	256	17	(	(	PUNCT
ma-53	256	18	x	x	X
ma-53	256	19	)	)	PUNCT
ma-53	256	20	=	=	SYM
ma-53	256	21	0.(ii	0.(ii	X
ma-53	256	22	)	)	PUNCT
ma-53	256	23	there	there	PRON
ma-53	256	24	exists	exist	VERB
ma-53	256	25	s̄	s̄	NOUN
ma-53	256	26	≥	≥	PRON
ma-53	256	27	s∗	s∗	VERB
ma-53	256	28	such	such	ADJ
ma-53	256	29	that	that	SCONJ
ma-53	256	30	l0(s̄	l0(s̄	PROPN
ma-53	256	31	+	+	CCONJ
ma-53	256	32	s∗	s∗	PROPN
ma-53	256	33	)	)	PUNCT
ma-53	256	34	<	<	X
ma-53	257	1	2	2	X
ma-53	257	2	.	.	X
ma-53	257	3	set	set	VERB
ma-53	257	4	ω1	ω1	PROPN
ma-53	257	5	=	=	SYM
ma-53	257	6	u[x0	u[x0	ADJ
ma-53	257	7	,	,	PUNCT
ma-53	257	8	s̄	s̄	NOUN
ma-53	257	9	]	]	PUNCT
ma-53	257	10	∩ω	∩ω	INTJ
ma-53	257	11	.	.	PUNCT
ma-53	258	1	then	then	ADV
ma-53	258	2	,	,	PUNCT
ma-53	258	3	the	the	DET
ma-53	258	4	only	only	ADJ
ma-53	258	5	solution	solution	NOUN
ma-53	258	6	of	of	ADP
ma-53	258	7	equation	equation	NOUN
ma-53	258	8	f	f	X
ma-53	258	9	(	(	PUNCT
ma-53	258	10	x	x	X
ma-53	258	11	)	)	PUNCT
ma-53	258	12	=	=	SYM
ma-53	258	13	0	0	NUM
ma-53	258	14	in	in	ADP
ma-53	258	15	the	the	DET
ma-53	258	16	region	region	NOUN
ma-53	258	17	ω1	ω1	PROPN
ma-53	258	18	is	be	AUX
ma-53	258	19	x∗.	x∗.	ADJ
ma-53	258	20	proof	proof	NOUN
ma-53	258	21	.	.	PUNCT
ma-53	259	1	let	let	VERB
ma-53	259	2	x̄	x̄	PRON
ma-53	259	3	∈	∈	PROPN
ma-53	259	4	ω1	ω1	PROPN
ma-53	259	5	with	with	ADP
ma-53	259	6	f	f	PROPN
ma-53	259	7	(	(	PUNCT
ma-53	259	8	x̄	x̄	PROPN
ma-53	259	9	)	)	PUNCT
ma-53	259	10	=	=	SYM
ma-53	260	1	0	0	X
ma-53	260	2	.	.	PUNCT
ma-53	260	3	let	let	VERB
ma-53	260	4	m	m	VERB
ma-53	260	5	=	=	SYM
ma-53	260	6	∫	∫	PROPN
ma-53	261	1	1	1	NUM
ma-53	261	2	0	0	NUM
ma-53	261	3	f	f	PROPN
ma-53	261	4	′(x̄	′(x̄	NOUN
ma-53	261	5	+	+	CCONJ
ma-53	261	6	θ(x∗	θ(x∗	NOUN
ma-53	261	7	−	−	PROPN
ma-53	261	8	x̄))dθ	x̄))dθ	ADJ
ma-53	261	9	.	.	PUNCT
ma-53	262	1	then	then	ADV
ma-53	262	2	,	,	PUNCT
ma-53	262	3	in	in	ADP
ma-53	262	4	view	view	NOUN
ma-53	262	5	of	of	ADP
ma-53	262	6	(	(	PUNCT
ma-53	262	7	a2	a2	PROPN
ma-53	262	8	)	)	PUNCT
ma-53	262	9	and(ii	and(ii	NOUN
ma-53	262	10	)	)	PUNCT
ma-53	262	11	,	,	PUNCT
ma-53	262	12	we	we	PRON
ma-53	262	13	obtain	obtain	VERB
ma-53	262	14	‖f	‖f	PRON
ma-53	262	15	′(x0)−1(m	′(x0)−1(m	PROPN
ma-53	262	16	−	−	PROPN
ma-53	262	17	f	f	PROPN
ma-53	262	18	′(x0))‖	′(x0))‖	NUM
ma-53	262	19	≤	≤	NUM
ma-53	262	20	l0	l0	PROPN
ma-53	262	21	∫	∫	PROPN
ma-53	262	22	1	1	NUM
ma-53	262	23	0	0	NUM
ma-53	263	1	[	[	X
ma-53	263	2	(	(	PUNCT
ma-53	263	3	1−	1−	NUM
ma-53	263	4	θ)‖x̄	θ)‖x̄	NUM
ma-53	263	5	−	−	X
ma-53	263	6	x0‖+	x0‖+	SYM
ma-53	263	7	θ‖x∗	θ‖x∗	NOUN
ma-53	263	8	−	−	PROPN
ma-53	263	9	x0‖]dθ	x0‖]dθ	SYM
ma-53	263	10	≤	≤	NUM
ma-53	263	11	l0	l0	NOUN
ma-53	263	12	2	2	NUM
ma-53	263	13	(	(	PUNCT
ma-53	263	14	s̄	s̄	NOUN
ma-53	263	15	+	+	CCONJ
ma-53	263	16	s∗	s∗	PROPN
ma-53	263	17	)	)	PUNCT
ma-53	263	18	<	<	X
ma-53	263	19	1	1	NUM
ma-53	263	20	,	,	PUNCT
ma-53	263	21	so	so	ADV
ma-53	263	22	x̄	x̄	PUNCT
ma-53	263	23	=	=	PUNCT
ma-53	263	24	x∗	x∗	PROPN
ma-53	263	25	since	since	SCONJ
ma-53	263	26	m−1	m−1	PROPN
ma-53	263	27	exists	exist	VERB
ma-53	263	28	and	and	CCONJ
ma-53	263	29	m(x∗	m(x∗	NOUN
ma-53	263	30	−	−	PROPN
ma-53	263	31	x̄	x̄	PROPN
ma-53	263	32	)	)	PUNCT
ma-53	263	33	=	=	SYM
ma-53	263	34	f	f	X
ma-53	263	35	(	(	PUNCT
ma-53	263	36	x∗)−	x∗)−	PROPN
ma-53	263	37	f	f	X
ma-53	263	38	(	(	PUNCT
ma-53	263	39	x̄	x̄	PROPN
ma-53	263	40	)	)	PUNCT
ma-53	263	41	=	=	PUNCT
ma-53	263	42	0−	0−	NUM
ma-53	263	43	0	0	NUM
ma-53	263	44	=	=	SYM
ma-53	263	45	0	0	X
ma-53	263	46	.	.	PUNCT
ma-53	263	47	�	�	PROPN
ma-53	263	48	remark	remark	VERB
ma-53	263	49	2.4	2.4	NUM
ma-53	263	50	.	.	PUNCT
ma-53	264	1	notice	notice	VERB
ma-53	264	2	that	that	SCONJ
ma-53	264	3	s∗∗	s∗∗	ADV
ma-53	264	4	given	give	VERB
ma-53	264	5	in	in	ADP
ma-53	264	6	closed	closed	ADJ
ma-53	264	7	form	form	NOUN
ma-53	264	8	can	can	AUX
ma-53	264	9	repalce	repalce	VERB
ma-53	264	10	s∗	s∗	PROPN
ma-53	264	11	in	in	ADP
ma-53	264	12	the	the	DET
ma-53	264	13	conditions	condition	NOUN
ma-53	264	14	of	of	ADP
ma-53	264	15	theorem	theorem	NOUN
ma-53	264	16	2.2	2.2	NUM
ma-53	264	17	.	.	PUNCT
ma-53	265	1	3	3	X
ma-53	265	2	.	.	X
ma-53	265	3	local	local	ADJ
ma-53	265	4	convergence	convergence	NOUN
ma-53	265	5	as	as	ADP
ma-53	265	6	in	in	ADP
ma-53	265	7	section	section	NOUN
ma-53	265	8	2	2	NUM
ma-53	265	9	we	we	PRON
ma-53	265	10	develop	develop	VERB
ma-53	265	11	some	some	DET
ma-53	265	12	functions	function	NOUN
ma-53	265	13	and	and	CCONJ
ma-53	265	14	parameters	parameter	NOUN
ma-53	265	15	.	.	PUNCT
ma-53	266	1	let	let	VERB
ma-53	266	2	li	li	PROPN
ma-53	266	3	,	,	PUNCT
ma-53	266	4	i	i	PRON
ma-53	266	5	=	=	NOUN
ma-53	266	6	0	0	NUM
ma-53	266	7	,	,	PUNCT
ma-53	266	8	1	1	NUM
ma-53	266	9	,	,	PUNCT
ma-53	266	10	2	2	NUM
ma-53	266	11	,	,	PUNCT
ma-53	266	12	3	3	NUM
ma-53	266	13	,	,	PUNCT
ma-53	266	14	4	4	NUM
ma-53	266	15	be	be	AUX
ma-53	266	16	givenparameters	givenparameter	NOUN
ma-53	266	17	.	.	PUNCT
ma-53	267	1	define	define	VERB
ma-53	267	2	function	function	NOUN
ma-53	267	3	ϕ1	ϕ1	NOUN
ma-53	267	4	on	on	ADP
ma-53	267	5	the	the	DET
ma-53	267	6	interval	interval	NOUN
ma-53	267	7	t	t	NOUN
ma-53	267	8	=	=	PUNCT
ma-53	268	1	[	[	X
ma-53	268	2	0	0	NUM
ma-53	268	3	,	,	PUNCT
ma-53	268	4	1l0	1l0	NUM
ma-53	268	5	)	)	PUNCT
ma-53	268	6	by	by	ADP
ma-53	268	7	ϕ1(t	ϕ1(t	PRON
ma-53	268	8	)	)	PUNCT
ma-53	268	9	=	=	VERB
ma-53	268	10	lt	lt	PRON
ma-53	268	11	2(1−	2(1−	NUM
ma-53	268	12	l0	l0	PROPN
ma-53	268	13	t	t	PROPN
ma-53	268	14	)	)	PUNCT
ma-53	268	15	.	.	PUNCT
ma-53	269	1	notice	notice	VERB
ma-53	269	2	that	that	SCONJ
ma-53	269	3	parameter	parameter	NOUN
ma-53	269	4	ra	ra	PROPN
ma-53	269	5	=	=	NOUN
ma-53	269	6	2	2	NUM
ma-53	269	7	2l0	2l0	NUM
ma-53	270	1	+	+	CCONJ
ma-53	270	2	l	l	X
ma-53	270	3	<	<	X
ma-53	270	4	1	1	NUM
ma-53	270	5	l0	l0	NOUN
ma-53	270	6	(	(	PUNCT
ma-53	270	7	3.1)solves	3.1)solves	NUM
ma-53	270	8	equation	equation	NOUN
ma-53	270	9	ϕ1(t	ϕ1(t	PRON
ma-53	270	10	)	)	PUNCT
ma-53	270	11	=	=	SYM
ma-53	270	12	1.define	1.define	NUM
ma-53	270	13	functions	function	NOUN
ma-53	270	14	on	on	ADP
ma-53	270	15	the	the	DET
ma-53	270	16	interval	interval	NOUN
ma-53	270	17	t	t	NOUN
ma-53	270	18	by	by	ADP
ma-53	270	19	q(t	q(t	PROPN
ma-53	270	20	)	)	PUNCT
ma-53	270	21	=	=	PRON
ma-53	270	22	l0ϕ1(t)t	l0ϕ1(t)t	VERB
ma-53	270	23	−	−	NUM
ma-53	270	24	1	1	NUM
ma-53	270	25	and	and	CCONJ
ma-53	270	26	p(t	p(t	NOUN
ma-53	270	27	)	)	PUNCT
ma-53	270	28	=	=	PUNCT
ma-53	270	29	(	(	PUNCT
ma-53	270	30	2l1(1	2l1(1	NUM
ma-53	270	31	+	+	NUM
ma-53	270	32	ϕ1(t	ϕ1(t	NUM
ma-53	270	33	)	)	PUNCT
ma-53	270	34	)	)	PUNCT
ma-53	271	1	+	+	CCONJ
ma-53	271	2	l)t	l)t	X
ma-53	271	3	.	.	PUNCT
ma-53	272	1	suppose	suppose	VERB
ma-53	272	2	that	that	SCONJ
ma-53	272	3	these	these	DET
ma-53	272	4	functions	function	NOUN
ma-53	272	5	have	have	VERB
ma-53	272	6	smallest	small	ADJ
ma-53	272	7	zeros	zero	NOUN
ma-53	272	8	rq	rq	NOUN
ma-53	272	9	and	and	CCONJ
ma-53	272	10	rp	rp	NOUN
ma-53	272	11	in	in	ADP
ma-53	272	12	(	(	PUNCT
ma-53	272	13	0	0	NUM
ma-53	272	14	,	,	PUNCT
ma-53	272	15	1l0	1l0	NUM
ma-53	272	16	)	)	PUNCT
ma-53	272	17	,	,	PUNCT
ma-53	272	18	respectively	respectively	ADV
ma-53	272	19	.	.	PUNCT
ma-53	273	1	let	let	VERB
ma-53	273	2	r1	r1	PROPN
ma-53	273	3	=	=	SYM
ma-53	273	4	min{rq	min{rq	PROPN
ma-53	273	5	,	,	PUNCT
ma-53	273	6	rp	rp	NOUN
ma-53	273	7	}	}	PUNCT
ma-53	273	8	and	and	CCONJ
ma-53	273	9	t0	t0	X
ma-53	273	10	=	=	PUNCT
ma-53	274	1	[	[	X
ma-53	274	2	0	0	NUM
ma-53	274	3	,	,	PUNCT
ma-53	274	4	r1	r1	PROPN
ma-53	274	5	)	)	PUNCT
ma-53	274	6	.	.	PUNCT
ma-53	275	1	define	define	VERB
ma-53	275	2	function	function	NOUN
ma-53	275	3	ϕ2	ϕ2	ADV
ma-53	275	4	on	on	ADP
ma-53	275	5	t0	t0	PROPN
ma-53	275	6	by	by	ADP
ma-53	275	7	ϕ2(t	ϕ2(t	PROPN
ma-53	275	8	)	)	PUNCT
ma-53	276	1	=	=	PRON
ma-53	276	2	[	[	PUNCT
ma-53	276	3	lϕ1(t	lϕ1(t	PROPN
ma-53	276	4	)	)	PUNCT
ma-53	276	5	2(1−	2(1−	NUM
ma-53	276	6	l0ϕ1(t)t	l0ϕ1(t)t	NOUN
ma-53	276	7	)	)	PUNCT
ma-53	277	1	+	+	CCONJ
ma-53	277	2	l4(l2	l4(l2	PRON
ma-53	277	3	+	+	CCONJ
ma-53	277	4	l3)(1	l3)(1	PROPN
ma-53	277	5	+	+	CCONJ
ma-53	277	6	ϕ1(t))ϕ1(t	ϕ1(t))ϕ1(t	X
ma-53	277	7	)	)	PUNCT
ma-53	277	8	(	(	PUNCT
ma-53	277	9	1−	1−	NUM
ma-53	277	10	l0ϕ1(t)t)(1−	l0ϕ1(t)t)(1−	NOUN
ma-53	277	11	p(t	p(t	NOUN
ma-53	277	12	)	)	PUNCT
ma-53	277	13	)	)	PUNCT
ma-53	277	14	]	]	PUNCT
ma-53	278	1	t.	t.	PROPN
ma-53	278	2	suppose	suppose	VERB
ma-53	278	3	that	that	SCONJ
ma-53	278	4	function	function	NOUN
ma-53	278	5	ϕ2(t)−	ϕ2(t)−	PROPN
ma-53	278	6	1	1	NUM
ma-53	278	7	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	278	8	eur	eur	NOUN
ma-53	278	9	.	.	PUNCT
ma-53	279	1	j.	j.	PROPN
ma-53	279	2	math	math	PROPN
ma-53	279	3	.	.	PUNCT
ma-53	280	1	anal	anal	PROPN
ma-53	280	2	.	.	PUNCT
ma-53	281	1	10.28924	10.28924	NUM
ma-53	281	2	/	/	SYM
ma-53	281	3	ada	ada	PROPN
ma-53	281	4	/	/	SYM
ma-53	281	5	ma.2.3	ma.2.3	PROPN
ma-53	281	6	11has	11has	CCONJ
ma-53	281	7	smallest	small	ADJ
ma-53	281	8	zero	zero	NUM
ma-53	281	9	r2	r2	PROPN
ma-53	281	10	∈	∈	PROPN
ma-53	281	11	(	(	PUNCT
ma-53	281	12	0	0	NUM
ma-53	281	13	,	,	PUNCT
ma-53	281	14	r1	r1	PROPN
ma-53	281	15	)	)	PUNCT
ma-53	281	16	.	.	PUNCT
ma-53	282	1	we	we	PRON
ma-53	282	2	shall	shall	AUX
ma-53	282	3	show	show	VERB
ma-53	282	4	that	that	DET
ma-53	282	5	parameter	parameter	NOUN
ma-53	282	6	r	r	NOUN
ma-53	282	7	=	=	PUNCT
ma-53	282	8	min{ra	min{ra	NOUN
ma-53	282	9	,	,	PUNCT
ma-53	282	10	r2	r2	PROPN
ma-53	282	11	}	}	PUNCT
ma-53	282	12	(	(	PUNCT
ma-53	282	13	3.2	3.2	NUM
ma-53	282	14	)	)	PUNCT
ma-53	282	15	is	be	AUX
ma-53	282	16	a	a	DET
ma-53	282	17	convergence	convergence	NOUN
ma-53	282	18	radius	radius	NOUN
ma-53	282	19	for	for	ADP
ma-53	282	20	method	method	NOUN
ma-53	282	21	(	(	PUNCT
ma-53	282	22	1.2	1.2	NUM
ma-53	282	23	)	)	PUNCT
ma-53	282	24	.	.	PUNCT
ma-53	283	1	let	let	VERB
ma-53	283	2	t1	t1	NOUN
ma-53	283	3	=	=	PUNCT
ma-53	284	1	[	[	X
ma-53	284	2	0	0	NUM
ma-53	284	3	,	,	PUNCT
ma-53	284	4	r	r	NOUN
ma-53	284	5	)	)	PUNCT
ma-53	284	6	.	.	PUNCT
ma-53	285	1	then	then	ADV
ma-53	285	2	,	,	PUNCT
ma-53	285	3	it	it	PRON
ma-53	285	4	follows	follow	VERB
ma-53	285	5	by	by	ADP
ma-53	285	6	these	these	DET
ma-53	285	7	definitions	definition	NOUN
ma-53	285	8	thatfor	thatfor	ADP
ma-53	285	9	each	each	DET
ma-53	285	10	t	t	PROPN
ma-53	285	11	∈	∈	PROPN
ma-53	285	12	t1	t1	PROPN
ma-53	285	13	l0	l0	PROPN
ma-53	285	14	t	t	PROPN
ma-53	285	15	<	<	X
ma-53	285	16	1	1	NUM
ma-53	285	17	(	(	PUNCT
ma-53	285	18	3.3	3.3	NUM
ma-53	285	19	)	)	PUNCT
ma-53	285	20	0	0	NUM
ma-53	285	21	≤	≤	NUM
ma-53	286	1	ϕ1(t	ϕ1(t	NUM
ma-53	286	2	)	)	PUNCT
ma-53	286	3	<	<	X
ma-53	286	4	1	1	NUM
ma-53	286	5	,	,	PUNCT
ma-53	286	6	(	(	PUNCT
ma-53	286	7	3.4	3.4	NUM
ma-53	286	8	)	)	PUNCT
ma-53	286	9	0	0	NUM
ma-53	287	1	≤	≤	NOUN
ma-53	287	2	ϕ1(t)t	ϕ1(t)t	NUM
ma-53	287	3	<	<	X
ma-53	287	4	1	1	NUM
ma-53	287	5	(	(	PUNCT
ma-53	287	6	3.5	3.5	NUM
ma-53	287	7	)	)	PUNCT
ma-53	287	8	0	0	NUM
ma-53	287	9	≤	≤	NUM
ma-53	287	10	p(t	p(t	NOUN
ma-53	287	11	)	)	PUNCT
ma-53	287	12	<	<	X
ma-53	287	13	1	1	NUM
ma-53	287	14	(	(	PUNCT
ma-53	287	15	3.6	3.6	NUM
ma-53	287	16	)	)	PUNCT
ma-53	287	17	and	and	CCONJ
ma-53	287	18	0	0	NUM
ma-53	287	19	≤	≤	X
ma-53	287	20	ϕ2(t)t	ϕ2(t)t	NUM
ma-53	287	21	<	<	X
ma-53	287	22	1	1	NUM
ma-53	287	23	(	(	PUNCT
ma-53	287	24	3.7)hold.the	3.7)hold.the	NUM
ma-53	287	25	conditions	condition	NOUN
ma-53	287	26	(	(	PUNCT
ma-53	287	27	h	h	NOUN
ma-53	287	28	)	)	PUNCT
ma-53	287	29	to	to	PART
ma-53	287	30	be	be	AUX
ma-53	287	31	used	use	VERB
ma-53	287	32	in	in	ADP
ma-53	287	33	the	the	DET
ma-53	287	34	local	local	ADJ
ma-53	287	35	convergence	convergence	NOUN
ma-53	287	36	of	of	ADP
ma-53	287	37	method	method	NOUN
ma-53	287	38	(	(	PUNCT
ma-53	287	39	1.2	1.2	NUM
ma-53	287	40	)	)	PUNCT
ma-53	287	41	are	be	AUX
ma-53	287	42	as	as	ADP
ma-53	287	43	follows.suppose:(h1	follows.suppose:(h1	PROPN
ma-53	287	44	)	)	PUNCT
ma-53	287	45	there	there	PRON
ma-53	287	46	exists	exist	VERB
ma-53	287	47	a	a	DET
ma-53	287	48	simple	simple	ADJ
ma-53	287	49	solution	solution	NOUN
ma-53	287	50	x∗	x∗	PROPN
ma-53	287	51	∈	∈	PROPN
ma-53	287	52	ω	ω	PROPN
ma-53	287	53	of	of	ADP
ma-53	287	54	equation	equation	NOUN
ma-53	287	55	f	f	X
ma-53	287	56	(	(	PUNCT
ma-53	287	57	x	x	X
ma-53	287	58	)	)	PUNCT
ma-53	287	59	=	=	SYM
ma-53	287	60	0.(h2	0.(h2	NOUN
ma-53	287	61	)	)	PUNCT
ma-53	287	62	for	for	ADP
ma-53	287	63	each	each	DET
ma-53	287	64	x	x	SYM
ma-53	287	65	∈	∈	PROPN
ma-53	287	66	ω	ω	PROPN
ma-53	287	67	‖f	‖f	ADP
ma-53	287	68	′(x)−1(f	′(x)−1(f	ADJ
ma-53	287	69	′(x)−	′(x)−	PROPN
ma-53	287	70	f	f	PROPN
ma-53	287	71	′(x∗))‖	′(x∗))‖	PROPN
ma-53	287	72	≤	≤	PUNCT
ma-53	287	73	l0‖x	l0‖x	NUM
ma-53	287	74	−	−	NOUN
ma-53	287	75	x∗‖.set	x∗‖.set	PUNCT
ma-53	287	76	ω0	ω0	PROPN
ma-53	287	77	=	=	SYM
ma-53	287	78	u[x∗	u[x∗	PROPN
ma-53	287	79	,	,	PUNCT
ma-53	287	80	1	1	NUM
ma-53	287	81	l0	l0	NOUN
ma-53	287	82	]	]	PUNCT
ma-53	287	83	∩ω.(h3	∩ω.(h3	PROPN
ma-53	287	84	)	)	PUNCT
ma-53	287	85	for	for	ADP
ma-53	287	86	each	each	DET
ma-53	287	87	x	x	NOUN
ma-53	287	88	,	,	PUNCT
ma-53	287	89	y	y	PROPN
ma-53	287	90	∈	∈	PROPN
ma-53	287	91	ω0	ω0	NOUN
ma-53	287	92	‖f	‖f	ADP
ma-53	287	93	′(x∗)−1(f	′(x∗)−1(f	NOUN
ma-53	287	94	′(y)−	′(y)−	VERB
ma-53	287	95	f	f	PROPN
ma-53	287	96	′(x))‖	′(x))‖	PROPN
ma-53	287	97	≤	≤	PROPN
ma-53	287	98	l‖y	l‖y	NOUN
ma-53	287	99	−	−	PROPN
ma-53	287	100	x‖	x‖	PROPN
ma-53	287	101	,	,	PUNCT
ma-53	287	102	‖f	‖f	PRON
ma-53	287	103	′(x∗)−1(f	′(x∗)−1(f	NOUN
ma-53	287	104	′(y)−	′(y)−	VERB
ma-53	287	105	f	f	PROPN
ma-53	287	106	′(x∗))‖	′(x∗))‖	PROPN
ma-53	287	107	≤	≤	X
ma-53	287	108	l1(‖y	l1(‖y	VERB
ma-53	287	109	−	−	PROPN
ma-53	287	110	x∗‖+	x∗‖+	PUNCT
ma-53	287	111	‖x	‖x	NOUN
ma-53	287	112	−	−	PROPN
ma-53	287	113	x∗‖	x∗‖	NUM
ma-53	287	114	)	)	PUNCT
ma-53	287	115	,	,	PUNCT
ma-53	287	116	‖f	‖f	ADP
ma-53	287	117	′(x∗)−1([y	′(x∗)−1([y	NOUN
ma-53	287	118	,	,	PUNCT
ma-53	287	119	x	x	PROPN
ma-53	287	120	;	;	PUNCT
ma-53	287	121	f	f	X
ma-53	287	122	]	]	X
ma-53	287	123	−	−	X
ma-53	287	124	f	f	PROPN
ma-53	287	125	′(x))‖	′(x))‖	PROPN
ma-53	287	126	≤	≤	VERB
ma-53	287	127	l2‖y	l2‖y	ADJ
ma-53	287	128	−	−	PROPN
ma-53	287	129	x‖	x‖	PROPN
ma-53	287	130	,	,	PUNCT
ma-53	287	131	‖f	‖f	ADP
ma-53	287	132	′(x∗)−1([y	′(x∗)−1([y	NOUN
ma-53	287	133	,	,	PUNCT
ma-53	287	134	x	x	PROPN
ma-53	287	135	;	;	PUNCT
ma-53	287	136	f	f	X
ma-53	287	137	]	]	X
ma-53	287	138	−	−	X
ma-53	287	139	f	f	PROPN
ma-53	287	140	′(y))‖	′(y))‖	PROPN
ma-53	287	141	≤	≤	PROPN
ma-53	287	142	l3‖y	l3‖y	VERB
ma-53	287	143	−	−	PROPN
ma-53	287	144	x‖and	x‖and	PUNCT
ma-53	288	1	‖f	‖f	ADP
ma-53	288	2	′(x∗)−1f	′(x∗)−1f	NOUN
ma-53	288	3	′(x)‖	′(x)‖	X
ma-53	288	4	≤	≤	NUM
ma-53	289	1	l4‖x	l4‖x	ADV
ma-53	289	2	−	−	PROPN
ma-53	289	3	x∗‖.and(h4	x∗‖.and(h4	PROPN
ma-53	289	4	)	)	PUNCT
ma-53	290	1	u[x∗	u[x∗	PROPN
ma-53	290	2	,	,	PUNCT
ma-53	290	3	r	r	NOUN
ma-53	290	4	]	]	PUNCT
ma-53	290	5	⊂	⊂	PROPN
ma-53	291	1	ω.in	ω.in	INTJ
ma-53	291	2	view	view	NOUN
ma-53	291	3	of	of	ADP
ma-53	291	4	conditions	condition	NOUN
ma-53	291	5	(	(	PUNCT
ma-53	291	6	h	h	NOUN
ma-53	291	7	)	)	PUNCT
ma-53	291	8	and	and	CCONJ
ma-53	291	9	the	the	DET
ma-53	291	10	developed	develop	VERB
ma-53	291	11	notation	notation	NOUN
ma-53	291	12	we	we	PRON
ma-53	291	13	can	can	AUX
ma-53	291	14	show	show	VERB
ma-53	291	15	the	the	DET
ma-53	291	16	local	local	ADJ
ma-53	291	17	convergence	convergence	NOUN
ma-53	291	18	result	result	NOUN
ma-53	291	19	formethod	formethod	NOUN
ma-53	291	20	(	(	PUNCT
ma-53	291	21	1.2	1.2	NUM
ma-53	291	22	)	)	PUNCT
ma-53	291	23	.	.	PUNCT
ma-53	292	1	theorem	theorem	VERB
ma-53	292	2	3.1	3.1	NUM
ma-53	292	3	.	.	PUNCT
ma-53	293	1	under	under	ADP
ma-53	293	2	the	the	DET
ma-53	293	3	conditions	condition	NOUN
ma-53	293	4	(	(	PUNCT
ma-53	293	5	h	h	NOUN
ma-53	293	6	)	)	PUNCT
ma-53	293	7	,	,	PUNCT
ma-53	293	8	further	far	ADV
ma-53	293	9	suppose	suppose	VERB
ma-53	293	10	that	that	SCONJ
ma-53	293	11	x0	x0	PROPN
ma-53	293	12	∈	∈	PROPN
ma-53	293	13	u(x∗	u(x∗	PROPN
ma-53	293	14	,	,	PUNCT
ma-53	293	15	r	r	NOUN
ma-53	293	16	)	)	PUNCT
ma-53	293	17	−	−	NOUN
ma-53	293	18	{	{	PUNCT
ma-53	293	19	x∗	x∗	PROPN
ma-53	293	20	}	}	PUNCT
ma-53	293	21	.	.	PUNCT
ma-53	294	1	then	then	ADV
ma-53	294	2	,	,	PUNCT
ma-53	294	3	sequence	sequence	NOUN
ma-53	294	4	{	{	PUNCT
ma-53	294	5	xk	xk	NOUN
ma-53	294	6	}	}	PUNCT
ma-53	294	7	,	,	PUNCT
ma-53	294	8	{	{	PUNCT
ma-53	294	9	yn	yn	NOUN
ma-53	294	10	}	}	PUNCT
ma-53	294	11	generated	generate	VERB
ma-53	294	12	by	by	ADP
ma-53	294	13	method	method	NOUN
ma-53	294	14	(	(	PUNCT
ma-53	294	15	1.2	1.2	NUM
ma-53	294	16	)	)	PUNCT
ma-53	294	17	is	be	AUX
ma-53	294	18	well	well	ADV
ma-53	294	19	defined	define	VERB
ma-53	294	20	in	in	ADP
ma-53	294	21	u(x∗	u(x∗	PROPN
ma-53	294	22	,	,	PUNCT
ma-53	294	23	r	r	NOUN
ma-53	294	24	)	)	PUNCT
ma-53	294	25	,	,	PUNCT
ma-53	294	26	remains	remain	VERB
ma-53	294	27	in	in	ADP
ma-53	294	28	u(x∗	u(x∗	PROPN
ma-53	294	29	,	,	PUNCT
ma-53	294	30	r	r	NOUN
ma-53	294	31	)	)	PUNCT
ma-53	294	32	for	for	ADP
ma-53	294	33	each	each	PRON
ma-53	294	34	k	k	NOUN
ma-53	294	35	=	=	SYM
ma-53	294	36	0	0	NUM
ma-53	294	37	,	,	PUNCT
ma-53	294	38	1	1	NUM
ma-53	294	39	,	,	PUNCT
ma-53	294	40	2	2	NUM
ma-53	294	41	,	,	PUNCT
ma-53	294	42	.	.	PUNCT
ma-53	294	43	.	.	PUNCT
ma-53	294	44	.	.	PUNCT
ma-53	295	1	and	and	CCONJ
ma-53	295	2	converges	converge	VERB
ma-53	295	3	to	to	ADP
ma-53	295	4	x∗.	x∗.	PROPN
ma-53	295	5	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	295	6	eur	eur	PROPN
ma-53	295	7	.	.	PUNCT
ma-53	296	1	j.	j.	PROPN
ma-53	296	2	math	math	PROPN
ma-53	296	3	.	.	PUNCT
ma-53	297	1	anal	anal	PROPN
ma-53	297	2	.	.	PUNCT
ma-53	298	1	10.28924	10.28924	NUM
ma-53	298	2	/	/	SYM
ma-53	298	3	ada	ada	PROPN
ma-53	298	4	/	/	SYM
ma-53	298	5	ma.2.3	ma.2.3	PROPN
ma-53	298	6	12	12	NUM
ma-53	298	7	proof	proof	NOUN
ma-53	298	8	.	.	PUNCT
ma-53	299	1	let	let	VERB
ma-53	299	2	u	u	PRON
ma-53	299	3	∈	∈	PROPN
ma-53	299	4	u(x∗	u(x∗	PROPN
ma-53	299	5	,	,	PUNCT
ma-53	299	6	r)−	r)−	PROPN
ma-53	299	7	{	{	PUNCT
ma-53	299	8	x∗	x∗	PROPN
ma-53	299	9	}	}	PUNCT
ma-53	299	10	.	.	PUNCT
ma-53	300	1	by	by	ADP
ma-53	300	2	(	(	PUNCT
ma-53	300	3	h1	h1	PROPN
ma-53	300	4	)	)	PUNCT
ma-53	300	5	and	and	CCONJ
ma-53	300	6	(	(	PUNCT
ma-53	300	7	h2	h2	NOUN
ma-53	300	8	)	)	PUNCT
ma-53	300	9	,	,	PUNCT
ma-53	300	10	we	we	PRON
ma-53	300	11	get	get	VERB
ma-53	300	12	in	in	ADP
ma-53	300	13	turn	turn	NOUN
ma-53	300	14	that	that	SCONJ
ma-53	300	15	‖f	‖f	PRON
ma-53	300	16	′(x∗)−1(f	′(x∗)−1(f	ADJ
ma-53	300	17	′(u)−	′(u)−	PROPN
ma-53	300	18	f	f	PROPN
ma-53	300	19	′(x∗))‖	′(x∗))‖	PROPN
ma-53	300	20	≤	≤	NUM
ma-53	300	21	l0‖u	l0‖u	NUM
ma-53	300	22	−	−	PROPN
ma-53	300	23	x∗‖	x∗‖	PROPN
ma-53	300	24	≤	≤	PROPN
ma-53	301	1	l0r	l0r	PROPN
ma-53	301	2	<	<	X
ma-53	301	3	1	1	NUM
ma-53	301	4	,	,	PUNCT
ma-53	301	5	so	so	ADV
ma-53	301	6	f	f	PROPN
ma-53	301	7	′(u	′(u	NOUN
ma-53	301	8	)	)	PUNCT
ma-53	301	9	is	be	AUX
ma-53	301	10	invertibale	invertibale	ADJ
ma-53	301	11	and	and	CCONJ
ma-53	301	12	‖f	‖f	ADJ
ma-53	301	13	′(u)−1f	′(u)−1f	NOUN
ma-53	301	14	′(x∗)‖	′(x∗)‖	ADP
ma-53	301	15	≤	≤	NUM
ma-53	301	16	1	1	NUM
ma-53	301	17	1−	1−	NUM
ma-53	301	18	l0‖u	l0‖u	NOUN
ma-53	301	19	−	−	PROPN
ma-53	302	1	x∗‖	x∗‖	PROPN
ma-53	302	2	.	.	PUNCT
ma-53	303	1	(	(	PUNCT
ma-53	303	2	3.8	3.8	NUM
ma-53	303	3	)	)	PUNCT
ma-53	303	4	iterate	iterate	NOUN
ma-53	303	5	y0	y0	PROPN
ma-53	303	6	is	be	AUX
ma-53	303	7	well	well	ADV
ma-53	303	8	defined	define	VERB
ma-53	303	9	by	by	ADP
ma-53	303	10	the	the	DET
ma-53	303	11	first	first	ADJ
ma-53	303	12	substep	substep	NOUN
ma-53	303	13	of	of	ADP
ma-53	303	14	method	method	NOUN
ma-53	303	15	(	(	PUNCT
ma-53	303	16	1.2	1.2	NUM
ma-53	303	17	)	)	PUNCT
ma-53	303	18	and	and	CCONJ
ma-53	303	19	(	(	PUNCT
ma-53	303	20	3.8	3.8	NUM
ma-53	303	21	)	)	PUNCT
ma-53	303	22	for	for	ADP
ma-53	303	23	u	u	NOUN
ma-53	303	24	=	=	NOUN
ma-53	303	25	x0	x0	PROPN
ma-53	303	26	.	.	PUNCT
ma-53	304	1	then	then	ADV
ma-53	304	2	,	,	PUNCT
ma-53	304	3	we	we	PRON
ma-53	304	4	canwrite	canwrite	VERB
ma-53	304	5	y0	y0	PROPN
ma-53	304	6	−	−	NOUN
ma-53	304	7	x∗	x∗	PROPN
ma-53	304	8	=	=	PUNCT
ma-53	305	1	x0	x0	PROPN
ma-53	305	2	−	−	PROPN
ma-53	305	3	x∗	x∗	PROPN
ma-53	306	1	−	−	PROPN
ma-53	306	2	f	f	PROPN
ma-53	306	3	′(x0)−1f	′(x0)−1f	PROPN
ma-53	306	4	(	(	PUNCT
ma-53	306	5	x0	x0	PROPN
ma-53	306	6	)	)	PUNCT
ma-53	306	7	=	=	PUNCT
ma-53	307	1	(	(	PUNCT
ma-53	307	2	f	f	PROPN
ma-53	307	3	′(x0	′(x0	NOUN
ma-53	307	4	)	)	PUNCT
ma-53	307	5	−1f	−1f	PROPN
ma-53	307	6	′(x∗	′(x∗	PROPN
ma-53	307	7	)	)	PUNCT
ma-53	307	8	)	)	PUNCT
ma-53	308	1	×	×	NOUN
ma-53	308	2	(	(	PUNCT
ma-53	308	3	∫	∫	PROPN
ma-53	308	4	1	1	NUM
ma-53	308	5	0	0	NUM
ma-53	308	6	f	f	PROPN
ma-53	308	7	′(x∗	′(x∗	PROPN
ma-53	308	8	)	)	PUNCT
ma-53	308	9	−1(f	−1(f	NUM
ma-53	308	10	′(x∗	′(x∗	NOUN
ma-53	308	11	+	+	NUM
ma-53	309	1	θ(x0	θ(x0	NOUN
ma-53	309	2	−	−	PROPN
ma-53	310	1	x∗))−	x∗))−	NUM
ma-53	310	2	f	f	X
ma-53	310	3	′(x0))dθ(x0	′(x0))dθ(x0	PROPN
ma-53	310	4	−	−	PROPN
ma-53	310	5	x∗	x∗	PROPN
ma-53	310	6	)	)	PUNCT
ma-53	310	7	.	.	PUNCT
ma-53	311	1	(	(	PUNCT
ma-53	311	2	3.9	3.9	NUM
ma-53	311	3	)	)	PUNCT
ma-53	311	4	by	by	ADP
ma-53	311	5	(	(	PUNCT
ma-53	311	6	3.2	3.2	NUM
ma-53	311	7	)	)	PUNCT
ma-53	311	8	,	,	PUNCT
ma-53	311	9	(	(	PUNCT
ma-53	311	10	3.4	3.4	NUM
ma-53	311	11	)	)	PUNCT
ma-53	311	12	,	,	PUNCT
ma-53	311	13	(	(	PUNCT
ma-53	311	14	h3	h3	NOUN
ma-53	311	15	)	)	PUNCT
ma-53	311	16	,	,	PUNCT
ma-53	311	17	(	(	PUNCT
ma-53	311	18	3.8	3.8	NUM
ma-53	311	19	)	)	PUNCT
ma-53	311	20	and	and	CCONJ
ma-53	311	21	(	(	PUNCT
ma-53	311	22	3.9	3.9	NUM
ma-53	311	23	)	)	PUNCT
ma-53	311	24	,	,	PUNCT
ma-53	311	25	we	we	PRON
ma-53	311	26	have	have	VERB
ma-53	311	27	in	in	ADP
ma-53	311	28	turn	turn	NOUN
ma-53	311	29	that	that	SCONJ
ma-53	311	30	‖y0	‖y0	VERB
ma-53	311	31	−	−	PROPN
ma-53	311	32	x∗‖	x∗‖	PROPN
ma-53	311	33	≤	≤	PROPN
ma-53	311	34	l0‖x0	l0‖x0	PUNCT
ma-53	312	1	−	−	PROPN
ma-53	312	2	x∗‖2	x∗‖2	PROPN
ma-53	312	3	2(1−	2(1−	NUM
ma-53	312	4	l0‖x0	l0‖x0	NUM
ma-53	312	5	−	−	PROPN
ma-53	312	6	x∗‖	x∗‖	PROPN
ma-53	312	7	≤	≤	PROPN
ma-53	312	8	l‖x0	l‖x0	VERB
ma-53	313	1	−	−	X
ma-53	313	2	x∗‖2	x∗‖2	PROPN
ma-53	313	3	2(1−	2(1−	NUM
ma-53	313	4	l0‖x0	l0‖x0	NUM
ma-53	313	5	−	−	PROPN
ma-53	313	6	x∗‖	x∗‖	PROPN
ma-53	313	7	)	)	PUNCT
ma-53	313	8	≤	≤	NOUN
ma-53	314	1	ϕ1(‖x0	ϕ1(‖x0	SCONJ
ma-53	314	2	−	−	PROPN
ma-53	315	1	x∗‖)‖x0	x∗‖)‖x0	NOUN
ma-53	315	2	−	−	PROPN
ma-53	315	3	x∗‖	x∗‖	PROPN
ma-53	315	4	≤	≤	NUM
ma-53	315	5	‖x0	‖x0	NOUN
ma-53	316	1	−	−	PROPN
ma-53	316	2	x∗‖	x∗‖	X
ma-53	316	3	<	<	X
ma-53	316	4	r	r	X
ma-53	316	5	(	(	PUNCT
ma-53	316	6	3.10	3.10	NUM
ma-53	316	7	)	)	PUNCT
ma-53	316	8	so	so	ADV
ma-53	316	9	y0	y0	PROPN
ma-53	316	10	∈	∈	PROPN
ma-53	316	11	u(x∗	u(x∗	NOUN
ma-53	316	12	,	,	PUNCT
ma-53	316	13	r	r	NOUN
ma-53	316	14	)	)	PUNCT
ma-53	316	15	.	.	PUNCT
ma-53	317	1	next	next	ADV
ma-53	317	2	,	,	PUNCT
ma-53	317	3	we	we	PRON
ma-53	317	4	show	show	VERB
ma-53	317	5	linear	linear	ADJ
ma-53	317	6	operator	operator	NOUN
ma-53	317	7	a	a	PRON
ma-53	317	8	)	)	PUNCT
ma-53	317	9	is	be	AUX
ma-53	317	10	invertible	invertible	ADJ
ma-53	317	11	.	.	PUNCT
ma-53	318	1	indeed	indeed	ADV
ma-53	318	2	,	,	PUNCT
ma-53	318	3	using	use	VERB
ma-53	318	4	(	(	PUNCT
ma-53	318	5	3.2	3.2	NUM
ma-53	318	6	)	)	PUNCT
ma-53	318	7	,	,	PUNCT
ma-53	318	8	(	(	PUNCT
ma-53	318	9	3.6	3.6	NUM
ma-53	318	10	)	)	PUNCT
ma-53	318	11	,	,	PUNCT
ma-53	318	12	(	(	PUNCT
ma-53	318	13	h3	h3	NOUN
ma-53	318	14	)	)	PUNCT
ma-53	318	15	and(3.10	and(3.10	NOUN
ma-53	318	16	)	)	PUNCT
ma-53	318	17	,	,	PUNCT
ma-53	318	18	we	we	PRON
ma-53	318	19	get	get	VERB
ma-53	318	20	in	in	ADP
ma-53	318	21	turn	turn	NOUN
ma-53	318	22	that	that	SCONJ
ma-53	318	23	‖f	‖f	ADP
ma-53	318	24	′(x∗)−1(a0	′(x∗)−1(a0	VERB
ma-53	318	25	−	−	PROPN
ma-53	318	26	f	f	PROPN
ma-53	318	27	′(x∗))‖	′(x∗))‖	NOUN
ma-53	318	28	≤	≤	NOUN
ma-53	318	29	‖f	‖f	PUNCT
ma-53	318	30	′(x∗)−1([y0	′(x∗)−1([y0	NOUN
ma-53	318	31	,	,	PUNCT
ma-53	318	32	x0;f	x0;f	PROPN
ma-53	318	33	]	]	X
ma-53	318	34	−	−	PROPN
ma-53	318	35	f	f	PROPN
ma-53	318	36	′(x∗))‖	′(x∗))‖	NOUN
ma-53	318	37	+	+	PROPN
ma-53	318	38	‖f	‖f	ADJ
ma-53	318	39	′(x∗)−1([y0	′(x∗)−1([y0	NOUN
ma-53	318	40	,	,	PUNCT
ma-53	318	41	x0;f	x0;f	PROPN
ma-53	318	42	]	]	X
ma-53	318	43	−	−	PROPN
ma-53	318	44	f	f	PROPN
ma-53	318	45	′(x∗))‖	′(x∗))‖	PROPN
ma-53	318	46	+	+	PROPN
ma-53	318	47	‖f	‖f	ADJ
ma-53	318	48	′(x∗)−1(f	′(x∗)−1(f	ADJ
ma-53	318	49	′(x0)−	′(x0)−	PROPN
ma-53	318	50	f	f	PROPN
ma-53	318	51	′(x∗))‖	′(x∗))‖	NOUN
ma-53	318	52	≤	≤	NOUN
ma-53	319	1	2l1(‖y0	2l1(‖y0	NUM
ma-53	319	2	−	−	PROPN
ma-53	319	3	x∗‖+	x∗‖+	PROPN
ma-53	319	4	‖x0	‖x0	NOUN
ma-53	319	5	−	−	NOUN
ma-53	319	6	x∗‖	x∗‖	NUM
ma-53	319	7	)	)	PUNCT
ma-53	319	8	+	+	CCONJ
ma-53	319	9	l0‖x0	l0‖x0	NUM
ma-53	319	10	−	−	PROPN
ma-53	319	11	x∗‖	x∗‖	PROPN
ma-53	319	12	≤	≤	NUM
ma-53	319	13	2l1(1	2l1(1	NUM
ma-53	319	14	+	+	CCONJ
ma-53	319	15	ϕ1(‖x0	ϕ1(‖x0	PROPN
ma-53	319	16	−	−	PROPN
ma-53	319	17	x∗‖))‖x0	x∗‖))‖x0	PUNCT
ma-53	319	18	−	−	PROPN
ma-53	319	19	x∗‖+	x∗‖+	PUNCT
ma-53	319	20	l‖x0	l‖x0	VERB
ma-53	319	21	−	−	PROPN
ma-53	319	22	x∗‖	x∗‖	PROPN
ma-53	319	23	≤	≤	NUM
ma-53	319	24	p(‖x0	p(‖x0	NOUN
ma-53	319	25	−	−	PROPN
ma-53	319	26	x∗‖	x∗‖	NUM
ma-53	319	27	)	)	PUNCT
ma-53	319	28	≤	≤	NOUN
ma-53	319	29	p(r	p(r	NOUN
ma-53	319	30	)	)	PUNCT
ma-53	319	31	<	<	X
ma-53	319	32	1	1	NUM
ma-53	319	33	,	,	PUNCT
ma-53	319	34	so	so	ADV
ma-53	319	35	‖a−10	‖a−10	PROPN
ma-53	319	36	f	f	PROPN
ma-53	319	37	′(x∗)‖	′(x∗)‖	PUNCT
ma-53	319	38	≤	≤	NUM
ma-53	319	39	1	1	NUM
ma-53	319	40	1−	1−	NUM
ma-53	319	41	p(‖x0	p(‖x0	NOUN
ma-53	319	42	−	−	PROPN
ma-53	319	43	x∗‖	x∗‖	NUM
ma-53	319	44	)	)	PUNCT
ma-53	319	45	(	(	PUNCT
ma-53	319	46	3.11	3.11	NUM
ma-53	319	47	)	)	PUNCT
ma-53	319	48	and	and	CCONJ
ma-53	319	49	iterate	iterate	NOUN
ma-53	319	50	x1	x1	PROPN
ma-53	319	51	is	be	AUX
ma-53	319	52	well	well	ADV
ma-53	319	53	defined	define	VERB
ma-53	319	54	by	by	ADP
ma-53	319	55	the	the	DET
ma-53	319	56	second	second	ADJ
ma-53	319	57	substep	substep	NOUN
ma-53	319	58	of	of	ADP
ma-53	319	59	method	method	NOUN
ma-53	319	60	(	(	PUNCT
ma-53	319	61	1.2	1.2	NUM
ma-53	319	62	)	)	PUNCT
ma-53	319	63	for	for	ADP
ma-53	319	64	n	n	NOUN
ma-53	319	65	=	=	SYM
ma-53	319	66	0	0	NUM
ma-53	319	67	.	.	PUNCT
ma-53	320	1	then	then	ADV
ma-53	320	2	,	,	PUNCT
ma-53	320	3	we	we	PRON
ma-53	320	4	can	can	AUX
ma-53	320	5	write	write	VERB
ma-53	320	6	x1	x1	PRON
ma-53	320	7	−	−	PROPN
ma-53	320	8	x∗	x∗	PROPN
ma-53	320	9	=	=	SYM
ma-53	320	10	(	(	PUNCT
ma-53	320	11	y0	y0	NOUN
ma-53	320	12	−	−	NOUN
ma-53	320	13	x∗	x∗	PROPN
ma-53	320	14	−	−	PROPN
ma-53	320	15	f	f	PROPN
ma-53	320	16	′(y0)−1f	′(y0)−1f	PROPN
ma-53	320	17	(	(	PUNCT
ma-53	320	18	y0	y0	NOUN
ma-53	320	19	)	)	PUNCT
ma-53	320	20	)	)	PUNCT
ma-53	321	1	+	+	CCONJ
ma-53	321	2	f	f	NOUN
ma-53	321	3	′(y0	′(y0	NOUN
ma-53	321	4	)	)	PUNCT
ma-53	321	5	−1(a	−1(a	ADP
ma-53	321	6	)	)	PUNCT
ma-53	322	1	−	−	PROPN
ma-53	322	2	f	f	PROPN
ma-53	322	3	′(y0))a−10	′(y0))a−10	PROPN
ma-53	322	4	f	f	PROPN
ma-53	322	5	(	(	PUNCT
ma-53	322	6	y0	y0	PROPN
ma-53	322	7	)	)	PUNCT
ma-53	322	8	.	.	PUNCT
ma-53	323	1	(	(	PUNCT
ma-53	323	2	3.12	3.12	NUM
ma-53	323	3	)	)	PUNCT
ma-53	323	4	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	323	5	eur	eur	NOUN
ma-53	323	6	.	.	PUNCT
ma-53	324	1	j.	j.	PROPN
ma-53	324	2	math	math	PROPN
ma-53	324	3	.	.	PUNCT
ma-53	325	1	anal	anal	PROPN
ma-53	325	2	.	.	PUNCT
ma-53	326	1	10.28924	10.28924	NUM
ma-53	326	2	/	/	SYM
ma-53	326	3	ada	ada	PROPN
ma-53	326	4	/	/	SYM
ma-53	326	5	ma.2.3	ma.2.3	PROPN
ma-53	326	6	13using	13using	PROPN
ma-53	326	7	(	(	PUNCT
ma-53	326	8	3.2	3.2	NUM
ma-53	326	9	)	)	PUNCT
ma-53	326	10	,	,	PUNCT
ma-53	326	11	(	(	PUNCT
ma-53	326	12	3.7	3.7	NUM
ma-53	326	13	)	)	PUNCT
ma-53	326	14	,	,	PUNCT
ma-53	326	15	(	(	PUNCT
ma-53	326	16	h3	h3	NOUN
ma-53	326	17	)	)	PUNCT
ma-53	326	18	,	,	PUNCT
ma-53	326	19	(	(	PUNCT
ma-53	326	20	3.8	3.8	NUM
ma-53	326	21	)	)	PUNCT
ma-53	326	22	(	(	PUNCT
ma-53	326	23	for	for	ADP
ma-53	326	24	u	u	NOUN
ma-53	326	25	=	=	NOUN
ma-53	326	26	x0	x0	PROPN
ma-53	326	27	,	,	PUNCT
ma-53	326	28	y0	y0	PROPN
ma-53	326	29	)	)	PUNCT
ma-53	326	30	and	and	CCONJ
ma-53	326	31	(	(	PUNCT
ma-53	326	32	3.10)–(3.13	3.10)–(3.13	NUM
ma-53	326	33	)	)	PUNCT
ma-53	326	34	,	,	PUNCT
ma-53	326	35	we	we	PRON
ma-53	326	36	obtain	obtain	VERB
ma-53	326	37	in	in	ADP
ma-53	326	38	turn	turn	NOUN
ma-53	326	39	that	that	DET
ma-53	326	40	‖x1	‖x1	NOUN
ma-53	326	41	−	−	PROPN
ma-53	326	42	x∗‖	x∗‖	PROPN
ma-53	326	43	≤	≤	NUM
ma-53	326	44	‖y0	‖y0	PUNCT
ma-53	327	1	−	−	NOUN
ma-53	327	2	x∗	x∗	PROPN
ma-53	327	3	−	−	PROPN
ma-53	327	4	f	f	PROPN
ma-53	327	5	′(y0)−1f	′(y0)−1f	PROPN
ma-53	327	6	(	(	PUNCT
ma-53	327	7	y0)‖	y0)‖	X
ma-53	327	8	+	+	NOUN
ma-53	327	9	‖f	‖f	PRON
ma-53	327	10	′(y0)−1f	′(y0)−1f	NOUN
ma-53	327	11	′(x∗)‖‖f	′(x∗)‖‖f	PROPN
ma-53	327	12	′(x∗)−1(a0	′(x∗)−1(a0	VERB
ma-53	327	13	−	−	PROPN
ma-53	327	14	f	f	PROPN
ma-53	327	15	′(y0))‖‖a−10	′(y0))‖‖a−10	PROPN
ma-53	327	16	f	f	PROPN
ma-53	327	17	′(x∗)‖‖f	′(x∗)‖‖f	PROPN
ma-53	327	18	′(x∗)−1f	′(x∗)−1f	NOUN
ma-53	327	19	(	(	PUNCT
ma-53	327	20	y0)‖	y0)‖	NOUN
ma-53	327	21	≤	≤	NOUN
ma-53	327	22	l‖y0	l‖y0	ADV
ma-53	327	23	−	−	PROPN
ma-53	327	24	x∗‖2	x∗‖2	PROPN
ma-53	328	1	2(1−	2(1−	NUM
ma-53	328	2	l0‖x0	l0‖x0	NUM
ma-53	328	3	−	−	PROPN
ma-53	328	4	x∗‖	x∗‖	PROPN
ma-53	328	5	+	+	CCONJ
ma-53	328	6	(	(	PUNCT
ma-53	328	7	l2	l2	VERB
ma-53	328	8	+	+	CCONJ
ma-53	328	9	l3)‖y0	l3)‖y0	PROPN
ma-53	328	10	−	−	PUNCT
ma-53	328	11	x0‖l4‖y0	x0‖l4‖y0	X
ma-53	328	12	−	−	PROPN
ma-53	328	13	x∗‖	x∗‖	X
ma-53	328	14	(	(	PUNCT
ma-53	328	15	1−	1−	NUM
ma-53	328	16	l0‖y0	l0‖y0	PROPN
ma-53	328	17	−	−	PROPN
ma-53	328	18	x∗‖)(1−	x∗‖)(1−	NOUN
ma-53	328	19	p(‖x0	p(‖x0	NOUN
ma-53	328	20	−	−	PROPN
ma-53	328	21	x∗‖	x∗‖	NUM
ma-53	328	22	)	)	PUNCT
ma-53	328	23	)	)	PUNCT
ma-53	329	1	≤	≤	PUNCT
ma-53	330	1	ϕ2(‖x0	ϕ2(‖x0	INTJ
ma-53	330	2	−	−	PROPN
ma-53	331	1	x∗‖)‖x0	x∗‖)‖x0	NOUN
ma-53	331	2	−	−	PROPN
ma-53	331	3	x∗‖	x∗‖	PROPN
ma-53	331	4	≤	≤	NUM
ma-53	331	5	‖x0	‖x0	NOUN
ma-53	332	1	−	−	PROPN
ma-53	332	2	x∗‖	x∗‖	X
ma-53	332	3	<	<	X
ma-53	332	4	r	r	NOUN
ma-53	332	5	,	,	PUNCT
ma-53	332	6	so	so	CCONJ
ma-53	332	7	x1	x1	PROPN
ma-53	332	8	∈	∈	PROPN
ma-53	332	9	u(x∗	u(x∗	NOUN
ma-53	332	10	,	,	PUNCT
ma-53	332	11	r	r	NOUN
ma-53	332	12	)	)	PUNCT
ma-53	332	13	,	,	PUNCT
ma-53	332	14	where	where	SCONJ
ma-53	332	15	we	we	PRON
ma-53	332	16	also	also	ADV
ma-53	332	17	used	use	VERB
ma-53	332	18	‖f	‖f	SCONJ
ma-53	332	19	′(x∗)−1(a0	′(x∗)−1(a0	VERB
ma-53	332	20	−	−	PROPN
ma-53	332	21	f	f	X
ma-53	332	22	′(y0))‖	′(y0))‖	X
ma-53	332	23	≤	≤	NOUN
ma-53	332	24	‖f	‖f	ADP
ma-53	332	25	′(x∗)−1([y0	′(x∗)−1([y0	NOUN
ma-53	332	26	,	,	PUNCT
ma-53	332	27	x0;f	x0;f	PROPN
ma-53	332	28	]	]	X
ma-53	332	29	−	−	PROPN
ma-53	332	30	f	f	PROPN
ma-53	332	31	′(x0))‖	′(x0))‖	NUM
ma-53	333	1	+	+	PROPN
ma-53	333	2	‖f	‖f	ADJ
ma-53	333	3	′(x∗)−1([y0	′(x∗)−1([y0	NOUN
ma-53	333	4	,	,	PUNCT
ma-53	333	5	x0;f	x0;f	PROPN
ma-53	333	6	]	]	X
ma-53	333	7	−	−	PROPN
ma-53	333	8	f	f	PROPN
ma-53	333	9	′(y0))‖	′(y0))‖	X
ma-53	333	10	≤	≤	NOUN
ma-53	333	11	(	(	PUNCT
ma-53	333	12	l2	l2	VERB
ma-53	333	13	+	+	CCONJ
ma-53	334	1	l3)‖y0	l3)‖y0	PROPN
ma-53	334	2	−	−	NUM
ma-53	334	3	x0‖	x0‖	PROPN
ma-53	334	4	≤	≤	PROPN
ma-53	334	5	(	(	PUNCT
ma-53	334	6	l2	l2	NOUN
ma-53	334	7	+	+	CCONJ
ma-53	334	8	l3)(‖y0	l3)(‖y0	NOUN
ma-53	334	9	−	−	PROPN
ma-53	334	10	x∗‖+	x∗‖+	PROPN
ma-53	334	11	‖x0	‖x0	NOUN
ma-53	334	12	−	−	NOUN
ma-53	334	13	x∗‖	x∗‖	PROPN
ma-53	334	14	)	)	PUNCT
ma-53	334	15	≤	≤	NOUN
ma-53	334	16	(	(	PUNCT
ma-53	334	17	l2	l2	VERB
ma-53	334	18	+	+	CCONJ
ma-53	334	19	l3)(1	l3)(1	PROPN
ma-53	334	20	+	+	CCONJ
ma-53	334	21	ϕ1(‖x0	ϕ1(‖x0	ADJ
ma-53	334	22	−	−	PROPN
ma-53	334	23	x∗‖))‖x0	x∗‖))‖x0	PROPN
ma-53	335	1	−	−	PROPN
ma-53	335	2	x∗‖	x∗‖	PROPN
ma-53	335	3	,	,	PUNCT
ma-53	335	4	and	and	CCONJ
ma-53	335	5	‖f	‖f	ADP
ma-53	335	6	′(x∗)−1f	′(x∗)−1f	X
ma-53	335	7	(	(	PUNCT
ma-53	335	8	y0)‖	y0)‖	X
ma-53	335	9	=	=	SYM
ma-53	335	10	‖	‖	PROPN
ma-53	335	11	∫	∫	PROPN
ma-53	335	12	1	1	NUM
ma-53	335	13	0	0	NUM
ma-53	335	14	f	f	PROPN
ma-53	335	15	′(x∗	′(x∗	PROPN
ma-53	335	16	)	)	PUNCT
ma-53	335	17	−1f	−1f	PROPN
ma-53	335	18	′(x∗	′(x∗	PROPN
ma-53	335	19	+	+	CCONJ
ma-53	335	20	θ(y0	θ(y0	PROPN
ma-53	335	21	−	−	PROPN
ma-53	335	22	x∗))dθ(y0	x∗))dθ(y0	PROPN
ma-53	335	23	−	−	PROPN
ma-53	335	24	x∗)‖	x∗)‖	SYM
ma-53	335	25	≤	≤	NUM
ma-53	335	26	l4‖y0	l4‖y0	VERB
ma-53	335	27	−	−	PROPN
ma-53	335	28	x∗‖2	x∗‖2	PROPN
ma-53	335	29	.	.	PUNCT
ma-53	336	1	so	so	ADV
ma-53	336	2	,	,	PUNCT
ma-53	336	3	far	far	ADV
ma-53	336	4	showed	show	VERB
ma-53	336	5	‖y0	‖y0	PROPN
ma-53	336	6	−	−	PROPN
ma-53	336	7	x∗‖	x∗‖	PROPN
ma-53	336	8	≤	≤	PROPN
ma-53	336	9	ϕ1(‖x0	ϕ1(‖x0	SCONJ
ma-53	337	1	−	−	PROPN
ma-53	338	1	x∗‖)‖x0	x∗‖)‖x0	NOUN
ma-53	338	2	−	−	PROPN
ma-53	339	1	x∗‖	x∗‖	X
ma-53	339	2	<	<	X
ma-53	339	3	r	r	NOUN
ma-53	339	4	and	and	CCONJ
ma-53	339	5	‖x1	‖x1	NOUN
ma-53	339	6	−	−	PROPN
ma-53	339	7	x∗‖	x∗‖	PROPN
ma-53	339	8	≤	≤	NOUN
ma-53	340	1	ϕ2(‖x0	ϕ2(‖x0	INTJ
ma-53	340	2	−	−	PROPN
ma-53	341	1	x∗‖)‖x0	x∗‖)‖x0	NOUN
ma-53	341	2	−	−	PROPN
ma-53	342	1	x∗‖	x∗‖	X
ma-53	342	2	<	<	X
ma-53	342	3	r.	r.	PROPN
ma-53	342	4	by	by	ADP
ma-53	342	5	simply	simply	ADV
ma-53	342	6	replacing	replace	VERB
ma-53	342	7	x0	x0	NUM
ma-53	342	8	,	,	PUNCT
ma-53	342	9	y0	y0	PROPN
ma-53	342	10	,	,	PUNCT
ma-53	342	11	x1	x1	NUM
ma-53	342	12	by	by	ADP
ma-53	342	13	xm	xm	PROPN
ma-53	342	14	,	,	PUNCT
ma-53	342	15	ym	ym	PROPN
ma-53	342	16	,	,	PUNCT
ma-53	342	17	xm+1	xm+1	PROPN
ma-53	342	18	in	in	ADP
ma-53	342	19	the	the	DET
ma-53	342	20	preceding	precede	VERB
ma-53	342	21	calculations	calculation	NOUN
ma-53	342	22	,	,	PUNCT
ma-53	342	23	we	we	PRON
ma-53	342	24	get	get	VERB
ma-53	342	25	‖ym	‖ym	NUM
ma-53	342	26	−	−	PROPN
ma-53	342	27	x∗‖	x∗‖	SYM
ma-53	342	28	≤	≤	NUM
ma-53	343	1	ϕ1(‖xm	ϕ1(‖xm	PUNCT
ma-53	343	2	−	−	PROPN
ma-53	343	3	x∗‖)‖xm	x∗‖)‖xm	PUNCT
ma-53	344	1	−	−	PROPN
ma-53	344	2	x∗‖	x∗‖	X
ma-53	344	3	<	<	X
ma-53	344	4	r	r	NOUN
ma-53	344	5	and	and	CCONJ
ma-53	344	6	‖xm+1	‖xm+1	NUM
ma-53	344	7	−	−	PROPN
ma-53	344	8	x∗‖	x∗‖	PROPN
ma-53	344	9	≤	≤	PROPN
ma-53	345	1	ϕ2(‖xm	ϕ2(‖xm	PUNCT
ma-53	346	1	−	−	PROPN
ma-53	346	2	x∗‖)‖xm	x∗‖)‖xm	PUNCT
ma-53	347	1	−	−	PROPN
ma-53	347	2	x∗‖	x∗‖	X
ma-53	347	3	<	<	X
ma-53	347	4	r.	r.	PROPN
ma-53	347	5	then	then	ADV
ma-53	347	6	,	,	PUNCT
ma-53	347	7	from	from	ADP
ma-53	347	8	the	the	DET
ma-53	347	9	estimation	estimation	NOUN
ma-53	347	10	‖xm+1	‖xm+1	PUNCT
ma-53	347	11	−	−	PROPN
ma-53	347	12	x∗‖	x∗‖	PROPN
ma-53	347	13	≤	≤	NOUN
ma-53	348	1	α‖xm	α‖xm	PROPN
ma-53	348	2	−	−	PROPN
ma-53	349	1	x∗‖	x∗‖	X
ma-53	349	2	<	<	X
ma-53	349	3	r	r	NOUN
ma-53	349	4	,	,	PUNCT
ma-53	349	5	(	(	PUNCT
ma-53	349	6	3.13	3.13	NUM
ma-53	349	7	)	)	PUNCT
ma-53	349	8	where	where	SCONJ
ma-53	349	9	α	α	NOUN
ma-53	349	10	=	=	PUNCT
ma-53	349	11	ϕ2(‖x0	ϕ2(‖x0	PROPN
ma-53	349	12	−	−	PROPN
ma-53	349	13	x∗‖	x∗‖	PROPN
ma-53	349	14	)	)	PUNCT
ma-53	349	15	∈	∈	PROPN
ma-53	350	1	[	[	X
ma-53	350	2	0	0	NUM
ma-53	350	3	,	,	PUNCT
ma-53	350	4	1	1	NUM
ma-53	350	5	)	)	PUNCT
ma-53	350	6	,	,	PUNCT
ma-53	350	7	limm−→∞	limm−→∞	NOUN
ma-53	350	8	xm	xm	PROPN
ma-53	350	9	=	=	PUNCT
ma-53	350	10	x∗	x∗	PROPN
ma-53	350	11	and	and	CCONJ
ma-53	350	12	ym	ym	PROPN
ma-53	350	13	,	,	PUNCT
ma-53	350	14	xm+1	xm+1	PROPN
ma-53	350	15	∈	∈	PROPN
ma-53	350	16	u(x∗	u(x∗	NOUN
ma-53	350	17	,	,	PUNCT
ma-53	350	18	r	r	NOUN
ma-53	350	19	)	)	PUNCT
ma-53	350	20	.	.	PUNCT
ma-53	351	1	�	�	PROPN
ma-53	351	2	remark	remark	VERB
ma-53	351	3	3.2	3.2	NUM
ma-53	351	4	.	.	PUNCT
ma-53	352	1	by	by	ADP
ma-53	352	2	the	the	DET
ma-53	352	3	definition	definition	NOUN
ma-53	352	4	of	of	ADP
ma-53	352	5	r	r	NOUN
ma-53	352	6	,	,	PUNCT
ma-53	352	7	we	we	PRON
ma-53	352	8	see	see	VERB
ma-53	352	9	that	that	SCONJ
ma-53	352	10	r	r	NOUN
ma-53	352	11	≤	≤	NUM
ma-53	352	12	ra	ra	NOUN
ma-53	352	13	.	.	PUNCT
ma-53	353	1	(	(	PUNCT
ma-53	353	2	3.14	3.14	NUM
ma-53	353	3	)	)	PUNCT
ma-53	353	4	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	353	5	eur	eur	NOUN
ma-53	353	6	.	.	PUNCT
ma-53	354	1	j.	j.	PROPN
ma-53	354	2	math	math	PROPN
ma-53	354	3	.	.	PUNCT
ma-53	355	1	anal	anal	PROPN
ma-53	355	2	.	.	PUNCT
ma-53	356	1	10.28924	10.28924	NUM
ma-53	356	2	/	/	SYM
ma-53	356	3	ada	ada	PROPN
ma-53	356	4	/	/	SYM
ma-53	356	5	ma.2.3	ma.2.3	PROPN
ma-53	356	6	14	14	NUM
ma-53	356	7	parameter	parameter	NOUN
ma-53	356	8	ra	ra	PROPN
ma-53	356	9	was	be	AUX
ma-53	356	10	shown	show	VERB
ma-53	356	11	in	in	ADP
ma-53	356	12	[	[	X
ma-53	356	13	4	4	NUM
ma-53	356	14	]	]	PUNCT
ma-53	356	15	to	to	PART
ma-53	356	16	be	be	AUX
ma-53	356	17	a	a	DET
ma-53	356	18	convergence	convergence	NOUN
ma-53	356	19	radius	radius	NOUN
ma-53	356	20	for	for	ADP
ma-53	356	21	newton	newton	PROPN
ma-53	356	22	’s	’s	PART
ma-53	356	23	method	method	NOUN
ma-53	356	24	.	.	PUNCT
ma-53	357	1	notice	notice	VERB
ma-53	357	2	the	the	DET
ma-53	357	3	radius	radius	NOUN
ma-53	357	4	of	of	ADP
ma-53	357	5	convergence	convergence	NOUN
ma-53	357	6	for	for	ADP
ma-53	357	7	newton	newton	PROPN
ma-53	357	8	’s	’s	PART
ma-53	357	9	method	method	NOUN
ma-53	357	10	given	give	VERB
ma-53	357	11	independently	independently	ADV
ma-53	357	12	by	by	ADP
ma-53	357	13	traub	traub	PROPN
ma-53	358	1	[	[	X
ma-53	358	2	35	35	NUM
ma-53	358	3	]	]	PUNCT
ma-53	359	1	and	and	CCONJ
ma-53	359	2	rheinbold	rheinbold	VERB
ma-53	359	3	[	[	PUNCT
ma-53	359	4	29	29	NUM
ma-53	359	5	]	]	PUNCT
ma-53	359	6	is	be	AUX
ma-53	359	7	rtr	rtr	NOUN
ma-53	359	8	=	=	SYM
ma-53	359	9	2	2	NUM
ma-53	359	10	3m1	3m1	NUM
ma-53	359	11	,	,	PUNCT
ma-53	359	12	where	where	SCONJ
ma-53	359	13	l1	l1	PROPN
ma-53	359	14	is	be	AUX
ma-53	359	15	the	the	DET
ma-53	359	16	lipschitz	lipschitz	NOUN
ma-53	359	17	constant	constant	ADJ
ma-53	359	18	on	on	ADP
ma-53	359	19	ω	ω	NUM
ma-53	359	20	.	.	PUNCT
ma-53	360	1	so	so	ADV
ma-53	360	2	,	,	PUNCT
ma-53	360	3	we	we	PRON
ma-53	360	4	have	have	VERB
ma-53	360	5	rtr	rtr	PROPN
ma-53	360	6	≤	≤	PROPN
ma-53	360	7	ra	ra	PROPN
ma-53	360	8	,	,	PUNCT
ma-53	360	9	since	since	SCONJ
ma-53	360	10	l	l	NOUN
ma-53	360	11	≤	≤	X
ma-53	360	12	l1	l1	PROPN
ma-53	360	13	and	and	CCONJ
ma-53	360	14	l0	l0	PROPN
ma-53	360	15	≤	≤	NOUN
ma-53	360	16	m1	m1	NOUN
ma-53	360	17	.	.	PUNCT
ma-53	361	1	4	4	X
ma-53	361	2	.	.	X
ma-53	361	3	numerical	numerical	ADJ
ma-53	361	4	experiments	experiment	NOUN
ma-53	361	5	we	we	PRON
ma-53	361	6	provide	provide	VERB
ma-53	361	7	some	some	DET
ma-53	361	8	examples	example	NOUN
ma-53	361	9	,	,	PUNCT
ma-53	361	10	showing	show	VERB
ma-53	361	11	that	that	SCONJ
ma-53	361	12	the	the	DET
ma-53	361	13	old	old	ADJ
ma-53	361	14	convergence	convergence	NOUN
ma-53	361	15	criteria	criterion	NOUN
ma-53	361	16	are	be	AUX
ma-53	361	17	not	not	PART
ma-53	361	18	verified	verify	VERB
ma-53	361	19	but	but	CCONJ
ma-53	361	20	oursare	oursare	PROPN
ma-53	361	21	.	.	PUNCT
ma-53	361	22	example	example	NOUN
ma-53	361	23	4.1	4.1	NUM
ma-53	361	24	.	.	PUNCT
ma-53	362	1	define	define	VERB
ma-53	362	2	function	function	NOUN
ma-53	362	3	f	f	PROPN
ma-53	362	4	(	(	PUNCT
ma-53	362	5	t	t	PROPN
ma-53	362	6	)	)	PUNCT
ma-53	362	7	=	=	SYM
ma-53	363	1	θ0	θ0	PROPN
ma-53	363	2	t	t	PROPN
ma-53	363	3	+	+	CCONJ
ma-53	363	4	θ1	θ1	NOUN
ma-53	363	5	+	+	CCONJ
ma-53	363	6	θ2	θ2	PROPN
ma-53	363	7	sin	sin	PROPN
ma-53	363	8	θ3	θ3	PROPN
ma-53	363	9	t	t	PROPN
ma-53	363	10	,	,	PUNCT
ma-53	363	11	t0	t0	X
ma-53	363	12	=	=	SYM
ma-53	363	13	0	0	PROPN
ma-53	363	14	,	,	PUNCT
ma-53	363	15	where	where	SCONJ
ma-53	363	16	θj	θj	ADV
ma-53	363	17	,	,	PUNCT
ma-53	363	18	j	j	PROPN
ma-53	363	19	=	=	SYM
ma-53	363	20	0	0	NUM
ma-53	363	21	,	,	PUNCT
ma-53	363	22	1	1	NUM
ma-53	363	23	,	,	PUNCT
ma-53	363	24	2	2	NUM
ma-53	363	25	,	,	PUNCT
ma-53	363	26	3	3	NUM
ma-53	363	27	are	be	AUX
ma-53	363	28	parameters	parameter	NOUN
ma-53	363	29	.	.	PUNCT
ma-53	364	1	then	then	ADV
ma-53	364	2	,	,	PUNCT
ma-53	364	3	clearly	clearly	ADV
ma-53	364	4	for	for	SCONJ
ma-53	364	5	θ3	θ3	PROPN
ma-53	364	6	large	large	ADJ
ma-53	364	7	and	and	CCONJ
ma-53	364	8	θ2	θ2	ADV
ma-53	364	9	small	small	ADJ
ma-53	364	10	,	,	PUNCT
ma-53	364	11	l0k	l0k	PROPN
ma-53	364	12	can	can	AUX
ma-53	364	13	be	be	AUX
ma-53	364	14	small	small	ADJ
ma-53	364	15	(	(	PUNCT
ma-53	364	16	arbitrarily	arbitrarily	ADV
ma-53	364	17	)	)	PUNCT
ma-53	364	18	.	.	PUNCT
ma-53	365	1	example	example	NOUN
ma-53	366	1	4.2	4.2	NUM
ma-53	366	2	.	.	PUNCT
ma-53	367	1	let	let	VERB
ma-53	367	2	b	b	NOUN
ma-53	367	3	=	=	SYM
ma-53	367	4	b1	b1	PROPN
ma-53	367	5	=	=	SYM
ma-53	367	6	u[0	u[0	PROPN
ma-53	367	7	,	,	PUNCT
ma-53	367	8	1	1	NUM
ma-53	367	9	]	]	PUNCT
ma-53	367	10	the	the	DET
ma-53	367	11	domain	domain	NOUN
ma-53	367	12	of	of	ADP
ma-53	367	13	functions	function	NOUN
ma-53	367	14	given	give	VERB
ma-53	367	15	on	on	ADP
ma-53	367	16	[	[	X
ma-53	367	17	0	0	NUM
ma-53	367	18	,	,	PUNCT
ma-53	367	19	1	1	NUM
ma-53	367	20	]	]	PUNCT
ma-53	367	21	which	which	PRON
ma-53	367	22	are	be	AUX
ma-53	367	23	continuous	continuous	ADJ
ma-53	367	24	.	.	PUNCT
ma-53	368	1	we	we	PRON
ma-53	368	2	consider	consider	VERB
ma-53	368	3	the	the	DET
ma-53	368	4	max	max	NOUN
ma-53	368	5	-	-	PUNCT
ma-53	368	6	norm	norm	NOUN
ma-53	368	7	.	.	PUNCT
ma-53	369	1	choose	choose	VERB
ma-53	369	2	ω	ω	NUM
ma-53	369	3	=	=	SYM
ma-53	369	4	b(0	b(0	PROPN
ma-53	369	5	,	,	PUNCT
ma-53	369	6	d	d	NOUN
ma-53	369	7	)	)	PUNCT
ma-53	369	8	,	,	PUNCT
ma-53	370	1	d	d	X
ma-53	370	2	>	>	X
ma-53	370	3	1	1	X
ma-53	370	4	.	.	PUNCT
ma-53	370	5	define	define	VERB
ma-53	370	6	f	f	PROPN
ma-53	370	7	on	on	ADP
ma-53	370	8	ω	ω	PROPN
ma-53	370	9	be	be	PROPN
ma-53	370	10	f	f	PROPN
ma-53	370	11	(	(	PUNCT
ma-53	370	12	x)(s	x)(s	PROPN
ma-53	370	13	)	)	PUNCT
ma-53	371	1	=	=	PUNCT
ma-53	371	2	x(s)−	x(s)−	PROPN
ma-53	371	3	w(s)−	w(s)−	VERB
ma-53	371	4	ξ	ξ	X
ma-53	371	5	∫	∫	PROPN
ma-53	371	6	1	1	NUM
ma-53	371	7	0	0	NUM
ma-53	371	8	k(s	k(s	PROPN
ma-53	371	9	,	,	PUNCT
ma-53	371	10	t)x3(t)dt	t)x3(t)dt	NOUN
ma-53	371	11	,	,	PUNCT
ma-53	371	12	(	(	PUNCT
ma-53	371	13	4.1	4.1	NUM
ma-53	371	14	)	)	PUNCT
ma-53	371	15	x	x	SYM
ma-53	371	16	∈	∈	PROPN
ma-53	371	17	b	b	PROPN
ma-53	371	18	,	,	PUNCT
ma-53	371	19	s	s	NOUN
ma-53	371	20	∈	∈	PROPN
ma-53	372	1	[	[	X
ma-53	372	2	0	0	NUM
ma-53	372	3	,	,	PUNCT
ma-53	372	4	1	1	NUM
ma-53	372	5	]	]	PUNCT
ma-53	372	6	,	,	PUNCT
ma-53	372	7	w	w	PROPN
ma-53	372	8	∈	∈	PROPN
ma-53	372	9	b	b	PROPN
ma-53	372	10	is	be	AUX
ma-53	372	11	given	give	VERB
ma-53	372	12	,	,	PUNCT
ma-53	372	13	ξ	ξ	X
ma-53	372	14	is	be	AUX
ma-53	372	15	a	a	DET
ma-53	372	16	parameter	parameter	NOUN
ma-53	372	17	and	and	CCONJ
ma-53	372	18	k	k	PROPN
ma-53	372	19	is	be	AUX
ma-53	372	20	the	the	DET
ma-53	372	21	green	green	PROPN
ma-53	372	22	’s	’s	PART
ma-53	372	23	kernel	kernel	NOUN
ma-53	372	24	given	give	VERB
ma-53	372	25	by	by	ADP
ma-53	372	26	k(s2	k(s2	PROPN
ma-53	372	27	,	,	PUNCT
ma-53	372	28	s1	s1	NOUN
ma-53	372	29	)	)	PUNCT
ma-53	373	1	=	=	SYM
ma-53	373	2	{	{	PUNCT
ma-53	373	3	(	(	PUNCT
ma-53	373	4	1−	1−	NUM
ma-53	373	5	s2)s1	s2)s1	NOUN
ma-53	373	6	,	,	PUNCT
ma-53	373	7	s1	s1	PROPN
ma-53	373	8	≤	≤	PROPN
ma-53	373	9	s2	s2	PROPN
ma-53	373	10	s2(1−	s2(1−	PROPN
ma-53	373	11	s1	s1	PROPN
ma-53	373	12	)	)	PUNCT
ma-53	373	13	,	,	PUNCT
ma-53	373	14	s2	s2	VERB
ma-53	373	15	≤	≤	NUM
ma-53	373	16	s1	s1	NOUN
ma-53	373	17	.	.	PUNCT
ma-53	374	1	by	by	ADP
ma-53	374	2	(	(	PUNCT
ma-53	374	3	4.1	4.1	NUM
ma-53	374	4	)	)	PUNCT
ma-53	374	5	,	,	PUNCT
ma-53	374	6	we	we	PRON
ma-53	374	7	have	have	VERB
ma-53	374	8	(	(	PUNCT
ma-53	374	9	f	f	PROPN
ma-53	374	10	′(x)(z))(s	′(x)(z))(s	NOUN
ma-53	374	11	)	)	PUNCT
ma-53	375	1	=	=	PUNCT
ma-53	376	1	z(s)−	z(s)−	PROPN
ma-53	376	2	3ξ	3ξ	NUM
ma-53	376	3	∫	∫	PROPN
ma-53	376	4	1	1	NUM
ma-53	376	5	0	0	NUM
ma-53	376	6	k(s	k(s	PROPN
ma-53	376	7	,	,	PUNCT
ma-53	376	8	t)x2(t)z(t)dt	t)x2(t)z(t)dt	NOUN
ma-53	376	9	,	,	PUNCT
ma-53	376	10	t	t	PROPN
ma-53	376	11	∈	∈	PROPN
ma-53	376	12	bs	bs	PROPN
ma-53	376	13	∈	∈	PROPN
ma-53	377	1	[	[	X
ma-53	377	2	0	0	NUM
ma-53	377	3	,	,	PUNCT
ma-53	377	4	1	1	NUM
ma-53	377	5	]	]	PUNCT
ma-53	377	6	.	.	PUNCT
ma-53	378	1	consider	consider	VERB
ma-53	378	2	x0(s	x0(s	PRON
ma-53	378	3	)	)	PUNCT
ma-53	378	4	=	=	SYM
ma-53	378	5	w(s	w(s	PROPN
ma-53	378	6	)	)	PUNCT
ma-53	378	7	=	=	SYM
ma-53	378	8	1	1	NUM
ma-53	378	9	and	and	CCONJ
ma-53	378	10	|ξ|	|ξ|	PROPN
ma-53	378	11	<	<	X
ma-53	378	12	8	8	NUM
ma-53	378	13	3	3	NUM
ma-53	378	14	.	.	PUNCT
ma-53	379	1	we	we	PRON
ma-53	379	2	get	get	VERB
ma-53	379	3	‖i	‖i	NOUN
ma-53	379	4	−	−	NOUN
ma-53	380	1	f	f	X
ma-53	381	1	′(x0)‖	′(x0)‖	NOUN
ma-53	381	2	<	<	X
ma-53	381	3	3	3	NUM
ma-53	381	4	8	8	NUM
ma-53	381	5	|ξ|	|ξ|	NOUN
ma-53	381	6	,	,	PUNCT
ma-53	381	7	f	f	PROPN
ma-53	381	8	′(x0)−1	′(x0)−1	PROPN
ma-53	381	9	∈	∈	PROPN
ma-53	381	10	l(b1b	l(b1b	X
ma-53	381	11	)	)	PUNCT
ma-53	381	12	,	,	PUNCT
ma-53	381	13	‖f	‖f	ADP
ma-53	381	14	′(x0)−1‖	′(x0)−1‖	VERB
ma-53	381	15	≤	≤	ADV
ma-53	381	16	8	8	NUM
ma-53	381	17	8−	8−	NUM
ma-53	381	18	3|ξ|	3|ξ|	NUM
ma-53	381	19	,	,	PUNCT
ma-53	381	20	η	η	PROPN
ma-53	381	21	=	=	SYM
ma-53	381	22	|ξ|	|ξ|	PROPN
ma-53	381	23	8−	8−	NUM
ma-53	381	24	3|ξ|	3|ξ|	NUM
ma-53	381	25	,	,	PUNCT
ma-53	381	26	l0	l0	PROPN
ma-53	381	27	=	=	SYM
ma-53	381	28	12|ξ|	12|ξ|	NUM
ma-53	381	29	8−	8−	NUM
ma-53	381	30	3|ξ|	3|ξ|	NUM
ma-53	381	31	,	,	PUNCT
ma-53	381	32	k	k	PROPN
ma-53	381	33	=	=	SYM
ma-53	381	34	6d	6d	NUM
ma-53	381	35	|ξ|	|ξ|	PROPN
ma-53	381	36	8−3|ξ|	8−3|ξ|	PROPN
ma-53	381	37	,	,	PUNCT
ma-53	381	38	k1	k1	NOUN
ma-53	381	39	=	=	SYM
ma-53	381	40	l0	l0	PROPN
ma-53	381	41	2	2	NUM
ma-53	381	42	and	and	CCONJ
ma-53	381	43	k2	k2	PROPN
ma-53	381	44	=	=	SYM
ma-53	381	45	k	k	PROPN
ma-53	381	46	2	2	X
ma-53	381	47	=	=	SYM
ma-53	381	48	k3	k3	VERB
ma-53	381	49	.	.	PUNCT
ma-53	382	1	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	382	2	eur	eur	PROPN
ma-53	382	3	.	.	PUNCT
ma-53	383	1	j.	j.	PROPN
ma-53	383	2	math	math	PROPN
ma-53	383	3	.	.	PUNCT
ma-53	384	1	anal	anal	PROPN
ma-53	384	2	.	.	PUNCT
ma-53	385	1	10.28924	10.28924	NUM
ma-53	385	2	/	/	SYM
ma-53	385	3	ada	ada	PROPN
ma-53	385	4	/	/	SYM
ma-53	385	5	ma.2.3	ma.2.3	PROPN
ma-53	385	6	15	15	NUM
ma-53	385	7	example	example	NOUN
ma-53	385	8	4.3	4.3	NUM
ma-53	385	9	.	.	PUNCT
ma-53	386	1	let	let	VERB
ma-53	386	2	b	b	NOUN
ma-53	386	3	=	=	SYM
ma-53	386	4	b1	b1	PROPN
ma-53	386	5	=	=	SYM
ma-53	386	6	r3	r3	PROPN
ma-53	386	7	and	and	CCONJ
ma-53	386	8	ω	ω	NUM
ma-53	386	9	be	be	AUX
ma-53	386	10	as	as	ADP
ma-53	386	11	in	in	ADP
ma-53	386	12	the	the	DET
ma-53	386	13	example	example	NOUN
ma-53	386	14	4.2	4.2	NUM
ma-53	386	15	.	.	PUNCT
ma-53	387	1	it	it	PRON
ma-53	387	2	is	be	AUX
ma-53	387	3	well	well	ADV
ma-53	387	4	known	know	VERB
ma-53	387	5	that	that	SCONJ
ma-53	387	6	the	the	DET
ma-53	387	7	boundary	boundary	ADJ
ma-53	387	8	value	value	NOUN
ma-53	387	9	problem	problem	NOUN
ma-53	387	10	[	[	X
ma-53	387	11	16	16	NUM
ma-53	387	12	]	]	X
ma-53	387	13	ψ(0	ψ(0	NOUN
ma-53	387	14	)	)	PUNCT
ma-53	387	15	=	=	SYM
ma-53	387	16	0	0	NUM
ma-53	387	17	,	,	PUNCT
ma-53	387	18	ψ(1	ψ(1	PROPN
ma-53	387	19	)	)	PUNCT
ma-53	387	20	=	=	SYM
ma-53	387	21	1	1	NUM
ma-53	387	22	,	,	PUNCT
ma-53	387	23	ψ′′	ψ′′	X
ma-53	387	24	=	=	SYM
ma-53	387	25	−ψ	−ψ	NOUN
ma-53	387	26	−	−	PROPN
ma-53	387	27	τψ2	τψ2	NOUN
ma-53	387	28	can	can	AUX
ma-53	387	29	be	be	AUX
ma-53	387	30	given	give	VERB
ma-53	387	31	as	as	ADP
ma-53	387	32	a	a	DET
ma-53	387	33	hammerstein	hammerstein	NOUN
ma-53	387	34	-	-	PUNCT
ma-53	387	35	like	like	ADJ
ma-53	387	36	nonlinear	nonlinear	ADJ
ma-53	387	37	integral	integral	ADJ
ma-53	387	38	equation	equation	NOUN
ma-53	387	39	ψ(s	ψ(s	PROPN
ma-53	387	40	)	)	PUNCT
ma-53	388	1	=	=	SYM
ma-53	388	2	s	s	PART
ma-53	389	1	+	+	NUM
ma-53	389	2	∫	∫	PROPN
ma-53	389	3	1	1	NUM
ma-53	389	4	0	0	NUM
ma-53	389	5	k(s	k(s	PROPN
ma-53	389	6	,	,	PUNCT
ma-53	389	7	t)(ψ3(t	t)(ψ3(t	X
ma-53	389	8	)	)	PUNCT
ma-53	390	1	+	+	NUM
ma-53	390	2	τψ2(t))dt	τψ2(t))dt	PROPN
ma-53	390	3	where	where	SCONJ
ma-53	390	4	τ	τ	PROPN
ma-53	390	5	is	be	AUX
ma-53	390	6	a	a	DET
ma-53	390	7	parameter	parameter	NOUN
ma-53	390	8	.	.	PUNCT
ma-53	391	1	then	then	ADV
ma-53	391	2	,	,	PUNCT
ma-53	391	3	define	define	VERB
ma-53	391	4	f	f	X
ma-53	391	5	:	:	PUNCT
ma-53	391	6	ω	ω	NUM
ma-53	391	7	−→	−→	NOUN
ma-53	391	8	t2	t2	NOUN
ma-53	391	9	by	by	ADP
ma-53	391	10	[	[	X
ma-53	391	11	f	f	X
ma-53	391	12	(	(	PUNCT
ma-53	391	13	x)](s	x)](s	PROPN
ma-53	391	14	)	)	PUNCT
ma-53	392	1	=	=	PUNCT
ma-53	393	1	x(s)−	x(s)−	PROPN
ma-53	393	2	s	s	PART
ma-53	393	3	−	−	NOUN
ma-53	393	4	∫	∫	PROPN
ma-53	393	5	1	1	NUM
ma-53	393	6	0	0	NUM
ma-53	393	7	k(s	k(s	PROPN
ma-53	393	8	,	,	PUNCT
ma-53	393	9	t)(x3(t	t)(x3(t	NUM
ma-53	393	10	)	)	PUNCT
ma-53	393	11	+	+	NUM
ma-53	393	12	τx2(t))dt	τx2(t))dt	NOUN
ma-53	393	13	.	.	PUNCT
ma-53	394	1	choose	choose	VERB
ma-53	394	2	x0(s	x0(s	X
ma-53	394	3	)	)	PUNCT
ma-53	394	4	=	=	SYM
ma-53	394	5	s	s	PROPN
ma-53	394	6	and	and	CCONJ
ma-53	394	7	ω	ω	NUM
ma-53	394	8	=	=	SYM
ma-53	394	9	u(x0	u(x0	PROPN
ma-53	394	10	,	,	PUNCT
ma-53	394	11	r0	r0	NOUN
ma-53	394	12	)	)	PUNCT
ma-53	394	13	.	.	PUNCT
ma-53	395	1	then	then	ADV
ma-53	395	2	,	,	PUNCT
ma-53	395	3	clearly	clearly	ADV
ma-53	395	4	u(x0	u(x0	NOUN
ma-53	395	5	,	,	PUNCT
ma-53	395	6	r0	r0	NOUN
ma-53	395	7	)	)	PUNCT
ma-53	395	8	⊂	⊂	PROPN
ma-53	396	1	u(0	u(0	PROPN
ma-53	396	2	,	,	PUNCT
ma-53	396	3	r0	r0	NOUN
ma-53	396	4	+	+	NOUN
ma-53	396	5	1	1	NUM
ma-53	396	6	)	)	PUNCT
ma-53	396	7	,	,	PUNCT
ma-53	396	8	since	since	SCONJ
ma-53	396	9	‖x0‖	‖x0‖	PROPN
ma-53	396	10	=	=	SYM
ma-53	396	11	1	1	X
ma-53	396	12	.	.	PUNCT
ma-53	396	13	suppose	suppose	VERB
ma-53	396	14	2τ	2τ	NUM
ma-53	396	15	<	<	X
ma-53	396	16	5	5	NUM
ma-53	396	17	.	.	PUNCT
ma-53	397	1	then	then	ADV
ma-53	397	2	,	,	PUNCT
ma-53	397	3	by	by	ADP
ma-53	397	4	conditions	condition	NOUN
ma-53	397	5	(	(	PUNCT
ma-53	397	6	a	a	X
ma-53	397	7	)	)	PUNCT
ma-53	397	8	are	be	AUX
ma-53	397	9	satisfied	satisfied	ADJ
ma-53	397	10	for	for	ADP
ma-53	397	11	l0	l0	NOUN
ma-53	397	12	=	=	SYM
ma-53	397	13	2τ+3r0	2τ+3r0	NUM
ma-53	397	14	+	+	PROPN
ma-53	397	15	6	6	NUM
ma-53	397	16	8	8	NUM
ma-53	397	17	,	,	PUNCT
ma-53	397	18	k	k	X
ma-53	397	19	=	=	PUNCT
ma-53	397	20	τ+6r0	τ+6r0	PROPN
ma-53	398	1	+	+	NOUN
ma-53	398	2	3	3	NUM
ma-53	398	3	4	4	NUM
ma-53	398	4	,	,	PUNCT
ma-53	398	5	k1	k1	NOUN
ma-53	398	6	=	=	SYM
ma-53	398	7	l0	l0	PROPN
ma-53	398	8	2	2	NUM
ma-53	398	9	and	and	CCONJ
ma-53	398	10	k2	k2	PROPN
ma-53	398	11	=	=	SYM
ma-53	398	12	k	k	PROPN
ma-53	398	13	2	2	X
ma-53	398	14	=	=	SYM
ma-53	398	15	k3	k3	PROPN
ma-53	398	16	.	.	PUNCT
ma-53	399	1	and	and	CCONJ
ma-53	399	2	η	η	PROPN
ma-53	399	3	=	=	PROPN
ma-53	399	4	1+τ	1+τ	NUM
ma-53	399	5	5−2τ	5−2τ	NUM
ma-53	399	6	.	.	PUNCT
ma-53	400	1	notice	notice	VERB
ma-53	400	2	that	that	SCONJ
ma-53	401	1	l0	l0	PROPN
ma-53	401	2	<	<	X
ma-53	401	3	k.	k.	PROPN
ma-53	402	1	the	the	DET
ma-53	402	2	rest	rest	NOUN
ma-53	402	3	of	of	ADP
ma-53	402	4	the	the	DET
ma-53	402	5	examples	example	NOUN
ma-53	402	6	are	be	AUX
ma-53	402	7	given	give	VERB
ma-53	402	8	for	for	ADP
ma-53	402	9	the	the	DET
ma-53	402	10	local	local	ADJ
ma-53	402	11	convergence	convergence	NOUN
ma-53	402	12	study	study	NOUN
ma-53	402	13	of	of	ADP
ma-53	402	14	newton	newton	PROPN
ma-53	402	15	’s	’s	PART
ma-53	402	16	method	method	NOUN
ma-53	402	17	.	.	PUNCT
ma-53	403	1	example	example	NOUN
ma-53	403	2	4.4	4.4	NUM
ma-53	403	3	.	.	PUNCT
ma-53	404	1	let	let	VERB
ma-53	404	2	b	b	NOUN
ma-53	404	3	=	=	SYM
ma-53	404	4	b1	b1	PROPN
ma-53	404	5	=	=	SYM
ma-53	404	6	r3	r3	PROPN
ma-53	404	7	,	,	PUNCT
ma-53	404	8	ω	ω	NOUN
ma-53	404	9	=	=	SYM
ma-53	404	10	u[0	u[0	PROPN
ma-53	404	11	,	,	PUNCT
ma-53	404	12	1	1	NUM
ma-53	404	13	]	]	PUNCT
ma-53	404	14	and	and	CCONJ
ma-53	404	15	x∗	x∗	PROPN
ma-53	404	16	=	=	SYM
ma-53	404	17	(	(	PUNCT
ma-53	404	18	0	0	NUM
ma-53	404	19	,	,	PUNCT
ma-53	404	20	0	0	NUM
ma-53	404	21	,	,	PUNCT
ma-53	404	22	0)tr	0)tr	PROPN
ma-53	404	23	.	.	PUNCT
ma-53	405	1	define	define	VERB
ma-53	405	2	mapping	mapping	NOUN
ma-53	405	3	e	e	NOUN
ma-53	405	4	on	on	ADP
ma-53	405	5	ω	ω	PROPN
ma-53	405	6	for	for	ADP
ma-53	405	7	λ	λ	PROPN
ma-53	405	8	=	=	SYM
ma-53	405	9	(	(	PUNCT
ma-53	405	10	λ1	λ1	ADJ
ma-53	405	11	,	,	PUNCT
ma-53	405	12	λ2	λ2	PROPN
ma-53	405	13	,	,	PUNCT
ma-53	405	14	λ3	λ3	PROPN
ma-53	405	15	)	)	PUNCT
ma-53	405	16	tr	tr	VERB
ma-53	405	17	as	as	ADP
ma-53	405	18	e(λ	e(λ	NOUN
ma-53	405	19	)	)	PUNCT
ma-53	405	20	=	=	SYM
ma-53	406	1	(	(	PUNCT
ma-53	406	2	eλ1	eλ1	NOUN
ma-53	406	3	−	−	PROPN
ma-53	406	4	1	1	NUM
ma-53	406	5	,	,	PUNCT
ma-53	406	6	e	e	NOUN
ma-53	406	7	−	−	PROPN
ma-53	406	8	1	1	NUM
ma-53	406	9	2	2	NUM
ma-53	406	10	λ22	λ22	NOUN
ma-53	406	11	+	+	X
ma-53	406	12	λ1	λ1	ADJ
ma-53	406	13	,	,	PUNCT
ma-53	406	14	λ3	λ3	PROPN
ma-53	406	15	)	)	PUNCT
ma-53	406	16	tr	tr	VERB
ma-53	406	17	.	.	PUNCT
ma-53	407	1	then	then	ADV
ma-53	407	2	,	,	PUNCT
ma-53	407	3	conditions	condition	NOUN
ma-53	407	4	(	(	PUNCT
ma-53	407	5	h	h	NOUN
ma-53	407	6	)	)	PUNCT
ma-53	407	7	hold	hold	NOUN
ma-53	407	8	provided	provide	VERB
ma-53	407	9	that	that	DET
ma-53	407	10	l0	l0	NOUN
ma-53	407	11	=	=	PUNCT
ma-53	408	1	e	e	NOUN
ma-53	408	2	−	−	PROPN
ma-53	408	3	1	1	NUM
ma-53	408	4	,	,	PUNCT
ma-53	408	5	l	l	NOUN
ma-53	408	6	=	=	PUNCT
ma-53	408	7	e	e	X
ma-53	408	8	1	1	NUM
ma-53	408	9	l0	l0	NOUN
ma-53	408	10	and	and	CCONJ
ma-53	408	11	m1	m1	PROPN
ma-53	408	12	=	=	SYM
ma-53	408	13	e	e	NOUN
ma-53	408	14	,	,	PUNCT
ma-53	408	15	since	since	SCONJ
ma-53	408	16	f	f	PROPN
ma-53	408	17	′(x∗)−1	′(x∗)−1	PROPN
ma-53	408	18	=	=	SYM
ma-53	408	19	f	f	PROPN
ma-53	408	20	′(x∗	′(x∗	NOUN
ma-53	408	21	)	)	PUNCT
ma-53	408	22	=	=	PRON
ma-53	409	1	diag{1	diag{1	ADJ
ma-53	409	2	,	,	PUNCT
ma-53	409	3	1	1	NUM
ma-53	409	4	,	,	PUNCT
ma-53	409	5	1	1	NUM
ma-53	409	6	,	,	PUNCT
ma-53	409	7	}	}	PUNCT
ma-53	409	8	.	.	PUNCT
ma-53	410	1	notice	notice	VERB
ma-53	410	2	that	that	SCONJ
ma-53	410	3	l0	l0	PROPN
ma-53	410	4	<	<	X
ma-53	410	5	l	l	X
ma-53	410	6	<	<	X
ma-53	410	7	m1	m1	PROPN
ma-53	410	8	,	,	PUNCT
ma-53	410	9	l1	l1	PROPN
ma-53	410	10	=	=	SYM
ma-53	410	11	l0	l0	PROPN
ma-53	410	12	2	2	NUM
ma-53	410	13	,	,	PUNCT
ma-53	410	14	l4	l4	PROPN
ma-53	410	15	=	=	SYM
ma-53	410	16	l	l	PROPN
ma-53	410	17	,	,	PUNCT
ma-53	410	18	l2	l2	NOUN
ma-53	410	19	=	=	SYM
ma-53	410	20	l3	l3	NOUN
ma-53	410	21	=	=	SYM
ma-53	410	22	l	l	PROPN
ma-53	410	23	2	2	NUM
ma-53	410	24	.	.	PUNCT
ma-53	411	1	rtr	rtr	PROPN
ma-53	411	2	=	=	PROPN
ma-53	411	3	0.2453	0.2453	NUM
ma-53	411	4	<	<	X
ma-53	411	5	ra	ra	PROPN
ma-53	411	6	=	=	SYM
ma-53	411	7	0.3827	0.3827	NUM
ma-53	411	8	,	,	PUNCT
ma-53	411	9	r	r	NOUN
ma-53	411	10	=	=	SYM
ma-53	411	11	0.2124	0.2124	NUM
ma-53	411	12	.	.	PUNCT
ma-53	412	1	hence	hence	ADV
ma-53	412	2	,	,	PUNCT
ma-53	412	3	our	our	PRON
ma-53	412	4	radius	radius	NOUN
ma-53	412	5	of	of	ADP
ma-53	412	6	convergence	convergence	NOUN
ma-53	412	7	is	be	AUX
ma-53	412	8	larger	large	ADJ
ma-53	412	9	.	.	PUNCT
ma-53	413	1	example	example	NOUN
ma-53	413	2	4.5	4.5	NUM
ma-53	413	3	.	.	PUNCT
ma-53	414	1	let	let	VERB
ma-53	414	2	b	b	NOUN
ma-53	414	3	=	=	SYM
ma-53	414	4	b1	b1	PROPN
ma-53	414	5	and	and	CCONJ
ma-53	414	6	ω	ω	NOUN
ma-53	414	7	be	be	VERB
ma-53	414	8	as	as	ADP
ma-53	414	9	in	in	ADP
ma-53	414	10	example	example	NOUN
ma-53	414	11	4.2	4.2	NUM
ma-53	414	12	.	.	PUNCT
ma-53	415	1	define	define	VERB
ma-53	415	2	f	f	PROPN
ma-53	415	3	on	on	ADP
ma-53	415	4	ω	ω	PROPN
ma-53	415	5	as	as	ADP
ma-53	415	6	f	f	PROPN
ma-53	415	7	(	(	PUNCT
ma-53	415	8	ϕ1)(x	ϕ1)(x	NOUN
ma-53	415	9	)	)	PUNCT
ma-53	416	1	=	=	NOUN
ma-53	417	1	ϕ1(x)−	ϕ1(x)−	PROPN
ma-53	417	2	∫	∫	PROPN
ma-53	417	3	1	1	NUM
ma-53	417	4	0	0	NUM
ma-53	417	5	xϕ1(j	xϕ1(j	PROPN
ma-53	417	6	)	)	PUNCT
ma-53	417	7	3dj	3dj	NOUN
ma-53	417	8	.	.	PUNCT
ma-53	418	1	then	then	ADV
ma-53	418	2	,	,	PUNCT
ma-53	418	3	we	we	PRON
ma-53	418	4	obtain	obtain	VERB
ma-53	418	5	f	f	PROPN
ma-53	418	6	′(ϕ1(ψ1))(x	′(ϕ1(ψ1))(x	NOUN
ma-53	418	7	)	)	PUNCT
ma-53	418	8	=	=	SYM
ma-53	419	1	ψ1(x)−	ψ1(x)−	PROPN
ma-53	419	2	3	3	NUM
ma-53	419	3	∫	∫	NOUN
ma-53	419	4	1	1	NUM
ma-53	419	5	0	0	NUM
ma-53	419	6	xjϕ1(j	xjϕ1(j	NUM
ma-53	419	7	)	)	PUNCT
ma-53	419	8	2ψ1(j)dj	2ψ1(j)dj	NOUN
ma-53	419	9	for	for	ADP
ma-53	419	10	all	all	DET
ma-53	419	11	ψ1	ψ1	PROPN
ma-53	419	12	∈	∈	PROPN
ma-53	419	13	ω	ω	NOUN
ma-53	419	14	.	.	PUNCT
ma-53	420	1	so	so	ADV
ma-53	420	2	,	,	PUNCT
ma-53	420	3	we	we	PRON
ma-53	420	4	can	can	AUX
ma-53	420	5	choose	choose	VERB
ma-53	420	6	l0	l0	PROPN
ma-53	420	7	=	=	NOUN
ma-53	420	8	1.5	1.5	NUM
ma-53	420	9	,	,	PUNCT
ma-53	421	1	l	l	NOUN
ma-53	421	2	=	=	SYM
ma-53	421	3	m1	m1	NOUN
ma-53	421	4	=	=	SYM
ma-53	421	5	3	3	X
ma-53	421	6	.	.	PUNCT
ma-53	421	7	l1	l1	PROPN
ma-53	421	8	=	=	SYM
ma-53	421	9	l0	l0	PROPN
ma-53	421	10	2	2	NUM
ma-53	421	11	,	,	PUNCT
ma-53	421	12	l4	l4	PROPN
ma-53	421	13	=	=	SYM
ma-53	421	14	l	l	PROPN
ma-53	421	15	,	,	PUNCT
ma-53	421	16	l2	l2	NOUN
ma-53	421	17	=	=	SYM
ma-53	421	18	l3	l3	NOUN
ma-53	421	19	=	=	SYM
ma-53	421	20	l	l	PROPN
ma-53	421	21	2	2	NUM
ma-53	421	22	.	.	PUNCT
ma-53	422	1	but	but	CCONJ
ma-53	422	2	then	then	ADV
ma-53	422	3	,	,	PUNCT
ma-53	422	4	we	we	PRON
ma-53	422	5	get	get	VERB
ma-53	422	6	again	again	ADV
ma-53	422	7	rtr	rtr	NOUN
ma-53	423	1	=	=	PUNCT
ma-53	424	1	0.2222	0.2222	NUM
ma-53	424	2	<	<	X
ma-53	424	3	ra	ra	PROPN
ma-53	424	4	=	=	PUNCT
ma-53	424	5	0.3333	0.3333	NUM
ma-53	424	6	,	,	PUNCT
ma-53	424	7	r	r	NOUN
ma-53	424	8	=	=	SYM
ma-53	424	9	0.2663	0.2663	NUM
ma-53	424	10	.	.	PUNCT
ma-53	425	1	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	425	2	eur	eur	PROPN
ma-53	425	3	.	.	PUNCT
ma-53	426	1	j.	j.	PROPN
ma-53	426	2	math	math	PROPN
ma-53	426	3	.	.	PUNCT
ma-53	427	1	anal	anal	PROPN
ma-53	427	2	.	.	PUNCT
ma-53	428	1	10.28924	10.28924	NUM
ma-53	428	2	/	/	SYM
ma-53	428	3	ada	ada	PROPN
ma-53	428	4	/	/	SYM
ma-53	428	5	ma.2.3	ma.2.3	PROPN
ma-53	428	6	165	165	NUM
ma-53	428	7	.	.	PUNCT
ma-53	429	1	conclusion	conclusion	NOUN
ma-53	429	2	ostrowski	ostrowski	PROPN
ma-53	429	3	’s	’s	PART
ma-53	429	4	method	method	NOUN
ma-53	429	5	was	be	AUX
ma-53	429	6	revisited	revisit	VERB
ma-53	429	7	and	and	CCONJ
ma-53	429	8	its	its	PRON
ma-53	429	9	applicability	applicability	NOUN
ma-53	429	10	was	be	AUX
ma-53	429	11	extended	extend	VERB
ma-53	429	12	in	in	ADP
ma-53	429	13	both	both	CCONJ
ma-53	429	14	the	the	DET
ma-53	429	15	semi	semi	ADJ
ma-53	429	16	-	-	ADJ
ma-53	429	17	localand	localand	ADJ
ma-53	429	18	local	local	ADJ
ma-53	429	19	convergence	convergence	NOUN
ma-53	429	20	case	case	NOUN
ma-53	429	21	.	.	PUNCT
ma-53	430	1	in	in	ADP
ma-53	430	2	particular	particular	ADJ
ma-53	430	3	,	,	PUNCT
ma-53	430	4	the	the	DET
ma-53	430	5	benefits	benefit	NOUN
ma-53	430	6	in	in	ADP
ma-53	430	7	the	the	DET
ma-53	430	8	semi	semi	ADJ
ma-53	430	9	-	-	ADJ
ma-53	430	10	local	local	ADJ
ma-53	430	11	convergence	convergence	NOUN
ma-53	430	12	case	case	NOUN
ma-53	430	13	include	include	VERB
ma-53	430	14	:	:	PUNCT
ma-53	430	15	weaker	weak	ADJ
ma-53	430	16	sufficient	sufficient	ADJ
ma-53	430	17	convergence	convergence	NOUN
ma-53	430	18	criteria	criterion	NOUN
ma-53	430	19	(	(	PUNCT
ma-53	430	20	i.e.	i.e.	X
ma-53	430	21	more	more	ADJ
ma-53	430	22	starters	starter	NOUN
ma-53	430	23	x0	x0	PROPN
ma-53	430	24	become	become	VERB
ma-53	430	25	available	available	ADJ
ma-53	430	26	)	)	PUNCT
ma-53	430	27	;	;	PUNCT
ma-53	430	28	tighter	tight	ADJ
ma-53	430	29	upper	upper	ADJ
ma-53	430	30	boundson	boundson	NOUN
ma-53	430	31	‖xk+1	‖xk+1	PUNCT
ma-53	430	32	−	−	PROPN
ma-53	430	33	xk‖	xk‖	PROPN
ma-53	430	34	,	,	PUNCT
ma-53	430	35	‖xk	‖xk	PROPN
ma-53	430	36	−	−	PROPN
ma-53	430	37	x∗‖	x∗‖	PROPN
ma-53	430	38	(	(	PUNCT
ma-53	430	39	i.e.	i.e.	X
ma-53	430	40	,	,	PUNCT
ma-53	430	41	fewer	few	ADJ
ma-53	430	42	iterates	iterate	NOUN
ma-53	430	43	are	be	AUX
ma-53	430	44	computed	compute	VERB
ma-53	430	45	to	to	PART
ma-53	430	46	reach	reach	VERB
ma-53	430	47	a	a	DET
ma-53	430	48	predecided	predecided	ADJ
ma-53	430	49	error	error	NOUN
ma-53	430	50	accuracy)and	accuracy)and	VERB
ma-53	430	51	the	the	DET
ma-53	430	52	information	information	NOUN
ma-53	430	53	on	on	ADP
ma-53	430	54	the	the	DET
ma-53	430	55	location	location	NOUN
ma-53	430	56	of	of	ADP
ma-53	430	57	x∗	x∗	PROPN
ma-53	430	58	is	be	AUX
ma-53	430	59	more	more	ADV
ma-53	430	60	precise.the	precise.the	DET
ma-53	430	61	results	result	NOUN
ma-53	430	62	are	be	AUX
ma-53	430	63	based	base	VERB
ma-53	430	64	on	on	ADP
ma-53	430	65	generalized	generalized	ADJ
ma-53	430	66	continuity	continuity	NOUN
ma-53	430	67	which	which	PRON
ma-53	430	68	is	be	AUX
ma-53	430	69	more	more	ADV
ma-53	430	70	general	general	ADJ
ma-53	430	71	than	than	SCONJ
ma-53	430	72	lipschitz	lipschitz	NOUN
ma-53	430	73	continuityused	continuityuse	VERB
ma-53	430	74	before	before	ADV
ma-53	430	75	.	.	PUNCT
ma-53	431	1	our	our	PRON
ma-53	431	2	two	two	NUM
ma-53	431	3	techniques	technique	NOUN
ma-53	431	4	are	be	AUX
ma-53	431	5	very	very	ADV
ma-53	431	6	general	general	ADJ
ma-53	431	7	and	and	CCONJ
ma-53	431	8	can	can	AUX
ma-53	431	9	be	be	AUX
ma-53	431	10	used	use	VERB
ma-53	431	11	to	to	PART
ma-53	431	12	extend	extend	VERB
ma-53	431	13	the	the	DET
ma-53	431	14	applicability	applicability	NOUN
ma-53	431	15	ofother	ofother	NOUN
ma-53	431	16	methods	method	NOUN
ma-53	431	17	.	.	PUNCT
ma-53	432	1	references	reference	NOUN
ma-53	432	2	[	[	X
ma-53	432	3	1	1	NUM
ma-53	432	4	]	]	X
ma-53	432	5	i.k	i.k	PROPN
ma-53	432	6	.	.	PROPN
ma-53	432	7	argyros	argyros	PROPN
ma-53	432	8	,	,	PUNCT
ma-53	432	9	on	on	ADP
ma-53	432	10	the	the	DET
ma-53	432	11	newton	newton	PROPN
ma-53	432	12	kantorovich	kantorovich	PROPN
ma-53	432	13	hypothesis	hypothesis	NOUN
ma-53	432	14	for	for	ADP
ma-53	432	15	solving	solve	VERB
ma-53	432	16	equations	equation	NOUN
ma-53	432	17	,	,	PUNCT
ma-53	432	18	j.	j.	PROPN
ma-53	432	19	comput	comput	PROPN
ma-53	432	20	.	.	PUNCT
ma-53	433	1	math	math	NOUN
ma-53	433	2	.	.	PUNCT
ma-53	434	1	169	169	NUM
ma-53	434	2	(	(	PUNCT
ma-53	434	3	2004	2004	NUM
ma-53	434	4	)	)	PUNCT
ma-53	435	1	315–332	315–332	NUM
ma-53	435	2	.	.	PUNCT
ma-53	436	1	https://doi.org/10.1007/s12190-008-0140-6.[2	https://doi.org/10.1007/s12190-008-0140-6.[2	NOUN
ma-53	436	2	]	]	X
ma-53	436	3	i.k	i.k	PROPN
ma-53	436	4	.	.	PROPN
ma-53	436	5	argyros	argyros	PROPN
ma-53	436	6	,	,	PUNCT
ma-53	436	7	computational	computational	ADJ
ma-53	436	8	theory	theory	NOUN
ma-53	436	9	of	of	ADP
ma-53	436	10	iterative	iterative	ADJ
ma-53	436	11	methods	method	NOUN
ma-53	436	12	,	,	PUNCT
ma-53	436	13	1st	1st	ADJ
ma-53	436	14	ed	ed	NOUN
ma-53	436	15	,	,	PUNCT
ma-53	436	16	elsevier	elsevier	NOUN
ma-53	436	17	,	,	PUNCT
ma-53	436	18	amsterdam	amsterdam	PROPN
ma-53	436	19	;	;	PUNCT
ma-53	436	20	london	london	PROPN
ma-53	436	21	,	,	PUNCT
ma-53	436	22	2007.[3	2007.[3	NUM
ma-53	436	23	]	]	X
ma-53	436	24	i.k	i.k	PROPN
ma-53	436	25	.	.	PROPN
ma-53	436	26	argyros	argyros	PROPN
ma-53	436	27	,	,	PUNCT
ma-53	436	28	convergence	convergence	NOUN
ma-53	436	29	and	and	CCONJ
ma-53	436	30	applications	application	NOUN
ma-53	436	31	of	of	ADP
ma-53	436	32	newton	newton	NOUN
ma-53	436	33	-	-	PUNCT
ma-53	436	34	type	type	NOUN
ma-53	436	35	iterations	iteration	NOUN
ma-53	436	36	,	,	PUNCT
ma-53	436	37	springer	springer	NOUN
ma-53	436	38	verlag	verlag	PROPN
ma-53	436	39	,	,	PUNCT
ma-53	436	40	berlin	berlin	PROPN
ma-53	436	41	,	,	PUNCT
ma-53	436	42	germany	germany	PROPN
ma-53	436	43	,	,	PUNCT
ma-53	436	44	(	(	PUNCT
ma-53	436	45	2008).[4	2008).[4	NOUN
ma-53	436	46	]	]	X
ma-53	436	47	i.k	i.k	PROPN
ma-53	436	48	.	.	PROPN
ma-53	436	49	argyros	argyros	PROPN
ma-53	436	50	,	,	PUNCT
ma-53	436	51	s.	s.	PROPN
ma-53	436	52	hilout	hilout	PROPN
ma-53	436	53	,	,	PUNCT
ma-53	436	54	weaker	weak	ADJ
ma-53	436	55	conditions	condition	NOUN
ma-53	436	56	for	for	ADP
ma-53	436	57	the	the	DET
ma-53	436	58	convergence	convergence	NOUN
ma-53	436	59	of	of	ADP
ma-53	436	60	newton	newton	PROPN
ma-53	436	61	’s	’s	PART
ma-53	436	62	method	method	NOUN
ma-53	436	63	,	,	PUNCT
ma-53	436	64	j.	j.	PROPN
ma-53	436	65	complexity	complexity	PROPN
ma-53	436	66	,	,	PUNCT
ma-53	436	67	28	28	NUM
ma-53	436	68	(	(	PUNCT
ma-53	436	69	2012	2012	NUM
ma-53	436	70	)	)	PUNCT
ma-53	436	71	364–387	364–387	NUM
ma-53	436	72	.	.	PUNCT
ma-53	437	1	https://doi.org/10.1016/j.jco.2011.12.003.[5	https://doi.org/10.1016/j.jco.2011.12.003.[5	PRON
ma-53	437	2	]	]	X
ma-53	437	3	i.k	i.k	PROPN
ma-53	437	4	.	.	PROPN
ma-53	437	5	argyros	argyros	PROPN
ma-53	437	6	,	,	PUNCT
ma-53	437	7	s.	s.	PROPN
ma-53	437	8	hilout	hilout	PROPN
ma-53	437	9	,	,	PUNCT
ma-53	437	10	on	on	ADP
ma-53	437	11	an	an	DET
ma-53	437	12	improved	improved	ADJ
ma-53	437	13	convergence	convergence	NOUN
ma-53	437	14	analysis	analysis	NOUN
ma-53	437	15	of	of	ADP
ma-53	437	16	newton	newton	PROPN
ma-53	437	17	’s	’s	PART
ma-53	437	18	method	method	NOUN
ma-53	437	19	,	,	PUNCT
ma-53	437	20	appl	appl	PROPN
ma-53	437	21	.	.	PROPN
ma-53	437	22	math	math	NOUN
ma-53	437	23	.	.	PUNCT
ma-53	438	1	comput	comput	NOUN
ma-53	438	2	.	.	PUNCT
ma-53	439	1	225	225	NUM
ma-53	439	2	(	(	PUNCT
ma-53	439	3	2013)372–386	2013)372–386	NUM
ma-53	439	4	.	.	PUNCT
ma-53	440	1	https://doi.org/10.1016/j.amc.2013.09.049.[6	https://doi.org/10.1016/j.amc.2013.09.049.[6	PUNCT
ma-53	440	2	]	]	X
ma-53	440	3	i.k	i.k	PROPN
ma-53	440	4	.	.	PROPN
ma-53	440	5	argyros	argyros	PROPN
ma-53	440	6	,	,	PUNCT
ma-53	440	7	a.a	a.a	PROPN
ma-53	440	8	.	.	PROPN
ma-53	440	9	magréñan	magréñan	PROPN
ma-53	440	10	,	,	PUNCT
ma-53	440	11	iterative	iterative	NOUN
ma-53	440	12	methods	method	NOUN
ma-53	440	13	and	and	CCONJ
ma-53	440	14	their	their	PRON
ma-53	440	15	dynamics	dynamic	NOUN
ma-53	440	16	with	with	ADP
ma-53	440	17	applications	application	NOUN
ma-53	440	18	,	,	PUNCT
ma-53	440	19	crc	crc	NOUN
ma-53	440	20	press	press	NOUN
ma-53	440	21	,	,	PUNCT
ma-53	440	22	new	new	PROPN
ma-53	440	23	york	york	PROPN
ma-53	440	24	,	,	PUNCT
ma-53	440	25	usa,2017.[7	usa,2017.[7	NOUN
ma-53	440	26	]	]	X
ma-53	440	27	i.k	i.k	PROPN
ma-53	440	28	.	.	PROPN
ma-53	440	29	argyros	argyros	PROPN
ma-53	440	30	,	,	PUNCT
ma-53	440	31	a.a	a.a	PROPN
ma-53	440	32	.	.	PROPN
ma-53	440	33	magréñan	magréñan	PROPN
ma-53	440	34	,	,	PUNCT
ma-53	440	35	a	a	DET
ma-53	440	36	contemporary	contemporary	ADJ
ma-53	440	37	study	study	NOUN
ma-53	440	38	of	of	ADP
ma-53	440	39	iterative	iterative	ADJ
ma-53	440	40	methods	method	NOUN
ma-53	440	41	,	,	PUNCT
ma-53	440	42	elsevier	elsevier	NOUN
ma-53	440	43	(	(	PUNCT
ma-53	440	44	academic	academic	ADJ
ma-53	440	45	press	press	NOUN
ma-53	440	46	)	)	PUNCT
ma-53	440	47	,	,	PUNCT
ma-53	440	48	new	new	PROPN
ma-53	440	49	york	york	PROPN
ma-53	440	50	,	,	PUNCT
ma-53	440	51	2018.[8	2018.[8	NUM
ma-53	440	52	]	]	X
ma-53	440	53	r.	r.	PROPN
ma-53	440	54	behl	behl	PROPN
ma-53	440	55	,	,	PUNCT
ma-53	440	56	p.	p.	PROPN
ma-53	440	57	maroju	maroju	PROPN
ma-53	440	58	,	,	PUNCT
ma-53	440	59	e.	e.	PROPN
ma-53	440	60	martinez	martinez	PROPN
ma-53	440	61	,	,	PUNCT
ma-53	440	62	s.	s.	PROPN
ma-53	440	63	singh	singh	PROPN
ma-53	440	64	,	,	PUNCT
ma-53	440	65	a	a	DET
ma-53	440	66	study	study	NOUN
ma-53	440	67	of	of	ADP
ma-53	440	68	the	the	DET
ma-53	440	69	local	local	ADJ
ma-53	440	70	convergence	convergence	NOUN
ma-53	440	71	of	of	ADP
ma-53	440	72	a	a	DET
ma-53	440	73	fifth	fifth	ADJ
ma-53	440	74	order	order	NOUN
ma-53	440	75	iterative	iterative	NOUN
ma-53	440	76	method	method	NOUN
ma-53	440	77	,	,	PUNCT
ma-53	440	78	indianj	indianj	ADJ
ma-53	440	79	.	.	PUNCT
ma-53	441	1	pure	pure	ADJ
ma-53	441	2	appl	appl	PROPN
ma-53	441	3	.	.	PUNCT
ma-53	441	4	math	math	NOUN
ma-53	441	5	.	.	PUNCT
ma-53	442	1	51	51	NUM
ma-53	442	2	(	(	PUNCT
ma-53	442	3	2020	2020	NUM
ma-53	442	4	)	)	PUNCT
ma-53	442	5	439	439	NUM
ma-53	442	6	-	-	SYM
ma-53	442	7	455	455	NUM
ma-53	442	8	.	.	PUNCT
ma-53	443	1	https://doi.org/10.1007/s13226-020-0409-5.[9	https://doi.org/10.1007/s13226-020-0409-5.[9	PRON
ma-53	443	2	]	]	X
ma-53	443	3	e.	e.	PROPN
ma-53	443	4	cătinaş	cătinaş	PROPN
ma-53	443	5	,	,	PUNCT
ma-53	443	6	the	the	DET
ma-53	443	7	inexact	inexact	ADJ
ma-53	443	8	,	,	PUNCT
ma-53	443	9	inexact	inexact	ADJ
ma-53	443	10	perturbed	perturb	VERB
ma-53	443	11	,	,	PUNCT
ma-53	443	12	and	and	CCONJ
ma-53	443	13	quasi	quasi	ADJ
ma-53	443	14	-	-	ADJ
ma-53	443	15	newton	newton	PROPN
ma-53	443	16	methods	method	NOUN
ma-53	443	17	are	be	AUX
ma-53	443	18	equivalent	equivalent	ADJ
ma-53	443	19	models	model	NOUN
ma-53	443	20	,	,	PUNCT
ma-53	443	21	math	math	NOUN
ma-53	443	22	.	.	PUNCT
ma-53	444	1	comp	comp	PROPN
ma-53	444	2	.	.	PUNCT
ma-53	445	1	74(2005	74(2005	NUM
ma-53	445	2	)	)	PUNCT
ma-53	446	1	291–301	291–301	NUM
ma-53	446	2	.	.	PUNCT
ma-53	447	1	https://doi.org/10.1090/s0025-5718-04-01646-1.[10	https://doi.org/10.1090/s0025-5718-04-01646-1.[10	PROPN
ma-53	447	2	]	]	PUNCT
ma-53	447	3	x.	x.	PROPN
ma-53	447	4	chen	chen	PROPN
ma-53	447	5	,	,	PUNCT
ma-53	447	6	t.	t.	PROPN
ma-53	447	7	yamamoto	yamamoto	PROPN
ma-53	447	8	,	,	PUNCT
ma-53	447	9	convergence	convergence	NOUN
ma-53	447	10	domains	domain	NOUN
ma-53	447	11	of	of	ADP
ma-53	447	12	certain	certain	ADJ
ma-53	447	13	iterative	iterative	NOUN
ma-53	447	14	methods	method	NOUN
ma-53	447	15	for	for	ADP
ma-53	447	16	solving	solve	VERB
ma-53	447	17	nonlinear	nonlinear	ADJ
ma-53	447	18	equations	equation	NOUN
ma-53	447	19	,	,	PUNCT
ma-53	447	20	numer.funct	numer.funct	PROPN
ma-53	447	21	.	.	PROPN
ma-53	447	22	anal	anal	PROPN
ma-53	447	23	.	.	PUNCT
ma-53	448	1	optim	optim	PROPN
ma-53	448	2	.	.	PUNCT
ma-53	449	1	10	10	NUM
ma-53	449	2	(	(	PUNCT
ma-53	449	3	1989	1989	NUM
ma-53	449	4	)	)	PUNCT
ma-53	450	1	37–48.[11	37–48.[11	NUM
ma-53	450	2	]	]	X
ma-53	450	3	j.e	j.e	PROPN
ma-53	450	4	.	.	PROPN
ma-53	450	5	dennis	dennis	PROPN
ma-53	450	6	,	,	PUNCT
ma-53	450	7	jr	jr	PROPN
ma-53	450	8	.	.	PROPN
ma-53	450	9	,	,	PUNCT
ma-53	450	10	on	on	ADP
ma-53	450	11	newton	newton	PROPN
ma-53	450	12	-	-	PUNCT
ma-53	450	13	like	like	ADJ
ma-53	450	14	methods	method	NOUN
ma-53	450	15	.	.	PUNCT
ma-53	451	1	numer	numer	PROPN
ma-53	451	2	.	.	PUNCT
ma-53	451	3	math	math	NOUN
ma-53	451	4	.	.	PUNCT
ma-53	452	1	11	11	NUM
ma-53	452	2	(	(	PUNCT
ma-53	452	3	1968	1968	NUM
ma-53	452	4	)	)	PUNCT
ma-53	453	1	324–330[12	324–330[12	PROPN
ma-53	453	2	]	]	X
ma-53	454	1	j.e	j.e	PROPN
ma-53	454	2	.	.	PROPN
ma-53	454	3	dennis	dennis	PROPN
ma-53	454	4	jr	jr	PROPN
ma-53	454	5	.	.	PROPN
ma-53	454	6	,	,	PUNCT
ma-53	454	7	r.b	r.b	PROPN
ma-53	454	8	.	.	PROPN
ma-53	454	9	schnabel	schnabel	PROPN
ma-53	454	10	,	,	PUNCT
ma-53	454	11	numerical	numerical	ADJ
ma-53	454	12	methods	method	NOUN
ma-53	454	13	for	for	ADP
ma-53	454	14	unconstrained	unconstrained	ADJ
ma-53	454	15	optimization	optimization	NOUN
ma-53	454	16	and	and	CCONJ
ma-53	454	17	nonlinear	nonlinear	ADJ
ma-53	454	18	equations	equation	NOUN
ma-53	454	19	,	,	PUNCT
ma-53	454	20	prentice	prentice	NOUN
ma-53	454	21	-	-	PUNCT
ma-53	454	22	hall	hall	NOUN
ma-53	454	23	,	,	PUNCT
ma-53	454	24	englewood	englewood	PROPN
ma-53	454	25	cliffs	cliffs	PROPN
ma-53	454	26	,	,	PUNCT
ma-53	454	27	1983.[13	1983.[13	NUM
ma-53	454	28	]	]	X
ma-53	454	29	p.	p.	NOUN
ma-53	454	30	deuflhard	deuflhard	NOUN
ma-53	454	31	,	,	PUNCT
ma-53	454	32	g.	g.	PROPN
ma-53	454	33	heindl	heindl	PROPN
ma-53	454	34	,	,	PUNCT
ma-53	454	35	affine	affine	NOUN
ma-53	454	36	invariant	invariant	ADJ
ma-53	454	37	convergence	convergence	NOUN
ma-53	454	38	theorems	theorem	NOUN
ma-53	454	39	for	for	ADP
ma-53	454	40	newton	newton	PROPN
ma-53	454	41	’s	’s	PART
ma-53	454	42	method	method	NOUN
ma-53	454	43	and	and	CCONJ
ma-53	454	44	extensions	extension	NOUN
ma-53	454	45	to	to	ADP
ma-53	454	46	relatedmethods	relatedmethod	NOUN
ma-53	454	47	.	.	PUNCT
ma-53	455	1	siam	siam	PROPN
ma-53	455	2	j.	j.	PROPN
ma-53	455	3	numer	numer	PROPN
ma-53	455	4	.	.	PUNCT
ma-53	456	1	anal	anal	PROPN
ma-53	456	2	.	.	PUNCT
ma-53	457	1	16	16	NUM
ma-53	457	2	(	(	PUNCT
ma-53	457	3	1979	1979	NUM
ma-53	457	4	)	)	PUNCT
ma-53	457	5	1–10	1–10	NOUN
ma-53	457	6	.	.	PUNCT
ma-53	458	1	https://doi.org/10.1137/0716001.[14	https://doi.org/10.1137/0716001.[14	PROPN
ma-53	458	2	]	]	PUNCT
ma-53	458	3	p.	p.	NOUN
ma-53	458	4	deuflhard	deuflhard	PROPN
ma-53	458	5	,	,	PUNCT
ma-53	458	6	newton	newton	PROPN
ma-53	458	7	methods	method	NOUN
ma-53	458	8	for	for	ADP
ma-53	458	9	nonlinear	nonlinear	ADJ
ma-53	458	10	problems	problem	NOUN
ma-53	458	11	.	.	PUNCT
ma-53	459	1	affine	affine	PROPN
ma-53	459	2	invariance	invariance	NOUN
ma-53	459	3	and	and	CCONJ
ma-53	459	4	adaptive	adaptive	ADJ
ma-53	459	5	algorithms	algorithm	NOUN
ma-53	459	6	,	,	PUNCT
ma-53	459	7	springer	springer	NOUN
ma-53	459	8	seriesin	seriesin	PROPN
ma-53	459	9	computational	computational	ADJ
ma-53	459	10	mathematics	mathematic	NOUN
ma-53	459	11	,	,	PUNCT
ma-53	459	12	35	35	NUM
ma-53	459	13	,	,	PUNCT
ma-53	459	14	springer	springer	NOUN
ma-53	459	15	-	-	PUNCT
ma-53	459	16	verlag	verlag	PROPN
ma-53	459	17	,	,	PUNCT
ma-53	459	18	berlin	berlin	PROPN
ma-53	459	19	.	.	PUNCT
ma-53	460	1	(	(	PUNCT
ma-53	460	2	2004).[15	2004).[15	NUM
ma-53	460	3	]	]	X
ma-53	460	4	j.a	j.a	PROPN
ma-53	460	5	.	.	PROPN
ma-53	460	6	ezquerro	ezquerro	PROPN
ma-53	460	7	,	,	PUNCT
ma-53	460	8	j.m	j.m	PROPN
ma-53	460	9	.	.	PROPN
ma-53	460	10	gutiérrez	gutiérrez	PROPN
ma-53	460	11	,	,	PUNCT
ma-53	460	12	m.a	m.a	PROPN
ma-53	460	13	.	.	PROPN
ma-53	460	14	hernández	hernández	PROPN
ma-53	460	15	,	,	PUNCT
ma-53	460	16	n.	n.	PROPN
ma-53	460	17	romero	romero	PROPN
ma-53	460	18	,	,	PUNCT
ma-53	460	19	m.j	m.j	PROPN
ma-53	460	20	.	.	PROPN
ma-53	460	21	rubio	rubio	PROPN
ma-53	460	22	,	,	PUNCT
ma-53	460	23	the	the	DET
ma-53	460	24	newton	newton	PROPN
ma-53	460	25	method	method	NOUN
ma-53	460	26	:	:	PUNCT
ma-53	460	27	from	from	ADP
ma-53	460	28	newton	newton	PROPN
ma-53	460	29	tokantorovich	tokantorovich	PROPN
ma-53	460	30	(	(	PUNCT
ma-53	460	31	spanish	spanish	ADJ
ma-53	460	32	)	)	PUNCT
ma-53	460	33	,	,	PUNCT
ma-53	460	34	gac	gac	PROPN
ma-53	460	35	.	.	PUNCT
ma-53	461	1	r.	r.	PROPN
ma-53	461	2	soc	soc	PROPN
ma-53	461	3	.	.	PUNCT
ma-53	462	1	mat	mat	PROPN
ma-53	462	2	.	.	PUNCT
ma-53	463	1	esp	esp	PROPN
ma-53	463	2	.	.	PUNCT
ma-53	464	1	13	13	NUM
ma-53	464	2	(	(	PUNCT
ma-53	464	3	2010	2010	NUM
ma-53	464	4	)	)	PUNCT
ma-53	464	5	53	53	NUM
ma-53	464	6	-	-	SYM
ma-53	464	7	76.[16	76.[16	NUM
ma-53	464	8	]	]	X
ma-53	464	9	j.a	j.a	PROPN
ma-53	464	10	.	.	PROPN
ma-53	464	11	ezquerro	ezquerro	PROPN
ma-53	464	12	,	,	PUNCT
ma-53	464	13	m.a	m.a	PROPN
ma-53	464	14	.	.	PROPN
ma-53	464	15	hernandez	hernandez	PROPN
ma-53	464	16	,	,	PUNCT
ma-53	464	17	newton	newton	PROPN
ma-53	464	18	’s	’s	PART
ma-53	464	19	method	method	NOUN
ma-53	464	20	:	:	PUNCT
ma-53	464	21	an	an	DET
ma-53	464	22	updated	update	VERB
ma-53	464	23	approach	approach	NOUN
ma-53	464	24	of	of	ADP
ma-53	464	25	kantorovich	kantorovich	PROPN
ma-53	464	26	’s	’s	PART
ma-53	464	27	theory	theory	NOUN
ma-53	464	28	,	,	PUNCT
ma-53	464	29	cham	cham	PROPN
ma-53	464	30	switzerland,(2018).[17	switzerland,(2018).[17	PROPN
ma-53	464	31	]	]	PUNCT
ma-53	464	32	m.	m.	NOUN
ma-53	464	33	grau	grau	PROPN
ma-53	464	34	-	-	PUNCT
ma-53	464	35	sánchez	sánchez	PROPN
ma-53	464	36	,	,	PUNCT
ma-53	464	37	a.	a.	PROPN
ma-53	464	38	grau	grau	PROPN
ma-53	464	39	,	,	PUNCT
ma-53	464	40	m.	m.	NOUN
ma-53	464	41	noguera	noguera	PROPN
ma-53	464	42	,	,	PUNCT
ma-53	464	43	ostrowski	ostrowski	ADJ
ma-53	464	44	type	type	NOUN
ma-53	464	45	methods	method	NOUN
ma-53	464	46	for	for	ADP
ma-53	464	47	solving	solve	VERB
ma-53	464	48	systems	system	NOUN
ma-53	464	49	of	of	ADP
ma-53	464	50	nonlinear	nonlinear	ADJ
ma-53	464	51	equations	equation	NOUN
ma-53	464	52	.	.	PUNCT
ma-53	465	1	appl.math	appl.math	NOUN
ma-53	465	2	.	.	PUNCT
ma-53	465	3	comput	comput	NOUN
ma-53	465	4	.	.	PUNCT
ma-53	466	1	281	281	NUM
ma-53	466	2	(	(	PUNCT
ma-53	466	3	2011	2011	NUM
ma-53	466	4	)	)	PUNCT
ma-53	466	5	2377	2377	NUM
ma-53	466	6	-	-	SYM
ma-53	466	7	2385	2385	NUM
ma-53	466	8	.	.	PUNCT
ma-53	467	1	https://doi.org/10.1016/j.amc.2011.08.011	https://doi.org/10.1016/j.amc.2011.08.011	PROPN
ma-53	467	2	.	.	PUNCT
ma-53	468	1	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	469	1	https://doi.org/10.1007/s12190-008-0140-6	https://doi.org/10.1007/s12190-008-0140-6	NUM
ma-53	469	2	https://doi.org/10.1016/j.jco.2011.12.003	https://doi.org/10.1016/j.jco.2011.12.003	NOUN
ma-53	469	3	https://doi.org/10.1016/j.amc.2013.09.049	https://doi.org/10.1016/j.amc.2013.09.049	NOUN
ma-53	469	4	https://doi.org/10.1007/s13226-020-0409-5	https://doi.org/10.1007/s13226-020-0409-5	NUM
ma-53	469	5	https://doi.org/10.1090/s0025-5718-04-01646-1	https://doi.org/10.1090/s0025-5718-04-01646-1	ADV
ma-53	469	6	https://doi.org/10.1137/0716001	https://doi.org/10.1137/0716001	ADJ
ma-53	469	7	eur	eur	PROPN
ma-53	469	8	.	.	PUNCT
ma-53	470	1	j.	j.	PROPN
ma-53	470	2	math	math	PROPN
ma-53	470	3	.	.	PUNCT
ma-53	471	1	anal	anal	PROPN
ma-53	471	2	.	.	PUNCT
ma-53	472	1	10.28924	10.28924	NUM
ma-53	472	2	/	/	SYM
ma-53	472	3	ada	ada	PROPN
ma-53	472	4	/	/	SYM
ma-53	472	5	ma.2.3	ma.2.3	PROPN
ma-53	472	6	17	17	NUM
ma-53	473	1	[	[	X
ma-53	473	2	18	18	NUM
ma-53	473	3	]	]	X
ma-53	473	4	j.m	j.m	PROPN
ma-53	473	5	.	.	PROPN
ma-53	473	6	gutiérrez	gutiérrez	PROPN
ma-53	473	7	,	,	PUNCT
ma-53	473	8	a.a	a.a	PROPN
ma-53	473	9	.	.	PROPN
ma-53	473	10	magreñán	magreñán	PROPN
ma-53	473	11	,	,	PUNCT
ma-53	473	12	n.	n.	PROPN
ma-53	473	13	romero	romero	PROPN
ma-53	473	14	,	,	PUNCT
ma-53	473	15	on	on	ADP
ma-53	473	16	the	the	DET
ma-53	473	17	semilocal	semilocal	ADJ
ma-53	473	18	convergence	convergence	NOUN
ma-53	473	19	of	of	ADP
ma-53	473	20	newton	newton	PROPN
ma-53	473	21	-	-	PUNCT
ma-53	473	22	kantorovich	kantorovich	PROPN
ma-53	473	23	method	method	NOUN
ma-53	473	24	undercenter	undercenter	ADJ
ma-53	473	25	-	-	PUNCT
ma-53	473	26	lipschitz	lipschitz	NOUN
ma-53	473	27	conditions	condition	NOUN
ma-53	473	28	,	,	PUNCT
ma-53	473	29	appl	appl	PROPN
ma-53	473	30	.	.	PROPN
ma-53	473	31	math	math	NOUN
ma-53	473	32	.	.	PUNCT
ma-53	474	1	comput	comput	NOUN
ma-53	474	2	.	.	PUNCT
ma-53	475	1	221	221	NUM
ma-53	475	2	(	(	PUNCT
ma-53	475	3	2013	2013	NUM
ma-53	475	4	)	)	PUNCT
ma-53	475	5	79	79	NUM
ma-53	475	6	-	-	SYM
ma-53	475	7	88	88	NUM
ma-53	475	8	.	.	PUNCT
ma-53	476	1	https://doi.org/10.1016/j.amc.2013.05	https://doi.org/10.1016/j.amc.2013.05	NOUN
ma-53	476	2	.	.	PUNCT
ma-53	477	1	078.[19	078.[19	PROPN
ma-53	477	2	]	]	X
ma-53	477	3	l.v	l.v	PROPN
ma-53	477	4	.	.	PROPN
ma-53	477	5	kantorovich	kantorovich	PROPN
ma-53	477	6	,	,	PUNCT
ma-53	477	7	g.p	g.p	PROPN
ma-53	477	8	.	.	PROPN
ma-53	477	9	akilov	akilov	PROPN
ma-53	477	10	,	,	PUNCT
ma-53	477	11	functional	functional	ADJ
ma-53	477	12	analysis	analysis	NOUN
ma-53	477	13	,	,	PUNCT
ma-53	477	14	pergamon	pergamon	PROPN
ma-53	477	15	press	press	PROPN
ma-53	477	16	,	,	PUNCT
ma-53	477	17	oxford	oxford	PROPN
ma-53	477	18	,	,	PUNCT
ma-53	477	19	(	(	PUNCT
ma-53	477	20	1982).[20	1982).[20	NUM
ma-53	477	21	]	]	X
ma-53	477	22	a.a	a.a	PROPN
ma-53	477	23	.	.	PROPN
ma-53	477	24	magréñan	magréñan	PROPN
ma-53	477	25	,	,	PUNCT
ma-53	477	26	i.k	i.k	PROPN
ma-53	477	27	.	.	PROPN
ma-53	477	28	argyros	argyros	PROPN
ma-53	477	29	,	,	PUNCT
ma-53	477	30	j.j	j.j	PROPN
ma-53	477	31	.	.	PROPN
ma-53	477	32	rainer	rainer	PROPN
ma-53	477	33	,	,	PUNCT
ma-53	477	34	j.a	j.a	PROPN
ma-53	477	35	.	.	PROPN
ma-53	477	36	sicilia	sicilia	PROPN
ma-53	477	37	,	,	PUNCT
ma-53	477	38	ball	ball	NOUN
ma-53	477	39	convergence	convergence	NOUN
ma-53	477	40	of	of	ADP
ma-53	477	41	a	a	DET
ma-53	477	42	sixth	sixth	ADJ
ma-53	477	43	-	-	PUNCT
ma-53	477	44	order	order	NOUN
ma-53	477	45	newton	newton	NOUN
ma-53	477	46	-	-	PUNCT
ma-53	477	47	like	like	NOUN
ma-53	477	48	methodbased	methodbase	VERB
ma-53	477	49	on	on	ADP
ma-53	477	50	means	mean	NOUN
ma-53	477	51	under	under	ADP
ma-53	477	52	weak	weak	ADJ
ma-53	477	53	conditions	condition	NOUN
ma-53	477	54	,	,	PUNCT
ma-53	477	55	j.	j.	PROPN
ma-53	477	56	mat	mat	PROPN
ma-53	477	57	.	.	PROPN
ma-53	477	58	chem	chem	NOUN
ma-53	477	59	.	.	PUNCT
ma-53	478	1	56	56	NUM
ma-53	478	2	(	(	PUNCT
ma-53	478	3	2018	2018	NUM
ma-53	478	4	)	)	PUNCT
ma-53	478	5	2117	2117	NUM
ma-53	478	6	-	-	SYM
ma-53	478	7	2131	2131	NUM
ma-53	478	8	.	.	PUNCT
ma-53	479	1	https://doi.org/10.1007/	https://doi.org/10.1007/	PROPN
ma-53	479	2	s10910	s10910	PROPN
ma-53	479	3	-	-	PUNCT
ma-53	479	4	018	018	NUM
ma-53	479	5	-	-	PUNCT
ma-53	479	6	0856	0856	NUM
ma-53	479	7	-	-	PUNCT
ma-53	479	8	y.[21	y.[21	PROPN
ma-53	479	9	]	]	PUNCT
ma-53	479	10	a.a	a.a	PROPN
ma-53	479	11	.	.	PROPN
ma-53	479	12	magréñan	magréñan	PROPN
ma-53	479	13	,	,	PUNCT
ma-53	479	14	j.m	j.m	PROPN
ma-53	479	15	.	.	PROPN
ma-53	479	16	gutiérrez	gutiérrez	PROPN
ma-53	479	17	,	,	PUNCT
ma-53	479	18	real	real	ADJ
ma-53	479	19	dynamics	dynamic	NOUN
ma-53	479	20	for	for	ADP
ma-53	479	21	damped	damped	PROPN
ma-53	479	22	newton	newton	PROPN
ma-53	479	23	’s	’s	PART
ma-53	479	24	method	method	NOUN
ma-53	479	25	applied	apply	VERB
ma-53	479	26	to	to	ADP
ma-53	479	27	cubic	cubic	ADJ
ma-53	479	28	polynomials	polynomial	NOUN
ma-53	479	29	,	,	PUNCT
ma-53	479	30	j.	j.	PROPN
ma-53	479	31	comput.appl	comput.appl	PROPN
ma-53	479	32	.	.	PUNCT
ma-53	479	33	math	math	NOUN
ma-53	479	34	.	.	PUNCT
ma-53	480	1	275	275	NUM
ma-53	480	2	(	(	PUNCT
ma-53	480	3	2015	2015	NUM
ma-53	480	4	)	)	PUNCT
ma-53	481	1	527–538	527–538	NUM
ma-53	481	2	.	.	PUNCT
ma-53	482	1	https://doi.org/10.1016/j.cam.2013.11.019.[22	https://doi.org/10.1016/j.cam.2013.11.019.[22	NOUN
ma-53	482	2	]	]	PUNCT
ma-53	482	3	m.z	m.z	PROPN
ma-53	482	4	.	.	PROPN
ma-53	482	5	nashed	nashe	VERB
ma-53	482	6	,	,	PUNCT
ma-53	482	7	x.	x.	PROPN
ma-53	482	8	chen	chen	PROPN
ma-53	482	9	,	,	PUNCT
ma-53	482	10	convergence	convergence	NOUN
ma-53	482	11	of	of	ADP
ma-53	482	12	newton	newton	PROPN
ma-53	482	13	-	-	PUNCT
ma-53	482	14	like	like	ADJ
ma-53	482	15	methods	method	NOUN
ma-53	482	16	for	for	ADP
ma-53	482	17	singular	singular	ADJ
ma-53	482	18	operator	operator	NOUN
ma-53	482	19	equations	equation	NOUN
ma-53	482	20	using	use	VERB
ma-53	482	21	outer	outer	ADJ
ma-53	482	22	inverses	inverse	NOUN
ma-53	482	23	,	,	PUNCT
ma-53	482	24	numer	numer	PROPN
ma-53	482	25	.	.	PROPN
ma-53	482	26	math	math	NOUN
ma-53	482	27	.	.	PUNCT
ma-53	483	1	66	66	NUM
ma-53	483	2	(	(	PUNCT
ma-53	483	3	1993	1993	NUM
ma-53	483	4	)	)	PUNCT
ma-53	483	5	235	235	NUM
ma-53	483	6	-	-	SYM
ma-53	483	7	257	257	NUM
ma-53	483	8	,	,	PUNCT
ma-53	483	9	https://doi.org/10.1007/bf01385696.[23	https://doi.org/10.1007/bf01385696.[23	NOUN
ma-53	483	10	]	]	X
ma-53	483	11	l.m	l.m	PROPN
ma-53	483	12	.	.	PROPN
ma-53	483	13	ortega	ortega	PROPN
ma-53	483	14	,	,	PUNCT
ma-53	483	15	w.c	w.c	PROPN
ma-53	483	16	.	.	PROPN
ma-53	483	17	rheinboldt	rheinboldt	ADJ
ma-53	483	18	,	,	PUNCT
ma-53	483	19	iterative	iterative	ADJ
ma-53	483	20	solution	solution	NOUN
ma-53	483	21	of	of	ADP
ma-53	483	22	nonlinear	nonlinear	ADJ
ma-53	483	23	equations	equation	NOUN
ma-53	483	24	in	in	ADP
ma-53	483	25	several	several	ADJ
ma-53	483	26	variables	variable	NOUN
ma-53	483	27	,	,	PUNCT
ma-53	483	28	academic	academic	ADJ
ma-53	483	29	press	press	NOUN
ma-53	483	30	,	,	PUNCT
ma-53	483	31	newyork	newyork	NOUN
ma-53	483	32	,	,	PUNCT
ma-53	483	33	(	(	PUNCT
ma-53	483	34	1970).[24	1970).[24	NUM
ma-53	483	35	]	]	X
ma-53	483	36	l.m	l.m	PROPN
ma-53	483	37	.	.	PROPN
ma-53	483	38	ortega	ortega	PROPN
ma-53	483	39	,	,	PUNCT
ma-53	483	40	w.c	w.c	PROPN
ma-53	483	41	.	.	PROPN
ma-53	483	42	rheinboldt	rheinboldt	ADJ
ma-53	483	43	,	,	PUNCT
ma-53	483	44	iterative	iterative	ADJ
ma-53	483	45	solution	solution	NOUN
ma-53	483	46	of	of	ADP
ma-53	483	47	nonlinear	nonlinear	ADJ
ma-53	483	48	equations	equation	NOUN
ma-53	483	49	in	in	ADP
ma-53	483	50	several	several	ADJ
ma-53	483	51	variables	variable	NOUN
ma-53	483	52	,	,	PUNCT
ma-53	483	53	siam	siam	ADJ
ma-53	483	54	publ	publ	NOUN
ma-53	483	55	.	.	PUNCT
ma-53	484	1	philadelphia,2000	philadelphia,2000	NOUN
ma-53	484	2	,	,	PUNCT
ma-53	484	3	first	first	ADV
ma-53	484	4	published	publish	VERB
ma-53	484	5	by	by	ADP
ma-53	484	6	academic	academic	ADJ
ma-53	484	7	press	press	NOUN
ma-53	484	8	,	,	PUNCT
ma-53	484	9	new	new	PROPN
ma-53	484	10	york	york	PROPN
ma-53	484	11	and	and	CCONJ
ma-53	484	12	london	london	PROPN
ma-53	484	13	,	,	PUNCT
ma-53	484	14	(	(	PUNCT
ma-53	484	15	1997).[25	1997).[25	X
ma-53	484	16	]	]	X
ma-53	484	17	m.a	m.a	PROPN
ma-53	484	18	.	.	PROPN
ma-53	484	19	ostrowski	ostrowski	PROPN
ma-53	484	20	,	,	PUNCT
ma-53	484	21	solution	solution	NOUN
ma-53	484	22	of	of	ADP
ma-53	484	23	equations	equation	NOUN
ma-53	484	24	in	in	ADP
ma-53	484	25	euclidean	euclidean	NOUN
ma-53	484	26	and	and	CCONJ
ma-53	484	27	banach	banach	NOUN
ma-53	484	28	spaces	space	NOUN
ma-53	484	29	,	,	PUNCT
ma-53	484	30	elsevier	elsevier	NOUN
ma-53	484	31	,	,	PUNCT
ma-53	484	32	1973.[26	1973.[26	NUM
ma-53	484	33	]	]	X
ma-53	484	34	f.a	f.a	PROPN
ma-53	484	35	.	.	PROPN
ma-53	484	36	potra	potra	PROPN
ma-53	484	37	,	,	PUNCT
ma-53	484	38	v.	v.	ADP
ma-53	484	39	pták	pták	ADJ
ma-53	484	40	,	,	PUNCT
ma-53	484	41	nondiscrete	nondiscrete	ADJ
ma-53	484	42	induction	induction	NOUN
ma-53	484	43	and	and	CCONJ
ma-53	484	44	iterative	iterative	NOUN
ma-53	484	45	processes	process	NOUN
ma-53	484	46	,	,	PUNCT
ma-53	484	47	research	research	NOUN
ma-53	484	48	notes	note	NOUN
ma-53	484	49	in	in	ADP
ma-53	484	50	mathematics	mathematic	NOUN
ma-53	484	51	,	,	PUNCT
ma-53	484	52	103	103	NUM
ma-53	484	53	.	.	PUNCT
ma-53	485	1	pitman(advanced	pitman(advance	VERB
ma-53	485	2	publishing	publishing	NOUN
ma-53	485	3	program	program	NOUN
ma-53	485	4	)	)	PUNCT
ma-53	485	5	,	,	PUNCT
ma-53	485	6	boston	boston	PROPN
ma-53	485	7	,	,	PUNCT
ma-53	485	8	ma	ma	PROPN
ma-53	485	9	.	.	PROPN
ma-53	486	1	(	(	PUNCT
ma-53	486	2	1984).[27	1984).[27	X
ma-53	486	3	]	]	X
ma-53	486	4	p.d	p.d	PROPN
ma-53	486	5	.	.	PROPN
ma-53	486	6	proinov	proinov	PROPN
ma-53	486	7	,	,	PUNCT
ma-53	486	8	general	general	ADJ
ma-53	486	9	local	local	ADJ
ma-53	486	10	convergence	convergence	NOUN
ma-53	486	11	theory	theory	NOUN
ma-53	486	12	for	for	ADP
ma-53	486	13	a	a	DET
ma-53	486	14	class	class	NOUN
ma-53	486	15	of	of	ADP
ma-53	486	16	iterative	iterative	NOUN
ma-53	486	17	processes	process	NOUN
ma-53	486	18	and	and	CCONJ
ma-53	486	19	its	its	PRON
ma-53	486	20	applications	application	NOUN
ma-53	486	21	to	to	PART
ma-53	486	22	newton’smethod	newton’smethod	VERB
ma-53	486	23	,	,	PUNCT
ma-53	486	24	j.	j.	PROPN
ma-53	486	25	complexity	complexity	PROPN
ma-53	486	26	,	,	PUNCT
ma-53	486	27	25	25	NUM
ma-53	486	28	(	(	PUNCT
ma-53	486	29	2009	2009	NUM
ma-53	486	30	)	)	PUNCT
ma-53	486	31	38	38	NUM
ma-53	486	32	-	-	SYM
ma-53	486	33	62	62	NUM
ma-53	486	34	.	.	PUNCT
ma-53	487	1	https://doi.org/10.1016/j.jco.2008.05.006.[28	https://doi.org/10.1016/j.jco.2008.05.006.[28	PROPN
ma-53	487	2	]	]	PUNCT
ma-53	487	3	p.d	p.d	PROPN
ma-53	487	4	.	.	PROPN
ma-53	487	5	proinov	proinov	PROPN
ma-53	487	6	,	,	PUNCT
ma-53	487	7	new	new	ADJ
ma-53	487	8	general	general	ADJ
ma-53	487	9	convergence	convergence	NOUN
ma-53	487	10	theory	theory	NOUN
ma-53	487	11	for	for	ADP
ma-53	487	12	iterative	iterative	NOUN
ma-53	487	13	processes	process	NOUN
ma-53	487	14	and	and	CCONJ
ma-53	487	15	its	its	PRON
ma-53	487	16	applications	application	NOUN
ma-53	487	17	to	to	ADP
ma-53	487	18	newton	newton	PROPN
ma-53	487	19	-	-	PUNCT
ma-53	487	20	kantorovichtype	kantorovichtype	NOUN
ma-53	487	21	theorems	theorems	PROPN
ma-53	487	22	,	,	PUNCT
ma-53	487	23	j.	j.	PROPN
ma-53	487	24	complexity	complexity	PROPN
ma-53	487	25	,	,	PUNCT
ma-53	487	26	26	26	NUM
ma-53	487	27	(	(	PUNCT
ma-53	487	28	2010	2010	NUM
ma-53	487	29	)	)	PUNCT
ma-53	487	30	3	3	NUM
ma-53	487	31	-	-	SYM
ma-53	487	32	42	42	NUM
ma-53	487	33	.	.	PUNCT
ma-53	488	1	https://doi.org/10.1016/j.jco.2009.05.001.[29	https://doi.org/10.1016/j.jco.2009.05.001.[29	PROPN
ma-53	488	2	]	]	PUNCT
ma-53	488	3	w.c	w.c	PROPN
ma-53	488	4	.	.	PROPN
ma-53	488	5	rheinboldt	rheinboldt	PROPN
ma-53	488	6	,	,	PUNCT
ma-53	488	7	an	an	DET
ma-53	488	8	adaptive	adaptive	ADJ
ma-53	488	9	continuation	continuation	NOUN
ma-53	488	10	process	process	NOUN
ma-53	488	11	of	of	ADP
ma-53	488	12	solving	solve	VERB
ma-53	488	13	systems	system	NOUN
ma-53	488	14	of	of	ADP
ma-53	488	15	nonlinear	nonlinear	ADJ
ma-53	488	16	equations	equation	NOUN
ma-53	488	17	.	.	PUNCT
ma-53	489	1	polish	polish	PROPN
ma-53	489	2	academy	academy	PROPN
ma-53	489	3	ofscience	ofscience	PROPN
ma-53	489	4	,	,	PUNCT
ma-53	489	5	banach	banach	PROPN
ma-53	489	6	ctr	ctr	PROPN
ma-53	489	7	.	.	PUNCT
ma-53	489	8	publ	publ	PROPN
ma-53	489	9	.	.	PUNCT
ma-53	490	1	3	3	NUM
ma-53	490	2	(	(	PUNCT
ma-53	490	3	1978	1978	NUM
ma-53	490	4	)	)	PUNCT
ma-53	490	5	129	129	NUM
ma-53	490	6	-	-	SYM
ma-53	490	7	142.[30	142.[30	NUM
ma-53	490	8	]	]	X
ma-53	490	9	s.m	s.m	PROPN
ma-53	490	10	.	.	PROPN
ma-53	490	11	shakhno	shakhno	PROPN
ma-53	490	12	,	,	PUNCT
ma-53	490	13	o.p	o.p	PROPN
ma-53	490	14	.	.	PROPN
ma-53	490	15	gnatyshyn	gnatyshyn	PROPN
ma-53	490	16	,	,	PUNCT
ma-53	490	17	on	on	ADP
ma-53	490	18	an	an	DET
ma-53	490	19	iterative	iterative	ADJ
ma-53	490	20	algorithm	algorithm	NOUN
ma-53	490	21	of	of	ADP
ma-53	490	22	order	order	NOUN
ma-53	490	23	1.839	1.839	NUM
ma-53	490	24	.	.	PUNCT
ma-53	490	25	.	.	PUNCT
ma-53	490	26	.	.	PUNCT
ma-53	491	1	for	for	ADP
ma-53	491	2	solving	solve	VERB
ma-53	491	3	nonlinear	nonlinear	ADJ
ma-53	491	4	operator	operator	NOUN
ma-53	491	5	equations	equation	NOUN
ma-53	491	6	,	,	PUNCT
ma-53	491	7	appl	appl	PROPN
ma-53	491	8	.	.	PROPN
ma-53	491	9	math	math	PROPN
ma-53	491	10	.	.	PUNCT
ma-53	491	11	appl	appl	PROPN
ma-53	491	12	.	.	PROPN
ma-53	491	13	161	161	NUM
ma-53	491	14	(	(	PUNCT
ma-53	491	15	2005	2005	NUM
ma-53	491	16	)	)	PUNCT
ma-53	491	17	253	253	NUM
ma-53	491	18	-	-	SYM
ma-53	491	19	264	264	NUM
ma-53	491	20	,	,	PUNCT
ma-53	491	21	https://doi.org/10.1016/j.amc.2003.12.025.[31	https://doi.org/10.1016/j.amc.2003.12.025.[31	PROPN
ma-53	491	22	]	]	PUNCT
ma-53	491	23	s.m	s.m	PROPN
ma-53	491	24	.	.	PROPN
ma-53	491	25	shakhno	shakhno	PROPN
ma-53	491	26	,	,	PUNCT
ma-53	491	27	r.p	r.p	PROPN
ma-53	491	28	.	.	PROPN
ma-53	491	29	iakymchuk	iakymchuk	PROPN
ma-53	491	30	,	,	PUNCT
ma-53	491	31	h.p	h.p	PROPN
ma-53	491	32	.	.	PROPN
ma-53	491	33	yarmola	yarmola	PROPN
ma-53	491	34	,	,	PUNCT
ma-53	491	35	convergence	convergence	NOUN
ma-53	491	36	analysis	analysis	NOUN
ma-53	491	37	of	of	ADP
ma-53	491	38	a	a	DET
ma-53	491	39	two	two	NUM
ma-53	491	40	step	step	NOUN
ma-53	491	41	method	method	NOUN
ma-53	491	42	for	for	ADP
ma-53	491	43	the	the	DET
ma-53	491	44	nonlinear	nonlinear	ADJ
ma-53	491	45	squaresproblem	squaresproblem	NOUN
ma-53	491	46	with	with	ADP
ma-53	491	47	decomposition	decomposition	NOUN
ma-53	491	48	of	of	ADP
ma-53	491	49	operator	operator	NOUN
ma-53	491	50	,	,	PUNCT
ma-53	491	51	j.	j.	PROPN
ma-53	491	52	numer	numer	PROPN
ma-53	491	53	.	.	PUNCT
ma-53	491	54	appl	appl	PROPN
ma-53	491	55	.	.	PROPN
ma-53	491	56	math	math	PROPN
ma-53	491	57	.	.	PUNCT
ma-53	492	1	128	128	NUM
ma-53	492	2	(	(	PUNCT
ma-53	492	3	2018	2018	NUM
ma-53	492	4	)	)	PUNCT
ma-53	492	5	82	82	NUM
ma-53	492	6	-	-	SYM
ma-53	492	7	95.[32	95.[32	PROPN
ma-53	492	8	]	]	X
ma-53	492	9	j.r	j.r	PROPN
ma-53	492	10	.	.	PROPN
ma-53	492	11	sharma	sharma	PROPN
ma-53	492	12	,	,	PUNCT
ma-53	492	13	r.k	r.k	PROPN
ma-53	492	14	.	.	PROPN
ma-53	492	15	guha	guha	PROPN
ma-53	492	16	,	,	PUNCT
ma-53	492	17	r.	r.	PROPN
ma-53	492	18	sharma	sharma	PROPN
ma-53	492	19	,	,	PUNCT
ma-53	492	20	an	an	DET
ma-53	492	21	efficient	efficient	ADJ
ma-53	492	22	fourth	fourth	ADJ
ma-53	492	23	order	order	NOUN
ma-53	492	24	weighted	weight	VERB
ma-53	492	25	newton	newton	PROPN
ma-53	492	26	method	method	NOUN
ma-53	492	27	for	for	ADP
ma-53	492	28	systems	system	NOUN
ma-53	492	29	of	of	ADP
ma-53	492	30	nonlinearequations	nonlinearequation	NOUN
ma-53	492	31	.	.	PUNCT
ma-53	493	1	numer	numer	PROPN
ma-53	493	2	.	.	PUNCT
ma-53	494	1	algorithms	algorithms	PROPN
ma-53	494	2	,	,	PUNCT
ma-53	494	3	62	62	NUM
ma-53	494	4	(	(	PUNCT
ma-53	494	5	2013	2013	NUM
ma-53	494	6	)	)	PUNCT
ma-53	494	7	307	307	NUM
ma-53	494	8	-	-	SYM
ma-53	494	9	323	323	NUM
ma-53	494	10	.	.	PUNCT
ma-53	495	1	https://doi.org/10.1007/s11075-012-9585-7.[33	https://doi.org/10.1007/s11075-012-9585-7.[33	PRON
ma-53	495	2	]	]	X
ma-53	495	3	f.	f.	PROPN
ma-53	495	4	soleymani	soleymani	PROPN
ma-53	495	5	,	,	PUNCT
ma-53	495	6	t.	t.	PROPN
ma-53	495	7	lotfi	lotfi	PROPN
ma-53	495	8	,	,	PUNCT
ma-53	495	9	p.	p.	NOUN
ma-53	495	10	bakhtiari	bakhtiari	PROPN
ma-53	495	11	,	,	PUNCT
ma-53	495	12	a	a	DET
ma-53	495	13	multi	multi	ADJ
ma-53	495	14	-	-	ADJ
ma-53	495	15	step	step	ADJ
ma-53	495	16	class	class	NOUN
ma-53	495	17	of	of	ADP
ma-53	495	18	iterative	iterative	ADJ
ma-53	495	19	methods	method	NOUN
ma-53	495	20	for	for	ADP
ma-53	495	21	nonlinear	nonlinear	ADJ
ma-53	495	22	systems	system	NOUN
ma-53	495	23	.	.	PUNCT
ma-53	496	1	optim	optim	ADJ
ma-53	496	2	.	.	PUNCT
ma-53	497	1	lett	lett	PROPN
ma-53	497	2	.	.	PUNCT
ma-53	498	1	8(2014	8(2014	NUM
ma-53	498	2	)	)	PUNCT
ma-53	498	3	1001	1001	NUM
ma-53	498	4	-	-	SYM
ma-53	498	5	1015	1015	NUM
ma-53	498	6	.	.	PUNCT
ma-53	499	1	https://doi.org/10.1007/s11590-013-0617-6.[34	https://doi.org/10.1007/s11590-013-0617-6.[34	ADJ
ma-53	499	2	]	]	X
ma-53	499	3	j.f	j.f	PROPN
ma-53	499	4	.	.	PROPN
ma-53	499	5	steffensen	steffensen	PROPN
ma-53	499	6	,	,	PUNCT
ma-53	499	7	remarks	remark	VERB
ma-53	499	8	on	on	ADP
ma-53	499	9	iteration	iteration	NOUN
ma-53	499	10	,	,	PUNCT
ma-53	499	11	skand	skand	VERB
ma-53	499	12	aktuar	aktuar	PROPN
ma-53	499	13	tidsr	tidsr	NOUN
ma-53	499	14	.	.	PUNCT
ma-53	500	1	16	16	NUM
ma-53	500	2	(	(	PUNCT
ma-53	500	3	1993	1993	NUM
ma-53	500	4	)	)	PUNCT
ma-53	500	5	64	64	NUM
ma-53	500	6	-	-	SYM
ma-53	500	7	72.[35	72.[35	PROPN
ma-53	500	8	]	]	X
ma-53	500	9	j.f	j.f	PROPN
ma-53	500	10	.	.	PROPN
ma-53	500	11	traub	traub	PROPN
ma-53	500	12	,	,	PUNCT
ma-53	500	13	iterative	iterative	NOUN
ma-53	500	14	methods	method	NOUN
ma-53	500	15	for	for	ADP
ma-53	500	16	the	the	DET
ma-53	500	17	solution	solution	NOUN
ma-53	500	18	of	of	ADP
ma-53	500	19	equations	equation	NOUN
ma-53	500	20	,	,	PUNCT
ma-53	500	21	prentice	prentice	NOUN
ma-53	500	22	hall	hall	PROPN
ma-53	500	23	,	,	PUNCT
ma-53	500	24	new	new	PROPN
ma-53	500	25	jersey	jersey	PROPN
ma-53	500	26	,	,	PUNCT
ma-53	500	27	u.s.a	u.s.a	PROPN
ma-53	500	28	.	.	PUNCT
ma-53	501	1	(	(	PUNCT
ma-53	501	2	1964).[36	1964).[36	NUM
ma-53	501	3	]	]	PUNCT
ma-53	501	4	t.	t.	PROPN
ma-53	501	5	yamamoto	yamamoto	PROPN
ma-53	501	6	,	,	PUNCT
ma-53	501	7	a	a	DET
ma-53	501	8	convergence	convergence	NOUN
ma-53	501	9	theorem	theorem	NOUN
ma-53	501	10	for	for	ADP
ma-53	501	11	newton	newton	PROPN
ma-53	501	12	-	-	PUNCT
ma-53	501	13	like	like	ADJ
ma-53	501	14	methods	method	NOUN
ma-53	501	15	in	in	ADP
ma-53	501	16	banach	banach	NOUN
ma-53	501	17	spaces	space	NOUN
ma-53	501	18	.	.	PUNCT
ma-53	502	1	numer	numer	PROPN
ma-53	502	2	.	.	PUNCT
ma-53	502	3	math	math	NOUN
ma-53	502	4	.	.	PUNCT
ma-53	503	1	51	51	NUM
ma-53	503	2	(	(	PUNCT
ma-53	503	3	1987	1987	NUM
ma-53	503	4	)	)	PUNCT
ma-53	504	1	545	545	NUM
ma-53	504	2	-	-	SYM
ma-53	504	3	557	557	NUM
ma-53	504	4	.	.	PUNCT
ma-53	504	5	https://doi.org/10.1007/bf01400355.[37	https://doi.org/10.1007/bf01400355.[37	PROPN
ma-53	504	6	]	]	PUNCT
ma-53	504	7	r.	r.	PROPN
ma-53	504	8	verma	verma	PROPN
ma-53	504	9	,	,	PUNCT
ma-53	504	10	new	new	ADJ
ma-53	504	11	trends	trend	NOUN
ma-53	504	12	in	in	ADP
ma-53	504	13	fractional	fractional	ADJ
ma-53	504	14	programming	programming	NOUN
ma-53	504	15	,	,	PUNCT
ma-53	504	16	nova	nova	PROPN
ma-53	504	17	science	science	NOUN
ma-53	504	18	publisher	publisher	NOUN
ma-53	504	19	,	,	PUNCT
ma-53	504	20	new	new	PROPN
ma-53	504	21	york	york	PROPN
ma-53	504	22	,	,	PUNCT
ma-53	504	23	usa	usa	PROPN
ma-53	504	24	.	.	PROPN
ma-53	505	1	(	(	PUNCT
ma-53	505	2	2019).[38	2019).[38	NUM
ma-53	505	3	]	]	X
ma-53	505	4	p.p	p.p	PROPN
ma-53	505	5	.	.	PROPN
ma-53	505	6	zabrejko	zabrejko	PROPN
ma-53	505	7	,	,	PUNCT
ma-53	505	8	d.f	d.f	PROPN
ma-53	505	9	.	.	PROPN
ma-53	505	10	nguen	nguen	PROPN
ma-53	505	11	,	,	PUNCT
ma-53	505	12	the	the	DET
ma-53	505	13	majorant	majorant	NOUN
ma-53	505	14	method	method	NOUN
ma-53	505	15	in	in	ADP
ma-53	505	16	the	the	DET
ma-53	505	17	theory	theory	NOUN
ma-53	505	18	of	of	ADP
ma-53	505	19	newton	newton	PROPN
ma-53	505	20	-	-	PUNCT
ma-53	505	21	kantorovich	kantorovich	PROPN
ma-53	505	22	approximations	approximation	NOUN
ma-53	505	23	and	and	CCONJ
ma-53	505	24	the	the	DET
ma-53	505	25	ptákerror	ptákerror	NOUN
ma-53	505	26	estimates	estimate	NOUN
ma-53	505	27	,	,	PUNCT
ma-53	505	28	numer	numer	PROPN
ma-53	505	29	.	.	PUNCT
ma-53	506	1	funct	funct	PROPN
ma-53	506	2	.	.	PUNCT
ma-53	507	1	anal	anal	PROPN
ma-53	507	2	.	.	PUNCT
ma-53	508	1	optim	optim	PROPN
ma-53	508	2	.	.	PUNCT
ma-53	509	1	9	9	NUM
ma-53	509	2	(	(	PUNCT
ma-53	509	3	1987	1987	NUM
ma-53	509	4	)	)	PUNCT
ma-53	509	5	671	671	NUM
ma-53	509	6	-	-	SYM
ma-53	509	7	684	684	NUM
ma-53	509	8	.	.	PUNCT
ma-53	510	1	https://doi.org/10.1080/01630568708816254	https://doi.org/10.1080/01630568708816254	ADJ
ma-53	510	2	.	.	PUNCT
ma-53	511	1	https://doi.org/10.28924/ada/ma.2.3	https://doi.org/10.28924/ada/ma.2.3	PROPN
ma-53	511	2	https://doi.org/10.1016/j.amc.2013.05.078	https://doi.org/10.1016/j.amc.2013.05.078	PROPN
ma-53	511	3	https://doi.org/10.1016/j.amc.2013.05.078	https://doi.org/10.1016/j.amc.2013.05.078	PROPN
ma-53	511	4	https://doi.org/10.1007/s10910-018-0856-y	https://doi.org/10.1007/s10910-018-0856-y	PROPN
ma-53	511	5	https://doi.org/10.1007/s10910-018-0856-y	https://doi.org/10.1007/s10910-018-0856-y	PROPN
ma-53	511	6	https://doi.org/10.1016/j.cam.2013.11.019	https://doi.org/10.1016/j.cam.2013.11.019	PROPN
ma-53	511	7	https://doi.org/10.1007/bf01385696	https://doi.org/10.1007/bf01385696	NOUN
ma-53	511	8	https://doi.org/10.1016/j.jco.2008.05.006	https://doi.org/10.1016/j.jco.2008.05.006	PROPN
ma-53	511	9	https://doi.org/10.1016/j.jco.2009.05.001	https://doi.org/10.1016/j.jco.2009.05.001	PROPN
ma-53	511	10	https://doi.org/10.1016/j.amc.2003.12.025	https://doi.org/10.1016/j.amc.2003.12.025	PROPN
ma-53	511	11	https://doi.org/10.1007/s11075-012-9585-7	https://doi.org/10.1007/s11075-012-9585-7	NUM
ma-53	511	12	https://doi.org/10.1007/s11590-013-0617-6	https://doi.org/10.1007/s11590-013-0617-6	NUM
ma-53	511	13	https://doi.org/10.1007/bf01400355	https://doi.org/10.1007/bf01400355	X
ma-53	511	14	https://doi.org/10.1080/01630568708816254	https://doi.org/10.1080/01630568708816254	PROPN
ma-53	512	1	1	1	NUM
ma-53	512	2	.	.	PUNCT
ma-53	512	3	introduction	introduction	NOUN
ma-53	512	4	2	2	NUM
ma-53	512	5	.	.	PUNCT
ma-53	512	6	semi	semi	ADJ
ma-53	512	7	-	-	ADJ
ma-53	512	8	local	local	ADJ
ma-53	512	9	convergence	convergence	NOUN
ma-53	512	10	3	3	NUM
ma-53	512	11	.	.	PUNCT
ma-53	512	12	local	local	ADJ
ma-53	512	13	convergence	convergence	NOUN
ma-53	512	14	4	4	NUM
ma-53	512	15	.	.	PUNCT
ma-53	512	16	numerical	numerical	ADJ
ma-53	512	17	experiments	experiment	NOUN
ma-53	512	18	5	5	NUM
ma-53	512	19	.	.	PUNCT
ma-53	513	1	conclusion	conclusion	NOUN
ma-53	513	2	references	reference	NOUN