id	sid	tid	token	lemma	pos
cana-3310	1	1	communications	communication	NOUN
cana-3310	1	2	on	on	ADP
cana-3310	1	3	applied	apply	VERB
cana-3310	1	4	nonlinear	nonlinear	ADJ
cana-3310	1	5	analysis	analysis	NOUN
cana-3310	1	6	issn	issn	NOUN
cana-3310	1	7	:	:	PUNCT
cana-3310	1	8	1074	1074	NUM
cana-3310	1	9	-	-	PUNCT
cana-3310	1	10	133x	133x	NUM
cana-3310	1	11	vol	vol	NOUN
cana-3310	1	12	32	32	NUM
cana-3310	1	13	no	no	NOUN
cana-3310	1	14	.	.	PUNCT
cana-3310	2	1	6s	6s	NUM
cana-3310	2	2	(	(	PUNCT
cana-3310	2	3	2025	2025	NUM
cana-3310	2	4	)	)	PUNCT
cana-3310	2	5	459	459	NUM
cana-3310	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3310	2	7	using	use	VERB
cana-3310	2	8	geogebra	geogebra	NOUN
cana-3310	2	9	and	and	CCONJ
cana-3310	2	10	sagemath	sagemath	NOUN
cana-3310	2	11	to	to	PART
cana-3310	2	12	assist	assist	VERB
cana-3310	2	13	engineering	engineering	NOUN
cana-3310	2	14	students	student	NOUN
cana-3310	2	15	in	in	ADP
cana-3310	2	16	tracing	trace	VERB
cana-3310	2	17	different	different	ADJ
cana-3310	2	18	curves	curve	NOUN
cana-3310	2	19	and	and	CCONJ
cana-3310	2	20	solving	solve	VERB
cana-3310	2	21	numerical	numerical	ADJ
cana-3310	2	22	methods	method	NOUN
cana-3310	2	23	bhavna1	bhavna1	PROPN
cana-3310	2	24	,	,	PUNCT
cana-3310	2	25	shubham	shubham	PROPN
cana-3310	2	26	ghodke2	ghodke2	PROPN
cana-3310	2	27	,	,	PUNCT
cana-3310	2	28	kamalkishor	kamalkishor	PROPN
cana-3310	2	29	maniyar3	maniyar3	PROPN
cana-3310	2	30	1,2,3first	1,2,3first	NUM
cana-3310	2	31	year	year	NOUN
cana-3310	2	32	engineering	engineering	NOUN
cana-3310	2	33	department	department	PROPN
cana-3310	2	34	,	,	PUNCT
cana-3310	2	35	dr	dr	PROPN
cana-3310	2	36	.	.	PROPN
cana-3310	2	37	d.	d.	PROPN
cana-3310	2	38	y.	y.	PROPN
cana-3310	2	39	patil	patil	PROPN
cana-3310	2	40	institute	institute	PROPN
cana-3310	2	41	of	of	ADP
cana-3310	2	42	technology	technology	NOUN
cana-3310	2	43	pimpri	pimpri	NOUN
cana-3310	2	44	,	,	PUNCT
cana-3310	2	45	pune-18	pune-18	NOUN
cana-3310	2	46	1bhavnasingh2092@gmail.com	1bhavnasingh2092@gmail.com	NUM
cana-3310	2	47	,	,	PUNCT
cana-3310	2	48	2	2	NUM
cana-3310	2	49	shubhamghodake000@gmail.com	shubhamghodake000@gmail.com	SYM
cana-3310	2	50	.3	.3	NUM
cana-3310	2	51	kkmaniyar2020@gmail.com	kkmaniyar2020@gmail.com	NOUN
cana-3310	3	1	article	article	NOUN
cana-3310	3	2	history	history	NOUN
cana-3310	3	3	:	:	PUNCT
cana-3310	3	4	received	receive	VERB
cana-3310	3	5	:	:	PUNCT
cana-3310	3	6	23	23	NUM
cana-3310	3	7	-	-	SYM
cana-3310	3	8	10	10	NUM
cana-3310	3	9	-	-	PUNCT
cana-3310	3	10	2024	2024	NUM
cana-3310	3	11	revised	revise	VERB
cana-3310	3	12	:	:	PUNCT
cana-3310	3	13	07	07	NUM
cana-3310	3	14	-	-	SYM
cana-3310	3	15	12	12	NUM
cana-3310	3	16	-	-	PUNCT
cana-3310	3	17	2024	2024	NUM
cana-3310	3	18	accepted	accept	VERB
cana-3310	3	19	:	:	PUNCT
cana-3310	3	20	14	14	NUM
cana-3310	3	21	-	-	SYM
cana-3310	3	22	12	12	NUM
cana-3310	3	23	-	-	PUNCT
cana-3310	3	24	2024	2024	NUM
cana-3310	3	25	abstract	abstract	NOUN
cana-3310	3	26	:	:	PUNCT
cana-3310	3	27	recently	recently	ADV
cana-3310	3	28	,	,	PUNCT
cana-3310	3	29	there	there	PRON
cana-3310	3	30	has	have	AUX
cana-3310	3	31	been	be	AUX
cana-3310	3	32	a	a	DET
cana-3310	3	33	growing	grow	VERB
cana-3310	3	34	emphasis	emphasis	NOUN
cana-3310	3	35	on	on	ADP
cana-3310	3	36	the	the	DET
cana-3310	3	37	use	use	NOUN
cana-3310	3	38	of	of	ADP
cana-3310	3	39	technology	technology	NOUN
cana-3310	3	40	and	and	CCONJ
cana-3310	3	41	software	software	NOUN
cana-3310	3	42	in	in	ADP
cana-3310	3	43	the	the	DET
cana-3310	3	44	educational	educational	ADJ
cana-3310	3	45	process	process	NOUN
cana-3310	3	46	.	.	PUNCT
cana-3310	4	1	the	the	DET
cana-3310	4	2	purpose	purpose	NOUN
cana-3310	4	3	of	of	ADP
cana-3310	4	4	this	this	DET
cana-3310	4	5	article	article	NOUN
cana-3310	4	6	is	be	AUX
cana-3310	4	7	to	to	PART
cana-3310	4	8	show	show	VERB
cana-3310	4	9	readers	reader	NOUN
cana-3310	4	10	how	how	SCONJ
cana-3310	4	11	to	to	PART
cana-3310	4	12	use	use	VERB
cana-3310	4	13	the	the	DET
cana-3310	4	14	software	software	NOUN
cana-3310	4	15	geogebra	geogebra	NOUN
cana-3310	4	16	and	and	CCONJ
cana-3310	4	17	sagemath	sagemath	NOUN
cana-3310	4	18	to	to	PART
cana-3310	4	19	trace	trace	VERB
cana-3310	4	20	the	the	DET
cana-3310	4	21	different	different	ADJ
cana-3310	4	22	curve	curve	NOUN
cana-3310	4	23	and	and	CCONJ
cana-3310	4	24	solving	solve	VERB
cana-3310	4	25	numerical	numerical	ADJ
cana-3310	4	26	methods	method	NOUN
cana-3310	4	27	.	.	PUNCT
cana-3310	5	1	this	this	DET
cana-3310	5	2	application	application	NOUN
cana-3310	5	3	serves	serve	VERB
cana-3310	5	4	as	as	ADP
cana-3310	5	5	a	a	DET
cana-3310	5	6	helpful	helpful	ADJ
cana-3310	5	7	tool	tool	NOUN
cana-3310	5	8	for	for	ADP
cana-3310	5	9	instructors	instructor	NOUN
cana-3310	5	10	and	and	CCONJ
cana-3310	5	11	students	student	NOUN
cana-3310	5	12	for	for	ADP
cana-3310	5	13	projects	project	NOUN
cana-3310	5	14	,	,	PUNCT
cana-3310	5	15	training	training	NOUN
cana-3310	5	16	,	,	PUNCT
cana-3310	5	17	lectures	lecture	NOUN
cana-3310	5	18	,	,	PUNCT
cana-3310	5	19	and	and	CCONJ
cana-3310	5	20	other	other	ADJ
cana-3310	5	21	purposes	purpose	NOUN
cana-3310	5	22	in	in	ADP
cana-3310	5	23	addition	addition	NOUN
cana-3310	5	24	to	to	ADP
cana-3310	5	25	being	be	AUX
cana-3310	5	26	a	a	DET
cana-3310	5	27	computer	computer	NOUN
cana-3310	5	28	environment	environment	NOUN
cana-3310	5	29	.	.	PUNCT
cana-3310	6	1	the	the	DET
cana-3310	6	2	author	author	NOUN
cana-3310	6	3	discusses	discuss	VERB
cana-3310	6	4	numerical	numerical	ADJ
cana-3310	6	5	techniques	technique	NOUN
cana-3310	6	6	,	,	PUNCT
cana-3310	6	7	how	how	SCONJ
cana-3310	6	8	to	to	PART
cana-3310	6	9	make	make	VERB
cana-3310	6	10	certain	certain	ADJ
cana-3310	6	11	graphical	graphical	ADJ
cana-3310	6	12	representations	representation	NOUN
cana-3310	6	13	in	in	ADP
cana-3310	6	14	this	this	DET
cana-3310	6	15	application	application	NOUN
cana-3310	6	16	,	,	PUNCT
cana-3310	6	17	and	and	CCONJ
cana-3310	6	18	how	how	SCONJ
cana-3310	6	19	to	to	PART
cana-3310	6	20	trace	trace	VERB
cana-3310	6	21	any	any	DET
cana-3310	6	22	curve	curve	NOUN
cana-3310	6	23	(	(	PUNCT
cana-3310	6	24	parametric	parametric	ADJ
cana-3310	6	25	,	,	PUNCT
cana-3310	6	26	polar	polar	ADJ
cana-3310	6	27	,	,	PUNCT
cana-3310	6	28	and	and	CCONJ
cana-3310	6	29	cartesian	cartesian	ADJ
cana-3310	6	30	)	)	PUNCT
cana-3310	6	31	using	use	VERB
cana-3310	6	32	geogebra	geogebra	NOUN
cana-3310	6	33	and	and	CCONJ
cana-3310	6	34	sagemath	sagemath	NOUN
cana-3310	6	35	.	.	PUNCT
cana-3310	7	1	traditional	traditional	ADJ
cana-3310	7	2	techniques	technique	NOUN
cana-3310	7	3	of	of	ADP
cana-3310	7	4	tracing	trace	VERB
cana-3310	7	5	the	the	DET
cana-3310	7	6	curve	curve	NOUN
cana-3310	7	7	and	and	CCONJ
cana-3310	7	8	solving	solve	VERB
cana-3310	7	9	numerical	numerical	ADJ
cana-3310	7	10	procedures	procedure	NOUN
cana-3310	7	11	,	,	PUNCT
cana-3310	7	12	such	such	ADJ
cana-3310	7	13	as	as	ADP
cana-3310	7	14	the	the	DET
cana-3310	7	15	gauss	gauss	ADJ
cana-3310	7	16	-	-	PUNCT
cana-3310	7	17	seidal	seidal	NOUN
cana-3310	7	18	and	and	CCONJ
cana-3310	7	19	gauss	gauss	PROPN
cana-3310	7	20	-	-	PUNCT
cana-3310	7	21	jordan	jordan	PROPN
cana-3310	7	22	methods	method	NOUN
cana-3310	7	23	,	,	PUNCT
cana-3310	7	24	might	might	AUX
cana-3310	7	25	be	be	AUX
cana-3310	7	26	time	time	NOUN
cana-3310	7	27	-	-	PUNCT
cana-3310	7	28	consuming	consume	VERB
cana-3310	7	29	.	.	PUNCT
cana-3310	8	1	for	for	ADP
cana-3310	8	2	this	this	DET
cana-3310	8	3	reason	reason	NOUN
cana-3310	8	4	,	,	PUNCT
cana-3310	8	5	tools	tool	NOUN
cana-3310	8	6	like	like	ADP
cana-3310	8	7	geogebra	geogebra	NOUN
cana-3310	8	8	and	and	CCONJ
cana-3310	8	9	sagemath	sagemath	NOUN
cana-3310	8	10	can	can	AUX
cana-3310	8	11	make	make	VERB
cana-3310	8	12	calculations	calculation	NOUN
cana-3310	8	13	for	for	ADP
cana-3310	8	14	students	student	NOUN
cana-3310	8	15	,	,	PUNCT
cana-3310	8	16	teachers	teacher	NOUN
cana-3310	8	17	,	,	PUNCT
cana-3310	8	18	engineers	engineer	NOUN
cana-3310	8	19	,	,	PUNCT
cana-3310	8	20	and	and	CCONJ
cana-3310	8	21	other	other	ADJ
cana-3310	8	22	professionals	professional	NOUN
cana-3310	8	23	go	go	VERB
cana-3310	8	24	more	more	ADV
cana-3310	8	25	quickly	quickly	ADV
cana-3310	8	26	.	.	PUNCT
cana-3310	9	1	in	in	ADP
cana-3310	9	2	an	an	DET
cana-3310	9	3	effort	effort	NOUN
cana-3310	9	4	to	to	PART
cana-3310	9	5	add	add	VERB
cana-3310	9	6	to	to	ADP
cana-3310	9	7	the	the	DET
cana-3310	9	8	specialized	specialized	ADJ
cana-3310	9	9	literature	literature	NOUN
cana-3310	9	10	,	,	PUNCT
cana-3310	9	11	the	the	DET
cana-3310	9	12	author	author	NOUN
cana-3310	9	13	developed	develop	VERB
cana-3310	9	14	fresh	fresh	ADJ
cana-3310	9	15	examples	example	NOUN
cana-3310	9	16	of	of	ADP
cana-3310	9	17	abstract	abstract	ADJ
cana-3310	9	18	usage	usage	NOUN
cana-3310	9	19	of	of	ADP
cana-3310	9	20	geogebra	geogebra	NOUN
cana-3310	9	21	and	and	CCONJ
cana-3310	9	22	sagemath	sagemath	NOUN
cana-3310	9	23	.	.	PUNCT
cana-3310	10	1	the	the	DET
cana-3310	10	2	article	article	NOUN
cana-3310	10	3	notes	note	VERB
cana-3310	10	4	the	the	DET
cana-3310	10	5	need	need	NOUN
cana-3310	10	6	for	for	SCONJ
cana-3310	10	7	the	the	DET
cana-3310	10	8	education	education	NOUN
cana-3310	10	9	system	system	NOUN
cana-3310	10	10	to	to	PART
cana-3310	10	11	integrate	integrate	VERB
cana-3310	10	12	blended	blended	ADJ
cana-3310	10	13	learning	learning	NOUN
cana-3310	10	14	and	and	CCONJ
cana-3310	10	15	teaching	teaching	NOUN
cana-3310	10	16	methods	method	NOUN
cana-3310	10	17	,	,	PUNCT
cana-3310	10	18	which	which	PRON
cana-3310	10	19	combine	combine	VERB
cana-3310	10	20	computer	computer	NOUN
cana-3310	10	21	programs	program	NOUN
cana-3310	10	22	with	with	ADP
cana-3310	10	23	traditional	traditional	ADJ
cana-3310	10	24	teaching	teaching	NOUN
cana-3310	10	25	methods	method	NOUN
cana-3310	10	26	to	to	PART
cana-3310	10	27	improve	improve	VERB
cana-3310	10	28	the	the	DET
cana-3310	10	29	teaching	teaching	NOUN
cana-3310	10	30	of	of	ADP
cana-3310	10	31	mathematics	mathematic	NOUN
cana-3310	10	32	.	.	PUNCT
cana-3310	11	1	it	it	PRON
cana-3310	11	2	recommends	recommend	VERB
cana-3310	11	3	using	use	VERB
cana-3310	11	4	geogebra	geogebra	NOUN
cana-3310	11	5	,	,	PUNCT
cana-3310	11	6	sagemath	sagemath	NOUN
cana-3310	11	7	programs	program	NOUN
cana-3310	11	8	for	for	ADP
cana-3310	11	9	assisted	assist	VERB
cana-3310	11	10	instruction	instruction	NOUN
cana-3310	11	11	in	in	ADP
cana-3310	11	12	the	the	DET
cana-3310	11	13	teaching	teaching	NOUN
cana-3310	11	14	and	and	CCONJ
cana-3310	11	15	learning	learning	NOUN
cana-3310	11	16	of	of	ADP
cana-3310	11	17	mathematics	mathematic	NOUN
cana-3310	11	18	,	,	PUNCT
cana-3310	11	19	specifically	specifically	ADV
cana-3310	11	20	for	for	ADP
cana-3310	11	21	cases	case	NOUN
cana-3310	11	22	of	of	ADP
cana-3310	11	23	numerical	numerical	ADJ
cana-3310	11	24	methods	method	NOUN
cana-3310	11	25	and	and	CCONJ
cana-3310	11	26	tracing	trace	VERB
cana-3310	11	27	the	the	DET
cana-3310	11	28	curve	curve	NOUN
cana-3310	11	29	,	,	PUNCT
cana-3310	11	30	respectively	respectively	ADV
cana-3310	11	31	.	.	PUNCT
cana-3310	12	1	keywords	keyword	NOUN
cana-3310	12	2	:	:	PUNCT
cana-3310	12	3	geogebra	geogebra	NOUN
cana-3310	12	4	,	,	PUNCT
cana-3310	12	5	sagemath	sagemath	NOUN
cana-3310	12	6	,	,	PUNCT
cana-3310	12	7	tracing	trace	VERB
cana-3310	12	8	curve	curve	NOUN
cana-3310	12	9	,	,	PUNCT
cana-3310	12	10	numerical	numerical	ADJ
cana-3310	12	11	methods	method	NOUN
cana-3310	12	12	.	.	PUNCT
cana-3310	13	1	1	1	X
cana-3310	13	2	.	.	X
cana-3310	13	3	introduction	introduction	NOUN
cana-3310	13	4	engineering	engineering	NOUN
cana-3310	13	5	is	be	AUX
cana-3310	13	6	defined	define	VERB
cana-3310	13	7	as	as	ADP
cana-3310	13	8	"	"	PUNCT
cana-3310	13	9	the	the	DET
cana-3310	13	10	application	application	NOUN
cana-3310	13	11	of	of	ADP
cana-3310	13	12	mathematics	mathematic	NOUN
cana-3310	13	13	and	and	CCONJ
cana-3310	13	14	sciences	science	NOUN
cana-3310	13	15	to	to	ADP
cana-3310	13	16	the	the	DET
cana-3310	13	17	construction	construction	NOUN
cana-3310	13	18	and	and	CCONJ
cana-3310	13	19	design	design	NOUN
cana-3310	13	20	of	of	ADP
cana-3310	13	21	projects	project	NOUN
cana-3310	13	22	for	for	ADP
cana-3310	13	23	societal	societal	ADJ
cana-3310	13	24	use	use	NOUN
cana-3310	13	25	”	"	PUNCT
cana-3310	13	26	.	.	PUNCT
cana-3310	14	1	mathematics	mathematic	NOUN
cana-3310	14	2	teaches	teach	VERB
cana-3310	14	3	rational	rational	ADJ
cana-3310	14	4	thinking	thinking	NOUN
cana-3310	14	5	and	and	CCONJ
cana-3310	14	6	provides	provide	VERB
cana-3310	14	7	tools	tool	NOUN
cana-3310	14	8	for	for	ADP
cana-3310	14	9	conducting	conduct	VERB
cana-3310	14	10	analyses	analysis	NOUN
cana-3310	14	11	to	to	PART
cana-3310	14	12	learn	learn	VERB
cana-3310	14	13	about	about	ADP
cana-3310	14	14	systems	system	NOUN
cana-3310	14	15	.	.	PUNCT
cana-3310	15	1	traditional	traditional	ADJ
cana-3310	15	2	curve	curve	NOUN
cana-3310	15	3	-	-	PUNCT
cana-3310	15	4	tracing	trace	VERB
cana-3310	15	5	and	and	CCONJ
cana-3310	15	6	numerical	numerical	ADJ
cana-3310	15	7	procedure	procedure	NOUN
cana-3310	15	8	-	-	PUNCT
cana-3310	15	9	solving	solve	VERB
cana-3310	15	10	approaches	approach	NOUN
cana-3310	15	11	,	,	PUNCT
cana-3310	15	12	such	such	ADJ
cana-3310	15	13	as	as	ADP
cana-3310	15	14	the	the	DET
cana-3310	15	15	gauss	gauss	ADJ
cana-3310	15	16	-	-	PUNCT
cana-3310	15	17	seidal	seidal	NOUN
cana-3310	15	18	and	and	CCONJ
cana-3310	15	19	gauss	gauss	PROPN
cana-3310	15	20	-	-	PUNCT
cana-3310	15	21	jordan	jordan	PROPN
cana-3310	15	22	methods	method	NOUN
cana-3310	15	23	,	,	PUNCT
cana-3310	15	24	may	may	AUX
cana-3310	15	25	be	be	AUX
cana-3310	15	26	time	time	NOUN
cana-3310	15	27	-	-	PUNCT
cana-3310	15	28	consuming	consume	VERB
cana-3310	15	29	.	.	PUNCT
cana-3310	16	1	as	as	ADP
cana-3310	16	2	a	a	DET
cana-3310	16	3	result	result	NOUN
cana-3310	16	4	,	,	PUNCT
cana-3310	16	5	geogebra	geogebra	NOUN
cana-3310	16	6	and	and	CCONJ
cana-3310	16	7	sagemath	sagemath	NOUN
cana-3310	16	8	will	will	AUX
cana-3310	16	9	accelerate	accelerate	VERB
cana-3310	16	10	computations	computation	NOUN
cana-3310	16	11	for	for	ADP
cana-3310	16	12	students	student	NOUN
cana-3310	16	13	,	,	PUNCT
cana-3310	16	14	instructors	instructor	NOUN
cana-3310	16	15	,	,	PUNCT
cana-3310	16	16	engineers	engineer	NOUN
cana-3310	16	17	,	,	PUNCT
cana-3310	16	18	and	and	CCONJ
cana-3310	16	19	other	other	ADJ
cana-3310	16	20	professionals	professional	NOUN
cana-3310	16	21	.	.	PUNCT
cana-3310	17	1	the	the	DET
cana-3310	17	2	article	article	NOUN
cana-3310	17	3	recommends	recommend	VERB
cana-3310	17	4	the	the	DET
cana-3310	17	5	use	use	NOUN
cana-3310	17	6	of	of	ADP
cana-3310	17	7	geogebra	geogebra	NOUN
cana-3310	17	8	,	,	PUNCT
cana-3310	17	9	sagemath	sagemath	NOUN
cana-3310	17	10	,	,	PUNCT
cana-3310	17	11	and	and	CCONJ
cana-3310	17	12	other	other	ADJ
cana-3310	17	13	related	related	ADJ
cana-3310	17	14	programs	program	NOUN
cana-3310	17	15	for	for	ADP
cana-3310	17	16	assisted	assist	VERB
cana-3310	17	17	instruction	instruction	NOUN
cana-3310	17	18	in	in	ADP
cana-3310	17	19	the	the	DET
cana-3310	17	20	teaching	teaching	NOUN
cana-3310	17	21	and	and	CCONJ
cana-3310	17	22	learning	learning	NOUN
cana-3310	17	23	of	of	ADP
cana-3310	17	24	mathematical	mathematical	ADJ
cana-3310	17	25	concepts	concept	NOUN
cana-3310	17	26	,	,	PUNCT
cana-3310	17	27	for	for	ADP
cana-3310	17	28	specific	specific	ADJ
cana-3310	17	29	cases	case	NOUN
cana-3310	17	30	of	of	ADP
cana-3310	17	31	numerical	numerical	ADJ
cana-3310	17	32	methods	method	NOUN
cana-3310	17	33	and	and	CCONJ
cana-3310	17	34	traces	trace	NOUN
cana-3310	17	35	,	,	PUNCT
cana-3310	17	36	respectively	respectively	ADV
cana-3310	17	37	;	;	PUNCT
cana-3310	17	38	and	and	CCONJ
cana-3310	17	39	notes	note	VERB
cana-3310	17	40	the	the	DET
cana-3310	17	41	need	need	NOUN
cana-3310	17	42	for	for	SCONJ
cana-3310	17	43	the	the	DET
cana-3310	17	44	education	education	NOUN
cana-3310	17	45	system	system	NOUN
cana-3310	17	46	to	to	PART
cana-3310	17	47	integrate	integrate	VERB
cana-3310	17	48	interconnected	interconnected	ADJ
cana-3310	17	49	teaching	teaching	NOUN
cana-3310	17	50	and	and	CCONJ
cana-3310	17	51	learning	learning	NOUN
cana-3310	17	52	methods	method	NOUN
cana-3310	17	53	,	,	PUNCT
cana-3310	17	54	in	in	ADP
cana-3310	17	55	which	which	PRON
cana-3310	17	56	software	software	NOUN
cana-3310	17	57	programs	program	NOUN
cana-3310	17	58	are	be	AUX
cana-3310	17	59	used	use	VERB
cana-3310	17	60	in	in	ADP
cana-3310	17	61	conjunction	conjunction	NOUN
cana-3310	17	62	with	with	ADP
cana-3310	17	63	traditional	traditional	ADJ
cana-3310	17	64	teaching	teaching	NOUN
cana-3310	17	65	methods	method	NOUN
cana-3310	17	66	,	,	PUNCT
cana-3310	17	67	to	to	PART
cana-3310	17	68	enhance	enhance	VERB
cana-3310	17	69	the	the	DET
cana-3310	17	70	teaching	teaching	NOUN
cana-3310	17	71	process	process	NOUN
cana-3310	17	72	for	for	ADP
cana-3310	17	73	mathematics	mathematic	NOUN
cana-3310	17	74	.	.	PUNCT
cana-3310	18	1	communications	communication	NOUN
cana-3310	18	2	on	on	ADP
cana-3310	18	3	applied	apply	VERB
cana-3310	18	4	nonlinear	nonlinear	ADJ
cana-3310	18	5	analysis	analysis	NOUN
cana-3310	18	6	issn	issn	NOUN
cana-3310	18	7	:	:	PUNCT
cana-3310	18	8	1074	1074	NUM
cana-3310	18	9	-	-	PUNCT
cana-3310	18	10	133x	133x	NUM
cana-3310	18	11	vol	vol	NOUN
cana-3310	18	12	32	32	NUM
cana-3310	18	13	no	no	NOUN
cana-3310	18	14	.	.	PUNCT
cana-3310	19	1	6s	6s	NUM
cana-3310	19	2	(	(	PUNCT
cana-3310	19	3	2025	2025	NUM
cana-3310	19	4	)	)	PUNCT
cana-3310	19	5	460	460	NUM
cana-3310	20	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3310	20	2	comparison	comparison	NOUN
cana-3310	20	3	between	between	ADP
cana-3310	20	4	geogebra	geogebra	NOUN
cana-3310	20	5	and	and	CCONJ
cana-3310	20	6	sagemath	sagemath	NOUN
cana-3310	20	7	:	:	PUNCT
cana-3310	20	8	although	although	SCONJ
cana-3310	20	9	geogebra	geogebra	NOUN
cana-3310	20	10	and	and	CCONJ
cana-3310	20	11	sagemath	sagemath	NOUN
cana-3310	20	12	share	share	VERB
cana-3310	20	13	many	many	ADJ
cana-3310	20	14	functions	function	NOUN
cana-3310	20	15	,	,	PUNCT
cana-3310	20	16	their	their	PRON
cana-3310	20	17	main	main	ADJ
cana-3310	20	18	differences	difference	NOUN
cana-3310	20	19	are	be	AUX
cana-3310	20	20	in	in	ADP
cana-3310	20	21	the	the	DET
cana-3310	20	22	use	use	NOUN
cana-3310	20	23	cases	case	NOUN
cana-3310	20	24	and	and	CCONJ
cana-3310	20	25	target	target	VERB
cana-3310	20	26	audiences	audience	NOUN
cana-3310	20	27	.	.	PUNCT
cana-3310	21	1	due	due	ADP
cana-3310	21	2	to	to	ADP
cana-3310	21	3	its	its	PRON
cana-3310	21	4	emphasis	emphasis	NOUN
cana-3310	21	5	on	on	ADP
cana-3310	21	6	dynamic	dynamic	ADJ
cana-3310	21	7	visuals	visual	NOUN
cana-3310	21	8	and	and	CCONJ
cana-3310	21	9	user	user	NOUN
cana-3310	21	10	-	-	PUNCT
cana-3310	21	11	friendly	friendly	ADJ
cana-3310	21	12	interface	interface	NOUN
cana-3310	21	13	,	,	PUNCT
cana-3310	21	14	geogebra	geogebra	NOUN
cana-3310	21	15	is	be	AUX
cana-3310	21	16	frequently	frequently	ADV
cana-3310	21	17	used	use	VERB
cana-3310	21	18	in	in	ADP
cana-3310	21	19	educational	educational	ADJ
cana-3310	21	20	settings	setting	NOUN
cana-3310	21	21	.	.	PUNCT
cana-3310	22	1	it	it	PRON
cana-3310	22	2	's	be	AUX
cana-3310	22	3	very	very	ADV
cana-3310	22	4	helpful	helpful	ADJ
cana-3310	22	5	for	for	ADP
cana-3310	22	6	interactively	interactively	ADV
cana-3310	22	7	teaching	teaching	NOUN
cana-3310	22	8	and	and	CCONJ
cana-3310	22	9	understanding	understand	VERB
cana-3310	22	10	mathematical	mathematical	ADJ
cana-3310	22	11	topics	topic	NOUN
cana-3310	22	12	.	.	PUNCT
cana-3310	23	1	sagemath	sagemath	NOUN
cana-3310	23	2	is	be	AUX
cana-3310	23	3	preferred	prefer	VERB
cana-3310	23	4	by	by	ADP
cana-3310	23	5	professionals	professional	NOUN
cana-3310	23	6	,	,	PUNCT
cana-3310	23	7	academics	academic	NOUN
cana-3310	23	8	,	,	PUNCT
cana-3310	23	9	and	and	CCONJ
cana-3310	23	10	individuals	individual	NOUN
cana-3310	23	11	who	who	PRON
cana-3310	23	12	need	need	VERB
cana-3310	23	13	a	a	DET
cana-3310	23	14	more	more	ADV
cana-3310	23	15	comprehensive	comprehensive	ADJ
cana-3310	23	16	computational	computational	ADJ
cana-3310	23	17	toolset	toolset	NOUN
cana-3310	23	18	due	due	ADP
cana-3310	23	19	to	to	ADP
cana-3310	23	20	its	its	PRON
cana-3310	23	21	open	open	ADJ
cana-3310	23	22	-	-	PUNCT
cana-3310	23	23	source	source	NOUN
cana-3310	23	24	nature	nature	NOUN
cana-3310	23	25	and	and	CCONJ
cana-3310	23	26	wider	wide	ADJ
cana-3310	23	27	mathematical	mathematical	ADJ
cana-3310	23	28	coverage	coverage	NOUN
cana-3310	23	29	.	.	PUNCT
cana-3310	24	1	in	in	ADP
cana-3310	24	2	conclusion	conclusion	NOUN
cana-3310	24	3	,	,	PUNCT
cana-3310	24	4	geogebra	geogebra	NOUN
cana-3310	24	5	and	and	CCONJ
cana-3310	24	6	sagemath	sagemath	NOUN
cana-3310	24	7	are	be	AUX
cana-3310	24	8	useful	useful	ADJ
cana-3310	24	9	mathematical	mathematical	ADJ
cana-3310	24	10	tools	tool	NOUN
cana-3310	24	11	that	that	PRON
cana-3310	24	12	meet	meet	VERB
cana-3310	24	13	the	the	DET
cana-3310	24	14	demands	demand	NOUN
cana-3310	24	15	and	and	CCONJ
cana-3310	24	16	preferences	preference	NOUN
cana-3310	24	17	of	of	ADP
cana-3310	24	18	many	many	ADJ
cana-3310	24	19	mathematical	mathematical	ADJ
cana-3310	24	20	communities	community	NOUN
cana-3310	24	21	.	.	PUNCT
cana-3310	25	1	while	while	SCONJ
cana-3310	25	2	sagemath	sagemath	NOUN
cana-3310	25	3	provides	provide	VERB
cana-3310	25	4	an	an	DET
cana-3310	25	5	open	open	ADJ
cana-3310	25	6	-	-	PUNCT
cana-3310	25	7	source	source	NOUN
cana-3310	25	8	platform	platform	NOUN
cana-3310	25	9	that	that	PRON
cana-3310	25	10	is	be	AUX
cana-3310	25	11	more	more	ADV
cana-3310	25	12	adaptable	adaptable	ADJ
cana-3310	25	13	and	and	CCONJ
cana-3310	25	14	appropriate	appropriate	ADJ
cana-3310	25	15	for	for	ADP
cana-3310	25	16	sophisticated	sophisticated	ADJ
cana-3310	25	17	mathematical	mathematical	ADJ
cana-3310	25	18	computations	computation	NOUN
cana-3310	25	19	and	and	CCONJ
cana-3310	25	20	research	research	NOUN
cana-3310	25	21	,	,	PUNCT
cana-3310	25	22	geogebra	geogebra	NOUN
cana-3310	25	23	shines	shine	VERB
cana-3310	25	24	in	in	ADP
cana-3310	25	25	interactive	interactive	ADJ
cana-3310	25	26	instruction	instruction	NOUN
cana-3310	25	27	and	and	CCONJ
cana-3310	25	28	visualization	visualization	NOUN
cana-3310	25	29	.	.	PUNCT
cana-3310	26	1	2	2	X
cana-3310	26	2	.	.	X
cana-3310	26	3	objectives	objective	NOUN
cana-3310	26	4	rapid	rapid	ADJ
cana-3310	26	5	visualization	visualization	NOUN
cana-3310	26	6	:	:	PUNCT
cana-3310	26	7	•	•	NOUN
cana-3310	26	8	without	without	ADP
cana-3310	26	9	the	the	DET
cana-3310	26	10	need	need	NOUN
cana-3310	26	11	for	for	ADP
cana-3310	26	12	tedious	tedious	ADJ
cana-3310	26	13	computations	computation	NOUN
cana-3310	26	14	,	,	PUNCT
cana-3310	26	15	students	student	NOUN
cana-3310	26	16	may	may	AUX
cana-3310	26	17	easily	easily	ADV
cana-3310	26	18	visualize	visualize	VERB
cana-3310	26	19	curves	curve	NOUN
cana-3310	26	20	and	and	CCONJ
cana-3310	26	21	geometric	geometric	ADJ
cana-3310	26	22	relationships	relationship	NOUN
cana-3310	26	23	by	by	ADP
cana-3310	26	24	swiftly	swiftly	ADV
cana-3310	26	25	creating	create	VERB
cana-3310	26	26	and	and	CCONJ
cana-3310	26	27	manipulating	manipulate	VERB
cana-3310	26	28	function	function	NOUN
cana-3310	26	29	graphs	graph	NOUN
cana-3310	26	30	with	with	ADP
cana-3310	26	31	geogebra	geogebra	NOUN
cana-3310	26	32	.	.	PUNCT
cana-3310	27	1	•	•	NUM
cana-3310	27	2	compared	compare	VERB
cana-3310	27	3	to	to	ADP
cana-3310	27	4	conventional	conventional	ADJ
cana-3310	27	5	approaches	approach	NOUN
cana-3310	27	6	,	,	PUNCT
cana-3310	27	7	sagemath	sagemath	NOUN
cana-3310	27	8	allows	allow	VERB
cana-3310	27	9	for	for	ADP
cana-3310	27	10	the	the	DET
cana-3310	27	11	quick	quick	ADJ
cana-3310	27	12	execution	execution	NOUN
cana-3310	27	13	of	of	ADP
cana-3310	27	14	complex	complex	ADJ
cana-3310	27	15	computations	computation	NOUN
cana-3310	27	16	,	,	PUNCT
cana-3310	27	17	saving	save	VERB
cana-3310	27	18	time	time	NOUN
cana-3310	27	19	while	while	SCONJ
cana-3310	27	20	solving	solve	VERB
cana-3310	27	21	problems	problem	NOUN
cana-3310	27	22	.	.	PUNCT
cana-3310	28	1	interactive	interactive	ADJ
cana-3310	28	2	learning	learning	NOUN
cana-3310	28	3	:	:	PUNCT
cana-3310	28	4	•	•	ADV
cana-3310	28	5	by	by	ADP
cana-3310	28	6	experimenting	experiment	VERB
cana-3310	28	7	with	with	ADP
cana-3310	28	8	parameters	parameter	NOUN
cana-3310	28	9	and	and	CCONJ
cana-3310	28	10	seeing	see	VERB
cana-3310	28	11	instant	instant	ADJ
cana-3310	28	12	outcomes	outcome	NOUN
cana-3310	28	13	,	,	PUNCT
cana-3310	28	14	students	student	NOUN
cana-3310	28	15	can	can	AUX
cana-3310	28	16	learn	learn	VERB
cana-3310	28	17	via	via	ADP
cana-3310	28	18	exploration	exploration	NOUN
cana-3310	28	19	and	and	CCONJ
cana-3310	28	20	discovery	discovery	NOUN
cana-3310	28	21	more	more	ADV
cana-3310	28	22	quickly	quickly	ADV
cana-3310	28	23	.	.	PUNCT
cana-3310	29	1	automated	automate	VERB
cana-3310	29	2	calculations	calculation	NOUN
cana-3310	29	3	:	:	PUNCT
cana-3310	29	4	•	•	NUM
cana-3310	29	5	sagemath	sagemath	NOUN
cana-3310	29	6	automates	automate	VERB
cana-3310	29	7	repetitive	repetitive	ADJ
cana-3310	29	8	calculations	calculation	NOUN
cana-3310	29	9	,	,	PUNCT
cana-3310	29	10	allowing	allow	VERB
cana-3310	29	11	students	student	NOUN
cana-3310	29	12	to	to	PART
cana-3310	29	13	focus	focus	VERB
cana-3310	29	14	on	on	ADP
cana-3310	29	15	understanding	understand	VERB
cana-3310	29	16	concepts	concept	NOUN
cana-3310	29	17	rather	rather	ADV
cana-3310	29	18	than	than	ADP
cana-3310	29	19	getting	getting	AUX
cana-3310	29	20	bogged	bogge	VERB
cana-3310	29	21	down	down	ADP
cana-3310	29	22	in	in	ADP
cana-3310	29	23	arithmetic	arithmetic	ADJ
cana-3310	29	24	.	.	PUNCT
cana-3310	30	1	streamlined	streamlined	ADJ
cana-3310	30	2	problem	problem	NOUN
cana-3310	30	3	-	-	PUNCT
cana-3310	30	4	solving	solving	NOUN
cana-3310	30	5	:	:	PUNCT
cana-3310	30	6	•	•	NOUN
cana-3310	30	7	both	both	DET
cana-3310	30	8	tools	tool	NOUN
cana-3310	30	9	offer	offer	VERB
cana-3310	30	10	built	build	VERB
cana-3310	30	11	-	-	PUNCT
cana-3310	30	12	in	in	ADP
cana-3310	30	13	functions	function	NOUN
cana-3310	30	14	for	for	ADP
cana-3310	30	15	numerical	numerical	ADJ
cana-3310	30	16	methods	method	NOUN
cana-3310	30	17	,	,	PUNCT
cana-3310	30	18	making	make	VERB
cana-3310	30	19	it	it	PRON
cana-3310	30	20	easier	easy	ADJ
cana-3310	30	21	to	to	PART
cana-3310	30	22	implement	implement	VERB
cana-3310	30	23	algorithms	algorithm	NOUN
cana-3310	30	24	like	like	ADP
cana-3310	30	25	newton	newton	PROPN
cana-3310	30	26	’s	’s	PART
cana-3310	30	27	method	method	NOUN
cana-3310	30	28	,	,	PUNCT
cana-3310	30	29	integration	integration	NOUN
cana-3310	30	30	,	,	PUNCT
cana-3310	30	31	and	and	CCONJ
cana-3310	30	32	differential	differential	ADJ
cana-3310	30	33	equations	equation	NOUN
cana-3310	30	34	.	.	PUNCT
cana-3310	31	1	precision	precision	NOUN
cana-3310	31	2	of	of	ADP
cana-3310	31	3	calculations	calculation	NOUN
cana-3310	31	4	:	:	PUNCT
cana-3310	31	5	•	•	NUM
cana-3310	31	6	sagemath	sagemath	NOUN
cana-3310	31	7	supports	support	VERB
cana-3310	31	8	high	high	ADJ
cana-3310	31	9	-	-	PUNCT
cana-3310	31	10	precision	precision	NOUN
cana-3310	31	11	arithmetic	arithmetic	NOUN
cana-3310	31	12	,	,	PUNCT
cana-3310	31	13	reducing	reduce	VERB
cana-3310	31	14	rounding	round	VERB
cana-3310	31	15	errors	error	NOUN
cana-3310	31	16	common	common	ADJ
cana-3310	31	17	in	in	ADP
cana-3310	31	18	numerical	numerical	ADJ
cana-3310	31	19	methods	method	NOUN
cana-3310	31	20	and	and	CCONJ
cana-3310	31	21	ensuring	ensure	VERB
cana-3310	31	22	that	that	SCONJ
cana-3310	31	23	results	result	NOUN
cana-3310	31	24	are	be	AUX
cana-3310	31	25	reliable	reliable	ADJ
cana-3310	31	26	.	.	PUNCT
cana-3310	32	1	graphical	graphical	ADJ
cana-3310	32	2	accuracy	accuracy	NOUN
cana-3310	32	3	:	:	PUNCT
cana-3310	32	4	•	•	NUM
cana-3310	32	5	geogebra	geogebra	NOUN
cana-3310	32	6	provides	provide	VERB
cana-3310	32	7	high	high	ADJ
cana-3310	32	8	-	-	PUNCT
cana-3310	32	9	quality	quality	NOUN
cana-3310	32	10	graphing	graphing	NOUN
cana-3310	32	11	capabilities	capability	NOUN
cana-3310	32	12	,	,	PUNCT
cana-3310	32	13	enabling	enable	VERB
cana-3310	32	14	precise	precise	ADJ
cana-3310	32	15	tracing	tracing	NOUN
cana-3310	32	16	of	of	ADP
cana-3310	32	17	curves	curve	NOUN
cana-3310	32	18	and	and	CCONJ
cana-3310	32	19	geometric	geometric	ADJ
cana-3310	32	20	figures	figure	NOUN
cana-3310	32	21	,	,	PUNCT
cana-3310	32	22	which	which	PRON
cana-3310	32	23	helps	help	VERB
cana-3310	32	24	in	in	ADP
cana-3310	32	25	visualizing	visualize	VERB
cana-3310	32	26	mathematical	mathematical	ADJ
cana-3310	32	27	concepts	concept	NOUN
cana-3310	32	28	accurately	accurately	ADV
cana-3310	32	29	.	.	PUNCT
cana-3310	33	1	communications	communication	NOUN
cana-3310	33	2	on	on	ADP
cana-3310	33	3	applied	apply	VERB
cana-3310	33	4	nonlinear	nonlinear	ADJ
cana-3310	33	5	analysis	analysis	NOUN
cana-3310	33	6	issn	issn	NOUN
cana-3310	33	7	:	:	PUNCT
cana-3310	33	8	1074	1074	NUM
cana-3310	33	9	-	-	PUNCT
cana-3310	33	10	133x	133x	NUM
cana-3310	33	11	vol	vol	NOUN
cana-3310	33	12	32	32	NUM
cana-3310	33	13	no	no	NOUN
cana-3310	33	14	.	.	PUNCT
cana-3310	34	1	6s	6s	NUM
cana-3310	34	2	(	(	PUNCT
cana-3310	34	3	2025	2025	NUM
cana-3310	34	4	)	)	PUNCT
cana-3310	34	5	461	461	NUM
cana-3310	34	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3310	34	7	3	3	X
cana-3310	34	8	.	.	PUNCT
cana-3310	34	9	methods	method	NOUN
cana-3310	34	10	geogebra	geogebra	NOUN
cana-3310	34	11	:	:	PUNCT
cana-3310	34	12	geogebra	geogebra	NOUN
cana-3310	34	13	is	be	AUX
cana-3310	34	14	a	a	DET
cana-3310	34	15	dynamic	dynamic	ADJ
cana-3310	34	16	mathematics	mathematic	NOUN
cana-3310	34	17	program	program	NOUN
cana-3310	34	18	that	that	PRON
cana-3310	34	19	integrates	integrate	VERB
cana-3310	34	20	algebra	algebra	NOUN
cana-3310	34	21	,	,	PUNCT
cana-3310	34	22	spreadsheets	spreadsheet	NOUN
cana-3310	34	23	,	,	PUNCT
cana-3310	34	24	graphing	graphing	NOUN
cana-3310	34	25	,	,	PUNCT
cana-3310	34	26	calculus	calculus	NOUN
cana-3310	34	27	,	,	PUNCT
cana-3310	34	28	geometry	geometry	NOUN
cana-3310	34	29	,	,	PUNCT
cana-3310	34	30	and	and	CCONJ
cana-3310	34	31	other	other	ADJ
cana-3310	34	32	subjects	subject	NOUN
cana-3310	34	33	in	in	ADP
cana-3310	34	34	an	an	DET
cana-3310	34	35	engaging	engaging	ADJ
cana-3310	34	36	and	and	CCONJ
cana-3310	34	37	integrated	integrated	ADJ
cana-3310	34	38	manner	manner	NOUN
cana-3310	34	39	.	.	PUNCT
cana-3310	35	1	it	it	PRON
cana-3310	35	2	was	be	AUX
cana-3310	35	3	created	create	VERB
cana-3310	35	4	to	to	PART
cana-3310	35	5	offer	offer	VERB
cana-3310	35	6	a	a	DET
cana-3310	35	7	flexible	flexible	ADJ
cana-3310	35	8	platform	platform	NOUN
cana-3310	35	9	that	that	PRON
cana-3310	35	10	would	would	AUX
cana-3310	35	11	enable	enable	VERB
cana-3310	35	12	experts	expert	NOUN
cana-3310	35	13	,	,	PUNCT
cana-3310	35	14	educators	educator	NOUN
cana-3310	35	15	,	,	PUNCT
cana-3310	35	16	and	and	CCONJ
cana-3310	35	17	students	student	NOUN
cana-3310	35	18	to	to	PART
cana-3310	35	19	investigate	investigate	VERB
cana-3310	35	20	and	and	CCONJ
cana-3310	35	21	illustrate	illustrate	VERB
cana-3310	35	22	mathematical	mathematical	ADJ
cana-3310	35	23	ideas	idea	NOUN
cana-3310	35	24	in	in	ADP
cana-3310	35	25	an	an	DET
cana-3310	35	26	interesting	interesting	ADJ
cana-3310	35	27	and	and	CCONJ
cana-3310	35	28	dynamic	dynamic	ADJ
cana-3310	35	29	way	way	NOUN
cana-3310	35	30	.	.	PUNCT
cana-3310	36	1	the	the	DET
cana-3310	36	2	name	name	NOUN
cana-3310	36	3	,	,	PUNCT
cana-3310	36	4	geogebra	geogebra	NOUN
cana-3310	36	5	,	,	PUNCT
cana-3310	36	6	is	be	AUX
cana-3310	36	7	a	a	DET
cana-3310	36	8	play	play	NOUN
cana-3310	36	9	on	on	ADP
cana-3310	36	10	the	the	DET
cana-3310	36	11	words	word	NOUN
cana-3310	36	12	"	"	PUNCT
cana-3310	36	13	geometry	geometry	NOUN
cana-3310	36	14	"	"	PUNCT
cana-3310	36	15	and	and	CCONJ
cana-3310	36	16	"	"	PUNCT
cana-3310	36	17	algebra	algebra	NOUN
cana-3310	36	18	,	,	PUNCT
cana-3310	36	19	"	"	PUNCT
cana-3310	36	20	signifying	signify	VERB
cana-3310	36	21	its	its	PRON
cana-3310	36	22	main	main	ADJ
cana-3310	36	23	goal	goal	NOUN
cana-3310	36	24	of	of	ADP
cana-3310	36	25	uniting	unite	VERB
cana-3310	36	26	these	these	DET
cana-3310	36	27	two	two	NUM
cana-3310	36	28	essential	essential	ADJ
cana-3310	36	29	areas	area	NOUN
cana-3310	36	30	of	of	ADP
cana-3310	36	31	mathematics	mathematic	NOUN
cana-3310	36	32	.	.	PUNCT
cana-3310	37	1	by	by	ADP
cana-3310	37	2	letting	let	VERB
cana-3310	37	3	users	user	NOUN
cana-3310	37	4	make	make	VERB
cana-3310	37	5	things	thing	NOUN
cana-3310	37	6	,	,	PUNCT
cana-3310	37	7	work	work	VERB
cana-3310	37	8	with	with	ADP
cana-3310	37	9	algebraic	algebraic	ADJ
cana-3310	37	10	formulas	formula	NOUN
cana-3310	37	11	,	,	PUNCT
cana-3310	37	12	and	and	CCONJ
cana-3310	37	13	see	see	VERB
cana-3310	37	14	the	the	DET
cana-3310	37	15	dynamic	dynamic	ADJ
cana-3310	37	16	interactions	interaction	NOUN
cana-3310	37	17	between	between	ADP
cana-3310	37	18	various	various	ADJ
cana-3310	37	19	mathematical	mathematical	ADJ
cana-3310	37	20	objects	object	NOUN
cana-3310	37	21	,	,	PUNCT
cana-3310	37	22	the	the	DET
cana-3310	37	23	program	program	NOUN
cana-3310	37	24	is	be	AUX
cana-3310	37	25	intended	intend	VERB
cana-3310	37	26	to	to	PART
cana-3310	37	27	support	support	VERB
cana-3310	37	28	experiential	experiential	ADJ
cana-3310	37	29	learning	learning	NOUN
cana-3310	37	30	and	and	CCONJ
cana-3310	37	31	mathematical	mathematical	ADJ
cana-3310	37	32	inquiry	inquiry	NOUN
cana-3310	37	33	.	.	PUNCT
cana-3310	38	1	due	due	ADP
cana-3310	38	2	to	to	ADP
cana-3310	38	3	its	its	PRON
cana-3310	38	4	ability	ability	NOUN
cana-3310	38	5	to	to	PART
cana-3310	38	6	support	support	VERB
cana-3310	38	7	active	active	ADJ
cana-3310	38	8	learning	learning	NOUN
cana-3310	38	9	,	,	PUNCT
cana-3310	38	10	visualization	visualization	NOUN
cana-3310	38	11	,	,	PUNCT
cana-3310	38	12	and	and	CCONJ
cana-3310	38	13	the	the	DET
cana-3310	38	14	growth	growth	NOUN
cana-3310	38	15	of	of	ADP
cana-3310	38	16	mathematical	mathematical	ADJ
cana-3310	38	17	reasoning	reasoning	NOUN
cana-3310	38	18	,	,	PUNCT
cana-3310	38	19	geogebra	geogebra	NOUN
cana-3310	38	20	is	be	AUX
cana-3310	38	21	extensively	extensively	ADV
cana-3310	38	22	utilized	utilize	VERB
cana-3310	38	23	in	in	ADP
cana-3310	38	24	education	education	NOUN
cana-3310	38	25	at	at	ADP
cana-3310	38	26	all	all	DET
cana-3310	38	27	levels	level	NOUN
cana-3310	38	28	,	,	PUNCT
cana-3310	38	29	from	from	ADP
cana-3310	38	30	elementary	elementary	ADJ
cana-3310	38	31	schools	school	NOUN
cana-3310	38	32	to	to	ADP
cana-3310	38	33	colleges	college	NOUN
cana-3310	38	34	.	.	PUNCT
cana-3310	39	1	while	while	SCONJ
cana-3310	39	2	students	student	NOUN
cana-3310	39	3	can	can	AUX
cana-3310	39	4	individually	individually	ADV
cana-3310	39	5	investigate	investigate	VERB
cana-3310	39	6	mathematical	mathematical	ADJ
cana-3310	39	7	subjects	subject	NOUN
cana-3310	39	8	via	via	ADP
cana-3310	39	9	practical	practical	ADJ
cana-3310	39	10	exercises	exercise	NOUN
cana-3310	39	11	and	and	CCONJ
cana-3310	39	12	experimentation	experimentation	NOUN
cana-3310	39	13	,	,	PUNCT
cana-3310	39	14	teachers	teacher	NOUN
cana-3310	39	15	frequently	frequently	ADV
cana-3310	39	16	utilize	utilize	VERB
cana-3310	39	17	geogebra	geogebra	NOUN
cana-3310	39	18	to	to	PART
cana-3310	39	19	construct	construct	VERB
cana-3310	39	20	interactive	interactive	ADJ
cana-3310	39	21	courses	course	NOUN
cana-3310	39	22	and	and	CCONJ
cana-3310	39	23	presentations	presentation	NOUN
cana-3310	39	24	.	.	PUNCT
cana-3310	40	1	tracing	trace	VERB
cana-3310	40	2	curves	curve	NOUN
cana-3310	40	3	with	with	ADP
cana-3310	40	4	the	the	DET
cana-3310	40	5	traditional	traditional	ADJ
cana-3310	40	6	approach	approach	NOUN
cana-3310	40	7	takes	take	VERB
cana-3310	40	8	a	a	DET
cana-3310	40	9	long	long	ADJ
cana-3310	40	10	time	time	NOUN
cana-3310	40	11	since	since	SCONJ
cana-3310	40	12	there	there	PRON
cana-3310	40	13	are	be	VERB
cana-3310	40	14	many	many	ADJ
cana-3310	40	15	rules	rule	NOUN
cana-3310	40	16	to	to	PART
cana-3310	40	17	follow	follow	VERB
cana-3310	40	18	,	,	PUNCT
cana-3310	40	19	such	such	ADJ
cana-3310	40	20	as	as	ADP
cana-3310	40	21	guaranteeing	guarantee	VERB
cana-3310	40	22	symmetry	symmetry	NOUN
cana-3310	40	23	,	,	PUNCT
cana-3310	40	24	identifying	identify	VERB
cana-3310	40	25	if	if	SCONJ
cana-3310	40	26	the	the	DET
cana-3310	40	27	curve	curve	NOUN
cana-3310	40	28	passes	pass	VERB
cana-3310	40	29	through	through	ADP
cana-3310	40	30	the	the	DET
cana-3310	40	31	origin	origin	NOUN
cana-3310	40	32	,	,	PUNCT
cana-3310	40	33	determining	determine	VERB
cana-3310	40	34	the	the	DET
cana-3310	40	35	region	region	NOUN
cana-3310	40	36	of	of	ADP
cana-3310	40	37	absence	absence	NOUN
cana-3310	40	38	,	,	PUNCT
cana-3310	40	39	asymptotes	asymptote	NOUN
cana-3310	40	40	,	,	PUNCT
cana-3310	40	41	and	and	CCONJ
cana-3310	40	42	points	point	NOUN
cana-3310	40	43	of	of	ADP
cana-3310	40	44	intersection	intersection	NOUN
cana-3310	40	45	,	,	PUNCT
cana-3310	40	46	among	among	ADP
cana-3310	40	47	others	other	NOUN
cana-3310	40	48	.	.	PUNCT
cana-3310	41	1	to	to	PART
cana-3310	41	2	further	far	ADV
cana-3310	41	3	understand	understand	VERB
cana-3310	41	4	,	,	PUNCT
cana-3310	41	5	we	we	PRON
cana-3310	41	6	may	may	AUX
cana-3310	41	7	trace	trace	VERB
cana-3310	41	8	any	any	DET
cana-3310	41	9	curve	curve	NOUN
cana-3310	41	10	using	use	VERB
cana-3310	41	11	the	the	DET
cana-3310	41	12	free	free	ADJ
cana-3310	41	13	and	and	CCONJ
cana-3310	41	14	open	open	ADJ
cana-3310	41	15	-	-	PUNCT
cana-3310	41	16	source	source	NOUN
cana-3310	41	17	geogebra	geogebra	NOUN
cana-3310	41	18	program	program	NOUN
cana-3310	41	19	.	.	PUNCT
cana-3310	42	1	example1	example1	PROPN
cana-3310	42	2	:	:	PUNCT
cana-3310	42	3	trace	trace	VERB
cana-3310	42	4	the	the	DET
cana-3310	42	5	curve	curve	NOUN
cana-3310	42	6	𝐶𝑢𝑟𝑣𝑒	𝐶𝑢𝑟𝑣𝑒	PROPN
cana-3310	42	7	:	:	PUNCT
cana-3310	42	8	𝑦2	𝑦2	PROPN
cana-3310	42	9	(	(	PUNCT
cana-3310	42	10	𝑎2	𝑎2	NOUN
cana-3310	42	11	+	+	NUM
cana-3310	42	12	𝑥2	𝑥2	NOUN
cana-3310	42	13	)	)	PUNCT
cana-3310	42	14	=	=	SYM
cana-3310	42	15	𝑥2	𝑥2	NOUN
cana-3310	42	16	(	(	PUNCT
cana-3310	42	17	𝑎2	𝑎2	NOUN
cana-3310	42	18	−	−	PROPN
cana-3310	42	19	𝑥2	𝑥2	NOUN
cana-3310	42	20	)	)	PUNCT
cana-3310	42	21	•	•	NOUN
cana-3310	42	22	firstly	firstly	ADV
cana-3310	42	23	declare	declare	VERB
cana-3310	42	24	the	the	DET
cana-3310	42	25	variable	variable	NOUN
cana-3310	42	26	‘	'	PUNCT
cana-3310	42	27	a	a	PRON
cana-3310	42	28	’	'	PUNCT
cana-3310	42	29	.	.	PUNCT
cana-3310	43	1	•	•	NOUN
cana-3310	43	2	write	write	VERB
cana-3310	43	3	the	the	DET
cana-3310	43	4	equation	equation	NOUN
cana-3310	43	5	of	of	ADP
cana-3310	43	6	curve	curve	NOUN
cana-3310	43	7	which	which	PRON
cana-3310	43	8	we	we	PRON
cana-3310	43	9	have	have	VERB
cana-3310	43	10	to	to	PART
cana-3310	43	11	trace	trace	VERB
cana-3310	43	12	.	.	PUNCT
cana-3310	44	1	•	•	NUM
cana-3310	44	2	draw	draw	VERB
cana-3310	44	3	a	a	DET
cana-3310	44	4	tangent	tangent	NOUN
cana-3310	44	5	.	.	PUNCT
cana-3310	45	1	geogebra	geogebra	NOUN
cana-3310	45	2	code	code	NOUN
cana-3310	45	3	:	:	PUNCT
cana-3310	45	4	•	•	NOUN
cana-3310	45	5	firstly	firstly	ADV
cana-3310	45	6	declare	declare	VERB
cana-3310	45	7	the	the	DET
cana-3310	45	8	variable	variable	NOUN
cana-3310	45	9	‘	'	PUNCT
cana-3310	45	10	a	a	PRON
cana-3310	45	11	’	'	PUNCT
cana-3310	45	12	.	.	PUNCT
cana-3310	46	1	•	•	NOUN
cana-3310	46	2	write	write	VERB
cana-3310	46	3	the	the	DET
cana-3310	46	4	equation	equation	NOUN
cana-3310	46	5	of	of	ADP
cana-3310	46	6	curve	curve	NOUN
cana-3310	46	7	which	which	PRON
cana-3310	46	8	we	we	PRON
cana-3310	46	9	have	have	VERB
cana-3310	46	10	to	to	PART
cana-3310	46	11	trace	trace	VERB
cana-3310	46	12	.	.	PUNCT
cana-3310	47	1	communications	communication	NOUN
cana-3310	47	2	on	on	ADP
cana-3310	47	3	applied	apply	VERB
cana-3310	47	4	nonlinear	nonlinear	ADJ
cana-3310	47	5	analysis	analysis	NOUN
cana-3310	47	6	issn	issn	NOUN
cana-3310	47	7	:	:	PUNCT
cana-3310	47	8	1074	1074	NUM
cana-3310	47	9	-	-	PUNCT
cana-3310	47	10	133x	133x	NUM
cana-3310	47	11	vol	vol	NOUN
cana-3310	47	12	32	32	NUM
cana-3310	47	13	no	no	NOUN
cana-3310	47	14	.	.	PUNCT
cana-3310	48	1	6s	6s	NUM
cana-3310	48	2	(	(	PUNCT
cana-3310	48	3	2025	2025	NUM
cana-3310	48	4	)	)	PUNCT
cana-3310	48	5	462	462	NUM
cana-3310	49	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3310	49	2	figure	figure	NOUN
cana-3310	49	3	1	1	NUM
cana-3310	49	4	:	:	PUNCT
cana-3310	49	5	plotting	plot	VERB
cana-3310	49	6	the	the	DET
cana-3310	49	7	curve	curve	NOUN
cana-3310	49	8	𝑦2	𝑦2	NOUN
cana-3310	49	9	(	(	PUNCT
cana-3310	49	10	𝑎2	𝑎2	NOUN
cana-3310	49	11	+	+	NUM
cana-3310	49	12	𝑥2	𝑥2	NOUN
cana-3310	49	13	)	)	PUNCT
cana-3310	49	14	=	=	SYM
cana-3310	49	15	𝑥2	𝑥2	NOUN
cana-3310	49	16	(	(	PUNCT
cana-3310	49	17	𝑎2	𝑎2	NOUN
cana-3310	49	18	−	−	PROPN
cana-3310	49	19	𝑥2	𝑥2	NOUN
cana-3310	49	20	)	)	PUNCT
cana-3310	49	21	•	•	PUNCT
cana-3310	49	22	draw	draw	VERB
cana-3310	49	23	a	a	DET
cana-3310	49	24	tangents	tangent	NOUN
cana-3310	49	25	tracing	trace	VERB
cana-3310	49	26	the	the	DET
cana-3310	49	27	curve	curve	NOUN
cana-3310	49	28	using	use	VERB
cana-3310	49	29	sagemath	sagemath	NOUN
cana-3310	49	30	communications	communication	NOUN
cana-3310	49	31	on	on	ADP
cana-3310	49	32	applied	apply	VERB
cana-3310	49	33	nonlinear	nonlinear	ADJ
cana-3310	49	34	analysis	analysis	NOUN
cana-3310	49	35	issn	issn	NOUN
cana-3310	49	36	:	:	PUNCT
cana-3310	49	37	1074	1074	NUM
cana-3310	49	38	-	-	PUNCT
cana-3310	49	39	133x	133x	NUM
cana-3310	49	40	vol	vol	NOUN
cana-3310	49	41	32	32	NUM
cana-3310	49	42	no	no	NOUN
cana-3310	49	43	.	.	PUNCT
cana-3310	50	1	6s	6s	NUM
cana-3310	50	2	(	(	PUNCT
cana-3310	50	3	2025	2025	NUM
cana-3310	50	4	)	)	PUNCT
cana-3310	50	5	463	463	NUM
cana-3310	50	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-3310	50	7	figure	figure	NOUN
cana-3310	50	8	2	2	NUM
cana-3310	50	9	:	:	PUNCT
cana-3310	50	10	plotting	plot	VERB
cana-3310	50	11	the	the	DET
cana-3310	50	12	curve	curve	NOUN
cana-3310	50	13	𝑦2	𝑦2	NOUN
cana-3310	50	14	(	(	PUNCT
cana-3310	50	15	𝑎2	𝑎2	NOUN
cana-3310	50	16	+	+	NUM
cana-3310	50	17	𝑥2	𝑥2	NOUN
cana-3310	50	18	)	)	PUNCT
cana-3310	50	19	=	=	SYM
cana-3310	50	20	𝑥2	𝑥2	NOUN
cana-3310	50	21	(	(	PUNCT
cana-3310	50	22	𝑎2	𝑎2	NOUN
cana-3310	50	23	−	−	PROPN
cana-3310	50	24	𝑥2	𝑥2	NOUN
cana-3310	50	25	)	)	PUNCT
cana-3310	50	26	using	use	VERB
cana-3310	50	27	sagemath	sagemath	NOUN
cana-3310	50	28	example	example	NOUN
cana-3310	50	29	2	2	NUM
cana-3310	50	30	:	:	PUNCT
cana-3310	50	31	trace	trace	VERB
cana-3310	50	32	the	the	DET
cana-3310	50	33	curve2	curve2	NOUN
cana-3310	50	34	𝑦2	𝑦2	NOUN
cana-3310	50	35	(	(	PUNCT
cana-3310	50	36	𝑎	𝑎	PROPN
cana-3310	50	37	+	+	NUM
cana-3310	50	38	𝑥	𝑥	NOUN
cana-3310	50	39	)	)	PUNCT
cana-3310	50	40	=	=	NOUN
cana-3310	50	41	𝑥2	𝑥2	NOUN
cana-3310	50	42	(	(	PUNCT
cana-3310	50	43	𝑎	𝑎	PROPN
cana-3310	50	44	−	−	PROPN
cana-3310	50	45	𝑥	𝑥	NOUN
cana-3310	50	46	)	)	PUNCT
cana-3310	50	47	figure	figure	NOUN
cana-3310	50	48	2	2	NUM
cana-3310	50	49	:	:	PUNCT
cana-3310	50	50	plotting	plot	VERB
cana-3310	50	51	the	the	DET
cana-3310	50	52	𝐶𝑢𝑟𝑣𝑒2	𝐶𝑢𝑟𝑣𝑒2	NOUN
cana-3310	50	53	:	:	PUNCT
cana-3310	50	54	𝑦2	𝑦2	NOUN
cana-3310	50	55	(	(	PUNCT
cana-3310	50	56	𝑎	𝑎	PROPN
cana-3310	50	57	+	+	NUM
cana-3310	50	58	𝑥	𝑥	NOUN
cana-3310	50	59	)	)	PUNCT
cana-3310	50	60	=	=	NOUN
cana-3310	50	61	𝑥2	𝑥2	NOUN
cana-3310	50	62	(	(	PUNCT
cana-3310	50	63	𝑎	𝑎	PROPN
cana-3310	50	64	−	−	PROPN
cana-3310	50	65	𝑥	𝑥	NOUN
cana-3310	50	66	)	)	PUNCT
cana-3310	50	67	sagemath	sagemath	NOUN
cana-3310	50	68	:	:	PUNCT
cana-3310	50	69	a	a	DET
cana-3310	50	70	powerful	powerful	ADJ
cana-3310	50	71	and	and	CCONJ
cana-3310	50	72	free	free	ADJ
cana-3310	50	73	mathematics	mathematics	PROPN
cana-3310	50	74	software	software	NOUN
cana-3310	50	75	program	program	NOUN
cana-3310	50	76	,	,	PUNCT
cana-3310	50	77	sagemath	sagemath	NOUN
cana-3310	50	78	—	—	PUNCT
cana-3310	50	79	often	often	ADV
cana-3310	50	80	just	just	ADV
cana-3310	50	81	called	call	VERB
cana-3310	50	82	sage	sage	NOUN
cana-3310	50	83	—	—	PUNCT
cana-3310	50	84	offers	offer	VERB
cana-3310	50	85	a	a	DET
cana-3310	50	86	thorough	thorough	ADJ
cana-3310	50	87	environment	environment	NOUN
cana-3310	50	88	for	for	ADP
cana-3310	50	89	mathematical	mathematical	ADJ
cana-3310	50	90	investigation	investigation	NOUN
cana-3310	50	91	,	,	PUNCT
cana-3310	50	92	study	study	NOUN
cana-3310	50	93	,	,	PUNCT
cana-3310	50	94	and	and	CCONJ
cana-3310	50	95	calculation	calculation	NOUN
cana-3310	50	96	.	.	PUNCT
cana-3310	51	1	mathematical	mathematical	ADJ
cana-3310	51	2	topics	topic	NOUN
cana-3310	51	3	communications	communication	NOUN
cana-3310	51	4	on	on	ADP
cana-3310	51	5	applied	apply	VERB
cana-3310	51	6	nonlinear	nonlinear	ADJ
cana-3310	51	7	analysis	analysis	NOUN
cana-3310	51	8	issn	issn	NOUN
cana-3310	51	9	:	:	PUNCT
cana-3310	51	10	1074	1074	NUM
cana-3310	51	11	-	-	PUNCT
cana-3310	51	12	133x	133x	NUM
cana-3310	51	13	vol	vol	NOUN
cana-3310	51	14	32	32	NUM
cana-3310	51	15	no	no	NOUN
cana-3310	51	16	.	.	PUNCT
cana-3310	52	1	6s	6s	NUM
cana-3310	52	2	(	(	PUNCT
cana-3310	52	3	2025	2025	NUM
cana-3310	52	4	)	)	PUNCT
cana-3310	52	5	464	464	NUM
cana-3310	52	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-3310	52	7	covered	cover	VERB
cana-3310	52	8	by	by	ADP
cana-3310	52	9	sagemath	sagemath	NOUN
cana-3310	52	10	include	include	VERB
cana-3310	52	11	algebra	algebra	NOUN
cana-3310	52	12	,	,	PUNCT
cana-3310	52	13	calculus	calculus	NOUN
cana-3310	52	14	,	,	PUNCT
cana-3310	52	15	number	number	NOUN
cana-3310	52	16	theory	theory	NOUN
cana-3310	52	17	,	,	PUNCT
cana-3310	52	18	combinatorics	combinatoric	NOUN
cana-3310	52	19	,	,	PUNCT
cana-3310	52	20	and	and	CCONJ
cana-3310	52	21	more	more	ADJ
cana-3310	52	22	.	.	PUNCT
cana-3310	53	1	it	it	PRON
cana-3310	53	2	was	be	AUX
cana-3310	53	3	created	create	VERB
cana-3310	53	4	as	as	ADP
cana-3310	53	5	an	an	DET
cana-3310	53	6	open	open	ADJ
cana-3310	53	7	-	-	PUNCT
cana-3310	53	8	source	source	NOUN
cana-3310	53	9	,	,	PUNCT
cana-3310	53	10	cost	cost	NOUN
cana-3310	53	11	-	-	PUNCT
cana-3310	53	12	effective	effective	ADJ
cana-3310	53	13	substitute	substitute	NOUN
cana-3310	53	14	for	for	ADP
cana-3310	53	15	proprietary	proprietary	ADJ
cana-3310	53	16	mathematical	mathematical	ADJ
cana-3310	53	17	software	software	NOUN
cana-3310	53	18	.	.	PUNCT
cana-3310	54	1	researchers	researcher	NOUN
cana-3310	54	2	and	and	CCONJ
cana-3310	54	3	professionals	professional	NOUN
cana-3310	54	4	working	work	VERB
cana-3310	54	5	in	in	ADP
cana-3310	54	6	mathematics	mathematic	NOUN
cana-3310	54	7	,	,	PUNCT
cana-3310	54	8	physics	physics	NOUN
cana-3310	54	9	,	,	PUNCT
cana-3310	54	10	and	and	CCONJ
cana-3310	54	11	related	related	ADJ
cana-3310	54	12	subjects	subject	NOUN
cana-3310	54	13	are	be	AUX
cana-3310	54	14	especially	especially	ADV
cana-3310	54	15	fond	fond	ADJ
cana-3310	54	16	of	of	ADP
cana-3310	54	17	sagemath	sagemath	NOUN
cana-3310	54	18	.	.	PUNCT
cana-3310	55	1	it	it	PRON
cana-3310	55	2	offers	offer	VERB
cana-3310	55	3	a	a	DET
cana-3310	55	4	strong	strong	ADJ
cana-3310	55	5	,	,	PUNCT
cana-3310	55	6	flexible	flexible	ADJ
cana-3310	55	7	,	,	PUNCT
cana-3310	55	8	and	and	CCONJ
cana-3310	55	9	cost	cost	NOUN
cana-3310	55	10	-	-	PUNCT
cana-3310	55	11	free	free	ADJ
cana-3310	55	12	substitute	substitute	NOUN
cana-3310	55	13	for	for	ADP
cana-3310	55	14	those	those	PRON
cana-3310	55	15	who	who	PRON
cana-3310	55	16	need	need	VERB
cana-3310	55	17	sophisticated	sophisticated	ADJ
cana-3310	55	18	computational	computational	ADJ
cana-3310	55	19	instruments	instrument	NOUN
cana-3310	55	20	for	for	ADP
cana-3310	55	21	their	their	PRON
cana-3310	55	22	tasks	task	NOUN
cana-3310	55	23	.	.	PUNCT
cana-3310	56	1	the	the	DET
cana-3310	56	2	software	software	NOUN
cana-3310	56	3	is	be	AUX
cana-3310	56	4	a	a	DET
cana-3310	56	5	great	great	ADJ
cana-3310	56	6	resource	resource	NOUN
cana-3310	56	7	for	for	ADP
cana-3310	56	8	mathematical	mathematical	ADJ
cana-3310	56	9	study	study	NOUN
cana-3310	56	10	and	and	CCONJ
cana-3310	56	11	teaching	teaching	NOUN
cana-3310	56	12	because	because	SCONJ
cana-3310	56	13	of	of	ADP
cana-3310	56	14	its	its	PRON
cana-3310	56	15	dedication	dedication	NOUN
cana-3310	56	16	to	to	ADP
cana-3310	56	17	transparency	transparency	NOUN
cana-3310	56	18	,	,	PUNCT
cana-3310	56	19	cooperation	cooperation	NOUN
cana-3310	56	20	,	,	PUNCT
cana-3310	56	21	and	and	CCONJ
cana-3310	56	22	adaptability	adaptability	NOUN
cana-3310	56	23	.	.	PUNCT
cana-3310	57	1	gauss	gauss	ADJ
cana-3310	57	2	jacobi	jacobi	PROPN
cana-3310	57	3	and	and	CCONJ
cana-3310	57	4	gauss	gauss	PROPN
cana-3310	57	5	siedel	siedel	PROPN
cana-3310	57	6	methods	method	NOUN
cana-3310	57	7	are	be	AUX
cana-3310	57	8	numerical	numerical	ADJ
cana-3310	57	9	mehods	mehod	NOUN
cana-3310	57	10	for	for	ADP
cana-3310	57	11	solving	solve	VERB
cana-3310	57	12	system	system	NOUN
cana-3310	57	13	of	of	ADP
cana-3310	57	14	linear	linear	PROPN
cana-3310	57	15	equations	equation	NOUN
cana-3310	57	16	.	.	PUNCT
cana-3310	58	1	this	this	DET
cana-3310	58	2	methods	method	NOUN
cana-3310	58	3	are	be	AUX
cana-3310	58	4	particularly	particularly	ADV
cana-3310	58	5	useful	useful	ADJ
cana-3310	58	6	for	for	ADP
cana-3310	58	7	solving	solve	VERB
cana-3310	58	8	systems	system	NOUN
cana-3310	58	9	of	of	ADP
cana-3310	58	10	linear	linear	PROPN
cana-3310	58	11	equations	equation	NOUN
cana-3310	58	12	when	when	SCONJ
cana-3310	58	13	the	the	DET
cana-3310	58	14	matrix	matrix	NOUN
cana-3310	58	15	is	be	AUX
cana-3310	58	16	sparse	sparse	ADJ
cana-3310	58	17	or	or	CCONJ
cana-3310	58	18	when	when	SCONJ
cana-3310	58	19	direct	direct	ADJ
cana-3310	58	20	methods	method	NOUN
cana-3310	58	21	like	like	ADP
cana-3310	58	22	gaussian	gaussian	ADJ
cana-3310	58	23	elimination	elimination	NOUN
cana-3310	58	24	are	be	AUX
cana-3310	58	25	computationally	computationally	ADV
cana-3310	58	26	expensive	expensive	ADJ
cana-3310	58	27	.	.	PUNCT
cana-3310	59	1	still	still	ADV
cana-3310	59	2	for	for	ADP
cana-3310	59	3	larger	large	ADJ
cana-3310	59	4	systems	system	NOUN
cana-3310	59	5	manual	manual	ADJ
cana-3310	59	6	computation	computation	NOUN
cana-3310	59	7	of	of	ADP
cana-3310	59	8	iterations	iteration	NOUN
cana-3310	59	9	can	can	AUX
cana-3310	59	10	be	be	AUX
cana-3310	59	11	very	very	ADV
cana-3310	59	12	difficult	difficult	ADJ
cana-3310	59	13	task.for	task.for	X
cana-3310	59	14	ease	ease	NOUN
cana-3310	59	15	of	of	ADP
cana-3310	59	16	computations	computation	NOUN
cana-3310	59	17	in	in	ADP
cana-3310	59	18	this	this	DET
cana-3310	59	19	article	article	NOUN
cana-3310	59	20	we	we	PRON
cana-3310	59	21	are	be	AUX
cana-3310	59	22	using	use	VERB
cana-3310	59	23	sagemath	sagemath	NOUN
cana-3310	59	24	which	which	PRON
cana-3310	59	25	is	be	AUX
cana-3310	59	26	open	open	ADJ
cana-3310	59	27	source	source	NOUN
cana-3310	59	28	mathematical	mathematical	ADJ
cana-3310	59	29	software	software	NOUN
cana-3310	59	30	and	and	CCONJ
cana-3310	59	31	easily	easily	ADV
cana-3310	59	32	available	available	ADJ
cana-3310	59	33	on	on	ADP
cana-3310	59	34	any	any	DET
cana-3310	59	35	device	device	NOUN
cana-3310	59	36	.	.	PUNCT
cana-3310	60	1	guass	guass	PROPN
cana-3310	60	2	jacobi	jacobi	PROPN
cana-3310	60	3	method	method	NOUN
cana-3310	60	4	we	we	PRON
cana-3310	60	5	write	write	VERB
cana-3310	60	6	the	the	DET
cana-3310	60	7	system	system	NOUN
cana-3310	60	8	as	as	ADP
cana-3310	60	9	𝐴𝑋	𝐴𝑋	NOUN
cana-3310	60	10	=	=	SYM
cana-3310	60	11	𝐵	𝐵	NOUN
cana-3310	60	12	𝐿𝑒𝑡	𝐿𝑒𝑡	PROPN
cana-3310	60	13	𝐴	𝐴	PROPN
cana-3310	60	14	=	=	SYM
cana-3310	60	15	𝐿	𝐿	PROPN
cana-3310	60	16	+	+	PROPN
cana-3310	60	17	𝐷	𝐷	PROPN
cana-3310	60	18	+	+	CCONJ
cana-3310	60	19	𝑈	𝑈	PROPN
cana-3310	60	20	…	…	PUNCT
cana-3310	60	21	(	(	PUNCT
cana-3310	60	22	1	1	X
cana-3310	60	23	)	)	PUNCT
cana-3310	60	24	where	where	SCONJ
cana-3310	60	25	l	l	NOUN
cana-3310	60	26	is	be	AUX
cana-3310	60	27	strictly	strictly	ADV
cana-3310	60	28	lower	low	ADJ
cana-3310	60	29	triangular	triangular	NOUN
cana-3310	60	30	matrix	matrix	NOUN
cana-3310	60	31	,	,	PUNCT
cana-3310	60	32	d	d	NOUN
cana-3310	60	33	is	be	AUX
cana-3310	60	34	diagonal	diagonal	ADJ
cana-3310	60	35	matrix	matrix	NOUN
cana-3310	60	36	and	and	CCONJ
cana-3310	60	37	u	u	NOUN
cana-3310	60	38	is	be	AUX
cana-3310	60	39	strictly	strictly	ADV
cana-3310	60	40	upper	upper	ADJ
cana-3310	60	41	triangular	triangular	NOUN
cana-3310	60	42	matrix	matrix	NOUN
cana-3310	60	43	then	then	ADV
cana-3310	60	44	𝐴𝑋	𝐴𝑋	PROPN
cana-3310	60	45	=	=	SYM
cana-3310	60	46	𝐵	𝐵	PROPN
cana-3310	60	47	𝑟𝑒𝑑𝑢𝑐𝑒𝑠	𝑟𝑒𝑑𝑢𝑐𝑒𝑠	ADV
cana-3310	60	48	𝑡𝑜	𝑡𝑜	PROPN
cana-3310	60	49	(	(	PUNCT
cana-3310	60	50	𝐿	𝐿	PROPN
cana-3310	60	51	+	+	PROPN
cana-3310	60	52	𝐷	𝐷	PROPN
cana-3310	60	53	+	+	CCONJ
cana-3310	60	54	𝑈)𝑋	𝑈)𝑋	NOUN
cana-3310	60	55	=	=	SYM
cana-3310	60	56	𝐵	𝐵	NOUN
cana-3310	60	57	𝑤ℎ𝑖𝑐ℎ	𝑤ℎ𝑖𝑐ℎ	NOUN
cana-3310	60	58	𝑐𝑎𝑛	𝑐𝑎𝑛	VERB
cana-3310	60	59	𝑏𝑒	𝑏𝑒	ADP
cana-3310	60	60	𝑤𝑟𝑖𝑡𝑡𝑒𝑛	𝑤𝑟𝑖𝑡𝑡𝑒𝑛	NOUN
cana-3310	60	61	𝑎𝑠	𝑎𝑠	PROPN
cana-3310	60	62	𝐷𝑋	𝐷𝑋	PROPN
cana-3310	60	63	=	=	SYM
cana-3310	60	64	−(𝐿	−(𝐿	PROPN
cana-3310	61	1	+	+	NUM
cana-3310	61	2	𝑈)𝑋	𝑈)𝑋	NOUN
cana-3310	61	3	+	+	CCONJ
cana-3310	61	4	𝐵.	𝐵.	PROPN
cana-3310	61	5	𝐼𝑓	𝐼𝑓	PROPN
cana-3310	61	6	𝐷	𝐷	PROPN
cana-3310	61	7	𝑖𝑠	𝑖𝑠	NOUN
cana-3310	61	8	𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑎𝑏𝑙𝑒	𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑎𝑏𝑙𝑒	VERB
cana-3310	61	9	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
cana-3310	61	10	𝑋	𝑋	PROPN
cana-3310	61	11	=	=	PUNCT
cana-3310	61	12	−𝐷−1(𝐿	−𝐷−1(𝐿	NOUN
cana-3310	61	13	+	+	NUM
cana-3310	61	14	𝑈)𝑋	𝑈)𝑋	NOUN
cana-3310	61	15	+	+	CCONJ
cana-3310	61	16	𝐷−1𝐵	𝐷−1𝐵	NOUN
cana-3310	61	17	𝐻𝑒𝑛𝑐𝑒	𝐻𝑒𝑛𝑐𝑒	PROPN
cana-3310	61	18	𝑡ℎ𝑒	𝑡ℎ𝑒	ADP
cana-3310	61	19	𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑣𝑒	𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑣𝑒	ADJ
cana-3310	61	20	𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒	𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒	PROPN
cana-3310	61	21	𝑖𝑠	𝑖𝑠	PROPN
cana-3310	61	22	𝑔𝑖𝑣𝑒𝑛	𝑔𝑖𝑣𝑒𝑛	ADJ
cana-3310	61	23	𝑏𝑦	𝑏𝑦	NOUN
cana-3310	61	24	𝑋(𝑘+1	𝑋(𝑘+1	NOUN
cana-3310	61	25	)	)	PUNCT
cana-3310	61	26	=	=	SYM
cana-3310	61	27	−𝐷−1(𝐿	−𝐷−1(𝐿	PRON
cana-3310	61	28	+	+	SYM
cana-3310	61	29	𝑈)𝑋(𝑘	𝑈)𝑋(𝑘	NUM
cana-3310	61	30	)	)	PUNCT
cana-3310	62	1	+	+	CCONJ
cana-3310	62	2	𝐷−1𝐵	𝐷−1𝐵	NOUN
cana-3310	62	3	…	…	PUNCT
cana-3310	62	4	…	…	PUNCT
cana-3310	62	5	(	(	PUNCT
cana-3310	62	6	2	2	X
cana-3310	62	7	)	)	PUNCT
cana-3310	62	8	𝑇ℎ𝑢𝑠	𝑇ℎ𝑢𝑠	PROPN
cana-3310	62	9	𝑡ℎ𝑒	𝑡ℎ𝑒	VERB
cana-3310	62	10	𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛	𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛	ADJ
cana-3310	62	11	𝑚𝑎𝑡𝑟𝑖𝑥	𝑚𝑎𝑡𝑟𝑖𝑥	ADJ
cana-3310	62	12	𝐻	𝐻	PROPN
cana-3310	62	13	=	=	PUNCT
cana-3310	62	14	−𝐷−1(𝐿	−𝐷−1(𝐿	PROPN
cana-3310	62	15	+	+	CCONJ
cana-3310	62	16	𝑈	𝑈	NOUN
cana-3310	62	17	)	)	PUNCT
cana-3310	62	18	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3310	62	19	𝐶	𝐶	PROPN
cana-3310	62	20	=	=	PUNCT
cana-3310	62	21	𝐷−1𝐵	𝐷−1𝐵	NOUN
cana-3310	62	22	…	…	PUNCT
cana-3310	62	23	..	..	PUNCT
cana-3310	62	24	(	(	PUNCT
cana-3310	62	25	3	3	X
cana-3310	62	26	)	)	PUNCT
cana-3310	62	27	guass	guass	PROPN
cana-3310	62	28	seidal	seidal	NOUN
cana-3310	62	29	method	method	VERB
cana-3310	62	30	𝐿𝑒𝑡	𝐿𝑒𝑡	PROPN
cana-3310	62	31	𝐴	𝐴	PROPN
cana-3310	62	32	=	=	SYM
cana-3310	62	33	𝐿	𝐿	PROPN
cana-3310	62	34	+	+	PROPN
cana-3310	62	35	𝐷	𝐷	PROPN
cana-3310	62	36	+	+	CCONJ
cana-3310	62	37	𝑈	𝑈	PROPN
cana-3310	62	38	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	ADJ
cana-3310	62	39	𝐷	𝐷	PROPN
cana-3310	62	40	𝑖𝑠	𝑖𝑠	ADP
cana-3310	62	41	𝑡ℎ𝑒	𝑡ℎ𝑒	INTJ
cana-3310	62	42	𝐷𝑖𝑔𝑜𝑛𝑎𝑙	𝐷𝑖𝑔𝑜𝑛𝑎𝑙	PROPN
cana-3310	62	43	𝑝𝑎𝑟𝑡	𝑝𝑎𝑟𝑡	PROPN
cana-3310	62	44	,	,	PUNCT
cana-3310	62	45	𝐿	𝐿	PROPN
cana-3310	62	46	is	be	AUX
cana-3310	62	47	strictly	strictly	ADV
cana-3310	62	48	lower	low	ADJ
cana-3310	62	49	triangular	triangular	NOUN
cana-3310	62	50	matrix	matrix	NOUN
cana-3310	62	51	and	and	CCONJ
cana-3310	62	52	u	u	NOUN
cana-3310	62	53	is	be	AUX
cana-3310	62	54	strictly	strictly	ADV
cana-3310	62	55	upper	upper	ADJ
cana-3310	62	56	triangular	triangular	NOUN
cana-3310	62	57	matrix	matrix	NOUN
cana-3310	62	58	.	.	PUNCT
cana-3310	63	1	then	then	ADV
cana-3310	63	2	ax	ax	NOUN
cana-3310	63	3	=	=	SYM
cana-3310	63	4	b	b	NOUN
cana-3310	63	5	reduces	reduce	VERB
cana-3310	63	6	to	to	ADP
cana-3310	63	7	(	(	PUNCT
cana-3310	63	8	l	l	NOUN
cana-3310	64	1	+	+	X
cana-3310	64	2	d	d	X
cana-3310	64	3	+	+	X
cana-3310	64	4	u)x	u)x	X
cana-3310	64	5	=	=	SYM
cana-3310	64	6	b	b	NOUN
cana-3310	64	7	which	which	PRON
cana-3310	64	8	can	can	AUX
cana-3310	64	9	be	be	AUX
cana-3310	64	10	written	write	VERB
cana-3310	64	11	as	as	ADP
cana-3310	64	12	(	(	PUNCT
cana-3310	64	13	l	l	NOUN
cana-3310	64	14	+	+	NOUN
cana-3310	64	15	d)x	d)x	NOUN
cana-3310	64	16	=	=	SYM
cana-3310	64	17	−ux	−ux	PROPN
cana-3310	64	18	+	+	CCONJ
cana-3310	64	19	b.	b.	NOUN
cana-3310	65	1	if	if	SCONJ
cana-3310	65	2	l	l	PROPN
cana-3310	65	3	+	+	CCONJ
cana-3310	65	4	d	d	NOUN
cana-3310	65	5	is	be	AUX
cana-3310	65	6	invertiabe	invertiabe	ADJ
cana-3310	65	7	.	.	PUNCT
cana-3310	66	1	then	then	ADV
cana-3310	66	2	we	we	PRON
cana-3310	66	3	have	have	VERB
cana-3310	66	4	x	x	X
cana-3310	66	5	=	=	PUNCT
cana-3310	66	6	−(l	−(l	NOUN
cana-3310	66	7	+	+	CCONJ
cana-3310	66	8	d)−1𝑈𝑋	d)−1𝑈𝑋	NOUN
cana-3310	66	9	+	+	CCONJ
cana-3310	66	10	(	(	PUNCT
cana-3310	66	11	l	l	NOUN
cana-3310	66	12	+	+	CCONJ
cana-3310	66	13	d)−1b	d)−1b	X
cana-3310	66	14	𝐻𝑒𝑛𝑐𝑒	𝐻𝑒𝑛𝑐𝑒	PROPN
cana-3310	66	15	𝑡ℎ𝑒	𝑡ℎ𝑒	VERB
cana-3310	66	16	𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑣𝑒	𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑣𝑒	ADJ
cana-3310	66	17	𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒	𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒	PROPN
cana-3310	66	18	𝑖𝑠	𝑖𝑠	PROPN
cana-3310	66	19	𝑔𝑖𝑣𝑒𝑛	𝑔𝑖𝑣𝑒𝑛	ADJ
cana-3310	66	20	𝑏𝑦	𝑏𝑦	NOUN
cana-3310	66	21	x	x	X
cana-3310	66	22	(	(	PUNCT
cana-3310	66	23	k+1	k+1	NOUN
cana-3310	66	24	)	)	PUNCT
cana-3310	66	25	=	=	SYM
cana-3310	66	26	−(l	−(l	NOUN
cana-3310	66	27	+	+	CCONJ
cana-3310	66	28	d)−1𝑈𝑋(𝑘	d)−1𝑈𝑋(𝑘	X
cana-3310	66	29	)	)	PUNCT
cana-3310	67	1	+	+	CCONJ
cana-3310	67	2	(	(	PUNCT
cana-3310	67	3	l	l	NOUN
cana-3310	67	4	+	+	X
cana-3310	67	5	d)−1b	d)−1b	X
cana-3310	67	6	…	…	PUNCT
cana-3310	67	7	.	.	PUNCT
cana-3310	68	1	(	(	PUNCT
cana-3310	68	2	4	4	X
cana-3310	68	3	)	)	PUNCT
cana-3310	68	4	communications	communication	NOUN
cana-3310	68	5	on	on	ADP
cana-3310	68	6	applied	apply	VERB
cana-3310	68	7	nonlinear	nonlinear	ADJ
cana-3310	68	8	analysis	analysis	NOUN
cana-3310	68	9	issn	issn	NOUN
cana-3310	68	10	:	:	PUNCT
cana-3310	68	11	1074	1074	NUM
cana-3310	68	12	-	-	PUNCT
cana-3310	68	13	133x	133x	NUM
cana-3310	68	14	vol	vol	NOUN
cana-3310	68	15	32	32	NUM
cana-3310	68	16	no	no	NOUN
cana-3310	68	17	.	.	PUNCT
cana-3310	69	1	6s	6s	NUM
cana-3310	69	2	(	(	PUNCT
cana-3310	69	3	2025	2025	NUM
cana-3310	69	4	)	)	PUNCT
cana-3310	69	5	465	465	NUM
cana-3310	70	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3310	70	2	𝑇ℎ𝑢𝑠	𝑇ℎ𝑢𝑠	PROPN
cana-3310	70	3	𝑡ℎ𝑒	𝑡ℎ𝑒	VERB
cana-3310	70	4	𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛	𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛	ADJ
cana-3310	70	5	𝑚𝑎𝑡𝑟𝑖𝑥	𝑚𝑎𝑡𝑟𝑖𝑥	ADJ
cana-3310	70	6	𝐻	𝐻	PROPN
cana-3310	70	7	=	=	NOUN
cana-3310	70	8	=	=	SYM
cana-3310	70	9	−(l	−(l	NOUN
cana-3310	70	10	+	+	X
cana-3310	70	11	d)−1𝑈	d)−1𝑈	ADP
cana-3310	70	12	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3310	70	13	𝐶	𝐶	PROPN
cana-3310	70	14	=	=	PUNCT
cana-3310	70	15	(	(	PUNCT
cana-3310	70	16	l	l	NOUN
cana-3310	70	17	+	+	X
cana-3310	70	18	d)−1b	d)−1b	X
cana-3310	70	19	…	…	PUNCT
cana-3310	70	20	.	.	PUNCT
cana-3310	71	1	(	(	PUNCT
cana-3310	71	2	5	5	NUM
cana-3310	71	3	)	)	PUNCT
cana-3310	71	4	example	example	NOUN
cana-3310	71	5	1	1	X
cana-3310	71	6	)	)	PUNCT
cana-3310	71	7	solve	solve	VERB
cana-3310	71	8	the	the	DET
cana-3310	71	9	system	system	NOUN
cana-3310	71	10	of	of	ADP
cana-3310	71	11	linear	linear	PROPN
cana-3310	71	12	equations	equation	NOUN
cana-3310	71	13	by	by	ADP
cana-3310	71	14	using	use	VERB
cana-3310	71	15	guass	guass	PROPN
cana-3310	71	16	jacobi	jacobi	PROPN
cana-3310	71	17	method	method	NOUN
cana-3310	71	18	and	and	CCONJ
cana-3310	71	19	gauss	gauss	PROPN
cana-3310	71	20	seidel	seidel	PROPN
cana-3310	71	21	methods	method	NOUN
cana-3310	71	22	𝟓𝒙	𝟓𝒙	NOUN
cana-3310	71	23	−	−	PROPN
cana-3310	71	24	𝟐𝒚	𝟐𝒚	NOUN
cana-3310	71	25	+	+	CCONJ
cana-3310	71	26	𝟑𝒛	𝟑𝒛	NOUN
cana-3310	71	27	=	=	SYM
cana-3310	71	28	−𝟏	−𝟏	NOUN
cana-3310	71	29	;	;	PUNCT
cana-3310	72	1	−𝟑𝒙	−𝟑𝒙	PROPN
cana-3310	72	2	+	+	NUM
cana-3310	72	3	𝟗𝒚	𝟗𝒚	NOUN
cana-3310	72	4	+	+	CCONJ
cana-3310	72	5	𝒛	𝒛	NOUN
cana-3310	72	6	=	=	SYM
cana-3310	72	7	𝟐	𝟐	NUM
cana-3310	72	8	;	;	PUNCT
cana-3310	72	9	𝟐𝒙	𝟐𝒙	NOUN
cana-3310	72	10	−	−	PROPN
cana-3310	73	1	𝒚	𝒚	PROPN
cana-3310	74	1	−	−	PROPN
cana-3310	74	2	𝟕𝒛	𝟕𝒛	NOUN
cana-3310	74	3	=	=	SYM
cana-3310	74	4	𝟑	𝟑	X
cana-3310	74	5	,	,	PUNCT
cana-3310	74	6	with	with	ADP
cana-3310	74	7	initial	initial	ADJ
cana-3310	74	8	approximation	approximation	NOUN
cana-3310	74	9	as	as	ADP
cana-3310	74	10	𝒙	𝒙	PROPN
cana-3310	74	11	=	=	SYM
cana-3310	74	12	𝟎	𝟎	PROPN
cana-3310	74	13	,	,	PUNCT
cana-3310	74	14	𝒚	𝒚	NOUN
cana-3310	74	15	=	=	SYM
cana-3310	74	16	𝟎	𝟎	PROPN
cana-3310	74	17	,	,	PUNCT
cana-3310	74	18	𝒛	𝒛	NOUN
cana-3310	74	19	=	=	NOUN
cana-3310	74	20	𝟎.	𝟎.	NOUN
cana-3310	74	21	solution	solution	NOUN
cana-3310	74	22	:	:	PUNCT
cana-3310	74	23	by	by	ADP
cana-3310	74	24	gauss	gauss	PROPN
cana-3310	74	25	jacobi	jacobi	PROPN
cana-3310	74	26	method	method	PROPN
cana-3310	74	27	:	:	PUNCT
cana-3310	74	28	firstly	firstly	ADV
cana-3310	74	29	define	define	VERB
cana-3310	74	30	function	function	NOUN
cana-3310	74	31	gauss_jacobi	gauss_jacobi	PROPN
cana-3310	74	32	which	which	PRON
cana-3310	74	33	takes	take	VERB
cana-3310	74	34	inputes	inpute	NOUN
cana-3310	74	35	as	as	ADP
cana-3310	74	36	coefficient	coefficient	NOUN
cana-3310	74	37	matrix	matrix	NOUN
cana-3310	74	38	a	a	DET
cana-3310	74	39	,	,	PUNCT
cana-3310	74	40	constant	constant	ADJ
cana-3310	74	41	matrix	matrix	NOUN
cana-3310	74	42	b	b	NOUN
cana-3310	74	43	,	,	PUNCT
cana-3310	74	44	initial	initial	ADJ
cana-3310	74	45	approximation	approximation	NOUN
cana-3310	74	46	column	column	NOUN
cana-3310	74	47	matrix	matrix	NOUN
cana-3310	74	48	x0	x0	PROPN
cana-3310	74	49	,	,	PUNCT
cana-3310	74	50	no	no	PRON
cana-3310	74	51	of	of	ADP
cana-3310	74	52	iteration	iteration	NOUN
cana-3310	74	53	n	n	CCONJ
cana-3310	74	54	,	,	PUNCT
cana-3310	74	55	and	and	CCONJ
cana-3310	74	56	tolerance	tolerance	NOUN
cana-3310	74	57	limit	limit	NOUN
cana-3310	74	58	.	.	PUNCT
cana-3310	75	1	initializing	initialize	VERB
cana-3310	75	2	matrices	matrix	NOUN
cana-3310	75	3	h	h	NOUN
cana-3310	75	4	,	,	PUNCT
cana-3310	75	5	c	c	NOUN
cana-3310	75	6	,	,	PUNCT
cana-3310	75	7	l	l	NOUN
cana-3310	75	8	,	,	PUNCT
cana-3310	75	9	d	d	NOUN
cana-3310	75	10	,	,	PUNCT
cana-3310	75	11	u	u	NOUN
cana-3310	75	12	and	and	CCONJ
cana-3310	75	13	column	column	NOUN
cana-3310	75	14	matrices	matrix	NOUN
cana-3310	75	15	x	x	INTJ
cana-3310	75	16	,	,	PUNCT
cana-3310	75	17	xold	xold	INTJ
cana-3310	75	18	separate	separate	ADJ
cana-3310	75	19	matrix	matrix	NOUN
cana-3310	75	20	a	a	PRON
cana-3310	75	21	as	as	SCONJ
cana-3310	75	22	mentioned	mention	VERB
cana-3310	75	23	in	in	ADP
cana-3310	75	24	equation	equation	NOUN
cana-3310	75	25	(	(	PUNCT
cana-3310	75	26	1),and	1),and	NUM
cana-3310	75	27	calculate	calculate	NOUN
cana-3310	75	28	d-1	d-1	NOUN
cana-3310	75	29	,	,	PUNCT
cana-3310	75	30	h	h	NOUN
cana-3310	75	31	,	,	PUNCT
cana-3310	75	32	c	c	PROPN
cana-3310	75	33	as	as	SCONJ
cana-3310	75	34	mentioned	mention	VERB
cana-3310	75	35	in	in	ADP
cana-3310	75	36	equation	equation	NOUN
cana-3310	75	37	(	(	PUNCT
cana-3310	75	38	2	2	X
cana-3310	75	39	)	)	PUNCT
cana-3310	75	40	lastly	lastly	ADV
cana-3310	75	41	we	we	PRON
cana-3310	75	42	apply	apply	VERB
cana-3310	75	43	iteration	iteration	NOUN
cana-3310	75	44	scheme	scheme	NOUN
cana-3310	75	45	as	as	SCONJ
cana-3310	75	46	mentioned	mention	VERB
cana-3310	75	47	in	in	ADP
cana-3310	75	48	equation	equation	NOUN
cana-3310	75	49	(	(	PUNCT
cana-3310	75	50	2	2	X
cana-3310	75	51	)	)	PUNCT
cana-3310	75	52	now	now	ADV
cana-3310	75	53	we	we	PRON
cana-3310	75	54	give	give	VERB
cana-3310	75	55	input	input	NOUN
cana-3310	75	56	as	as	SCONJ
cana-3310	75	57	mentioned	mention	VERB
cana-3310	75	58	in	in	ADP
cana-3310	75	59	example	example	NOUN
cana-3310	75	60	with	with	ADP
cana-3310	75	61	the	the	DET
cana-3310	75	62	initial	initial	ADJ
cana-3310	75	63	approximation	approximation	NOUN
cana-3310	75	64	and	and	CCONJ
cana-3310	75	65	tolerance	tolerance	NOUN
cana-3310	75	66	limit	limit	NOUN
cana-3310	75	67	0.0001	0.0001	NUM
cana-3310	75	68	.	.	PUNCT
cana-3310	76	1	communications	communication	NOUN
cana-3310	76	2	on	on	ADP
cana-3310	76	3	applied	apply	VERB
cana-3310	76	4	nonlinear	nonlinear	ADJ
cana-3310	76	5	analysis	analysis	NOUN
cana-3310	76	6	issn	issn	NOUN
cana-3310	76	7	:	:	PUNCT
cana-3310	76	8	1074	1074	NUM
cana-3310	76	9	-	-	PUNCT
cana-3310	76	10	133x	133x	NUM
cana-3310	76	11	vol	vol	NOUN
cana-3310	76	12	32	32	NUM
cana-3310	76	13	no	no	NOUN
cana-3310	76	14	.	.	PUNCT
cana-3310	77	1	6s	6s	NUM
cana-3310	77	2	(	(	PUNCT
cana-3310	77	3	2025	2025	NUM
cana-3310	77	4	)	)	PUNCT
cana-3310	77	5	466	466	NUM
cana-3310	77	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3310	77	7	output	output	NOUN
cana-3310	77	8	:	:	PUNCT
cana-3310	77	9	hence	hence	ADV
cana-3310	77	10	the	the	DET
cana-3310	77	11	solution	solution	NOUN
cana-3310	77	12	is	be	AUX
cana-3310	77	13	𝒙	𝒙	PROPN
cana-3310	77	14	=	=	PUNCT
cana-3310	77	15	𝟎.	𝟎.	NOUN
cana-3310	77	16	𝟏𝟖𝟔𝟏	𝟏𝟖𝟔𝟏	NUM
cana-3310	77	17	,	,	PUNCT
cana-3310	77	18	𝒚	𝒚	PROPN
cana-3310	78	1	=	=	PUNCT
cana-3310	78	2	𝟎.	𝟎.	X
cana-3310	78	3	𝟑𝟑𝟏𝟐	𝟑𝟑𝟏𝟐	NUM
cana-3310	78	4	,	,	PUNCT
cana-3310	78	5	𝒛	𝒛	NOUN
cana-3310	78	6	=	=	PUNCT
cana-3310	78	7	−𝟎.	−𝟎.	X
cana-3310	78	8	𝟒𝟐𝟐𝟕	𝟒𝟐𝟐𝟕	PROPN
cana-3310	78	9	by	by	ADP
cana-3310	78	10	gauss	gauss	PROPN
cana-3310	78	11	seidel	seidel	PROPN
cana-3310	78	12	method	method	PROPN
cana-3310	78	13	:	:	PUNCT
cana-3310	78	14	function	function	NOUN
cana-3310	78	15	:	:	PUNCT
cana-3310	78	16	communications	communication	NOUN
cana-3310	78	17	on	on	ADP
cana-3310	78	18	applied	apply	VERB
cana-3310	78	19	nonlinear	nonlinear	ADJ
cana-3310	78	20	analysis	analysis	NOUN
cana-3310	78	21	issn	issn	NOUN
cana-3310	78	22	:	:	PUNCT
cana-3310	78	23	1074	1074	NUM
cana-3310	78	24	-	-	PUNCT
cana-3310	78	25	133x	133x	NUM
cana-3310	78	26	vol	vol	NOUN
cana-3310	78	27	32	32	NUM
cana-3310	78	28	no	no	NOUN
cana-3310	78	29	.	.	PUNCT
cana-3310	79	1	6s	6s	NUM
cana-3310	79	2	(	(	PUNCT
cana-3310	79	3	2025	2025	NUM
cana-3310	79	4	)	)	PUNCT
cana-3310	79	5	467	467	NUM
cana-3310	79	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3310	79	7	input	input	NOUN
cana-3310	79	8	:	:	PUNCT
cana-3310	79	9	output	output	NOUN
cana-3310	79	10	:	:	PUNCT
cana-3310	79	11	4	4	X
cana-3310	79	12	.	.	NOUN
cana-3310	79	13	results	result	NOUN
cana-3310	79	14	by	by	ADP
cana-3310	79	15	leveraging	leverage	VERB
cana-3310	79	16	the	the	DET
cana-3310	79	17	efficiency	efficiency	NOUN
cana-3310	79	18	of	of	ADP
cana-3310	79	19	geogebra	geogebra	NOUN
cana-3310	79	20	and	and	CCONJ
cana-3310	79	21	sagemath	sagemath	NOUN
cana-3310	79	22	,	,	PUNCT
cana-3310	79	23	engineering	engineering	NOUN
cana-3310	79	24	students	student	NOUN
cana-3310	79	25	can	can	AUX
cana-3310	79	26	enhance	enhance	VERB
cana-3310	79	27	their	their	PRON
cana-3310	79	28	learning	learning	NOUN
cana-3310	79	29	experience	experience	NOUN
cana-3310	79	30	,	,	PUNCT
cana-3310	79	31	leading	lead	VERB
cana-3310	79	32	to	to	ADP
cana-3310	79	33	quicker	quick	ADJ
cana-3310	79	34	mastery	mastery	NOUN
cana-3310	79	35	of	of	ADP
cana-3310	79	36	complex	complex	ADJ
cana-3310	79	37	concepts	concept	NOUN
cana-3310	79	38	.	.	PUNCT
cana-3310	80	1	the	the	DET
cana-3310	80	2	accuracy	accuracy	NOUN
cana-3310	80	3	provided	provide	VERB
cana-3310	80	4	by	by	ADP
cana-3310	80	5	these	these	DET
cana-3310	80	6	tools	tool	NOUN
cana-3310	80	7	ensures	ensure	VERB
cana-3310	80	8	that	that	SCONJ
cana-3310	80	9	students	student	NOUN
cana-3310	80	10	can	can	AUX
cana-3310	80	11	trust	trust	VERB
cana-3310	80	12	their	their	PRON
cana-3310	80	13	results	result	NOUN
cana-3310	80	14	,	,	PUNCT
cana-3310	80	15	which	which	PRON
cana-3310	80	16	is	be	AUX
cana-3310	80	17	crucial	crucial	ADJ
cana-3310	80	18	for	for	ADP
cana-3310	80	19	their	their	PRON
cana-3310	80	20	future	future	ADJ
cana-3310	80	21	work	work	NOUN
cana-3310	80	22	in	in	ADP
cana-3310	80	23	engineering	engineering	NOUN
cana-3310	80	24	.	.	PUNCT
cana-3310	81	1	together	together	ADV
cana-3310	81	2	,	,	PUNCT
cana-3310	81	3	these	these	DET
cana-3310	81	4	tools	tool	NOUN
cana-3310	81	5	create	create	VERB
cana-3310	81	6	a	a	DET
cana-3310	81	7	powerful	powerful	ADJ
cana-3310	81	8	educational	educational	ADJ
cana-3310	81	9	environment	environment	NOUN
cana-3310	81	10	that	that	PRON
cana-3310	81	11	fosters	foster	VERB
cana-3310	81	12	both	both	DET
cana-3310	81	13	understanding	understanding	NOUN
cana-3310	81	14	and	and	CCONJ
cana-3310	81	15	application	application	NOUN
cana-3310	81	16	of	of	ADP
cana-3310	81	17	mathematical	mathematical	ADJ
cana-3310	81	18	principles	principle	NOUN
cana-3310	81	19	.	.	PUNCT
cana-3310	82	1	the	the	DET
cana-3310	82	2	visualization	visualization	NOUN
cana-3310	82	3	of	of	ADP
cana-3310	82	4	difficult	difficult	ADJ
cana-3310	82	5	mathematical	mathematical	ADJ
cana-3310	82	6	concepts	concept	NOUN
cana-3310	82	7	and	and	CCONJ
cana-3310	82	8	the	the	DET
cana-3310	82	9	use	use	NOUN
cana-3310	82	10	of	of	ADP
cana-3310	82	11	numerical	numerical	ADJ
cana-3310	82	12	techniques	technique	NOUN
cana-3310	82	13	are	be	AUX
cana-3310	82	14	two	two	NUM
cana-3310	82	15	major	major	ADJ
cana-3310	82	16	benefits	benefit	NOUN
cana-3310	82	17	of	of	ADP
cana-3310	82	18	integrating	integrate	VERB
cana-3310	82	19	geogebra	geogebra	NOUN
cana-3310	82	20	and	and	CCONJ
cana-3310	82	21	sagemath	sagemath	NOUN
cana-3310	82	22	into	into	ADP
cana-3310	82	23	engineering	engineering	NOUN
cana-3310	82	24	education	education	NOUN
cana-3310	82	25	.	.	PUNCT
cana-3310	83	1	with	with	ADP
cana-3310	83	2	the	the	DET
cana-3310	83	3	help	help	NOUN
cana-3310	83	4	of	of	ADP
cana-3310	83	5	geogebra	geogebra	NOUN
cana-3310	83	6	's	's	PART
cana-3310	83	7	dynamic	dynamic	ADJ
cana-3310	83	8	geometry	geometry	NOUN
cana-3310	83	9	features	feature	NOUN
cana-3310	83	10	,	,	PUNCT
cana-3310	83	11	students	student	NOUN
cana-3310	83	12	may	may	AUX
cana-3310	83	13	change	change	VERB
cana-3310	83	14	and	and	CCONJ
cana-3310	83	15	study	study	VERB
cana-3310	83	16	curves	curve	NOUN
cana-3310	83	17	with	with	ADP
cana-3310	83	18	ease	ease	NOUN
cana-3310	83	19	,	,	PUNCT
cana-3310	83	20	developing	develop	VERB
cana-3310	83	21	a	a	DET
cana-3310	83	22	greater	great	ADJ
cana-3310	83	23	grasp	grasp	NOUN
cana-3310	83	24	of	of	ADP
cana-3310	83	25	their	their	PRON
cana-3310	83	26	characteristics	characteristic	NOUN
cana-3310	83	27	.	.	PUNCT
cana-3310	84	1	in	in	ADP
cana-3310	84	2	the	the	DET
cana-3310	84	3	meantime	meantime	NOUN
cana-3310	84	4	,	,	PUNCT
cana-3310	84	5	sagemath	sagemath	NOUN
cana-3310	84	6	offers	offer	VERB
cana-3310	84	7	a	a	DET
cana-3310	84	8	strong	strong	ADJ
cana-3310	84	9	foundation	foundation	NOUN
cana-3310	84	10	for	for	ADP
cana-3310	84	11	carrying	carry	VERB
cana-3310	84	12	out	out	ADP
cana-3310	84	13	numerical	numerical	ADJ
cana-3310	84	14	analysis	analysis	NOUN
cana-3310	84	15	and	and	CCONJ
cana-3310	84	16	symbolic	symbolic	ADJ
cana-3310	84	17	computations	computation	NOUN
cana-3310	84	18	,	,	PUNCT
cana-3310	84	19	empowering	empower	VERB
cana-3310	84	20	students	student	NOUN
cana-3310	84	21	to	to	PART
cana-3310	84	22	efficiently	efficiently	ADV
cana-3310	84	23	solve	solve	VERB
cana-3310	84	24	equations	equation	NOUN
cana-3310	84	25	and	and	CCONJ
cana-3310	84	26	simulate	simulate	VERB
cana-3310	84	27	real	real	ADJ
cana-3310	84	28	-	-	PUNCT
cana-3310	84	29	world	world	NOUN
cana-3310	84	30	issues	issue	NOUN
cana-3310	84	31	.	.	PUNCT
cana-3310	85	1	using	use	VERB
cana-3310	85	2	these	these	DET
cana-3310	85	3	tools	tool	NOUN
cana-3310	85	4	,	,	PUNCT
cana-3310	85	5	students	student	NOUN
cana-3310	85	6	can	can	AUX
cana-3310	85	7	investigate	investigate	VERB
cana-3310	85	8	the	the	DET
cana-3310	85	9	connections	connection	NOUN
cana-3310	85	10	between	between	ADP
cana-3310	85	11	various	various	ADJ
cana-3310	85	12	mathematical	mathematical	ADJ
cana-3310	85	13	entities	entity	NOUN
cana-3310	85	14	,	,	PUNCT
cana-3310	85	15	see	see	VERB
cana-3310	85	16	how	how	SCONJ
cana-3310	85	17	their	their	PRON
cana-3310	85	18	calculations	calculation	NOUN
cana-3310	85	19	are	be	AUX
cana-3310	85	20	shown	show	VERB
cana-3310	85	21	,	,	PUNCT
cana-3310	85	22	and	and	CCONJ
cana-3310	85	23	get	get	VERB
cana-3310	85	24	practical	practical	ADJ
cana-3310	85	25	experiences	experience	NOUN
cana-3310	85	26	addressing	address	VERB
cana-3310	85	27	engineering	engineering	NOUN
cana-3310	85	28	challenges	challenge	NOUN
cana-3310	85	29	.	.	PUNCT
cana-3310	86	1	in	in	ADP
cana-3310	86	2	addition	addition	NOUN
cana-3310	86	3	to	to	ADP
cana-3310	86	4	enhancing	enhance	VERB
cana-3310	86	5	their	their	PRON
cana-3310	86	6	analytical	analytical	ADJ
cana-3310	86	7	abilities	ability	NOUN
cana-3310	86	8	,	,	PUNCT
cana-3310	86	9	this	this	DET
cana-3310	86	10	combination	combination	NOUN
cana-3310	86	11	gets	get	VERB
cana-3310	86	12	them	they	PRON
cana-3310	86	13	ready	ready	ADJ
cana-3310	86	14	for	for	ADP
cana-3310	86	15	real	real	ADJ
cana-3310	86	16	-	-	PUNCT
cana-3310	86	17	world	world	NOUN
cana-3310	86	18	work	work	NOUN
cana-3310	86	19	in	in	ADP
cana-3310	86	20	the	the	DET
cana-3310	86	21	future	future	NOUN
cana-3310	86	22	.	.	PUNCT
cana-3310	87	1	references	reference	NOUN
cana-3310	87	2	[	[	X
cana-3310	87	3	1	1	X
cana-3310	87	4	]	]	PUNCT
cana-3310	87	5	http://www.jstor.org/stable/10.5951/mathteacher.106.3.0228	http://www.jstor.org/stable/10.5951/mathteacher.106.3.0228	PROPN
cana-3310	88	1	[	[	X
cana-3310	88	2	2	2	X
cana-3310	88	3	]	]	X
cana-3310	88	4	https://www.geogebra.org/.	https://www.geogebra.org/.	NOUN
cana-3310	89	1	[	[	X
cana-3310	89	2	3	3	NUM
cana-3310	89	3	]	]	PUNCT
cana-3310	89	4	https://issuu.com/ijasrweditor/docs/calculations_ofsometrigonometricinterpolationpolyn	https://issuu.com/ijasrweditor/docs/calculations_ofsometrigonometricinterpolationpolyn	NOUN