id	sid	tid	token	lemma	pos
ma-297	1	1	2025	2025	NUM
ma-297	1	2	ada	ada	PROPN
ma-297	1	3	academica	academica	PROPN
ma-297	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-297	1	5	.	.	PUNCT
ma-297	2	1	j.	j.	PROPN
ma-297	2	2	math	math	PROPN
ma-297	2	3	.	.	PUNCT
ma-297	3	1	anal	anal	ADJ
ma-297	3	2	.	.	PUNCT
ma-297	4	1	5	5	NUM
ma-297	4	2	(	(	PUNCT
ma-297	4	3	2025	2025	NUM
ma-297	4	4	)	)	PUNCT
ma-297	4	5	11doi	11doi	NUM
ma-297	4	6	:	:	PUNCT
ma-297	4	7	10.28924	10.28924	NUM
ma-297	4	8	/	/	SYM
ma-297	4	9	ada	ada	NOUN
ma-297	4	10	/	/	SYM
ma-297	4	11	ma.5.11	ma.5.11	VERB
ma-297	4	12	majorizing	majorize	VERB
ma-297	4	13	sequences	sequence	NOUN
ma-297	4	14	for	for	ADP
ma-297	4	15	newton	newton	PROPN
ma-297	4	16	-	-	PUNCT
ma-297	4	17	like	like	ADJ
ma-297	4	18	method	method	NOUN
ma-297	4	19	and	and	CCONJ
ma-297	4	20	their	their	PRON
ma-297	4	21	limit	limit	NOUN
ma-297	4	22	points	point	NOUN
ma-297	5	1	ioannis	ioannis	PROPN
ma-297	5	2	k.	k.	PROPN
ma-297	5	3	argyros1,∗	argyros1,∗	PROPN
ma-297	5	4	,	,	PUNCT
ma-297	5	5	santhosh	santhosh	PROPN
ma-297	5	6	george2	george2	PROPN
ma-297	5	7	,	,	PUNCT
ma-297	5	8	michael	michael	PROPN
ma-297	5	9	argyros3	argyros3	PROPN
ma-297	5	10	1department	1department	NUM
ma-297	5	11	of	of	ADP
ma-297	5	12	mathematical	mathematical	ADJ
ma-297	5	13	sciences	sciences	PROPN
ma-297	5	14	,	,	PUNCT
ma-297	5	15	cameron	cameron	PROPN
ma-297	5	16	university	university	PROPN
ma-297	5	17	,	,	PUNCT
ma-297	5	18	lawton	lawton	PROPN
ma-297	5	19	,	,	PUNCT
ma-297	5	20	ok	ok	PROPN
ma-297	5	21	73505	73505	NUM
ma-297	5	22	,	,	PUNCT
ma-297	5	23	usa	usa	PROPN
ma-297	5	24	iargyros@cameron.edu	iargyros@cameron.edu	PROPN
ma-297	6	1	2department	2department	NUM
ma-297	6	2	of	of	ADP
ma-297	6	3	mathematical	mathematical	ADJ
ma-297	6	4	and	and	CCONJ
ma-297	6	5	computational	computational	ADJ
ma-297	6	6	sciences	science	NOUN
ma-297	6	7	,	,	PUNCT
ma-297	6	8	national	national	PROPN
ma-297	6	9	institute	institute	PROPN
ma-297	6	10	of	of	ADP
ma-297	6	11	technology	technology	PROPN
ma-297	6	12	karnataka	karnataka	PROPN
ma-297	6	13	,	,	PUNCT
ma-297	6	14	india-575	india-575	ADJ
ma-297	6	15	025	025	NUM
ma-297	6	16	sgeorge@nitk.edu.in	sgeorge@nitk.edu.in	NOUN
ma-297	6	17	3department	3department	NUM
ma-297	6	18	of	of	ADP
ma-297	6	19	computer	computer	NOUN
ma-297	6	20	sciences	sciences	PROPN
ma-297	6	21	,	,	PUNCT
ma-297	6	22	franklin	franklin	PROPN
ma-297	6	23	university	university	PROPN
ma-297	6	24	,	,	PUNCT
ma-297	6	25	ohio	ohio	PROPN
ma-297	6	26	,	,	PUNCT
ma-297	6	27	usa	usa	PROPN
ma-297	6	28	argyro01@email.franklin.edu	argyro01@email.franklin.edu	PROPN
ma-297	6	29	∗correspondence	∗correspondence	NOUN
ma-297	6	30	:	:	PUNCT
ma-297	6	31	iargyros@cameron.edu	iargyros@cameron.edu	X
ma-297	7	1	abstract	abstract	ADJ
ma-297	7	2	.	.	PUNCT
ma-297	8	1	a	a	DET
ma-297	8	2	plethora	plethora	NOUN
ma-297	8	3	of	of	ADP
ma-297	8	4	problems	problem	NOUN
ma-297	8	5	from	from	ADP
ma-297	8	6	diverse	diverse	ADJ
ma-297	8	7	disciplines	discipline	NOUN
ma-297	8	8	of	of	ADP
ma-297	8	9	mathematics	mathematic	NOUN
ma-297	8	10	,	,	PUNCT
ma-297	8	11	mathematical	mathematical	ADJ
ma-297	8	12	biology	biology	NOUN
ma-297	8	13	,	,	PUNCT
ma-297	8	14	chemistry	chemistry	NOUN
ma-297	8	15	,	,	PUNCT
ma-297	8	16	medicine	medicine	NOUN
ma-297	8	17	,	,	PUNCT
ma-297	8	18	physics	physics	NOUN
ma-297	8	19	and	and	CCONJ
ma-297	8	20	engineering	engineering	NOUN
ma-297	8	21	to	to	PART
ma-297	8	22	mention	mention	VERB
ma-297	8	23	a	a	DET
ma-297	8	24	few	few	ADJ
ma-297	8	25	reduce	reduce	NOUN
ma-297	8	26	to	to	ADP
ma-297	8	27	solving	solve	VERB
ma-297	8	28	nonlinear	nonlinear	ADJ
ma-297	8	29	equationsor	equationsor	NOUN
ma-297	8	30	systems	system	NOUN
ma-297	8	31	of	of	ADP
ma-297	8	32	equations	equation	NOUN
ma-297	8	33	usually	usually	ADV
ma-297	8	34	in	in	ADP
ma-297	8	35	the	the	DET
ma-297	8	36	finite	finite	ADJ
ma-297	8	37	dimensional	dimensional	ADJ
ma-297	8	38	euclidean	euclidean	NOUN
ma-297	8	39	or	or	CCONJ
ma-297	8	40	more	more	ADV
ma-297	8	41	general	general	ADJ
ma-297	8	42	spaces	space	NOUN
ma-297	8	43	.	.	PUNCT
ma-297	9	1	thesolutions	thesolution	NOUN
ma-297	9	2	of	of	ADP
ma-297	9	3	such	such	ADJ
ma-297	9	4	equations	equation	NOUN
ma-297	9	5	are	be	AUX
ma-297	9	6	numbers	number	NOUN
ma-297	9	7	or	or	CCONJ
ma-297	9	8	vectors	vector	NOUN
ma-297	9	9	of	of	ADP
ma-297	9	10	functions	function	NOUN
ma-297	9	11	and	and	CCONJ
ma-297	9	12	can	can	AUX
ma-297	9	13	be	be	AUX
ma-297	9	14	found	find	VERB
ma-297	9	15	in	in	ADP
ma-297	9	16	closed	closed	ADJ
ma-297	9	17	form	form	NOUN
ma-297	9	18	onlyin	onlyin	X
ma-297	9	19	special	special	ADJ
ma-297	9	20	cases	case	NOUN
ma-297	9	21	.	.	PUNCT
ma-297	10	1	that	that	PRON
ma-297	10	2	is	be	AUX
ma-297	10	3	why	why	SCONJ
ma-297	10	4	researchers	researcher	NOUN
ma-297	10	5	and	and	CCONJ
ma-297	10	6	practitioners	practitioner	NOUN
ma-297	10	7	develop	develop	VERB
ma-297	10	8	mostly	mostly	ADV
ma-297	10	9	iterative	iterative	ADJ
ma-297	10	10	methods	method	NOUN
ma-297	10	11	whichgenerate	whichgenerate	VERB
ma-297	10	12	sequences	sequence	NOUN
ma-297	10	13	approximating	approximate	VERB
ma-297	10	14	the	the	DET
ma-297	10	15	solutions	solution	NOUN
ma-297	10	16	.	.	PUNCT
ma-297	11	1	the	the	DET
ma-297	11	2	least	least	ADJ
ma-297	11	3	number	number	NOUN
ma-297	11	4	of	of	ADP
ma-297	11	5	iterations	iteration	NOUN
ma-297	11	6	to	to	PART
ma-297	11	7	be	be	AUX
ma-297	11	8	carried	carry	VERB
ma-297	11	9	outin	outin	NOUN
ma-297	11	10	order	order	NOUN
ma-297	11	11	to	to	PART
ma-297	11	12	obtain	obtain	VERB
ma-297	11	13	a	a	DET
ma-297	11	14	pre	pre	ADJ
ma-297	11	15	-	-	ADJ
ma-297	11	16	decided	decide	VERB
ma-297	11	17	error	error	NOUN
ma-297	11	18	tolerance	tolerance	NOUN
ma-297	11	19	on	on	ADP
ma-297	11	20	the	the	DET
ma-297	11	21	distances	distance	NOUN
ma-297	11	22	between	between	ADP
ma-297	11	23	consecutive	consecutive	ADJ
ma-297	11	24	iterates	iterate	NOUN
ma-297	11	25	aswell	aswell	ADJ
ma-297	11	26	as	as	ADP
ma-297	11	27	the	the	DET
ma-297	11	28	choice	choice	NOUN
ma-297	11	29	of	of	ADP
ma-297	11	30	initial	initial	ADJ
ma-297	11	31	points	point	NOUN
ma-297	11	32	ensuring	ensure	VERB
ma-297	11	33	the	the	DET
ma-297	11	34	convergence	convergence	NOUN
ma-297	11	35	of	of	ADP
ma-297	11	36	the	the	DET
ma-297	11	37	methods	method	NOUN
ma-297	11	38	is	be	AUX
ma-297	11	39	very	very	ADV
ma-297	11	40	important	important	ADJ
ma-297	11	41	.	.	PUNCT
ma-297	12	1	thesetwo	thesetwo	NUM
ma-297	12	2	objectives	objective	NOUN
ma-297	12	3	can	can	AUX
ma-297	12	4	be	be	AUX
ma-297	12	5	achieved	achieve	VERB
ma-297	12	6	by	by	ADP
ma-297	12	7	introducing	introduce	VERB
ma-297	12	8	real	real	ADJ
ma-297	12	9	majorizing	majorize	VERB
ma-297	12	10	sequences	sequence	NOUN
ma-297	12	11	which	which	PRON
ma-297	12	12	control	control	VERB
ma-297	12	13	the	the	DET
ma-297	12	14	behaviourof	behaviourof	NOUN
ma-297	12	15	the	the	DET
ma-297	12	16	iterates	iterate	NOUN
ma-297	12	17	.	.	PUNCT
ma-297	13	1	moreover	moreover	ADV
ma-297	13	2	,	,	PUNCT
ma-297	13	3	the	the	DET
ma-297	13	4	closed	closed	ADJ
ma-297	13	5	form	form	NOUN
ma-297	13	6	of	of	ADP
ma-297	13	7	the	the	DET
ma-297	13	8	limits	limit	NOUN
ma-297	13	9	of	of	ADP
ma-297	13	10	the	the	DET
ma-297	13	11	real	real	ADJ
ma-297	13	12	sequences	sequence	NOUN
ma-297	13	13	determine	determine	VERB
ma-297	13	14	the	the	DET
ma-297	13	15	radiusof	radiusof	NOUN
ma-297	13	16	the	the	DET
ma-297	13	17	ball	ball	NOUN
ma-297	13	18	that	that	PRON
ma-297	13	19	contains	contain	VERB
ma-297	13	20	the	the	DET
ma-297	13	21	initial	initial	ADJ
ma-297	13	22	points	point	NOUN
ma-297	13	23	.	.	PUNCT
ma-297	14	1	in	in	ADP
ma-297	14	2	this	this	DET
ma-297	14	3	paper	paper	NOUN
ma-297	14	4	we	we	PRON
ma-297	14	5	contribute	contribute	VERB
ma-297	14	6	by	by	ADP
ma-297	14	7	introducing	introduce	VERB
ma-297	14	8	more	more	ADJ
ma-297	14	9	precisemajorizing	precisemajorizing	ADJ
ma-297	14	10	sequences	sequence	NOUN
ma-297	14	11	and	and	CCONJ
ma-297	14	12	limit	limit	VERB
ma-297	14	13	points	point	NOUN
ma-297	14	14	.	.	PUNCT
ma-297	15	1	1	1	X
ma-297	15	2	.	.	X
ma-297	15	3	introduction	introduction	NOUN
ma-297	15	4	majorizing	majorize	VERB
ma-297	15	5	sequences	sequence	NOUN
ma-297	15	6	have	have	AUX
ma-297	15	7	been	be	AUX
ma-297	15	8	used	use	VERB
ma-297	15	9	extensively	extensively	ADV
ma-297	15	10	to	to	PART
ma-297	15	11	study	study	VERB
ma-297	15	12	the	the	DET
ma-297	15	13	semi	semi	ADJ
ma-297	15	14	-	-	ADJ
ma-297	15	15	local	local	ADJ
ma-297	15	16	convergence	convergence	NOUN
ma-297	15	17	of	of	ADP
ma-297	15	18	new	new	ADJ
ma-297	15	19	-	-	PUNCT
ma-297	15	20	ton	ton	NOUN
ma-297	15	21	’s	’s	PART
ma-297	15	22	method	method	NOUN
ma-297	15	23	defined	define	VERB
ma-297	15	24	for	for	ADP
ma-297	15	25	x0	x0	PROPN
ma-297	15	26	∈	∈	PROPN
ma-297	15	27	d	d	NOUN
ma-297	15	28	and	and	CCONJ
ma-297	15	29	each	each	DET
ma-297	15	30	n	n	NOUN
ma-297	15	31	=	=	SYM
ma-297	15	32	0	0	NUM
ma-297	15	33	,	,	PUNCT
ma-297	15	34	1	1	NUM
ma-297	15	35	,	,	PUNCT
ma-297	15	36	2	2	NUM
ma-297	15	37	,	,	PUNCT
ma-297	15	38	...	...	PUNCT
ma-297	15	39	by	by	ADP
ma-297	15	40	xn+1	xn+1	PROPN
ma-297	16	1	=	=	SYM
ma-297	16	2	xn	xn	PROPN
ma-297	17	1	−	−	PROPN
ma-297	17	2	f	f	PROPN
ma-297	17	3	′(xn)−1f	′(xn)−1f	PROPN
ma-297	17	4	(	(	PUNCT
ma-297	17	5	xn	xn	PROPN
ma-297	17	6	)	)	PUNCT
ma-297	17	7	,	,	PUNCT
ma-297	17	8	(	(	PUNCT
ma-297	17	9	1.1	1.1	NUM
ma-297	17	10	)	)	PUNCT
ma-297	17	11	where	where	SCONJ
ma-297	17	12	f	f	NOUN
ma-297	17	13	:	:	PUNCT
ma-297	18	1	d	d	PROPN
ma-297	18	2	⊂	⊂	PROPN
ma-297	18	3	b1	b1	PROPN
ma-297	18	4	→	→	SYM
ma-297	18	5	b2	b2	NOUN
ma-297	18	6	is	be	AUX
ma-297	18	7	a	a	DET
ma-297	18	8	fŕechetdifferentiable	fŕechetdifferentiable	ADJ
ma-297	18	9	operator	operator	NOUN
ma-297	18	10	between	between	ADP
ma-297	18	11	banach	banach	NOUN
ma-297	18	12	spaces	space	NOUN
ma-297	18	13	b1	b1	NOUN
ma-297	18	14	,	,	PUNCT
ma-297	18	15	b2	b2	NOUN
ma-297	18	16	and	and	CCONJ
ma-297	18	17	d	d	NOUN
ma-297	18	18	is	be	AUX
ma-297	18	19	an	an	DET
ma-297	18	20	open	open	ADJ
ma-297	18	21	and	and	CCONJ
ma-297	18	22	convex	convex	NOUN
ma-297	18	23	set	set	NOUN
ma-297	18	24	[	[	X
ma-297	18	25	1	1	NUM
ma-297	18	26	,	,	PUNCT
ma-297	18	27	3	3	NUM
ma-297	18	28	,	,	PUNCT
ma-297	18	29	4	4	NUM
ma-297	18	30	,	,	PUNCT
ma-297	18	31	6	6	NUM
ma-297	18	32	,	,	PUNCT
ma-297	18	33	7	7	NUM
ma-297	18	34	]	]	PUNCT
ma-297	18	35	.	.	PUNCT
ma-297	19	1	the	the	DET
ma-297	19	2	usually	usually	ADV
ma-297	19	3	sufficient	sufficient	ADJ
ma-297	19	4	semi	semi	ADJ
ma-297	19	5	-	-	ADJ
ma-297	19	6	local	local	ADJ
ma-297	19	7	convergence	convergence	NOUN
ma-297	19	8	conditionsdiffer	conditionsdiffer	NOUN
ma-297	19	9	in	in	ADP
ma-297	19	10	general	general	ADJ
ma-297	19	11	as	as	ADV
ma-297	19	12	well	well	ADV
ma-297	19	13	as	as	ADP
ma-297	19	14	the	the	DET
ma-297	19	15	majorizing	majorize	VERB
ma-297	19	16	sequences	sequence	NOUN
ma-297	19	17	and	and	CCONJ
ma-297	19	18	their	their	PRON
ma-297	19	19	limit	limit	NOUN
ma-297	19	20	points	point	NOUN
ma-297	19	21	.	.	PUNCT
ma-297	20	1	we	we	PRON
ma-297	20	2	try	try	VERB
ma-297	20	3	to	to	PART
ma-297	20	4	relate	relate	VERB
ma-297	20	5	theseconditions	thesecondition	NOUN
ma-297	20	6	,	,	PUNCT
ma-297	20	7	sequences	sequence	NOUN
ma-297	20	8	and	and	CCONJ
ma-297	20	9	limit	limit	VERB
ma-297	20	10	points	point	NOUN
ma-297	20	11	in	in	ADP
ma-297	20	12	a	a	DET
ma-297	20	13	unified	unified	ADJ
ma-297	20	14	way	way	NOUN
ma-297	20	15	without	without	ADP
ma-297	20	16	additional	additional	ADJ
ma-297	20	17	hypotheses	hypothesis	NOUN
ma-297	20	18	.	.	PUNCT
ma-297	21	1	received	receive	VERB
ma-297	21	2	:	:	PUNCT
ma-297	21	3	10	10	NUM
ma-297	21	4	nov	nov	PROPN
ma-297	21	5	2024	2024	NUM
ma-297	21	6	.	.	PUNCT
ma-297	22	1	key	key	ADJ
ma-297	22	2	words	word	NOUN
ma-297	22	3	and	and	CCONJ
ma-297	22	4	phrases	phrase	NOUN
ma-297	22	5	.	.	PUNCT
ma-297	23	1	newton	newton	PROPN
ma-297	23	2	-	-	PUNCT
ma-297	23	3	like	like	ADJ
ma-297	23	4	method	method	NOUN
ma-297	23	5	;	;	PUNCT
ma-297	23	6	majorizing	majorize	VERB
ma-297	23	7	sequences	sequence	NOUN
ma-297	23	8	;	;	PUNCT
ma-297	23	9	fréchet	fréchet	NOUN
ma-297	23	10	derivative	derivative	NOUN
ma-297	23	11	;	;	PUNCT
ma-297	23	12	banach	banach	NOUN
ma-297	23	13	spaces.1	spaces.1	NOUN
ma-297	23	14	https://adac.ee	https://adac.ee	PROPN
ma-297	23	15	https://doi.org/10.28924/ada/ma.5.11	https://doi.org/10.28924/ada/ma.5.11	X
ma-297	23	16	https://orcid.org/0000-0002-9189-9298	https://orcid.org/0000-0002-9189-9298	PROPN
ma-297	23	17	https://orcid.org/0000-0002-3530-5539	https://orcid.org/0000-0002-3530-5539	PROPN
ma-297	23	18	eur	eur	PROPN
ma-297	23	19	.	.	PUNCT
ma-297	24	1	j.	j.	PROPN
ma-297	24	2	math	math	PROPN
ma-297	24	3	.	.	PUNCT
ma-297	25	1	anal	anal	PROPN
ma-297	25	2	.	.	PUNCT
ma-297	26	1	10.28924	10.28924	NUM
ma-297	26	2	/	/	SYM
ma-297	26	3	ada	ada	NOUN
ma-297	26	4	/	/	SYM
ma-297	26	5	ma.5.11	ma.5.11	ADJ
ma-297	26	6	22	22	NUM
ma-297	26	7	.	.	PUNCT
ma-297	27	1	lipschitz	lipschitz	NOUN
ma-297	27	2	conditions	condition	NOUN
ma-297	27	3	the	the	DET
ma-297	27	4	symbols	symbol	NOUN
ma-297	27	5	l(b1	l(b1	VERB
ma-297	27	6	,	,	PUNCT
ma-297	27	7	b2	b2	NOUN
ma-297	27	8	)	)	PUNCT
ma-297	27	9	,	,	PUNCT
ma-297	27	10	u(x	u(x	PROPN
ma-297	27	11	,	,	PUNCT
ma-297	27	12	r	r	NOUN
ma-297	27	13	)	)	PUNCT
ma-297	27	14	are	be	AUX
ma-297	27	15	used	use	VERB
ma-297	27	16	to	to	PART
ma-297	27	17	denote	denote	VERB
ma-297	27	18	the	the	DET
ma-297	27	19	space	space	NOUN
ma-297	27	20	of	of	ADP
ma-297	27	21	bounded	bounded	ADJ
ma-297	27	22	linear	linear	PROPN
ma-297	27	23	operators	operator	NOUN
ma-297	27	24	from	from	ADP
ma-297	27	25	b1	b1	NOUN
ma-297	27	26	into	into	ADP
ma-297	27	27	b2	b2	NOUN
ma-297	27	28	and	and	CCONJ
ma-297	28	1	the	the	DET
ma-297	28	2	open	open	ADJ
ma-297	28	3	ball	ball	NOUN
ma-297	28	4	centered	center	VERB
ma-297	28	5	at	at	ADP
ma-297	28	6	x	x	PROPN
ma-297	28	7	∈	∈	PROPN
ma-297	28	8	b1	b1	NOUN
ma-297	28	9	and	and	CCONJ
ma-297	28	10	of	of	ADP
ma-297	28	11	radius	radius	NOUN
ma-297	29	1	r	r	PROPN
ma-297	29	2	>	>	X
ma-297	29	3	0	0	NUM
ma-297	29	4	,	,	PUNCT
ma-297	29	5	respectively.we	respectively.we	NOUN
ma-297	29	6	introduce	introduce	VERB
ma-297	29	7	lipscits	lipscit	NOUN
ma-297	29	8	conditions	condition	NOUN
ma-297	29	9	used	use	VERB
ma-297	29	10	to	to	PART
ma-297	29	11	control	control	VERB
ma-297	29	12	f	f	PROPN
ma-297	29	13	′.	′.	PROPN
ma-297	29	14	then	then	ADV
ma-297	29	15	,	,	PUNCT
ma-297	29	16	we	we	PRON
ma-297	29	17	copare	copare	VERB
ma-297	29	18	them	they	PRON
ma-297	29	19	to	to	ADP
ma-297	29	20	each	each	DET
ma-297	29	21	other	other	ADJ
ma-297	29	22	.	.	PUNCT
ma-297	30	1	definition	definition	NOUN
ma-297	30	2	2.1	2.1	NUM
ma-297	30	3	.	.	PUNCT
ma-297	30	4	suppose	suppose	VERB
ma-297	30	5	m	m	VERB
ma-297	30	6	∈	∈	PROPN
ma-297	30	7	l	l	NOUN
ma-297	30	8	(	(	PUNCT
ma-297	30	9	b1	b1	NOUN
ma-297	30	10	,	,	PUNCT
ma-297	30	11	b2	b2	NOUN
ma-297	30	12	)	)	PUNCT
ma-297	30	13	is	be	AUX
ma-297	30	14	an	an	DET
ma-297	30	15	invertible	invertible	ADJ
ma-297	30	16	operator	operator	NOUN
ma-297	30	17	and	and	CCONJ
ma-297	31	1	x0	x0	PROPN
ma-297	31	2	∈	∈	PROPN
ma-297	31	3	d.	d.	NOUN
ma-297	31	4	we	we	PRON
ma-297	31	5	say	say	VERB
ma-297	31	6	that	that	SCONJ
ma-297	31	7	f	f	PROPN
ma-297	31	8	′	′	NOUN
ma-297	31	9	is	be	AUX
ma-297	31	10	center	center	ADJ
ma-297	31	11	-	-	PUNCT
ma-297	31	12	lipschitz	lipschitz	NOUN
ma-297	31	13	continuous	continuous	ADJ
ma-297	31	14	if	if	SCONJ
ma-297	31	15	there	there	PRON
ma-297	31	16	exists	exist	VERB
ma-297	31	17	l0	l0	PROPN
ma-297	31	18	>	>	X
ma-297	31	19	0	0	NUM
ma-297	32	1	such	such	ADJ
ma-297	32	2	that	that	SCONJ
ma-297	32	3	m−1(f	m−1(f	PROPN
ma-297	32	4	′(x)−m)‖	′(x)−m)‖	PROPN
ma-297	32	5	≤	≤	ADV
ma-297	32	6	l0‖x	l0‖x	NUM
ma-297	32	7	−	−	PROPN
ma-297	32	8	x0‖	x0‖	PROPN
ma-297	32	9	for	for	ADP
ma-297	32	10	each	each	DET
ma-297	32	11	x	x	SYM
ma-297	32	12	∈	∈	PROPN
ma-297	32	13	d0	d0	NOUN
ma-297	32	14	.	.	PUNCT
ma-297	33	1	(	(	PUNCT
ma-297	33	2	2.1	2.1	NUM
ma-297	33	3	)	)	PUNCT
ma-297	33	4	define	define	VERB
ma-297	33	5	the	the	DET
ma-297	33	6	region	region	NOUN
ma-297	33	7	d0	d0	NOUN
ma-297	33	8	=	=	SYM
ma-297	34	1	d	d	PROPN
ma-297	34	2	∩	∩	ADJ
ma-297	34	3	u(x0	u(x0	NOUN
ma-297	34	4	,	,	PUNCT
ma-297	34	5	1	1	NUM
ma-297	34	6	l0	l0	NOUN
ma-297	34	7	)	)	PUNCT
ma-297	34	8	.	.	PUNCT
ma-297	35	1	(	(	PUNCT
ma-297	35	2	2.2	2.2	NUM
ma-297	35	3	)	)	PUNCT
ma-297	35	4	definition	definition	NOUN
ma-297	35	5	2.2	2.2	NUM
ma-297	35	6	.	.	PUNCT
ma-297	35	7	suppose	suppose	VERB
ma-297	35	8	m	m	PROPN
ma-297	35	9	∈	∈	PROPN
ma-297	35	10	l	l	NOUN
ma-297	35	11	(	(	PUNCT
ma-297	35	12	b1	b1	NOUN
ma-297	35	13	,	,	PUNCT
ma-297	35	14	b2	b2	NOUN
ma-297	35	15	)	)	PUNCT
ma-297	35	16	is	be	AUX
ma-297	35	17	an	an	DET
ma-297	35	18	invertible	invertible	ADJ
ma-297	35	19	operator	operator	NOUN
ma-297	35	20	.	.	PUNCT
ma-297	36	1	we	we	PRON
ma-297	36	2	say	say	VERB
ma-297	36	3	that	that	SCONJ
ma-297	36	4	f	f	PROPN
ma-297	36	5	′	′	NOUN
ma-297	36	6	is	be	AUX
ma-297	36	7	restricted	restrict	VERB
ma-297	36	8	lipschitz	lipschitz	NOUN
ma-297	36	9	continuous	continuous	ADJ
ma-297	36	10	if	if	SCONJ
ma-297	36	11	there	there	PRON
ma-297	36	12	exists	exist	VERB
ma-297	36	13	l	l	NOUN
ma-297	36	14	>	>	X
ma-297	36	15	0	0	NUM
ma-297	36	16	such	such	ADJ
ma-297	36	17	that	that	SCONJ
ma-297	36	18	‖m−1(f	‖m−1(f	PROPN
ma-297	36	19	′(y)−	′(y)−	VERB
ma-297	36	20	f	f	PROPN
ma-297	36	21	′(x))‖	′(x))‖	PROPN
ma-297	36	22	≤	≤	PROPN
ma-297	36	23	l‖y	l‖y	NOUN
ma-297	36	24	−	−	PROPN
ma-297	36	25	x‖	x‖	PROPN
ma-297	36	26	for	for	ADP
ma-297	36	27	each	each	DET
ma-297	36	28	x	x	NOUN
ma-297	36	29	,	,	PUNCT
ma-297	36	30	y	y	PROPN
ma-297	36	31	∈	∈	PROPN
ma-297	36	32	d0	d0	NOUN
ma-297	36	33	.	.	PUNCT
ma-297	37	1	(	(	PUNCT
ma-297	37	2	2.3	2.3	NUM
ma-297	37	3	)	)	PUNCT
ma-297	37	4	definition	definition	NOUN
ma-297	37	5	2.3	2.3	NUM
ma-297	37	6	.	.	PUNCT
ma-297	38	1	suppose	suppose	VERB
ma-297	38	2	m	m	VERB
ma-297	38	3	∈	∈	PROPN
ma-297	38	4	l	l	NOUN
ma-297	38	5	(	(	PUNCT
ma-297	38	6	b1	b1	NOUN
ma-297	38	7	,	,	PUNCT
ma-297	38	8	b2	b2	NOUN
ma-297	38	9	)	)	PUNCT
ma-297	38	10	is	be	AUX
ma-297	38	11	an	an	DET
ma-297	38	12	invertible	invertible	ADJ
ma-297	38	13	operator	operator	NOUN
ma-297	38	14	.	.	PUNCT
ma-297	39	1	we	we	PRON
ma-297	39	2	say	say	VERB
ma-297	39	3	that	that	SCONJ
ma-297	39	4	f	f	PROPN
ma-297	39	5	′	′	NOUN
ma-297	39	6	is	be	AUX
ma-297	39	7	lipschitz	lipschitz	VERB
ma-297	39	8	continuous	continuous	ADJ
ma-297	39	9	if	if	SCONJ
ma-297	39	10	there	there	PRON
ma-297	39	11	exists	exist	VERB
ma-297	39	12	l1	l1	PROPN
ma-297	39	13	>	>	X
ma-297	39	14	0	0	NUM
ma-297	40	1	such	such	ADJ
ma-297	40	2	that	that	SCONJ
ma-297	40	3	‖m−1(f	‖m−1(f	PROPN
ma-297	40	4	′(y)−	′(y)−	VERB
ma-297	40	5	f	f	PROPN
ma-297	40	6	′(x))‖	′(x))‖	PROPN
ma-297	40	7	≤	≤	X
ma-297	40	8	l1‖y	l1‖y	NUM
ma-297	40	9	−	−	NOUN
ma-297	40	10	x‖	x‖	PROPN
ma-297	40	11	for	for	ADP
ma-297	40	12	each	each	DET
ma-297	40	13	x	x	NOUN
ma-297	40	14	,	,	PUNCT
ma-297	40	15	y	y	PROPN
ma-297	40	16	∈	∈	PROPN
ma-297	40	17	d.	d.	PROPN
ma-297	40	18	(	(	PUNCT
ma-297	40	19	2.4	2.4	NUM
ma-297	40	20	)	)	PUNCT
ma-297	40	21	remark	remark	NOUN
ma-297	40	22	2.4	2.4	NUM
ma-297	40	23	.	.	PUNCT
ma-297	41	1	it	it	PRON
ma-297	41	2	follows	follow	VERB
ma-297	41	3	by	by	ADP
ma-297	41	4	these	these	DET
ma-297	41	5	definitions	definition	NOUN
ma-297	41	6	that	that	SCONJ
ma-297	41	7	since	since	SCONJ
ma-297	41	8	d0	d0	NOUN
ma-297	41	9	⊆	⊆	NUM
ma-297	41	10	d	d	NOUN
ma-297	41	11	,	,	PUNCT
ma-297	41	12	we	we	PRON
ma-297	41	13	have	have	VERB
ma-297	41	14	l0	l0	NOUN
ma-297	41	15	≤	≤	PROPN
ma-297	41	16	l1	l1	PROPN
ma-297	41	17	(	(	PUNCT
ma-297	41	18	2.5	2.5	NUM
ma-297	41	19	)	)	PUNCT
ma-297	41	20	and	and	CCONJ
ma-297	41	21	l	l	NOUN
ma-297	41	22	≤	≤	PROPN
ma-297	41	23	l1	l1	PROPN
ma-297	41	24	.	.	PUNCT
ma-297	42	1	(	(	PUNCT
ma-297	42	2	2.6	2.6	NUM
ma-297	42	3	)	)	PUNCT
ma-297	42	4	it	it	PRON
ma-297	42	5	is	be	AUX
ma-297	42	6	worth	worth	ADJ
ma-297	42	7	noting	note	VERB
ma-297	42	8	that	that	SCONJ
ma-297	42	9	l0	l0	PROPN
ma-297	42	10	and	and	CCONJ
ma-297	42	11	l1	l1	PROPN
ma-297	42	12	depend	depend	VERB
ma-297	42	13	on	on	ADP
ma-297	42	14	x0	x0	PROPN
ma-297	42	15	,	,	PUNCT
ma-297	42	16	f	f	PROPN
ma-297	42	17	′	′	NOUN
ma-297	42	18	and	and	CCONJ
ma-297	42	19	d.	d.	PROPN
ma-297	42	20	but	but	CCONJ
ma-297	42	21	l	l	PROPN
ma-297	42	22	depends	depend	VERB
ma-297	42	23	on	on	ADP
ma-297	42	24	x0	x0	PROPN
ma-297	42	25	,	,	PUNCT
ma-297	42	26	f	f	PROPN
ma-297	42	27	′	′	NOUN
ma-297	42	28	and	and	CCONJ
ma-297	42	29	d0.moreover	d0.moreover	NOUN
ma-297	42	30	,	,	PUNCT
ma-297	42	31	in	in	ADP
ma-297	42	32	practice	practice	NOUN
ma-297	42	33	the	the	DET
ma-297	42	34	computation	computation	NOUN
ma-297	42	35	of	of	ADP
ma-297	42	36	l1	l1	PROPN
ma-297	42	37	requires	require	VERB
ma-297	42	38	that	that	PRON
ma-297	42	39	of	of	ADP
ma-297	42	40	l0	l0	PROPN
ma-297	42	41	and	and	CCONJ
ma-297	42	42	l	l	NOUN
ma-297	42	43	as	as	ADP
ma-297	42	44	special	special	ADJ
ma-297	42	45	cases	case	NOUN
ma-297	42	46	.	.	PUNCT
ma-297	43	1	theseconstants	theseconstant	NOUN
ma-297	43	2	are	be	AUX
ma-297	43	3	related	relate	VERB
ma-297	43	4	to	to	ADP
ma-297	43	5	majorizing	majorize	VERB
ma-297	43	6	sequences	sequence	NOUN
ma-297	43	7	in	in	ADP
ma-297	43	8	section	section	NOUN
ma-297	43	9	3	3	NUM
ma-297	43	10	.	.	NOUN
ma-297	43	11	3	3	X
ma-297	44	1	.	.	X
ma-297	44	2	convergence	convergence	NOUN
ma-297	44	3	of	of	ADP
ma-297	44	4	majorizing	majorize	VERB
ma-297	44	5	sequences	sequence	NOUN
ma-297	44	6	.	.	PUNCT
ma-297	45	1	let	let	VERB
ma-297	45	2	ω	ω	NUM
ma-297	45	3	≥	≥	NOUN
ma-297	45	4	0	0	NUM
ma-297	45	5	,	,	PUNCT
ma-297	45	6	l1	l1	PROPN
ma-297	45	7	>	>	X
ma-297	45	8	0	0	PUNCT
ma-297	46	1	and	and	CCONJ
ma-297	46	2	λ	λ	PROPN
ma-297	46	3	≥	≥	NOUN
ma-297	46	4	1	1	NUM
ma-297	46	5	be	be	AUX
ma-297	46	6	parameters	parameter	NOUN
ma-297	46	7	.	.	PUNCT
ma-297	47	1	define	define	VERB
ma-297	47	2	µ	µ	PROPN
ma-297	47	3	and	and	CCONJ
ma-297	47	4	β	β	X
ma-297	47	5	by	by	ADP
ma-297	47	6	µ	µ	NOUN
ma-297	47	7	=	=	SYM
ma-297	47	8	λω	λω	PROPN
ma-297	47	9	and	and	CCONJ
ma-297	47	10	β	β	X
ma-297	47	11	=	=	SYM
ma-297	47	12	λ	λ	NOUN
ma-297	47	13	1	1	NUM
ma-297	48	1	+	+	CCONJ
ma-297	48	2	(	(	PUNCT
ma-297	48	3	λ−	λ−	PROPN
ma-297	48	4	1)l1µ	1)l1µ	NUM
ma-297	48	5	.	.	PUNCT
ma-297	49	1	(	(	PUNCT
ma-297	49	2	3.1	3.1	NUM
ma-297	49	3	)	)	PUNCT
ma-297	49	4	moreover	moreover	ADV
ma-297	49	5	,	,	PUNCT
ma-297	49	6	define	define	VERB
ma-297	49	7	the	the	DET
ma-297	49	8	quadratic	quadratic	ADJ
ma-297	49	9	majorizing	majorizing	NOUN
ma-297	49	10	function	function	NOUN
ma-297	49	11	f1	f1	NOUN
ma-297	49	12	by	by	ADP
ma-297	49	13	f1(t	f1(t	PROPN
ma-297	49	14	)	)	PUNCT
ma-297	49	15	=	=	PUNCT
ma-297	50	1	βl1	βl1	AUX
ma-297	50	2	t	t	NOUN
ma-297	50	3	2	2	NUM
ma-297	50	4	2	2	NUM
ma-297	50	5	−	−	NOUN
ma-297	50	6	t	t	NOUN
ma-297	50	7	+	+	NUM
ma-297	50	8	µ.	µ.	NOUN
ma-297	50	9	(	(	PUNCT
ma-297	50	10	3.2	3.2	NUM
ma-297	50	11	)	)	PUNCT
ma-297	50	12	further	far	ADV
ma-297	50	13	more	more	ADV
ma-297	50	14	,	,	PUNCT
ma-297	50	15	define	define	VERB
ma-297	50	16	the	the	DET
ma-297	50	17	scalar	scalar	ADJ
ma-297	50	18	sequence	sequence	NOUN
ma-297	50	19	{	{	PUNCT
ma-297	50	20	vn	vn	NOUN
ma-297	50	21	}	}	PUNCT
ma-297	50	22	for	for	ADP
ma-297	50	23	v0	v0	NOUN
ma-297	50	24	=	=	SYM
ma-297	50	25	0	0	PUNCT
ma-297	50	26	and	and	CCONJ
ma-297	50	27	each	each	DET
ma-297	50	28	n	n	NOUN
ma-297	50	29	=	=	SYM
ma-297	50	30	0	0	NUM
ma-297	50	31	,	,	PUNCT
ma-297	50	32	1	1	NUM
ma-297	50	33	,	,	PUNCT
ma-297	50	34	2	2	NUM
ma-297	50	35	,	,	PUNCT
ma-297	50	36	...	...	PUNCT
ma-297	50	37	by	by	ADP
ma-297	50	38	vn+1	vn+1	PROPN
ma-297	50	39	=	=	SYM
ma-297	50	40	vn	vn	PROPN
ma-297	50	41	−	−	PROPN
ma-297	50	42	f1(vn	f1(vn	PROPN
ma-297	50	43	)	)	PUNCT
ma-297	50	44	f	f	PROPN
ma-297	50	45	′1(vn	′1(vn	PROPN
ma-297	50	46	)	)	PUNCT
ma-297	50	47	.	.	PUNCT
ma-297	51	1	(	(	PUNCT
ma-297	51	2	3.3	3.3	NUM
ma-297	51	3	)	)	PUNCT
ma-297	51	4	https://doi.org/10.28924/ada/ma.5.11	https://doi.org/10.28924/ada/ma.5.11	NUM
ma-297	51	5	eur	eur	NOUN
ma-297	51	6	.	.	PUNCT
ma-297	52	1	j.	j.	PROPN
ma-297	52	2	math	math	PROPN
ma-297	52	3	.	.	PUNCT
ma-297	53	1	anal	anal	PROPN
ma-297	53	2	.	.	PUNCT
ma-297	54	1	10.28924	10.28924	NUM
ma-297	54	2	/	/	SYM
ma-297	54	3	ada	ada	NOUN
ma-297	54	4	/	/	SYM
ma-297	54	5	ma.5.11	ma.5.11	VERB
ma-297	54	6	3an	3an	ADJ
ma-297	54	7	auxiliary	auxiliary	ADJ
ma-297	54	8	result	result	NOUN
ma-297	54	9	is	be	AUX
ma-297	54	10	needed	need	VERB
ma-297	54	11	for	for	ADP
ma-297	54	12	the	the	DET
ma-297	54	13	convergence	convergence	NOUN
ma-297	54	14	of	of	ADP
ma-297	54	15	the	the	DET
ma-297	54	16	sequence	sequence	NOUN
ma-297	54	17	{	{	PUNCT
ma-297	54	18	vn	vn	NOUN
ma-297	54	19	}	}	PUNCT
ma-297	54	20	.	.	PUNCT
ma-297	55	1	lemma	lemma	PROPN
ma-297	55	2	3.1	3.1	NUM
ma-297	55	3	.	.	PUNCT
ma-297	55	4	suppose	suppose	VERB
ma-297	55	5	h1	h1	PROPN
ma-297	55	6	=	=	SYM
ma-297	55	7	2βl1µ	2βl1µ	NUM
ma-297	55	8	≤	≤	NUM
ma-297	55	9	1	1	NUM
ma-297	55	10	.	.	PUNCT
ma-297	56	1	(	(	PUNCT
ma-297	56	2	3.4	3.4	NUM
ma-297	56	3	)	)	PUNCT
ma-297	56	4	then	then	ADV
ma-297	56	5	,	,	PUNCT
ma-297	56	6	the	the	DET
ma-297	56	7	following	follow	VERB
ma-297	56	8	assertions	assertion	NOUN
ma-297	56	9	hold(i	hold(i	NUM
ma-297	56	10	)	)	PUNCT
ma-297	56	11	the	the	DET
ma-297	56	12	zeros	zero	NOUN
ma-297	56	13	of	of	ADP
ma-297	56	14	the	the	DET
ma-297	56	15	function	function	NOUN
ma-297	56	16	f1	f1	NOUN
ma-297	56	17	are	be	AUX
ma-297	56	18	real	real	ADJ
ma-297	56	19	and	and	CCONJ
ma-297	56	20	given	give	VERB
ma-297	56	21	by	by	ADP
ma-297	56	22	v∗	v∗	PROPN
ma-297	56	23	=	=	PROPN
ma-297	56	24	1−	1−	NUM
ma-297	56	25	√	√	NUM
ma-297	56	26	1−	1−	NUM
ma-297	57	1	2βl1µ	2βl1µ	NUM
ma-297	57	2	βl1	βl1	NOUN
ma-297	57	3	and	and	CCONJ
ma-297	57	4	v∗∗	v∗∗	VERB
ma-297	57	5	=	=	SYM
ma-297	57	6	1	1	NUM
ma-297	57	7	+	+	CCONJ
ma-297	57	8	√	√	PROPN
ma-297	57	9	1−	1−	NUM
ma-297	57	10	2βl1µ	2βl1µ	NUM
ma-297	57	11	βl1	βl1	NOUN
ma-297	57	12	.	.	PUNCT
ma-297	58	1	(	(	PUNCT
ma-297	58	2	3.5	3.5	NUM
ma-297	58	3	)	)	PUNCT
ma-297	58	4	(	(	PUNCT
ma-297	58	5	ii	ii	NOUN
ma-297	58	6	)	)	PUNCT
ma-297	58	7	vn+1	vn+1	PROPN
ma-297	59	1	−	−	PROPN
ma-297	59	2	vn	vn	PROPN
ma-297	59	3	=	=	SYM
ma-297	59	4	βl1(vn	βl1(vn	PUNCT
ma-297	59	5	−	−	PROPN
ma-297	59	6	vn−1)2	vn−1)2	VERB
ma-297	59	7	2(1−	2(1−	NUM
ma-297	59	8	βl1vn	βl1vn	NUM
ma-297	59	9	)	)	PUNCT
ma-297	59	10	=	=	SYM
ma-297	60	1	−	−	PROPN
ma-297	60	2	f1(vn	f1(vn	PROPN
ma-297	60	3	)	)	PUNCT
ma-297	60	4	f	f	PROPN
ma-297	60	5	′1(vn	′1(vn	PROPN
ma-297	60	6	)	)	PUNCT
ma-297	60	7	.	.	PUNCT
ma-297	61	1	(	(	PUNCT
ma-297	61	2	3.6	3.6	NUM
ma-297	61	3	)	)	PUNCT
ma-297	61	4	(	(	PUNCT
ma-297	61	5	iii	iii	X
ma-297	61	6	)	)	PUNCT
ma-297	61	7	the	the	DET
ma-297	61	8	sequence	sequence	NOUN
ma-297	61	9	{	{	PUNCT
ma-297	61	10	vn	vn	NOUN
ma-297	61	11	}	}	PUNCT
ma-297	61	12	is	be	AUX
ma-297	61	13	increasingly	increasingly	ADV
ma-297	61	14	convergent	convergent	ADJ
ma-297	61	15	to	to	ADP
ma-297	61	16	v∗	v∗	VERB
ma-297	61	17	and	and	CCONJ
ma-297	61	18	can	can	AUX
ma-297	61	19	also	also	ADV
ma-297	61	20	be	be	AUX
ma-297	61	21	written	write	VERB
ma-297	61	22	in	in	ADP
ma-297	61	23	closed	close	VERB
ma-297	61	24	for	for	ADP
ma-297	61	25	as	as	ADP
ma-297	61	26	vn	vn	PROPN
ma-297	61	27	=	=	SYM
ma-297	61	28	∑2n−2	∑2n−2	PROPN
ma-297	61	29	j=0	j=0	PROPN
ma-297	61	30	qj1∑2n−1	qj1∑2n−1	PROPN
ma-297	61	31	j=0	j=0	PROPN
ma-297	61	32	qj1	qj1	PROPN
ma-297	61	33	v∗	v∗	NOUN
ma-297	61	34	,	,	PUNCT
ma-297	61	35	n	n	NOUN
ma-297	61	36	=	=	SYM
ma-297	61	37	1	1	NUM
ma-297	61	38	,	,	PUNCT
ma-297	61	39	2	2	NUM
ma-297	61	40	,	,	PUNCT
ma-297	61	41	...	...	PUNCT
ma-297	61	42	,	,	PUNCT
ma-297	61	43	(	(	PUNCT
ma-297	61	44	3.7	3.7	NUM
ma-297	61	45	)	)	PUNCT
ma-297	61	46	where	where	SCONJ
ma-297	61	47	,	,	PUNCT
ma-297	61	48	q1	q1	PROPN
ma-297	61	49	=	=	PUNCT
ma-297	61	50	v∗	v∗	PROPN
ma-297	61	51	v∗∗	v∗∗	NOUN
ma-297	61	52	=	=	SYM
ma-297	62	1	1−	1−	NUM
ma-297	63	1	√	√	NUM
ma-297	63	2	1−	1−	NUM
ma-297	64	1	2βl1µ	2βl1µ	NUM
ma-297	64	2	1	1	NUM
ma-297	64	3	+	+	CCONJ
ma-297	64	4	√	√	PROPN
ma-297	64	5	1−	1−	NUM
ma-297	64	6	2βl1µ	2βl1µ	NUM
ma-297	64	7	.	.	PUNCT
ma-297	65	1	(	(	PUNCT
ma-297	65	2	3.8	3.8	NUM
ma-297	65	3	)	)	PUNCT
ma-297	65	4	proof	proof	NOUN
ma-297	65	5	.	.	PUNCT
ma-297	66	1	(	(	PUNCT
ma-297	66	2	i	i	NOUN
ma-297	66	3	)	)	PUNCT
ma-297	66	4	the	the	DET
ma-297	66	5	zeros	zero	NOUN
ma-297	66	6	of	of	ADP
ma-297	66	7	the	the	DET
ma-297	66	8	function	function	NOUN
ma-297	66	9	f	f	PROPN
ma-297	66	10	are	be	AUX
ma-297	66	11	real	real	ADJ
ma-297	66	12	by	by	ADP
ma-297	66	13	(	(	PUNCT
ma-297	66	14	3.4).by	3.4).by	NUM
ma-297	66	15	setting	set	VERB
ma-297	66	16	f	f	X
ma-297	66	17	(	(	PUNCT
ma-297	66	18	t	t	PROPN
ma-297	66	19	)	)	PUNCT
ma-297	66	20	=	=	SYM
ma-297	66	21	0	0	PUNCT
ma-297	67	1	and	and	CCONJ
ma-297	67	2	using	use	VERB
ma-297	67	3	the	the	DET
ma-297	67	4	quadratic	quadratic	ADJ
ma-297	67	5	formula	formula	NOUN
ma-297	67	6	we	we	PRON
ma-297	67	7	obtain	obtain	VERB
ma-297	67	8	v∗	v∗	NOUN
ma-297	67	9	and	and	CCONJ
ma-297	67	10	v∗∗.(ii	v∗∗.(ii	ADP
ma-297	67	11	)	)	PUNCT
ma-297	67	12	let	let	VERB
ma-297	67	13	vn+1	vn+1	X
ma-297	67	14	=	=	SYM
ma-297	67	15	g1(vn	g1(vn	PROPN
ma-297	67	16	)	)	PUNCT
ma-297	67	17	,	,	PUNCT
ma-297	67	18	v0	v0	NOUN
ma-297	67	19	=	=	SYM
ma-297	67	20	0	0	NUM
ma-297	67	21	,	,	PUNCT
ma-297	67	22	n	n	NOUN
ma-297	67	23	=	=	SYM
ma-297	67	24	0	0	NUM
ma-297	67	25	,	,	PUNCT
ma-297	67	26	1	1	NUM
ma-297	67	27	,	,	PUNCT
ma-297	67	28	...	...	PUNCT
ma-297	67	29	(	(	PUNCT
ma-297	67	30	3.9)where	3.9)where	NUM
ma-297	67	31	,	,	PUNCT
ma-297	67	32	g1(t	g1(t	X
ma-297	67	33	)	)	PUNCT
ma-297	67	34	=	=	SYM
ma-297	67	35	1	1	NUM
ma-297	67	36	2βl1	2βl1	NUM
ma-297	67	37	t	t	NOUN
ma-297	67	38	2	2	NUM
ma-297	67	39	−	−	PROPN
ma-297	67	40	µ	µ	X
ma-297	67	41	βl1	βl1	NOUN
ma-297	67	42	t	t	NOUN
ma-297	67	43	−	−	NOUN
ma-297	67	44	1	1	NUM
ma-297	67	45	.	.	PUNCT
ma-297	68	1	(	(	PUNCT
ma-297	68	2	3.10	3.10	NUM
ma-297	68	3	)	)	PUNCT
ma-297	68	4	multiply	multiply	ADV
ma-297	68	5	(	(	PUNCT
ma-297	68	6	3.9	3.9	NUM
ma-297	68	7	)	)	PUNCT
ma-297	68	8	by	by	ADP
ma-297	68	9	(	(	PUNCT
ma-297	68	10	1−	1−	NUM
ma-297	68	11	βl1vn	βl1vn	NUM
ma-297	68	12	)	)	PUNCT
ma-297	68	13	and	and	CCONJ
ma-297	68	14	simplify	simplify	VERB
ma-297	68	15	to	to	PART
ma-297	68	16	get	get	VERB
ma-297	68	17	(	(	PUNCT
ma-297	68	18	1−	1−	NUM
ma-297	68	19	βl1vn)vn+1	βl1vn)vn+1	NUM
ma-297	68	20	=	=	SYM
ma-297	68	21	µ−	µ−	PROPN
ma-297	68	22	1	1	NUM
ma-297	68	23	2	2	NUM
ma-297	68	24	βl1v	βl1v	NUM
ma-297	68	25	2	2	NUM
ma-297	68	26	n	n	NOUN
ma-297	68	27	,	,	PUNCT
ma-297	68	28	or	or	CCONJ
ma-297	68	29	(	(	PUNCT
ma-297	68	30	3.11	3.11	NUM
ma-297	68	31	)	)	PUNCT
ma-297	68	32	1	1	NUM
ma-297	68	33	2	2	NUM
ma-297	68	34	βl1v	βl1v	NUM
ma-297	68	35	2	2	NUM
ma-297	68	36	n	n	PRON
ma-297	68	37	−	−	NOUN
ma-297	68	38	βl1vnvn+1	βl1vnvn+1	NOUN
ma-297	69	1	+	+	CCONJ
ma-297	69	2	1	1	NUM
ma-297	69	3	2	2	NUM
ma-297	69	4	βl1v	βl1v	NUM
ma-297	69	5	2	2	NUM
ma-297	69	6	n+1	n+1	NOUN
ma-297	69	7	=	=	SYM
ma-297	69	8	1	1	NUM
ma-297	69	9	2	2	NUM
ma-297	69	10	βl1v	βl1v	NUM
ma-297	69	11	2	2	NUM
ma-297	69	12	n+1	n+1	NUM
ma-297	69	13	−	−	PROPN
ma-297	69	14	vn+1	vn+1	PROPN
ma-297	69	15	+	+	CCONJ
ma-297	69	16	µ	µ	NOUN
ma-297	69	17	,	,	PUNCT
ma-297	69	18	so	so	ADV
ma-297	69	19	1	1	NUM
ma-297	69	20	2	2	NUM
ma-297	69	21	βl1(vn+1	βl1(vn+1	NOUN
ma-297	69	22	−	−	NOUN
ma-297	69	23	vn)2	vn)2	PROPN
ma-297	69	24	=	=	SYM
ma-297	69	25	1	1	NUM
ma-297	69	26	2	2	NUM
ma-297	69	27	βl1v	βl1v	NUM
ma-297	69	28	2	2	NUM
ma-297	69	29	n+1	n+1	NUM
ma-297	69	30	−	−	PROPN
ma-297	69	31	vn+1	vn+1	NOUN
ma-297	69	32	+	+	CCONJ
ma-297	69	33	µ.thus	µ.thus	NUM
ma-297	69	34	,	,	PUNCT
ma-297	69	35	we	we	PRON
ma-297	69	36	can	can	AUX
ma-297	69	37	write	write	VERB
ma-297	69	38	vn+1	vn+1	PROPN
ma-297	69	39	−	−	PROPN
ma-297	69	40	vn	vn	PROPN
ma-297	69	41	=	=	SYM
ma-297	69	42	1	1	NUM
ma-297	69	43	2βl1(vn	2βl1(vn	NUM
ma-297	69	44	−	−	NOUN
ma-297	69	45	vn−1)2	vn−1)2	NOUN
ma-297	69	46	1−	1−	NUM
ma-297	69	47	βl1vn	βl1vn	PROPN
ma-297	69	48	=	=	SYM
ma-297	69	49	−	−	PROPN
ma-297	69	50	f1(vn	f1(vn	PROPN
ma-297	69	51	)	)	PUNCT
ma-297	69	52	f	f	PROPN
ma-297	69	53	′1(vn	′1(vn	PROPN
ma-297	69	54	)	)	PUNCT
ma-297	69	55	.	.	PUNCT
ma-297	70	1	(	(	PUNCT
ma-297	70	2	3.12	3.12	NUM
ma-297	70	3	)	)	PUNCT
ma-297	70	4	(	(	PUNCT
ma-297	70	5	iii	iii	X
ma-297	70	6	)	)	PUNCT
ma-297	70	7	the	the	DET
ma-297	70	8	proof	proof	NOUN
ma-297	70	9	can	can	AUX
ma-297	70	10	be	be	AUX
ma-297	70	11	found	find	VERB
ma-297	70	12	in	in	ADP
ma-297	70	13	[	[	X
ma-297	70	14	5	5	NUM
ma-297	70	15	]	]	PUNCT
ma-297	70	16	.	.	PUNCT
ma-297	71	1	�	�	PROPN
ma-297	71	2	https://doi.org/10.28924/ada/ma.5.11	https://doi.org/10.28924/ada/ma.5.11	NUM
ma-297	71	3	eur	eur	PROPN
ma-297	71	4	.	.	PUNCT
ma-297	72	1	j.	j.	PROPN
ma-297	72	2	math	math	PROPN
ma-297	72	3	.	.	PUNCT
ma-297	73	1	anal	anal	PROPN
ma-297	73	2	.	.	PUNCT
ma-297	74	1	10.28924	10.28924	NUM
ma-297	74	2	/	/	SYM
ma-297	74	3	ada	ada	NOUN
ma-297	74	4	/	/	SYM
ma-297	74	5	ma.5.11	ma.5.11	ADJ
ma-297	74	6	4	4	NUM
ma-297	74	7	remark	remark	NOUN
ma-297	74	8	3.2	3.2	NUM
ma-297	74	9	.	.	PUNCT
ma-297	75	1	(	(	PUNCT
ma-297	75	2	i	i	NOUN
ma-297	75	3	)	)	PUNCT
ma-297	75	4	in	in	ADP
ma-297	75	5	view	view	NOUN
ma-297	75	6	of	of	ADP
ma-297	75	7	(	(	PUNCT
ma-297	75	8	3.1	3.1	NUM
ma-297	75	9	)	)	PUNCT
ma-297	75	10	the	the	DET
ma-297	75	11	results	result	NOUN
ma-297	75	12	of	of	ADP
ma-297	75	13	the	the	DET
ma-297	75	14	lemma	lemma	PROPN
ma-297	75	15	3.1	3.1	NUM
ma-297	75	16	can	can	AUX
ma-297	75	17	be	be	AUX
ma-297	75	18	given	give	VERB
ma-297	75	19	without	without	ADP
ma-297	75	20	β	β	X
ma-297	75	21	.	.	PUNCT
ma-297	76	1	for	for	ADP
ma-297	76	2	example	example	NOUN
ma-297	76	3	(	(	PUNCT
ma-297	76	4	3.4	3.4	NUM
ma-297	76	5	)	)	PUNCT
ma-297	76	6	becomes	become	VERB
ma-297	76	7	µl1(λ+	µl1(λ+	ADP
ma-297	76	8	1	1	NUM
ma-297	76	9	)	)	PUNCT
ma-297	76	10	≤	≤	NUM
ma-297	76	11	1	1	NUM
ma-297	76	12	.	.	PUNCT
ma-297	77	1	(	(	PUNCT
ma-297	77	2	3.13	3.13	NUM
ma-297	77	3	)	)	PUNCT
ma-297	77	4	(	(	PUNCT
ma-297	77	5	ii	ii	NOUN
ma-297	77	6	)	)	PUNCT
ma-297	77	7	let	let	AUX
ma-297	77	8	l	l	NOUN
ma-297	77	9	>	>	X
ma-297	77	10	0	0	X
ma-297	77	11	.	.	PUNCT
ma-297	77	12	define	define	VERB
ma-297	77	13	the	the	DET
ma-297	77	14	quadratic	quadratic	ADJ
ma-297	77	15	majorizing	majorizing	NOUN
ma-297	77	16	function	function	NOUN
ma-297	77	17	f	f	PROPN
ma-297	77	18	by	by	ADP
ma-297	77	19	f	f	PROPN
ma-297	77	20	(	(	PUNCT
ma-297	77	21	t	t	PROPN
ma-297	77	22	)	)	PUNCT
ma-297	77	23	=	=	SYM
ma-297	78	1	βlt2	βlt2	ADJ
ma-297	78	2	2	2	NUM
ma-297	78	3	−	−	NOUN
ma-297	78	4	t	t	PROPN
ma-297	78	5	+	+	CCONJ
ma-297	78	6	µ	µ	NUM
ma-297	78	7	,	,	PUNCT
ma-297	78	8	(	(	PUNCT
ma-297	78	9	3.14	3.14	NUM
ma-297	78	10	)	)	PUNCT
ma-297	78	11	and	and	CCONJ
ma-297	78	12	the	the	DET
ma-297	78	13	scalar	scalar	ADJ
ma-297	78	14	sequence	sequence	NOUN
ma-297	78	15	{	{	PUNCT
ma-297	78	16	un}f	un}f	PROPN
ma-297	78	17	oru0	oru0	PROPN
ma-297	78	18	=	=	SYM
ma-297	78	19	0	0	NUM
ma-297	78	20	and	and	CCONJ
ma-297	78	21	each	each	PRON
ma-297	78	22	n	n	NOUN
ma-297	78	23	=	=	SYM
ma-297	78	24	0	0	NUM
ma-297	78	25	,	,	PUNCT
ma-297	78	26	1	1	NUM
ma-297	78	27	,	,	PUNCT
ma-297	78	28	2	2	NUM
ma-297	78	29	,	,	PUNCT
ma-297	78	30	...	...	PUNCT
ma-297	78	31	by	by	ADP
ma-297	78	32	un+1	un+1	PROPN
ma-297	78	33	=	=	SYM
ma-297	78	34	un	un	PROPN
ma-297	78	35	−	−	PROPN
ma-297	78	36	f	f	PROPN
ma-297	78	37	(	(	PUNCT
ma-297	78	38	un	un	PROPN
ma-297	78	39	)	)	PUNCT
ma-297	78	40	f	f	PROPN
ma-297	78	41	′(un	′(un	PROPN
ma-297	78	42	)	)	PUNCT
ma-297	78	43	.	.	PUNCT
ma-297	79	1	(	(	PUNCT
ma-297	79	2	3.15	3.15	NUM
ma-297	79	3	)	)	PUNCT
ma-297	79	4	denote	denote	VERB
ma-297	79	5	the	the	DET
ma-297	79	6	corresponding	correspond	VERB
ma-297	79	7	zeros	zero	NOUN
ma-297	79	8	of	of	ADP
ma-297	79	9	f	f	PROPN
ma-297	79	10	(	(	PUNCT
ma-297	79	11	t	t	PROPN
ma-297	79	12	)	)	PUNCT
ma-297	79	13	=	=	SYM
ma-297	79	14	0	0	NUM
ma-297	79	15	by	by	ADP
ma-297	79	16	v∗	v∗	PROPN
ma-297	79	17	and	and	CCONJ
ma-297	79	18	v∗∗	v∗∗	NOUN
ma-297	79	19	,	,	PUNCT
ma-297	79	20	respectvely	respectvely	ADV
ma-297	79	21	provided	provide	VERB
ma-297	79	22	that	that	DET
ma-297	79	23	h2	h2	NOUN
ma-297	80	1	=	=	PUNCT
ma-297	80	2	2βlµ	2βlµ	PROPN
ma-297	80	3	≤	≤	NUM
ma-297	80	4	1	1	NUM
ma-297	80	5	(	(	PUNCT
ma-297	80	6	3.16	3.16	NUM
ma-297	80	7	)	)	PUNCT
ma-297	80	8	clearly	clearly	ADV
ma-297	80	9	,	,	PUNCT
ma-297	80	10	the	the	DET
ma-297	80	11	results	result	NOUN
ma-297	80	12	of	of	ADP
ma-297	80	13	the	the	DET
ma-297	80	14	lemma	lemma	PROPN
ma-297	80	15	3.1	3.1	NUM
ma-297	80	16	hold	hold	NOUN
ma-297	80	17	,	,	PUNCT
ma-297	80	18	if	if	SCONJ
ma-297	80	19	l	l	NOUN
ma-297	80	20	replaces	replace	VERB
ma-297	80	21	l1	l1	PROPN
ma-297	80	22	and	and	CCONJ
ma-297	80	23	un+1	un+1	PROPN
ma-297	80	24	−	−	PROPN
ma-297	80	25	un	un	PROPN
ma-297	80	26	=	=	PROPN
ma-297	80	27	βl(un	βl(un	PROPN
ma-297	80	28	−	−	PROPN
ma-297	80	29	un−1)2	un−1)2	VERB
ma-297	80	30	2(1−	2(1−	PROPN
ma-297	80	31	βlun	βlun	ADJ
ma-297	80	32	)	)	PUNCT
ma-297	80	33	.	.	PUNCT
ma-297	81	1	(	(	PUNCT
ma-297	81	2	3.17	3.17	NUM
ma-297	81	3	)	)	PUNCT
ma-297	81	4	let	let	VERB
ma-297	81	5	l	l	NOUN
ma-297	81	6	>	>	X
ma-297	81	7	0	0	X
ma-297	81	8	.	.	PUNCT
ma-297	81	9	define	define	VERB
ma-297	81	10	the	the	DET
ma-297	81	11	sequence	sequence	NOUN
ma-297	81	12	{	{	PUNCT
ma-297	81	13	sn	sn	NOUN
ma-297	81	14	}	}	PUNCT
ma-297	81	15	for	for	ADP
ma-297	81	16	0	0	NUM
ma-297	81	17	=	=	SYM
ma-297	81	18	0	0	NUM
ma-297	81	19	,	,	PUNCT
ma-297	81	20	s1	s1	PROPN
ma-297	81	21	=	=	SYM
ma-297	81	22	µ	µ	PROPN
ma-297	81	23	,	,	PUNCT
ma-297	81	24	s2	s2	NOUN
ma-297	81	25	=	=	SYM
ma-297	81	26	s1	s1	PROPN
ma-297	81	27	+	+	CCONJ
ma-297	81	28	βl0(s1	βl0(s1	PUNCT
ma-297	81	29	−	−	PROPN
ma-297	81	30	s0)2	s0)2	CCONJ
ma-297	81	31	2(1−	2(1−	PROPN
ma-297	81	32	l0βs1	l0βs1	NOUN
ma-297	81	33	)	)	PUNCT
ma-297	81	34	and	and	CCONJ
ma-297	81	35	(	(	PUNCT
ma-297	81	36	3.18	3.18	NUM
ma-297	81	37	)	)	PUNCT
ma-297	82	1	sn+1	sn+1	PROPN
ma-297	83	1	=	=	SYM
ma-297	83	2	sn	sn	PROPN
ma-297	83	3	+	+	CCONJ
ma-297	83	4	βl(sn	βl(sn	ADJ
ma-297	83	5	−	−	PROPN
ma-297	83	6	sn−1)2	sn−1)2	ADJ
ma-297	83	7	2(1−	2(1−	NOUN
ma-297	83	8	l0βsn	l0βsn	NUM
ma-297	83	9	)	)	PUNCT
ma-297	83	10	.	.	PUNCT
ma-297	84	1	next	next	ADV
ma-297	84	2	,	,	PUNCT
ma-297	84	3	we	we	PRON
ma-297	84	4	compare	compare	VERB
ma-297	84	5	the	the	DET
ma-297	84	6	sequences	sequence	NOUN
ma-297	84	7	{	{	PUNCT
ma-297	84	8	vn	vn	NOUN
ma-297	84	9	}	}	PUNCT
ma-297	84	10	,	,	PUNCT
ma-297	84	11	{	{	PUNCT
ma-297	84	12	un	un	PROPN
ma-297	84	13	}	}	PUNCT
ma-297	84	14	,	,	PUNCT
ma-297	84	15	and	and	CCONJ
ma-297	84	16	{	{	PUNCT
ma-297	84	17	sn	sn	NOUN
ma-297	84	18	}	}	PUNCT
ma-297	84	19	.	.	PUNCT
ma-297	85	1	lemma	lemma	PROPN
ma-297	85	2	3.3	3.3	NUM
ma-297	85	3	.	.	PUNCT
ma-297	86	1	suppose	suppose	VERB
ma-297	86	2	(	(	PUNCT
ma-297	86	3	2.5),(2.6	2.5),(2.6	NUM
ma-297	86	4	)	)	PUNCT
ma-297	86	5	and	and	CCONJ
ma-297	86	6	(	(	PUNCT
ma-297	86	7	3.4	3.4	NUM
ma-297	86	8	)	)	PUNCT
ma-297	86	9	hold	hold	NOUN
ma-297	86	10	.	.	PUNCT
ma-297	87	1	then	then	ADV
ma-297	87	2	,	,	PUNCT
ma-297	87	3	the	the	DET
ma-297	87	4	following	follow	VERB
ma-297	87	5	assertions	assertion	NOUN
ma-297	87	6	hold	hold	VERB
ma-297	87	7	0	0	NUM
ma-297	87	8	≤	≤	NUM
ma-297	87	9	sn	sn	PROPN
ma-297	87	10	≤	≤	NOUN
ma-297	87	11	sn+1	sn+1	PROPN
ma-297	87	12	,	,	PUNCT
ma-297	87	13	0	0	NUM
ma-297	87	14	≤	≤	NUM
ma-297	87	15	un	un	PROPN
ma-297	87	16	≤	≤	PROPN
ma-297	87	17	un+1	un+1	PROPN
ma-297	87	18	0	0	SYM
ma-297	87	19	≤	≤	NUM
ma-297	87	20	vn	vn	VERB
ma-297	87	21	≤	≤	PROPN
ma-297	87	22	vn+1	vn+1	PROPN
ma-297	87	23	,	,	PUNCT
ma-297	87	24	0	0	NUM
ma-297	87	25	≤	≤	NUM
ma-297	87	26	sn	sn	PROPN
ma-297	87	27	≤	≤	PROPN
ma-297	87	28	un	un	PROPN
ma-297	87	29	≤	≤	PROPN
ma-297	87	30	vn	vn	PROPN
ma-297	87	31	and	and	CCONJ
ma-297	87	32	0	0	NUM
ma-297	87	33	≤	≤	NUM
ma-297	87	34	s∗	s∗	PROPN
ma-297	87	35	=	=	PROPN
ma-297	87	36	lim	lim	PROPN
ma-297	87	37	n→+∞	n→+∞	VERB
ma-297	87	38	≤	≤	PUNCT
ma-297	87	39	u∗	u∗	NOUN
ma-297	87	40	=	=	SYM
ma-297	87	41	lim	lim	PROPN
ma-297	87	42	n→+∞	n→+∞	PROPN
ma-297	87	43	=	=	SYM
ma-297	87	44	1−	1−	NUM
ma-297	87	45	√	√	NUM
ma-297	87	46	1−	1−	NUM
ma-297	87	47	2βlµ	2βlµ	NUM
ma-297	87	48	βl	βl	ADP
ma-297	88	1	≤	≤	PROPN
ma-297	88	2	v∗.	v∗.	ADP
ma-297	88	3	proof	proof	NOUN
ma-297	88	4	.	.	PUNCT
ma-297	89	1	it	it	PRON
ma-297	89	2	follows	follow	VERB
ma-297	89	3	by	by	ADP
ma-297	89	4	simple	simple	ADJ
ma-297	89	5	induction	induction	NOUN
ma-297	89	6	(	(	PUNCT
ma-297	89	7	2.5),(2.6	2.5),(2.6	NUM
ma-297	89	8	)	)	PUNCT
ma-297	89	9	and	and	CCONJ
ma-297	89	10	the	the	DET
ma-297	89	11	definition	definition	NOUN
ma-297	89	12	of	of	ADP
ma-297	89	13	these	these	DET
ma-297	89	14	sequences	sequence	NOUN
ma-297	89	15	.	.	PUNCT
ma-297	90	1	�	�	PROPN
ma-297	90	2	in	in	ADP
ma-297	90	3	the	the	DET
ma-297	90	4	next	next	ADJ
ma-297	90	5	section	section	NOUN
ma-297	90	6	,	,	PUNCT
ma-297	90	7	we	we	PRON
ma-297	90	8	relate	relate	VERB
ma-297	90	9	sequences	sequence	NOUN
ma-297	90	10	{	{	PUNCT
ma-297	90	11	vn	vn	NOUN
ma-297	90	12	}	}	PUNCT
ma-297	90	13	,	,	PUNCT
ma-297	90	14	{	{	PUNCT
ma-297	90	15	un	un	PROPN
ma-297	90	16	}	}	PUNCT
ma-297	90	17	and	and	CCONJ
ma-297	90	18	{	{	PUNCT
ma-297	90	19	sn	sn	NOUN
ma-297	90	20	}	}	PUNCT
ma-297	90	21	to	to	ADP
ma-297	90	22	{	{	PUNCT
ma-297	90	23	xn	xn	NUM
ma-297	90	24	}	}	PUNCT
ma-297	90	25	.	.	PUNCT
ma-297	91	1	https://doi.org/10.28924/ada/ma.5.11	https://doi.org/10.28924/ada/ma.5.11	NUM
ma-297	91	2	eur	eur	PROPN
ma-297	91	3	.	.	PUNCT
ma-297	92	1	j.	j.	PROPN
ma-297	92	2	math	math	PROPN
ma-297	92	3	.	.	PUNCT
ma-297	93	1	anal	anal	PROPN
ma-297	93	2	.	.	PUNCT
ma-297	94	1	10.28924	10.28924	NUM
ma-297	94	2	/	/	SYM
ma-297	94	3	ada	ada	NOUN
ma-297	94	4	/	/	SYM
ma-297	94	5	ma.5.11	ma.5.11	ADJ
ma-297	94	6	54	54	NUM
ma-297	94	7	.	.	PUNCT
ma-297	95	1	convergence	convergence	NOUN
ma-297	95	2	of	of	ADP
ma-297	95	3	newton	newton	PROPN
ma-297	95	4	’s	’s	PART
ma-297	95	5	method	method	NOUN
ma-297	95	6	the	the	DET
ma-297	95	7	celebrated	celebrated	ADJ
ma-297	95	8	newton	newton	PROPN
ma-297	95	9	-	-	PUNCT
ma-297	95	10	kantorovich	kantorovich	PROPN
ma-297	95	11	theorem	theorem	NOUN
ma-297	95	12	for	for	ADP
ma-297	95	13	solving	solve	VERB
ma-297	95	14	nonlinear	nonlinear	ADJ
ma-297	95	15	equations	equation	NOUN
ma-297	95	16	using	use	VERB
ma-297	95	17	newton’smethod	newton’smethod	NOUN
ma-297	95	18	is	be	AUX
ma-297	95	19	stated	state	VERB
ma-297	95	20	next	next	ADV
ma-297	95	21	.	.	PUNCT
ma-297	96	1	the	the	DET
ma-297	96	2	proof	proof	NOUN
ma-297	96	3	can	can	AUX
ma-297	96	4	be	be	AUX
ma-297	96	5	found	find	VERB
ma-297	96	6	in	in	ADP
ma-297	96	7	[	[	X
ma-297	96	8	3	3	NUM
ma-297	96	9	,	,	PUNCT
ma-297	96	10	6	6	NUM
ma-297	96	11	]	]	PUNCT
ma-297	96	12	for	for	ADP
ma-297	96	13	m	m	PROPN
ma-297	96	14	=	=	SYM
ma-297	96	15	f	f	PROPN
ma-297	96	16	′(x0	′(x0	NOUN
ma-297	96	17	)	)	PUNCT
ma-297	96	18	.	.	PUNCT
ma-297	97	1	moreover	moreover	ADV
ma-297	97	2	,	,	PUNCT
ma-297	97	3	the	the	DET
ma-297	97	4	proof	proof	NOUN
ma-297	97	5	forgeneral	forgeneral	NOUN
ma-297	97	6	m	m	VERB
ma-297	97	7	follows	follow	VERB
ma-297	97	8	by	by	ADP
ma-297	97	9	simply	simply	ADV
ma-297	97	10	using	use	VERB
ma-297	97	11	m	m	PRON
ma-297	97	12	instead	instead	ADV
ma-297	97	13	of	of	ADP
ma-297	97	14	f	f	PROPN
ma-297	97	15	′(x0	′(x0	NOUN
ma-297	97	16	)	)	PUNCT
ma-297	97	17	in	in	ADP
ma-297	97	18	the	the	DET
ma-297	97	19	newton	newton	PROPN
ma-297	97	20	-	-	PUNCT
ma-297	97	21	kantorovich	kantorovich	PROPN
ma-297	97	22	theorem	theorem	PROPN
ma-297	97	23	.	.	PUNCT
ma-297	97	24	theorem	theorem	VERB
ma-297	97	25	4.1	4.1	NUM
ma-297	97	26	.	.	PUNCT
ma-297	98	1	suppose	suppose	VERB
ma-297	98	2	that	that	SCONJ
ma-297	98	3	(	(	PUNCT
ma-297	98	4	2.4	2.4	NUM
ma-297	98	5	)	)	PUNCT
ma-297	98	6	and	and	CCONJ
ma-297	98	7	(	(	PUNCT
ma-297	98	8	3.4	3.4	NUM
ma-297	98	9	)	)	PUNCT
ma-297	98	10	hold	hold	VERB
ma-297	98	11	for	for	ADP
ma-297	98	12	λ	λ	NOUN
ma-297	98	13	=	=	SYM
ma-297	98	14	1	1	NUM
ma-297	98	15	,	,	PUNCT
ma-297	98	16	µ	µ	X
ma-297	98	17	=	=	SYM
ma-297	98	18	ω	ω	PROPN
ma-297	98	19	and	and	CCONJ
ma-297	98	20	ω	ω	NUM
ma-297	98	21	≥	≥	NUM
ma-297	98	22	‖f	‖f	PRON
ma-297	98	23	′(x0)−1f	′(x0)−1f	NOUN
ma-297	98	24	(	(	PUNCT
ma-297	98	25	x0)‖.	x0)‖.	VERB
ma-297	98	26	then	then	ADV
ma-297	98	27	,	,	PUNCT
ma-297	98	28	the	the	DET
ma-297	98	29	sequence	sequence	NOUN
ma-297	98	30	{	{	PUNCT
ma-297	98	31	xn	xn	PROPN
ma-297	98	32	}	}	PUNCT
ma-297	98	33	generated	generate	VERB
ma-297	98	34	by	by	ADP
ma-297	98	35	newton	newton	PROPN
ma-297	98	36	’s	’s	PART
ma-297	98	37	method	method	NOUN
ma-297	98	38	(	(	PUNCT
ma-297	98	39	1.1	1.1	NUM
ma-297	98	40	)	)	PUNCT
ma-297	98	41	is	be	AUX
ma-297	98	42	well	well	ADV
ma-297	98	43	defined	define	VERB
ma-297	98	44	in	in	ADP
ma-297	98	45	u(x0	u(x0	NOUN
ma-297	98	46	,	,	PUNCT
ma-297	98	47	v	v	NOUN
ma-297	98	48	∗	∗	NOUN
ma-297	98	49	)	)	PUNCT
ma-297	98	50	,	,	PUNCT
ma-297	98	51	remains	remain	VERB
ma-297	98	52	in	in	ADP
ma-297	98	53	u(x0	u(x0	NOUN
ma-297	98	54	,	,	PUNCT
ma-297	98	55	v	v	NOUN
ma-297	98	56	∗	∗	NOUN
ma-297	98	57	)	)	PUNCT
ma-297	98	58	for	for	ADP
ma-297	98	59	each	each	DET
ma-297	98	60	n	n	NOUN
ma-297	98	61	=	=	SYM
ma-297	98	62	0	0	NUM
ma-297	98	63	,	,	PUNCT
ma-297	98	64	1	1	NUM
ma-297	98	65	,	,	PUNCT
ma-297	98	66	2	2	NUM
ma-297	98	67	,	,	PUNCT
ma-297	98	68	...	...	PUNCT
ma-297	98	69	and	and	CCONJ
ma-297	98	70	converges	converge	VERB
ma-297	98	71	to	to	ADP
ma-297	98	72	a	a	DET
ma-297	98	73	unique	unique	ADJ
ma-297	98	74	solution	solution	NOUN
ma-297	98	75	x∗	x∗	PROPN
ma-297	98	76	∈	∈	PROPN
ma-297	98	77	u(x0	u(x0	NOUN
ma-297	98	78	,	,	PUNCT
ma-297	98	79	r	r	NOUN
ma-297	98	80	∗	∗	NOUN
ma-297	98	81	)	)	PUNCT
ma-297	98	82	of	of	ADP
ma-297	98	83	the	the	DET
ma-297	98	84	equation	equation	NOUN
ma-297	99	1	f	f	X
ma-297	99	2	(	(	PUNCT
ma-297	99	3	x	x	X
ma-297	99	4	)	)	PUNCT
ma-297	99	5	=	=	SYM
ma-297	99	6	0	0	X
ma-297	99	7	.	.	PUNCT
ma-297	100	1	moreover	moreover	ADV
ma-297	100	2	,	,	PUNCT
ma-297	100	3	the	the	DET
ma-297	100	4	sequence	sequence	NOUN
ma-297	100	5	{	{	PUNCT
ma-297	100	6	vn	vn	NOUN
ma-297	100	7	}	}	PUNCT
ma-297	100	8	majorizes	majorize	NOUN
ma-297	100	9	{	{	PUNCT
ma-297	100	10	xn	xn	NUM
ma-297	100	11	}	}	PUNCT
ma-297	100	12	,	,	PUNCT
ma-297	100	13	‖xn+1	‖xn+1	NUM
ma-297	100	14	−	−	NOUN
ma-297	100	15	xn‖	xn‖	PROPN
ma-297	100	16	≤	≤	PROPN
ma-297	100	17	vn+1	vn+1	PROPN
ma-297	100	18	−	−	PROPN
ma-297	100	19	vn	vn	PROPN
ma-297	100	20	(	(	PUNCT
ma-297	100	21	4.1	4.1	NUM
ma-297	100	22	)	)	PUNCT
ma-297	100	23	and	and	CCONJ
ma-297	100	24	‖x∗	‖x∗	NUM
ma-297	100	25	−	−	NOUN
ma-297	100	26	xn‖	xn‖	PROPN
ma-297	100	27	≤	≤	PROPN
ma-297	100	28	v∗	v∗	PROPN
ma-297	100	29	−	−	PROPN
ma-297	100	30	vn	vn	PROPN
ma-297	100	31	.	.	PUNCT
ma-297	101	1	(	(	PUNCT
ma-297	101	2	4.2	4.2	NUM
ma-297	101	3	)	)	PUNCT
ma-297	101	4	furthermore	furthermore	ADV
ma-297	101	5	,	,	PUNCT
ma-297	101	6	if	if	SCONJ
ma-297	101	7	there	there	PRON
ma-297	101	8	exists	exist	VERB
ma-297	101	9	v̄	v̄	PROPN
ma-297	101	10	≥	≥	NOUN
ma-297	101	11	v∗	v∗	VERB
ma-297	101	12	such	such	ADJ
ma-297	101	13	that	that	DET
ma-297	101	14	l1	l1	PROPN
ma-297	101	15	2	2	NUM
ma-297	101	16	(	(	PUNCT
ma-297	101	17	v∗	v∗	NOUN
ma-297	101	18	+	+	CCONJ
ma-297	101	19	v̄	v̄	NOUN
ma-297	101	20	)	)	PUNCT
ma-297	101	21	<	<	X
ma-297	101	22	1	1	NUM
ma-297	101	23	,	,	PUNCT
ma-297	101	24	(	(	PUNCT
ma-297	101	25	4.3	4.3	NUM
ma-297	101	26	)	)	PUNCT
ma-297	101	27	then	then	ADV
ma-297	101	28	the	the	DET
ma-297	101	29	solution	solution	NOUN
ma-297	101	30	x∗	x∗	PRON
ma-297	101	31	is	be	AUX
ma-297	101	32	more	more	ADV
ma-297	101	33	unique	unique	ADJ
ma-297	101	34	in	in	ADP
ma-297	101	35	u[x0	u[x0	NOUN
ma-297	101	36	,	,	PUNCT
ma-297	101	37	2	2	NUM
ma-297	101	38	l1	l1	PROPN
ma-297	101	39	−	−	PROPN
ma-297	101	40	v∗	v∗	PROPN
ma-297	101	41	]	]	PUNCT
ma-297	101	42	,	,	PUNCT
ma-297	101	43	where	where	SCONJ
ma-297	101	44	u[x0	u[x0	ADV
ma-297	101	45	,	,	PUNCT
ma-297	101	46	r	r	NOUN
ma-297	101	47	]	]	PUNCT
ma-297	101	48	is	be	AUX
ma-297	101	49	the	the	DET
ma-297	101	50	closure	closure	NOUN
ma-297	101	51	of	of	ADP
ma-297	101	52	u(x0	u(x0	NOUN
ma-297	101	53	,	,	PUNCT
ma-297	101	54	r	r	NOUN
ma-297	101	55	)	)	PUNCT
ma-297	101	56	.	.	PUNCT
ma-297	102	1	remark	remark	VERB
ma-297	102	2	4.2	4.2	NUM
ma-297	102	3	.	.	PUNCT
ma-297	103	1	in	in	ADP
ma-297	103	2	view	view	NOUN
ma-297	103	3	of	of	ADP
ma-297	103	4	(	(	PUNCT
ma-297	103	5	2.6	2.6	NUM
ma-297	103	6	)	)	PUNCT
ma-297	103	7	theorem	theorem	NOUN
ma-297	103	8	(	(	PUNCT
ma-297	103	9	4.1	4.1	NUM
ma-297	103	10	)	)	PUNCT
ma-297	103	11	holds	holds	AUX
ma-297	103	12	provided	provide	VERB
ma-297	103	13	that	that	DET
ma-297	103	14	l	l	NOUN
ma-297	103	15	,	,	PUNCT
ma-297	103	16	{	{	PUNCT
ma-297	103	17	un	un	ADJ
ma-297	103	18	}	}	PUNCT
ma-297	103	19	replace	replace	VERB
ma-297	103	20	l1	l1	PROPN
ma-297	103	21	,	,	PUNCT
ma-297	103	22	{	{	PUNCT
ma-297	103	23	vn	vn	NOUN
ma-297	103	24	}	}	PUNCT
ma-297	103	25	,	,	PUNCT
ma-297	103	26	respectively	respectively	ADV
ma-297	103	27	.	.	PUNCT
ma-297	104	1	by	by	ADP
ma-297	104	2	lemma	lemma	PROPN
ma-297	104	3	3.1	3.1	NUM
ma-297	104	4	and	and	CCONJ
ma-297	104	5	3.3	3.3	NUM
ma-297	104	6	the	the	DET
ma-297	104	7	sequence	sequence	NOUN
ma-297	104	8	{	{	PUNCT
ma-297	104	9	un	un	PROPN
ma-297	104	10	}	}	PUNCT
ma-297	104	11	is	be	AUX
ma-297	104	12	tighter	tight	ADJ
ma-297	104	13	than	than	ADP
ma-297	104	14	{	{	PUNCT
ma-297	104	15	vn	vn	NOUN
ma-297	104	16	}	}	PUNCT
ma-297	104	17	and	and	CCONJ
ma-297	104	18	the	the	DET
ma-297	104	19	limit	limit	NOUN
ma-297	104	20	point	point	NOUN
ma-297	104	21	u∗	u∗	ADV
ma-297	104	22	is	be	AUX
ma-297	104	23	atleast	atleast	VERB
ma-297	104	24	as	as	ADV
ma-297	104	25	small	small	ADJ
ma-297	104	26	as	as	ADP
ma-297	104	27	v∗.	v∗.	PRON
ma-297	104	28	moreover	moreover	ADV
ma-297	104	29	,	,	PUNCT
ma-297	104	30	they	they	PRON
ma-297	104	31	are	be	AUX
ma-297	104	32	given	give	VERB
ma-297	104	33	in	in	ADP
ma-297	104	34	closed	closed	ADJ
ma-297	104	35	form	form	NOUN
ma-297	104	36	.	.	PUNCT
ma-297	105	1	this	this	PRON
ma-297	105	2	is	be	AUX
ma-297	105	3	not	not	PART
ma-297	105	4	however	however	ADV
ma-297	105	5	the	the	DET
ma-297	105	6	case	case	NOUN
ma-297	105	7	for	for	ADP
ma-297	105	8	s∗.	s∗.	ADJ
ma-297	105	9	the	the	DET
ma-297	105	10	convergence	convergence	NOUN
ma-297	105	11	condition	condition	NOUN
ma-297	105	12	for	for	ADP
ma-297	105	13	{	{	PUNCT
ma-297	105	14	sn	sn	NOUN
ma-297	105	15	}	}	PUNCT
ma-297	105	16	given	give	VERB
ma-297	105	17	in	in	ADP
ma-297	105	18	[	[	X
ma-297	105	19	2	2	NUM
ma-297	105	20	]	]	PUNCT
ma-297	105	21	for	for	ADP
ma-297	105	22	m	m	PROPN
ma-297	105	23	=	=	SYM
ma-297	105	24	f	f	PROPN
ma-297	105	25	′(x0	′(x0	NOUN
ma-297	105	26	)	)	PUNCT
ma-297	105	27	,	,	PUNCT
ma-297	105	28	λ	λ	X
ma-297	105	29	=	=	SYM
ma-297	105	30	1	1	NUM
ma-297	105	31	h3	h3	NOUN
ma-297	105	32	=	=	SYM
ma-297	105	33	2l̄µ	2l̄µ	NOUN
ma-297	106	1	≤	≤	NUM
ma-297	106	2	1	1	NUM
ma-297	106	3	,	,	PUNCT
ma-297	106	4	(	(	PUNCT
ma-297	106	5	4.4	4.4	NUM
ma-297	106	6	)	)	PUNCT
ma-297	106	7	where	where	SCONJ
ma-297	106	8	l̄	l̄	NOUN
ma-297	106	9	=	=	NOUN
ma-297	106	10	1	1	NUM
ma-297	106	11	8	8	NUM
ma-297	106	12	(	(	PUNCT
ma-297	106	13	4l0	4l0	NUM
ma-297	106	14	+	+	CCONJ
ma-297	106	15	√	√	ADJ
ma-297	106	16	l0l+	l0l+	ADJ
ma-297	106	17	8l20	8l20	NOUN
ma-297	106	18	+	+	CCONJ
ma-297	106	19	√	√	PROPN
ma-297	106	20	l0l	l0l	NOUN
ma-297	106	21	)	)	PUNCT
ma-297	106	22	.	.	PUNCT
ma-297	107	1	notice	notice	VERB
ma-297	107	2	that	that	PRON
ma-297	107	3	h1	h1	VERB
ma-297	107	4	≤	≤	ADJ
ma-297	107	5	1	1	NUM
ma-297	107	6	=	=	NOUN
ma-297	107	7	⇒	⇒	NOUN
ma-297	107	8	h2	h2	NOUN
ma-297	107	9	≤	≤	ADV
ma-297	107	10	1	1	NUM
ma-297	107	11	and	and	CCONJ
ma-297	107	12	h3	h3	VERB
ma-297	107	13	≤	≤	NUM
ma-297	107	14	1	1	NUM
ma-297	107	15	(	(	PUNCT
ma-297	107	16	4.5	4.5	NUM
ma-297	107	17	)	)	PUNCT
ma-297	107	18	but	but	CCONJ
ma-297	107	19	not	not	PART
ma-297	107	20	necessarily	necessarily	ADV
ma-297	107	21	vice	vice	ADV
ma-297	107	22	versa	versa	ADV
ma-297	107	23	unless	unless	SCONJ
ma-297	107	24	if	if	SCONJ
ma-297	107	25	l0	l0	PROPN
ma-297	107	26	=	=	SYM
ma-297	107	27	l	l	NOUN
ma-297	107	28	=	=	SYM
ma-297	107	29	l1	l1	PROPN
ma-297	107	30	.	.	PUNCT
ma-297	108	1	moreover	moreover	ADV
ma-297	108	2	,	,	PUNCT
ma-297	108	3	h3	h3	VERB
ma-297	108	4	h1	h1	PROPN
ma-297	108	5	→	→	SYM
ma-297	108	6	0	0	PUNCT
ma-297	108	7	as	as	ADP
ma-297	108	8	l0	l0	PROPN
ma-297	108	9	l1	l1	PROPN
ma-297	108	10	→	→	PROPN
ma-297	108	11	0	0	PROPN
ma-297	108	12	.	.	PUNCT
ma-297	108	13	(	(	PUNCT
ma-297	108	14	4.6	4.6	NUM
ma-297	108	15	)	)	PUNCT
ma-297	108	16	h3	h3	NOUN
ma-297	108	17	h1	h1	NOUN
ma-297	108	18	→	→	SYM
ma-297	108	19	0	0	PUNCT
ma-297	108	20	as	as	ADP
ma-297	108	21	l0	l0	PROPN
ma-297	108	22	l	l	NOUN
ma-297	108	23	→	→	X
ma-297	108	24	0	0	NUM
ma-297	108	25	.	.	PUNCT
ma-297	109	1	(	(	PUNCT
ma-297	109	2	4.7	4.7	NUM
ma-297	109	3	)	)	PUNCT
ma-297	109	4	in	in	ADP
ma-297	109	5	view	view	NOUN
ma-297	109	6	of	of	ADP
ma-297	109	7	(	(	PUNCT
ma-297	109	8	4.5)-(4.7	4.5)-(4.7	NOUN
ma-297	109	9	)	)	PUNCT
ma-297	109	10	and	and	CCONJ
ma-297	109	11	the	the	DET
ma-297	109	12	lemma	lemma	PROPN
ma-297	109	13	3.3	3.3	NUM
ma-297	109	14	the	the	DET
ma-297	109	15	results	result	NOUN
ma-297	109	16	using	use	VERB
ma-297	109	17	(	(	PUNCT
ma-297	109	18	4.4	4.4	NUM
ma-297	109	19	)	)	PUNCT
ma-297	109	20	improve	improve	VERB
ma-297	109	21	the	the	DET
ma-297	109	22	ones	one	NOUN
ma-297	109	23	by	by	ADP
ma-297	109	24	theorem	theorem	ADJ
ma-297	109	25	4.1	4.1	NUM
ma-297	109	26	infinitely	infinitely	ADV
ma-297	109	27	many	many	ADJ
ma-297	109	28	times	time	NOUN
ma-297	109	29	.	.	PUNCT
ma-297	110	1	however	however	ADV
ma-297	110	2	,	,	PUNCT
ma-297	110	3	s∗	s∗	PROPN
ma-297	110	4	is	be	AUX
ma-297	110	5	not	not	PART
ma-297	110	6	given	give	VERB
ma-297	110	7	in	in	ADP
ma-297	110	8	closed	closed	ADJ
ma-297	110	9	form	form	NOUN
ma-297	110	10	.	.	PUNCT
ma-297	111	1	but	but	CCONJ
ma-297	111	2	we	we	PRON
ma-297	111	3	have	have	AUX
ma-297	111	4	s∗	s∗	VERB
ma-297	111	5	≤	≤	NUM
ma-297	111	6	s̄	s̄	NOUN
ma-297	111	7	,	,	PUNCT
ma-297	111	8	(	(	PUNCT
ma-297	111	9	4.8	4.8	NUM
ma-297	111	10	)	)	PUNCT
ma-297	111	11	https://doi.org/10.28924/ada/ma.5.11	https://doi.org/10.28924/ada/ma.5.11	NUM
ma-297	111	12	eur	eur	NOUN
ma-297	111	13	.	.	PUNCT
ma-297	112	1	j.	j.	PROPN
ma-297	112	2	math	math	PROPN
ma-297	112	3	.	.	PUNCT
ma-297	113	1	anal	anal	PROPN
ma-297	113	2	.	.	PUNCT
ma-297	114	1	10.28924	10.28924	NUM
ma-297	114	2	/	/	SYM
ma-297	114	3	ada	ada	NOUN
ma-297	114	4	/	/	SYM
ma-297	114	5	ma.5.11	ma.5.11	NOUN
ma-297	114	6	6	6	NUM
ma-297	114	7	where	where	SCONJ
ma-297	114	8	s̄	s̄	NOUN
ma-297	114	9	=	=	PUNCT
ma-297	114	10	µ+	µ+	X
ma-297	114	11	l0µ	l0µ	ADJ
ma-297	114	12	2	2	NUM
ma-297	114	13	2(1−	2(1−	NUM
ma-297	114	14	α)(1−	α)(1−	NUM
ma-297	114	15	l0µ	l0µ	NOUN
ma-297	114	16	)	)	PUNCT
ma-297	114	17	,	,	PUNCT
ma-297	114	18	(	(	PUNCT
ma-297	114	19	4.9	4.9	NUM
ma-297	114	20	)	)	PUNCT
ma-297	114	21	where	where	SCONJ
ma-297	114	22	α	α	NOUN
ma-297	114	23	=	=	SYM
ma-297	114	24	2l	2l	X
ma-297	114	25	l+	l+	NOUN
ma-297	114	26	√	√	PUNCT
ma-297	114	27	l2	l2	NOUN
ma-297	114	28	+	+	CCONJ
ma-297	114	29	8l0l	8l0l	NOUN
ma-297	114	30	.	.	PUNCT
ma-297	115	1	(	(	PUNCT
ma-297	115	2	4.10	4.10	NUM
ma-297	115	3	)	)	PUNCT
ma-297	115	4	next	next	ADV
ma-297	115	5	,	,	PUNCT
ma-297	115	6	we	we	PRON
ma-297	115	7	shall	shall	AUX
ma-297	115	8	find	find	VERB
ma-297	115	9	an	an	DET
ma-297	115	10	upper	upper	ADJ
ma-297	115	11	bound	bind	VERB
ma-297	115	12	on	on	ADP
ma-297	115	13	s∗	s∗	PROPN
ma-297	115	14	which	which	PRON
ma-297	115	15	is	be	AUX
ma-297	115	16	given	give	VERB
ma-297	115	17	in	in	ADP
ma-297	115	18	closed	closed	ADJ
ma-297	115	19	form	form	NOUN
ma-297	115	20	and	and	CCONJ
ma-297	115	21	may	may	AUX
ma-297	115	22	be	be	AUX
ma-297	115	23	tighter	tight	ADJ
ma-297	115	24	than	than	ADP
ma-297	115	25	s̄	s̄	NOUN
ma-297	115	26	.	.	PUNCT
ma-297	116	1	let	let	VERB
ma-297	116	2	a	a	DET
ma-297	116	3	=	=	SYM
ma-297	116	4	1	1	NUM
ma-297	116	5	8l	8l	NOUN
ma-297	116	6	(	(	PUNCT
ma-297	116	7	4l0	4l0	NUM
ma-297	116	8	+	+	CCONJ
ma-297	116	9	√	√	ADJ
ma-297	116	10	l0l+	l0l+	ADJ
ma-297	116	11	8l20	8l20	NOUN
ma-297	116	12	+	+	CCONJ
ma-297	116	13	√	√	PROPN
ma-297	116	14	l0l	l0l	NOUN
ma-297	116	15	)	)	PUNCT
ma-297	116	16	(	(	PUNCT
ma-297	116	17	4.11	4.11	NUM
ma-297	116	18	)	)	PUNCT
ma-297	116	19	then	then	ADV
ma-297	116	20	,	,	PUNCT
ma-297	116	21	the	the	DET
ma-297	116	22	condition	condition	NOUN
ma-297	116	23	(	(	PUNCT
ma-297	116	24	4.4	4.4	NUM
ma-297	116	25	)	)	PUNCT
ma-297	116	26	is	be	AUX
ma-297	116	27	equivalent	equivalent	ADJ
ma-297	116	28	to	to	ADP
ma-297	116	29	h	h	NOUN
ma-297	116	30	=	=	SYM
ma-297	116	31	2alµ	2alµ	PROPN
ma-297	116	32	≤	≤	NUM
ma-297	116	33	1	1	NUM
ma-297	116	34	(	(	PUNCT
ma-297	116	35	4.12	4.12	NUM
ma-297	116	36	)	)	PUNCT
ma-297	116	37	then	then	ADV
ma-297	116	38	,	,	PUNCT
ma-297	116	39	the	the	DET
ma-297	116	40	corresponding	corresponding	ADJ
ma-297	116	41	theorem	theorem	NOUN
ma-297	116	42	in	in	ADP
ma-297	116	43	[	[	X
ma-297	116	44	2	2	NUM
ma-297	116	45	]	]	PUNCT
ma-297	116	46	can	can	AUX
ma-297	116	47	be	be	AUX
ma-297	116	48	written	write	VERB
ma-297	116	49	as	as	ADP
ma-297	116	50	theorem	theorem	VERB
ma-297	116	51	4.3	4.3	NUM
ma-297	116	52	.	.	PUNCT
ma-297	117	1	suppose	suppose	VERB
ma-297	117	2	for	for	ADP
ma-297	117	3	µ	µ	PRON
ma-297	117	4	≥	≥	NUM
ma-297	117	5	‖f	‖f	PRON
ma-297	117	6	′(x0)−1f	′(x0)−1f	NOUN
ma-297	117	7	(	(	PUNCT
ma-297	117	8	x0)‖	x0)‖	PROPN
ma-297	117	9	conditions	condition	NOUN
ma-297	117	10	(	(	PUNCT
ma-297	117	11	2.1	2.1	NUM
ma-297	117	12	)	)	PUNCT
ma-297	117	13	,	,	PUNCT
ma-297	117	14	(	(	PUNCT
ma-297	117	15	2.2	2.2	NUM
ma-297	117	16	)	)	PUNCT
ma-297	117	17	,	,	PUNCT
ma-297	117	18	(	(	PUNCT
ma-297	117	19	4.12	4.12	NUM
ma-297	117	20	)	)	PUNCT
ma-297	117	21	and	and	CCONJ
ma-297	117	22	ū[x0	ū[x0	ADV
ma-297	117	23	,	,	PUNCT
ma-297	117	24	s	s	NOUN
ma-297	117	25	∗	∗	NOUN
ma-297	117	26	]	]	X
ma-297	118	1	⊂	⊂	PROPN
ma-297	118	2	d.	d.	PROPN
ma-297	118	3	then	then	ADV
ma-297	118	4	,	,	PUNCT
ma-297	118	5	the	the	DET
ma-297	118	6	sequences	sequence	NOUN
ma-297	118	7	{	{	PUNCT
ma-297	118	8	xn	xn	PROPN
ma-297	118	9	}	}	PUNCT
ma-297	118	10	generated	generate	VERB
ma-297	118	11	by	by	ADP
ma-297	118	12	newton	newton	PROPN
ma-297	118	13	’s	’s	PART
ma-297	118	14	method	method	NOUN
ma-297	118	15	(	(	PUNCT
ma-297	118	16	1.1	1.1	NUM
ma-297	118	17	)	)	PUNCT
ma-297	118	18	is	be	AUX
ma-297	118	19	well	well	ADV
ma-297	118	20	defined	define	VERB
ma-297	118	21	in	in	ADP
ma-297	118	22	u(x0	u(x0	NOUN
ma-297	118	23	,	,	PUNCT
ma-297	118	24	s	s	PART
ma-297	118	25	∗	∗	NOUN
ma-297	118	26	)	)	PUNCT
ma-297	118	27	,	,	PUNCT
ma-297	118	28	remains	remain	VERB
ma-297	118	29	in	in	ADP
ma-297	118	30	u(x0	u(x0	NOUN
ma-297	118	31	,	,	PUNCT
ma-297	118	32	s	s	PART
ma-297	118	33	∗	∗	NOUN
ma-297	118	34	)	)	PUNCT
ma-297	118	35	for	for	ADP
ma-297	118	36	each	each	DET
ma-297	118	37	n	n	NOUN
ma-297	118	38	=	=	SYM
ma-297	118	39	0	0	NUM
ma-297	118	40	,	,	PUNCT
ma-297	118	41	1	1	NUM
ma-297	118	42	,	,	PUNCT
ma-297	118	43	2	2	NUM
ma-297	118	44	,	,	PUNCT
ma-297	118	45	...	...	PUNCT
ma-297	118	46	and	and	CCONJ
ma-297	118	47	is	be	AUX
ma-297	118	48	convergent	convergent	ADJ
ma-297	118	49	to	to	ADP
ma-297	118	50	a	a	DET
ma-297	118	51	solution	solution	NOUN
ma-297	118	52	x∗	x∗	PROPN
ma-297	118	53	∈	∈	PROPN
ma-297	118	54	u[x0	u[x0	NOUN
ma-297	118	55	,	,	PUNCT
ma-297	118	56	s	s	PART
ma-297	118	57	∗	∗	NOUN
ma-297	118	58	]	]	PUNCT
ma-297	118	59	of	of	ADP
ma-297	118	60	the	the	DET
ma-297	118	61	equation	equation	NOUN
ma-297	119	1	f	f	X
ma-297	119	2	(	(	PUNCT
ma-297	119	3	x	x	X
ma-297	119	4	)	)	PUNCT
ma-297	119	5	=	=	SYM
ma-297	119	6	0	0	X
ma-297	119	7	.	.	PUNCT
ma-297	120	1	moreover	moreover	ADV
ma-297	120	2	,	,	PUNCT
ma-297	120	3	the	the	DET
ma-297	120	4	following	follow	VERB
ma-297	120	5	error	error	NOUN
ma-297	120	6	estimates	estimate	NOUN
ma-297	120	7	hold	hold	VERB
ma-297	120	8	‖xn+1	‖xn+1	PUNCT
ma-297	120	9	−	−	NOUN
ma-297	120	10	xn‖	xn‖	PROPN
ma-297	120	11	≤	≤	X
ma-297	120	12	sn+1	sn+1	VERB
ma-297	120	13	−	−	PROPN
ma-297	120	14	sn	sn	PROPN
ma-297	120	15	(	(	PUNCT
ma-297	120	16	4.13	4.13	NUM
ma-297	120	17	)	)	PUNCT
ma-297	120	18	and	and	CCONJ
ma-297	120	19	‖x∗	‖x∗	NUM
ma-297	121	1	−	−	NOUN
ma-297	121	2	xn‖	xn‖	PROPN
ma-297	121	3	≤	≤	PROPN
ma-297	121	4	s∗	s∗	VERB
ma-297	121	5	−	−	PROPN
ma-297	121	6	sn	sn	PROPN
ma-297	121	7	.	.	PUNCT
ma-297	122	1	(	(	PUNCT
ma-297	122	2	4.14	4.14	NUM
ma-297	122	3	)	)	PUNCT
ma-297	122	4	additionally	additionally	ADV
ma-297	122	5	,	,	PUNCT
ma-297	122	6	if	if	SCONJ
ma-297	122	7	for	for	ADP
ma-297	122	8	some	some	DET
ma-297	122	9	b	b	NOUN
ma-297	122	10	≥	≥	NOUN
ma-297	122	11	s∗	s∗	VERB
ma-297	122	12	l0(s	l0(s	PROPN
ma-297	122	13	∗	∗	NOUN
ma-297	122	14	+	+	SYM
ma-297	123	1	b	b	X
ma-297	123	2	)	)	PUNCT
ma-297	123	3	<	<	X
ma-297	123	4	1	1	NUM
ma-297	123	5	(	(	PUNCT
ma-297	123	6	4.15	4.15	NUM
ma-297	123	7	)	)	PUNCT
ma-297	123	8	then	then	ADV
ma-297	123	9	,	,	PUNCT
ma-297	123	10	the	the	DET
ma-297	123	11	solution	solution	NOUN
ma-297	123	12	x∗	x∗	PROPN
ma-297	123	13	is	be	AUX
ma-297	123	14	unique	unique	ADJ
ma-297	123	15	in	in	ADP
ma-297	123	16	the	the	DET
ma-297	123	17	region	region	NOUN
ma-297	123	18	d	d	PROPN
ma-297	123	19	∩	∩	X
ma-297	123	20	u[x0	u[x0	X
ma-297	123	21	,	,	PUNCT
ma-297	123	22	b	b	NOUN
ma-297	123	23	]	]	PUNCT
ma-297	123	24	.	.	PUNCT
ma-297	124	1	proof	proof	NOUN
ma-297	124	2	.	.	PUNCT
ma-297	125	1	simply	simply	ADV
ma-297	125	2	notice	notice	VERB
ma-297	125	3	that	that	SCONJ
ma-297	125	4	(	(	PUNCT
ma-297	125	5	4.12	4.12	NUM
ma-297	125	6	)	)	PUNCT
ma-297	125	7	is	be	AUX
ma-297	125	8	equivalent	equivalent	ADJ
ma-297	125	9	to	to	ADP
ma-297	125	10	(	(	PUNCT
ma-297	125	11	4.4	4.4	NUM
ma-297	125	12	)	)	PUNCT
ma-297	125	13	used	use	VERB
ma-297	125	14	in	in	ADP
ma-297	125	15	[	[	X
ma-297	125	16	2	2	NUM
ma-297	125	17	]	]	PUNCT
ma-297	125	18	.	.	PUNCT
ma-297	126	1	�	�	PROPN
ma-297	126	2	remark	remark	VERB
ma-297	126	3	4.4	4.4	NUM
ma-297	126	4	.	.	PUNCT
ma-297	127	1	by	by	ADP
ma-297	127	2	the	the	DET
ma-297	127	3	definition	definition	NOUN
ma-297	127	4	of	of	ADP
ma-297	127	5	a	a	PRON
ma-297	127	6	it	it	PRON
ma-297	127	7	follows	follow	VERB
ma-297	127	8	that	that	SCONJ
ma-297	127	9	0	0	PUNCT
ma-297	127	10	<	<	X
ma-297	127	11	a	a	DET
ma-297	127	12	≤	≤	NUM
ma-297	127	13	1	1	NUM
ma-297	127	14	if	if	SCONJ
ma-297	127	15	l0	l0	NOUN
ma-297	127	16	≤	≤	NUM
ma-297	127	17	l	l	NOUN
ma-297	127	18	(	(	PUNCT
ma-297	127	19	4.16	4.16	NUM
ma-297	127	20	)	)	PUNCT
ma-297	127	21	and	and	CCONJ
ma-297	127	22	a	a	DET
ma-297	127	23	≥	≥	NOUN
ma-297	127	24	1	1	NUM
ma-297	127	25	if	if	SCONJ
ma-297	127	26	l	l	NOUN
ma-297	127	27	≤	≤	NUM
ma-297	127	28	l0	l0	NOUN
ma-297	127	29	.	.	PUNCT
ma-297	128	1	(	(	PUNCT
ma-297	128	2	4.17	4.17	NUM
ma-297	128	3	)	)	PUNCT
ma-297	128	4	define	define	VERB
ma-297	128	5	the	the	DET
ma-297	128	6	function	function	NOUN
ma-297	128	7	f	f	PROPN
ma-297	128	8	by	by	ADP
ma-297	128	9	f	f	PROPN
ma-297	128	10	(	(	PUNCT
ma-297	128	11	t	t	PROPN
ma-297	128	12	)	)	PUNCT
ma-297	128	13	=	=	SYM
ma-297	128	14	alt2	alt2	NOUN
ma-297	128	15	2	2	NUM
ma-297	128	16	−	−	NOUN
ma-297	128	17	t	t	NOUN
ma-297	128	18	+	+	X
ma-297	128	19	µ	µ	X
ma-297	128	20	(	(	PUNCT
ma-297	128	21	4.18	4.18	NUM
ma-297	128	22	)	)	PUNCT
ma-297	128	23	https://doi.org/10.28924/ada/ma.5.11	https://doi.org/10.28924/ada/ma.5.11	NUM
ma-297	128	24	eur	eur	NOUN
ma-297	128	25	.	.	PUNCT
ma-297	129	1	j.	j.	PROPN
ma-297	129	2	math	math	PROPN
ma-297	129	3	.	.	PUNCT
ma-297	130	1	anal	anal	PROPN
ma-297	130	2	.	.	PUNCT
ma-297	131	1	10.28924	10.28924	NUM
ma-297	131	2	/	/	SYM
ma-297	131	3	ada	ada	NOUN
ma-297	131	4	/	/	SYM
ma-297	131	5	ma.5.11	ma.5.11	X
ma-297	131	6	7	7	NUM
ma-297	131	7	and	and	CCONJ
ma-297	131	8	the	the	DET
ma-297	131	9	sequence	sequence	NOUN
ma-297	131	10	{	{	PUNCT
ma-297	131	11	s̄n	s̄n	PROPN
ma-297	131	12	}	}	PUNCT
ma-297	131	13	for	for	ADP
ma-297	131	14	s̄0	s̄0	X
ma-297	131	15	=	=	SYM
ma-297	131	16	0	0	NUM
ma-297	131	17	,	,	PUNCT
ma-297	131	18	s̄1	s̄1	VERB
ma-297	131	19	=	=	SYM
ma-297	131	20	µ	µ	X
ma-297	131	21	,	,	PUNCT
ma-297	131	22	s̄2	s̄2	NUM
ma-297	131	23	=	=	PUNCT
ma-297	131	24	s̄1	s̄1	NOUN
ma-297	131	25	+	+	CCONJ
ma-297	131	26	l0(s̄1	l0(s̄1	PROPN
ma-297	131	27	−	−	PROPN
ma-297	131	28	s̄0)2	s̄0)2	NOUN
ma-297	131	29	2(1−	2(1−	NUM
ma-297	131	30	l0s̄1	l0s̄1	NOUN
ma-297	131	31	)	)	PUNCT
ma-297	131	32	,	,	PUNCT
ma-297	131	33	(	(	PUNCT
ma-297	131	34	4.19	4.19	NUM
ma-297	131	35	)	)	PUNCT
ma-297	131	36	s̄n+1	s̄n+1	NOUN
ma-297	131	37	=	=	SYM
ma-297	131	38	s̄n+1	s̄n+1	PROPN
ma-297	131	39	−	−	PROPN
ma-297	131	40	f	f	PROPN
ma-297	131	41	(	(	PUNCT
ma-297	131	42	s̄n+1	s̄n+1	PROPN
ma-297	131	43	)	)	PUNCT
ma-297	131	44	f	f	PROPN
ma-297	131	45	′(s̄n+1	′(s̄n+1	X
ma-297	131	46	)	)	PUNCT
ma-297	131	47	,	,	PUNCT
ma-297	131	48	n	n	NOUN
ma-297	131	49	=	=	SYM
ma-297	131	50	1	1	NUM
ma-297	131	51	,	,	PUNCT
ma-297	131	52	2	2	NUM
ma-297	131	53	,	,	PUNCT
ma-297	131	54	...	...	PUNCT
ma-297	131	55	(	(	PUNCT
ma-297	131	56	4.20	4.20	NUM
ma-297	131	57	)	)	PUNCT
ma-297	131	58	proposition	proposition	NOUN
ma-297	131	59	4.5	4.5	NUM
ma-297	131	60	.	.	PUNCT
ma-297	131	61	suppose	suppose	VERB
ma-297	131	62	that	that	SCONJ
ma-297	131	63	the	the	DET
ma-297	131	64	conditions	condition	NOUN
ma-297	131	65	of	of	ADP
ma-297	131	66	theorem	theorem	ADJ
ma-297	131	67	4.3	4.3	NUM
ma-297	131	68	hold	hold	NOUN
ma-297	131	69	.	.	PUNCT
ma-297	132	1	then	then	ADV
ma-297	132	2	,	,	PUNCT
ma-297	132	3	we	we	PRON
ma-297	132	4	have	have	VERB
ma-297	132	5	that	that	DET
ma-297	132	6	smallest	small	ADJ
ma-297	132	7	solution	solution	NOUN
ma-297	132	8	denoted	denote	VERB
ma-297	132	9	by	by	ADP
ma-297	132	10	s̄∗	s̄∗	PROPN
ma-297	132	11	of	of	ADP
ma-297	132	12	the	the	DET
ma-297	132	13	equation	equation	NOUN
ma-297	132	14	f	f	X
ma-297	132	15	(	(	PUNCT
ma-297	132	16	t	t	PROPN
ma-297	132	17	)	)	PUNCT
ma-297	132	18	=	=	SYM
ma-297	132	19	0	0	NUM
ma-297	132	20	,	,	PUNCT
ma-297	132	21	i.e.	i.e.	X
ma-297	132	22	s̄∗	s̄∗	PROPN
ma-297	132	23	=	=	SYM
ma-297	132	24	1−	1−	NUM
ma-297	132	25	√	√	NUM
ma-297	132	26	1−	1−	NUM
ma-297	132	27	2alµ	2alµ	PROPN
ma-297	132	28	al	al	PROPN
ma-297	132	29	(	(	PUNCT
ma-297	132	30	4.21	4.21	NUM
ma-297	132	31	)	)	PUNCT
ma-297	132	32	is	be	AUX
ma-297	132	33	an	an	DET
ma-297	132	34	upper	upper	ADJ
ma-297	132	35	bound	bind	VERB
ma-297	132	36	in	in	ADP
ma-297	132	37	closed	closed	ADJ
ma-297	132	38	form	form	NOUN
ma-297	132	39	of	of	ADP
ma-297	132	40	the	the	DET
ma-297	132	41	sequence	sequence	NOUN
ma-297	132	42	{	{	PUNCT
ma-297	132	43	s̄n	s̄n	PROPN
ma-297	132	44	}	}	PUNCT
ma-297	132	45	and	and	CCONJ
ma-297	132	46	0	0	NUM
ma-297	132	47	≤	≤	NUM
ma-297	132	48	sn	sn	NOUN
ma-297	132	49	≤	≤	NOUN
ma-297	132	50	s̄n	s̄n	PROPN
ma-297	132	51	,	,	PUNCT
ma-297	132	52	(	(	PUNCT
ma-297	132	53	4.22	4.22	NUM
ma-297	132	54	)	)	PUNCT
ma-297	132	55	0	0	NUM
ma-297	133	1	≤	≤	NOUN
ma-297	133	2	sn+1	sn+1	VERB
ma-297	133	3	−	−	PROPN
ma-297	133	4	sn	sn	PROPN
ma-297	133	5	≤	≤	PROPN
ma-297	133	6	s̄n+1	s̄n+1	NOUN
ma-297	133	7	−	−	PROPN
ma-297	133	8	sn	sn	PROPN
ma-297	133	9	(	(	PUNCT
ma-297	133	10	4.23	4.23	NUM
ma-297	133	11	)	)	PUNCT
ma-297	133	12	and	and	CCONJ
ma-297	133	13	s∗	s∗	VERB
ma-297	133	14	≤	≤	NUM
ma-297	133	15	s̄∗.	s̄∗.	NOUN
ma-297	133	16	(	(	PUNCT
ma-297	133	17	4.24	4.24	NUM
ma-297	133	18	)	)	PUNCT
ma-297	133	19	proof	proof	NOUN
ma-297	133	20	.	.	PUNCT
ma-297	134	1	indeed	indeed	ADV
ma-297	134	2	,	,	PUNCT
ma-297	134	3	this	this	PRON
ma-297	134	4	is	be	AUX
ma-297	134	5	clear	clear	ADJ
ma-297	134	6	under	under	ADP
ma-297	134	7	(	(	PUNCT
ma-297	134	8	4.16	4.16	NUM
ma-297	134	9	)	)	PUNCT
ma-297	134	10	,	,	PUNCT
ma-297	134	11	whereas	whereas	SCONJ
ma-297	134	12	if	if	SCONJ
ma-297	134	13	(	(	PUNCT
ma-297	134	14	4.17	4.17	NUM
ma-297	134	15	)	)	PUNCT
ma-297	134	16	holds	hold	VERB
ma-297	134	17	the	the	PRON
ma-297	134	18	,	,	PUNCT
ma-297	134	19	we	we	PRON
ma-297	134	20	have	have	AUX
ma-297	134	21	from	from	ADP
ma-297	134	22	‖m−1(f	‖m−1(f	PROPN
ma-297	134	23	(	(	PUNCT
ma-297	134	24	xn+1	xn+1	PROPN
ma-297	134	25	−	−	PROPN
ma-297	134	26	f	f	PROPN
ma-297	134	27	(	(	PUNCT
ma-297	134	28	xn)−	xn)−	NOUN
ma-297	135	1	f	f	X
ma-297	135	2	′(xn)(xn+1	′(xn)(xn+1	PROPN
ma-297	135	3	−	−	PROPN
ma-297	135	4	xn)‖	xn)‖	PROPN
ma-297	135	5	≤	≤	PROPN
ma-297	135	6	l̃‖xn+1	l̃‖xn+1	PROPN
ma-297	135	7	−	−	PROPN
ma-297	136	1	xn‖2	xn‖2	PROPN
ma-297	136	2	2	2	NUM
ma-297	136	3	l̃	l̃	PROPN
ma-297	136	4	2	2	NUM
ma-297	136	5	(	(	PUNCT
ma-297	136	6	s̄n+1	s̄n+1	NOUN
ma-297	136	7	−	−	PROPN
ma-297	136	8	s̄n)2	s̄n)2	PROPN
ma-297	136	9	=	=	SYM
ma-297	136	10	f	f	PROPN
ma-297	136	11	(	(	PUNCT
ma-297	136	12	s̄n+1	s̄n+1	VERB
ma-297	136	13	a	a	PRON
ma-297	136	14	,	,	PUNCT
ma-297	136	15	where	where	SCONJ
ma-297	136	16	l̃	l̃	PROPN
ma-297	136	17	=	=	SYM
ma-297	136	18	{	{	PUNCT
ma-297	136	19	l0	l0	PROPN
ma-297	136	20	,	,	PUNCT
ma-297	136	21	n	n	NOUN
ma-297	136	22	=	=	SYM
ma-297	136	23	0	0	NUM
ma-297	136	24	l	l	NOUN
ma-297	136	25	,	,	PUNCT
ma-297	136	26	n	n	NOUN
ma-297	136	27	=	=	SYM
ma-297	136	28	1	1	NUM
ma-297	136	29	,	,	PUNCT
ma-297	136	30	2	2	NUM
ma-297	136	31	,	,	PUNCT
ma-297	136	32	...	...	PUNCT
ma-297	136	33	,	,	PUNCT
ma-297	136	34	‖f	‖f	ADP
ma-297	136	35	′(xn+1)−1m‖	′(xn+1)−1m‖	VERB
ma-297	136	36	≤	≤	NUM
ma-297	136	37	1	1	NUM
ma-297	136	38	1−	1−	NUM
ma-297	136	39	l0‖xn+1	l0‖xn+1	NOUN
ma-297	136	40	−	−	PROPN
ma-297	136	41	x0‖	x0‖	PROPN
ma-297	136	42	≤	≤	NUM
ma-297	136	43	1	1	NUM
ma-297	136	44	1−	1−	NUM
ma-297	136	45	l0s̄n+1so	l0s̄n+1so	SYM
ma-297	136	46	‖xn+1	‖xn+1	PUNCT
ma-297	136	47	−	−	PROPN
ma-297	137	1	xn+1‖	xn+1‖	PROPN
ma-297	137	2	≤	≤	NOUN
ma-297	137	3	‖f	‖f	ADP
ma-297	137	4	′(x−1n+1m‖‖m	′(x−1n+1m‖‖m	NOUN
ma-297	137	5	−1f	−1f	PROPN
ma-297	137	6	(	(	PUNCT
ma-297	137	7	xn+1)‖	xn+1)‖	PROPN
ma-297	137	8	≤	≤	PROPN
ma-297	137	9	f	f	PROPN
ma-297	137	10	(	(	PUNCT
ma-297	137	11	s̄n+1	s̄n+1	NOUN
ma-297	137	12	)	)	PUNCT
ma-297	137	13	a(1−	a(1−	NOUN
ma-297	137	14	l0s̄n+1	l0s̄n+1	NOUN
ma-297	137	15	)	)	PUNCT
ma-297	137	16	≤	≤	NOUN
ma-297	137	17	−	−	PROPN
ma-297	137	18	f	f	PROPN
ma-297	137	19	(	(	PUNCT
ma-297	137	20	s̄n+1	s̄n+1	PROPN
ma-297	137	21	)	)	PUNCT
ma-297	137	22	f	f	NOUN
ma-297	137	23	′(s̄n+1	′(s̄n+1	ADJ
ma-297	137	24	,	,	PUNCT
ma-297	137	25	since	since	SCONJ
ma-297	137	26	1	1	NUM
ma-297	137	27	a(1−	a(1−	NOUN
ma-297	137	28	l0s̄n+1	l0s̄n+1	NOUN
ma-297	137	29	)	)	PUNCT
ma-297	137	30	≤	≤	NUM
ma-297	137	31	1	1	NUM
ma-297	137	32	1−	1−	NUM
ma-297	137	33	als̄n+1	als̄n+1	PROPN
ma-297	137	34	=	=	SYM
ma-297	137	35	−	−	PROPN
ma-297	137	36	1	1	NUM
ma-297	137	37	f	f	NOUN
ma-297	137	38	′(s̄n+1	′(s̄n+1	ADJ
ma-297	137	39	)	)	PUNCT
ma-297	137	40	.	.	PUNCT
ma-297	138	1	�	�	PROPN
ma-297	138	2	5	5	NUM
ma-297	138	3	.	.	PUNCT
ma-297	139	1	a	a	DET
ma-297	139	2	numerical	numerical	ADJ
ma-297	139	3	example	example	NOUN
ma-297	139	4	the	the	DET
ma-297	139	5	convergence	convergence	NOUN
ma-297	139	6	conditions	condition	NOUN
ma-297	139	7	,	,	PUNCT
ma-297	139	8	majorizing	majorize	VERB
ma-297	139	9	sequences	sequence	NOUN
ma-297	139	10	and	and	CCONJ
ma-297	139	11	limit	limit	NOUN
ma-297	139	12	points	point	NOUN
ma-297	139	13	are	be	AUX
ma-297	139	14	compared	compare	VERB
ma-297	139	15	with	with	ADP
ma-297	139	16	each	each	DET
ma-297	139	17	other	other	ADJ
ma-297	139	18	.	.	PUNCT
ma-297	140	1	example	example	NOUN
ma-297	140	2	5.1	5.1	NUM
ma-297	140	3	.	.	PUNCT
ma-297	141	1	let	let	VERB
ma-297	141	2	b1	b1	NOUN
ma-297	141	3	=	=	SYM
ma-297	141	4	b2	b2	NOUN
ma-297	141	5	=	=	SYM
ma-297	141	6	r	r	NOUN
ma-297	141	7	,	,	PUNCT
ma-297	141	8	d	d	NOUN
ma-297	141	9	=	=	SYM
ma-297	141	10	u(x0	u(x0	NOUN
ma-297	141	11	,	,	PUNCT
ma-297	141	12	1	1	NUM
ma-297	141	13	−	−	PROPN
ma-297	141	14	p	p	X
ma-297	141	15	)	)	PUNCT
ma-297	141	16	,	,	PUNCT
ma-297	141	17	p	p	PROPN
ma-297	141	18	∈	∈	PROPN
ma-297	141	19	(	(	PUNCT
ma-297	141	20	0	0	NUM
ma-297	141	21	,	,	PUNCT
ma-297	141	22	1	1	NUM
ma-297	141	23	)	)	PUNCT
ma-297	141	24	and	and	CCONJ
ma-297	141	25	x0	x0	NUM
ma-297	141	26	=	=	SYM
ma-297	142	1	1	1	X
ma-297	142	2	.	.	X
ma-297	142	3	define	define	VERB
ma-297	142	4	the	the	DET
ma-297	142	5	function	function	NOUN
ma-297	142	6	ψ	ψ	NOUN
ma-297	142	7	:	:	PUNCT
ma-297	142	8	d	d	X
ma-297	142	9	→	→	SYM
ma-297	142	10	r	r	NOUN
ma-297	142	11	by	by	ADP
ma-297	142	12	https://doi.org/10.28924/ada/ma.5.11	https://doi.org/10.28924/ada/ma.5.11	NUM
ma-297	142	13	eur	eur	NOUN
ma-297	142	14	.	.	PUNCT
ma-297	143	1	j.	j.	PROPN
ma-297	143	2	math	math	PROPN
ma-297	143	3	.	.	PUNCT
ma-297	144	1	anal	anal	PROPN
ma-297	144	2	.	.	PUNCT
ma-297	145	1	10.28924	10.28924	NUM
ma-297	145	2	/	/	SYM
ma-297	145	3	ada	ada	NOUN
ma-297	145	4	/	/	SYM
ma-297	145	5	ma.5.11	ma.5.11	ADJ
ma-297	145	6	8	8	NUM
ma-297	145	7	ψ(t	ψ(t	NOUN
ma-297	145	8	)	)	PUNCT
ma-297	145	9	=	=	SYM
ma-297	146	1	x3	x3	NOUN
ma-297	146	2	−	−	PROPN
ma-297	147	1	p	p	X
ma-297	147	2	(	(	PUNCT
ma-297	147	3	5.1	5.1	NUM
ma-297	147	4	)	)	PUNCT
ma-297	147	5	then	then	ADV
ma-297	147	6	,	,	PUNCT
ma-297	147	7	for	for	ADP
ma-297	147	8	λ	λ	PROPN
ma-297	147	9	=	=	SYM
ma-297	147	10	1	1	NUM
ma-297	147	11	,	,	PUNCT
ma-297	147	12	µ	µ	X
ma-297	147	13	=	=	SYM
ma-297	147	14	1	1	NUM
ma-297	147	15	3(1−	3(1−	NUM
ma-297	147	16	p	p	NOUN
ma-297	147	17	)	)	PUNCT
ma-297	147	18	and	and	CCONJ
ma-297	147	19	β	β	X
ma-297	147	20	=	=	SYM
ma-297	147	21	1	1	X
ma-297	147	22	.	.	PUNCT
ma-297	148	1	moreover	moreover	ADV
ma-297	148	2	,	,	PUNCT
ma-297	148	3	the	the	DET
ma-297	148	4	definitions(2.1)-(2.3	definitions(2.1)-(2.3	NOUN
ma-297	148	5	)	)	PUNCT
ma-297	148	6	hold	hold	VERB
ma-297	148	7	if	if	SCONJ
ma-297	148	8	l0	l0	PROPN
ma-297	148	9	=	=	PUNCT
ma-297	148	10	3−	3−	NUM
ma-297	148	11	p	p	NOUN
ma-297	148	12	,	,	PUNCT
ma-297	148	13	l	l	NOUN
ma-297	148	14	=	=	SYM
ma-297	148	15	2(1	2(1	NUM
ma-297	148	16	+	+	CCONJ
ma-297	148	17	1	1	NUM
ma-297	148	18	3−p	3−p	NUM
ma-297	148	19	)	)	PUNCT
ma-297	148	20	and	and	CCONJ
ma-297	148	21	l1	l1	PROPN
ma-297	148	22	=	=	PUNCT
ma-297	149	1	2(2−	2(2−	NUM
ma-297	149	2	p	p	NOUN
ma-297	149	3	)	)	PUNCT
ma-297	149	4	.	.	PUNCT
ma-297	150	1	notice	notice	VERB
ma-297	150	2	that	that	SCONJ
ma-297	150	3	l0	l0	PROPN
ma-297	150	4	<	<	X
ma-297	150	5	l1	l1	PROPN
ma-297	150	6	,	,	PUNCT
ma-297	150	7	l	l	X
ma-297	150	8	<	<	X
ma-297	150	9	l1	l1	PROPN
ma-297	150	10	for	for	ADP
ma-297	150	11	each	each	DET
ma-297	150	12	p	p	PROPN
ma-297	150	13	∈	∈	PROPN
ma-297	150	14	(	(	PUNCT
ma-297	150	15	0	0	NUM
ma-297	150	16	,	,	PUNCT
ma-297	150	17	1	1	NUM
ma-297	150	18	)	)	PUNCT
ma-297	150	19	.	.	PUNCT
ma-297	151	1	we	we	PRON
ma-297	151	2	also	also	ADV
ma-297	151	3	have	have	VERB
ma-297	151	4	that	that	DET
ma-297	151	5	l	l	NOUN
ma-297	151	6	≤	≤	NOUN
ma-297	151	7	l0	l0	NOUN
ma-297	152	1	if	if	SCONJ
ma-297	152	2	p	p	X
ma-297	152	3	∈	∈	PROPN
ma-297	152	4	(	(	PUNCT
ma-297	152	5	0	0	NUM
ma-297	152	6	,	,	PUNCT
ma-297	152	7	2−	2−	NUM
ma-297	152	8	√	√	NOUN
ma-297	152	9	3	3	NUM
ma-297	152	10	]	]	PUNCT
ma-297	152	11	and	and	CCONJ
ma-297	152	12	l0	l0	NOUN
ma-297	152	13	≤	≤	NUM
ma-297	152	14	l	l	NOUN
ma-297	153	1	if	if	SCONJ
ma-297	153	2	p	p	X
ma-297	153	3	∈	∈	PROPN
ma-297	154	1	[	[	X
ma-297	154	2	2−	2−	NUM
ma-297	154	3	√	√	NUM
ma-297	154	4	3	3	NUM
ma-297	154	5	,	,	PUNCT
ma-297	154	6	1	1	NUM
ma-297	154	7	)	)	PUNCT
ma-297	154	8	.	.	PUNCT
ma-297	155	1	let	let	VERB
ma-297	155	2	us	we	PRON
ma-297	155	3	restrict	restrict	VERB
ma-297	155	4	p	p	NOUN
ma-297	155	5	∈	∈	PROPN
ma-297	155	6	(	(	PUNCT
ma-297	155	7	0	0	NUM
ma-297	155	8	,	,	PUNCT
ma-297	155	9	12	12	NUM
ma-297	155	10	)	)	PUNCT
ma-297	155	11	then	then	ADV
ma-297	155	12	,	,	PUNCT
ma-297	155	13	the	the	DET
ma-297	155	14	newton	newton	PROPN
ma-297	155	15	-	-	PUNCT
ma-297	155	16	kantorovich	kantorovich	PROPN
ma-297	155	17	condition	condition	NOUN
ma-297	155	18	(	(	PUNCT
ma-297	155	19	3.4	3.4	NUM
ma-297	155	20	)	)	PUNCT
ma-297	156	1	[	[	X
ma-297	156	2	3	3	NUM
ma-297	156	3	,	,	PUNCT
ma-297	156	4	6	6	NUM
ma-297	156	5	]	]	PUNCT
ma-297	156	6	does	do	AUX
ma-297	156	7	not	not	PART
ma-297	156	8	hold	hold	VERB
ma-297	156	9	,	,	PUNCT
ma-297	156	10	since	since	SCONJ
ma-297	156	11	(	(	PUNCT
ma-297	156	12	3.4	3.4	NUM
ma-297	156	13	)	)	PUNCT
ma-297	156	14	is	be	AUX
ma-297	156	15	not	not	PART
ma-297	156	16	satisfied	satisfied	ADJ
ma-297	156	17	for	for	ADP
ma-297	156	18	any	any	DET
ma-297	156	19	p	p	NOUN
ma-297	156	20	∈	∈	PROPN
ma-297	156	21	(	(	PUNCT
ma-297	156	22	0	0	NUM
ma-297	156	23	,	,	PUNCT
ma-297	156	24	12	12	NUM
ma-297	156	25	)	)	PUNCT
ma-297	156	26	.	.	PUNCT
ma-297	157	1	however	however	ADV
ma-297	157	2	,	,	PUNCT
ma-297	157	3	our	our	PRON
ma-297	157	4	condition	condition	NOUN
ma-297	157	5	(	(	PUNCT
ma-297	157	6	4.12	4.12	NUM
ma-297	157	7	)	)	PUNCT
ma-297	157	8	hold	hold	NOUN
ma-297	157	9	provided	provide	VERB
ma-297	157	10	that	that	SCONJ
ma-297	157	11	p	p	PROPN
ma-297	157	12	∈	∈	PROPN
ma-297	157	13	(	(	PUNCT
ma-297	157	14	.46	.46	NUM
ma-297	157	15	,	,	PUNCT
ma-297	157	16	12	12	NUM
ma-297	157	17	)	)	PUNCT
ma-297	157	18	.	.	PUNCT
ma-297	158	1	thus	thus	ADV
ma-297	158	2	,	,	PUNCT
ma-297	158	3	the	the	DET
ma-297	158	4	old	old	ADJ
ma-297	158	5	results	result	NOUN
ma-297	158	6	[	[	X
ma-297	158	7	6	6	NUM
ma-297	158	8	]	]	PUNCT
ma-297	158	9	can	can	AUX
ma-297	158	10	not	not	PART
ma-297	158	11	guarantee	guarantee	VERB
ma-297	158	12	the	the	DET
ma-297	158	13	convergence	convergence	NOUN
ma-297	158	14	of	of	ADP
ma-297	158	15	newton	newton	PROPN
ma-297	158	16	’s	’s	PART
ma-297	158	17	method	method	NOUN
ma-297	158	18	for	for	ADP
ma-297	158	19	any	any	DET
ma-297	158	20	p	p	NOUN
ma-297	158	21	∈	∈	PROPN
ma-297	158	22	(	(	PUNCT
ma-297	158	23	0	0	NUM
ma-297	158	24	,	,	PUNCT
ma-297	158	25	12	12	NUM
ma-297	158	26	)	)	PUNCT
ma-297	158	27	.	.	PUNCT
ma-297	159	1	however	however	ADV
ma-297	159	2	,	,	PUNCT
ma-297	159	3	newton	newton	PROPN
ma-297	159	4	’s	’s	PART
ma-297	159	5	method	method	PROPN
ma-297	159	6	converges	converge	VERB
ma-297	159	7	to	to	ADP
ma-297	159	8	x∗	x∗	PROPN
ma-297	159	9	=	=	PUNCT
ma-297	160	1	3	3	NUM
ma-297	160	2	√	√	NOUN
ma-297	160	3	p	p	NOUN
ma-297	161	1	if	if	SCONJ
ma-297	161	2	we	we	PRON
ma-297	161	3	say	say	VERB
ma-297	161	4	p	p	NOUN
ma-297	161	5	=	=	NOUN
ma-297	161	6	0.48	0.48	NUM
ma-297	161	7	.	.	PUNCT
ma-297	162	1	in	in	ADP
ma-297	162	2	order	order	NOUN
ma-297	162	3	to	to	PART
ma-297	162	4	compare	compare	VERB
ma-297	162	5	sequences	sequence	NOUN
ma-297	162	6	and	and	CCONJ
ma-297	162	7	limit	limit	VERB
ma-297	162	8	points	point	NOUN
ma-297	162	9	.	.	PUNCT
ma-297	163	1	let	let	VERB
ma-297	163	2	p	p	NOUN
ma-297	163	3	=	=	NOUN
ma-297	163	4	0.7	0.7	NUM
ma-297	163	5	.	.	PUNCT
ma-297	164	1	then	then	ADV
ma-297	164	2	,	,	PUNCT
ma-297	164	3	both	both	PRON
ma-297	164	4	(	(	PUNCT
ma-297	164	5	3.4	3.4	NUM
ma-297	164	6	)	)	PUNCT
ma-297	164	7	and	and	CCONJ
ma-297	164	8	(	(	PUNCT
ma-297	164	9	4.12	4.12	NUM
ma-297	164	10	)	)	PUNCT
ma-297	164	11	hold	hold	NOUN
ma-297	164	12	.	.	PUNCT
ma-297	165	1	thus	thus	ADV
ma-297	165	2	,	,	PUNCT
ma-297	165	3	the	the	DET
ma-297	165	4	old	old	ADJ
ma-297	165	5	results	result	NOUN
ma-297	165	6	[	[	X
ma-297	165	7	3	3	NUM
ma-297	165	8	,	,	PUNCT
ma-297	165	9	5–7	5–7	NOUN
ma-297	165	10	]	]	PUNCT
ma-297	165	11	can	can	AUX
ma-297	165	12	not	not	PART
ma-297	165	13	guarantee	guarantee	VERB
ma-297	165	14	the	the	DET
ma-297	165	15	convergence	convergence	NOUN
ma-297	165	16	of	of	ADP
ma-297	165	17	newton	newton	PROPN
ma-297	165	18	’s	’s	PART
ma-297	165	19	method	method	NOUN
ma-297	165	20	for	for	ADP
ma-297	165	21	any	any	DET
ma-297	165	22	p	p	NOUN
ma-297	165	23	∈	∈	PROPN
ma-297	165	24	(	(	PUNCT
ma-297	165	25	0	0	NUM
ma-297	165	26	,	,	PUNCT
ma-297	165	27	12	12	NUM
ma-297	165	28	)	)	PUNCT
ma-297	165	29	.	.	PUNCT
ma-297	166	1	however	however	ADV
ma-297	166	2	,	,	PUNCT
ma-297	166	3	newton	newton	PROPN
ma-297	166	4	’s	’s	PART
ma-297	166	5	method	method	PROPN
ma-297	166	6	converges	converge	VERB
ma-297	166	7	to	to	ADP
ma-297	166	8	x∗	x∗	PROPN
ma-297	166	9	=	=	SYM
ma-297	167	1	3	3	NUM
ma-297	167	2	√	√	NUM
ma-297	168	1	p.	p.	NOUN
ma-297	168	2	if	if	SCONJ
ma-297	168	3	say	say	VERB
ma-297	168	4	p	p	X
ma-297	168	5	=	=	NOUN
ma-297	168	6	0.48	0.48	NUM
ma-297	168	7	.	.	PUNCT
ma-297	169	1	in	in	ADP
ma-297	169	2	order	order	NOUN
ma-297	169	3	to	to	PART
ma-297	169	4	compare	compare	VERB
ma-297	169	5	sequences	sequence	NOUN
ma-297	169	6	and	and	CCONJ
ma-297	169	7	limit	limit	VERB
ma-297	169	8	points	point	NOUN
ma-297	169	9	.	.	PUNCT
ma-297	170	1	let	let	VERB
ma-297	170	2	p	p	NOUN
ma-297	170	3	=	=	NOUN
ma-297	170	4	0.7	0.7	NUM
ma-297	170	5	.	.	PUNCT
ma-297	171	1	then	then	ADV
ma-297	171	2	,	,	PUNCT
ma-297	171	3	both	both	PRON
ma-297	171	4	(	(	PUNCT
ma-297	171	5	3.4	3.4	NUM
ma-297	171	6	)	)	PUNCT
ma-297	171	7	and	and	CCONJ
ma-297	171	8	(	(	PUNCT
ma-297	171	9	4.12	4.12	NUM
ma-297	171	10	)	)	PUNCT
ma-297	171	11	hold	hold	NOUN
ma-297	171	12	.	.	PUNCT
ma-297	172	1	then	then	ADV
ma-297	172	2	,	,	PUNCT
ma-297	172	3	we	we	PRON
ma-297	172	4	have	have	VERB
ma-297	172	5	s̄	s̄	NOUN
ma-297	172	6	=	=	SYM
ma-297	172	7	0.3965	0.3965	NUM
ma-297	172	8	,	,	PUNCT
ma-297	172	9	v̄	v̄	NOUN
ma-297	172	10	=	=	SYM
ma-297	172	11	0.6511	0.6511	NUM
ma-297	172	12	,	,	PUNCT
ma-297	172	13	b	b	NOUN
ma-297	172	14	=	=	SYM
ma-297	172	15	0.3194	0.3194	PROPN
ma-297	172	16	.	.	PUNCT
ma-297	173	1	therefore	therefore	ADV
ma-297	173	2	,	,	PUNCT
ma-297	173	3	the	the	DET
ma-297	173	4	new	new	ADJ
ma-297	173	5	error	error	NOUN
ma-297	173	6	bounds	bound	NOUN
ma-297	173	7	and	and	CCONJ
ma-297	173	8	limit	limit	NOUN
ma-297	173	9	points	point	NOUN
ma-297	173	10	are	be	AUX
ma-297	173	11	tighter	tight	ADJ
ma-297	173	12	than	than	ADP
ma-297	173	13	the	the	DET
ma-297	173	14	ones	one	NOUN
ma-297	173	15	given	give	VERB
ma-297	173	16	before	before	ADP
ma-297	173	17	[	[	X
ma-297	173	18	3	3	NUM
ma-297	173	19	,	,	PUNCT
ma-297	173	20	5–7	5–7	NOUN
ma-297	173	21	]	]	PUNCT
ma-297	173	22	and	and	CCONJ
ma-297	173	23	under	under	ADP
ma-297	173	24	weaker	weak	ADJ
ma-297	173	25	sufficient	sufficient	ADJ
ma-297	173	26	semi	semi	ADJ
ma-297	173	27	-	-	ADJ
ma-297	173	28	local	local	ADJ
ma-297	173	29	convergence	convergence	NOUN
ma-297	173	30	criteria	criterion	NOUN
ma-297	173	31	.	.	PUNCT
ma-297	174	1	table	table	NOUN
ma-297	174	2	1	1	NUM
ma-297	174	3	.	.	PUNCT
ma-297	174	4	comparison	comparison	NOUN
ma-297	174	5	between	between	ADP
ma-297	174	6	majorizing	majorize	VERB
ma-297	174	7	sequences	sequence	NOUN
ma-297	174	8	and	and	CCONJ
ma-297	174	9	their	their	PRON
ma-297	174	10	limit	limit	NOUN
ma-297	174	11	points	point	VERB
ma-297	175	1	n	n	ADP
ma-297	175	2	vn	vn	PROPN
ma-297	175	3	vn+1	vn+1	PROPN
ma-297	176	1	−	−	PROPN
ma-297	176	2	vn	vn	PROPN
ma-297	176	3	sn	sn	PROPN
ma-297	176	4	sn+1	sn+1	VERB
ma-297	176	5	−	−	PROPN
ma-297	176	6	sn	sn	PROPN
ma-297	176	7	s̄n	s̄n	VERB
ma-297	176	8	s̄n+1	s̄n+1	NOUN
ma-297	176	9	−	−	ADP
ma-297	176	10	s̄n	s̄n	NOUN
ma-297	176	11	0	0	NUM
ma-297	176	12	0	0	NUM
ma-297	176	13	0	0	NUM
ma-297	176	14	0	0	NUM
ma-297	176	15	0	0	NUM
ma-297	176	16	0	0	NUM
ma-297	176	17	0	0	NUM
ma-297	176	18	1	1	NUM
ma-297	176	19	0.1000	0.1000	NUM
ma-297	176	20	0.1000	0.1000	NUM
ma-297	176	21	0.1000	0.1000	NUM
ma-297	176	22	0.1000	0.1000	NUM
ma-297	176	23	0.1000	0.1000	NUM
ma-297	176	24	0.1000	0.1000	NUM
ma-297	176	25	2	2	NUM
ma-297	176	26	0.1176	0.1176	NUM
ma-297	176	27	0.0176	0.0176	NUM
ma-297	176	28	0.1149	0.1149	NUM
ma-297	176	29	0.4350e-03	0.4350e-03	NUM
ma-297	176	30	0.1149	0.1149	NUM
ma-297	176	31	0.7619e-03	0.7619e-03	NOUN
ma-297	176	32	3	3	NUM
ma-297	176	33	0.1181	0.1181	NUM
ma-297	176	34	=	=	SYM
ma-297	176	35	v∗	v∗	PROPN
ma-297	176	36	0.0006	0.0006	NUM
ma-297	176	37	0.1154	0.1154	NUM
ma-297	176	38	=	=	PUNCT
ma-297	176	39	s∗	s∗	PROPN
ma-297	176	40	0.0004e-03	0.0004e-03	NUM
ma-297	176	41	0.1157	0.1157	NUM
ma-297	176	42	=	=	SYM
ma-297	176	43	s̄∗	s̄∗	PROPN
ma-297	176	44	0.0009e-03	0.0009e-03	NUM
ma-297	176	45	references	reference	NOUN
ma-297	176	46	[	[	X
ma-297	177	1	1	1	NUM
ma-297	177	2	]	]	X
ma-297	177	3	i.k	i.k	PROPN
ma-297	177	4	.	.	PROPN
ma-297	177	5	argyros	argyros	PROPN
ma-297	177	6	,	,	PUNCT
ma-297	177	7	the	the	DET
ma-297	177	8	theory	theory	NOUN
ma-297	177	9	and	and	CCONJ
ma-297	177	10	applications	application	NOUN
ma-297	177	11	of	of	ADP
ma-297	177	12	iteration	iteration	NOUN
ma-297	177	13	methods	method	NOUN
ma-297	177	14	,	,	PUNCT
ma-297	177	15	second	second	ADJ
ma-297	177	16	edition	edition	NOUN
ma-297	177	17	,	,	PUNCT
ma-297	177	18	crc	crc	NOUN
ma-297	177	19	press	press	PROPN
ma-297	177	20	,	,	PUNCT
ma-297	177	21	boca	boca	PROPN
ma-297	177	22	raton	raton	PROPN
ma-297	177	23	,	,	PUNCT
ma-297	177	24	2022.[2	2022.[2	PROPN
ma-297	177	25	]	]	X
ma-297	177	26	i.k	i.k	PROPN
ma-297	177	27	.	.	PROPN
ma-297	177	28	argyros	argyros	PROPN
ma-297	177	29	,	,	PUNCT
ma-297	177	30	s.	s.	PROPN
ma-297	177	31	hilout	hilout	PROPN
ma-297	177	32	,	,	PUNCT
ma-297	177	33	weaker	weak	ADJ
ma-297	177	34	conditions	condition	NOUN
ma-297	177	35	for	for	ADP
ma-297	177	36	the	the	DET
ma-297	177	37	convergence	convergence	NOUN
ma-297	177	38	of	of	ADP
ma-297	177	39	newton	newton	PROPN
ma-297	177	40	’s	’s	PART
ma-297	177	41	method	method	NOUN
ma-297	177	42	,	,	PUNCT
ma-297	177	43	j.	j.	PROPN
ma-297	177	44	complex	complex	PROPN
ma-297	177	45	.	.	PUNCT
ma-297	178	1	28	28	NUM
ma-297	178	2	(	(	PUNCT
ma-297	178	3	2012	2012	NUM
ma-297	178	4	)	)	PUNCT
ma-297	178	5	,	,	PUNCT
ma-297	178	6	364–387	364–387	NUM
ma-297	178	7	.	.	PUNCT
ma-297	179	1	https://doi.org/10.1016/j.jco.2011.12.003	https://doi.org/10.1016/j.jco.2011.12.003	NOUN
ma-297	179	2	.	.	PUNCT
ma-297	180	1	https://doi.org/10.28924/ada/ma.5.11	https://doi.org/10.28924/ada/ma.5.11	NUM
ma-297	180	2	https://doi.org/10.1016/j.jco.2011.12.003	https://doi.org/10.1016/j.jco.2011.12.003	PROPN
ma-297	180	3	eur	eur	NOUN
ma-297	180	4	.	.	PUNCT
ma-297	181	1	j.	j.	PROPN
ma-297	181	2	math	math	PROPN
ma-297	181	3	.	.	PUNCT
ma-297	182	1	anal	anal	PROPN
ma-297	182	2	.	.	PUNCT
ma-297	183	1	10.28924	10.28924	NUM
ma-297	183	2	/	/	SYM
ma-297	183	3	ada	ada	NOUN
ma-297	183	4	/	/	SYM
ma-297	183	5	ma.5.11	ma.5.11	ADJ
ma-297	183	6	9	9	NUM
ma-297	183	7	[	[	SYM
ma-297	183	8	3	3	NUM
ma-297	183	9	]	]	X
ma-297	183	10	p.	p.	NOUN
ma-297	183	11	deuflhard	deuflhard	NOUN
ma-297	183	12	,	,	PUNCT
ma-297	183	13	g.	g.	PROPN
ma-297	183	14	heindl	heindl	PROPN
ma-297	183	15	,	,	PUNCT
ma-297	183	16	affine	affine	NOUN
ma-297	183	17	invariant	invariant	ADJ
ma-297	183	18	convergence	convergence	NOUN
ma-297	183	19	theorems	theorem	NOUN
ma-297	183	20	for	for	ADP
ma-297	183	21	newton	newton	PROPN
ma-297	183	22	’s	’s	PART
ma-297	183	23	method	method	NOUN
ma-297	183	24	and	and	CCONJ
ma-297	183	25	extensions	extension	NOUN
ma-297	183	26	to	to	ADP
ma-297	183	27	relatedmethods	relatedmethod	NOUN
ma-297	183	28	,	,	PUNCT
ma-297	183	29	siam	siam	PROPN
ma-297	183	30	j.	j.	PROPN
ma-297	183	31	numer	numer	PROPN
ma-297	183	32	.	.	PUNCT
ma-297	184	1	anal	anal	PROPN
ma-297	184	2	.	.	PUNCT
ma-297	185	1	16	16	NUM
ma-297	185	2	(	(	PUNCT
ma-297	185	3	1979	1979	NUM
ma-297	185	4	)	)	PUNCT
ma-297	185	5	,	,	PUNCT
ma-297	185	6	1–10	1–10	NOUN
ma-297	185	7	.	.	PUNCT
ma-297	186	1	https://doi.org/10.1137/0716001.[4	https://doi.org/10.1137/0716001.[4	X
ma-297	186	2	]	]	X
ma-297	186	3	p.	p.	NOUN
ma-297	186	4	deuflhard	deuflhard	PROPN
ma-297	186	5	,	,	PUNCT
ma-297	186	6	newton	newton	PROPN
ma-297	186	7	methods	method	NOUN
ma-297	186	8	for	for	ADP
ma-297	186	9	nonlinear	nonlinear	ADJ
ma-297	186	10	problems	problem	NOUN
ma-297	186	11	:	:	PUNCT
ma-297	186	12	affine	affine	VERB
ma-297	186	13	invariance	invariance	NOUN
ma-297	186	14	and	and	CCONJ
ma-297	186	15	adaptive	adaptive	ADJ
ma-297	186	16	algorithms	algorithm	NOUN
ma-297	186	17	,	,	PUNCT
ma-297	186	18	springer	springer	NOUN
ma-297	186	19	,	,	PUNCT
ma-297	186	20	berlin,2004	berlin,2004	NOUN
ma-297	186	21	.	.	PUNCT
ma-297	187	1	https://doi.org/10.1007/978-3-642-23899-4.[5	https://doi.org/10.1007/978-3-642-23899-4.[5	PRON
ma-297	187	2	]	]	X
ma-297	188	1	w.b	w.b	PROPN
ma-297	188	2	.	.	PROPN
ma-297	188	3	gragg	gragg	PROPN
ma-297	188	4	,	,	PUNCT
ma-297	188	5	r.a	r.a	PROPN
ma-297	188	6	.	.	PROPN
ma-297	188	7	tapia	tapia	PROPN
ma-297	188	8	,	,	PUNCT
ma-297	188	9	optimal	optimal	ADJ
ma-297	188	10	error	error	NOUN
ma-297	188	11	bounds	bound	NOUN
ma-297	188	12	for	for	ADP
ma-297	188	13	the	the	DET
ma-297	188	14	newton	newton	PROPN
ma-297	188	15	–	–	PUNCT
ma-297	188	16	kantorovich	kantorovich	PROPN
ma-297	188	17	theorem	theorem	PROPN
ma-297	188	18	,	,	PUNCT
ma-297	188	19	siam	siam	PROPN
ma-297	188	20	j.	j.	PROPN
ma-297	188	21	numer	numer	PROPN
ma-297	188	22	.	.	PUNCT
ma-297	189	1	anal	anal	PROPN
ma-297	189	2	.	.	PUNCT
ma-297	190	1	11(1974	11(1974	NUM
ma-297	190	2	)	)	PUNCT
ma-297	190	3	,	,	PUNCT
ma-297	191	1	10–13	10–13	NUM
ma-297	191	2	.	.	PUNCT
ma-297	192	1	https://doi.org/10.1137/0711002.[6	https://doi.org/10.1137/0711002.[6	X
ma-297	192	2	]	]	X
ma-297	192	3	l.v	l.v	PROPN
ma-297	192	4	.	.	PROPN
ma-297	192	5	kantorovich	kantorovich	PROPN
ma-297	192	6	,	,	PUNCT
ma-297	192	7	g.p	g.p	PROPN
ma-297	192	8	.	.	PROPN
ma-297	192	9	akilov	akilov	PROPN
ma-297	192	10	,	,	PUNCT
ma-297	192	11	functional	functional	ADJ
ma-297	192	12	analysis	analysis	NOUN
ma-297	192	13	,	,	PUNCT
ma-297	192	14	pergamon	pergamon	PROPN
ma-297	192	15	press	press	PROPN
ma-297	192	16	,	,	PUNCT
ma-297	192	17	oxford	oxford	PROPN
ma-297	192	18	,	,	PUNCT
ma-297	192	19	1982.[7	1982.[7	NUM
ma-297	192	20	]	]	PUNCT
ma-297	192	21	a.m.	a.m.	NOUN
ma-297	193	1	ostrowski	ostrowski	PROPN
ma-297	193	2	,	,	PUNCT
ma-297	193	3	solutions	solution	NOUN
ma-297	193	4	of	of	ADP
ma-297	193	5	equations	equation	NOUN
ma-297	193	6	in	in	ADP
ma-297	193	7	euclidean	euclidean	NOUN
ma-297	193	8	and	and	CCONJ
ma-297	193	9	banach	banach	NOUN
ma-297	193	10	spaces	space	NOUN
ma-297	193	11	,	,	PUNCT
ma-297	193	12	academic	academic	ADJ
ma-297	193	13	press	press	NOUN
ma-297	193	14	,	,	PUNCT
ma-297	193	15	new	new	PROPN
ma-297	193	16	york	york	PROPN
ma-297	193	17	,	,	PUNCT
ma-297	193	18	1973	1973	NUM
ma-297	193	19	.	.	PUNCT
ma-297	194	1	https://doi.org/10.28924/ada/ma.5.11	https://doi.org/10.28924/ada/ma.5.11	NUM
ma-297	194	2	https://doi.org/10.1137/0716001	https://doi.org/10.1137/0716001	ADJ
ma-297	194	3	https://doi.org/10.1007/978-3-642-23899-4	https://doi.org/10.1007/978-3-642-23899-4	NOUN
ma-297	194	4	https://doi.org/10.1137/0711002	https://doi.org/10.1137/0711002	ADJ
ma-297	194	5	1	1	NUM
ma-297	194	6	.	.	PUNCT
ma-297	194	7	introduction	introduction	NOUN
ma-297	194	8	2	2	NUM
ma-297	194	9	.	.	NOUN
ma-297	194	10	lipschitz	lipschitz	NOUN
ma-297	194	11	conditions	condition	NOUN
ma-297	194	12	3	3	NUM
ma-297	194	13	.	.	X
ma-297	194	14	convergence	convergence	NOUN
ma-297	194	15	of	of	ADP
ma-297	194	16	majorizing	majorize	VERB
ma-297	194	17	sequences	sequence	NOUN
ma-297	194	18	.	.	PUNCT
ma-297	195	1	4	4	X
ma-297	195	2	.	.	X
ma-297	195	3	convergence	convergence	NOUN
ma-297	195	4	of	of	ADP
ma-297	195	5	newton	newton	PROPN
ma-297	195	6	's	's	PART
ma-297	195	7	method	method	NOUN
ma-297	195	8	5	5	NUM
ma-297	195	9	.	.	PUNCT
ma-297	196	1	a	a	DET
ma-297	196	2	numerical	numerical	ADJ
ma-297	196	3	example	example	NOUN
ma-297	196	4	references	reference	NOUN