Numericka matematika Numericka matematika Numericka matematika

❯♥✐✈❡r③✐t❡t ✉ ❚✉③❧✐
Pr✐r♦❞♥♦ ✲ ♠❛t❡♠❛t✐↔❦✐ ❢❛❦✉❧t❡t
❖❞s❥❡❦✿ ▼❛t❡♠❛t✐❦❛
Pr❡❞♠❡t✿ ◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛

❙❊▼■◆❆❘❙❑■ ❘❆❉

▼❡♥t♦r✿

❙t✉❞❡♥t✿
❉❛✈♦r ❇❡❣❛♥♦✈✐➣

❉r✳s❝✳❊♥❡s ❉✉✈♥❥❛❦♦✈✐➣✱ ✈❛♥r✳♣r♦❢✳

✶✷✳ sr♣♥❥❛ ✷✵✶✼✳

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛

❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✶

❙❛❞r➸❛❥

✶

✷

✸

❩❛❞❛t❛❦ ✶✳

✷

✶✳✶

▼❡t♦❞❛ s❥❡↔✐❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

✶✳✷

Pr✐♠❥❡r

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✲✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

▼❛t❤❡♠❛t✐❝❛

✶✳✸

❑♦❞ ✉

✶✳✹

❑♦♠❡♥t❛r

❩❛❞❛t❛❦ ✷✳

✻

✷✳✶

◆❡✇t♦♥✲♦✈❛ ♠❡t♦❞❛ ③❛ r❥❡➨❛✈❛♥❥❡ s✐st❡♠❛ ♥❡❧✐♥❡❛r♥✐❤ ❛❧❣❡❜❛rs❦✐❤ ❥❡❞♥❛↔✐♥❛

✻

✷✳✷

Pr✐♠❥❡r

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✲✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

▼❛t❤❡♠❛t✐❝❛

✷✳✸

❑♦❞ ✉

✷✳✹

❑♦♠❡♥t❛r

❩❛❞❛t❛❦ ✸✳

✾

✸✳✶

▼❡t♦❞❛ ❘✉♥❣❡✲❑✉tt❛ ③❛ r❥❡➨❛✈❛♥❥❡ ❞✐❢❡r❡♥❝✐❥❛❧♥✐❤ ❥❡❞♥❛↔✐♥❛

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✸✳✷

Pr✐♠❥❡r

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✲✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

▼❛t❤❡♠❛t✐❝❛

✸✳✸

❑♦❞ ✉

✸✳✹

❑♦♠❡♥t❛r

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛
✶

❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✷

❩❛❞❛t❛❦ ✶✳

✶✳✶

▼❡t♦❞❛ s❥❡↔✐❝❡

▼❡t♦❞❛ s❥❡↔✐❝❡

♣r❡t♣♦st❛✈❦❡✿

❥❡ ♠♦❞✐✜❦❛❝✐❥❛

◆❡✇t♦♥✲♦✈❡ ♠❡t♦❞❡✳

❩❛ ♠❡t♦❞✉ s❥❡↔✐❝❡ ✈r✐❥❡❞❡ s❧❥❡❞❡➣❡

• ◆❡❦❛ ♣♦st♦❥✐ x ∈ [a, b] t❛❦❛✈ ❞❛ ❥❡ f (x) = 0✳
• ◆❡❦❛ ❥❡ ❢✉♥❦❝✐❥❛ f ♥❡♣r❡❦✐❞♥❛ ♥❛ [a, b]✳
• ◆❡❦❛ ❥❡ f (a) · f (b) < 0✳

❯③♠✐♠♦ s❛❞❛ ❞✈✐❥❡ t❛↔❦❡ x0 ✐ x1 ✐③ [a, b]✱ t❛❞❛ ✐♠❛♠♦ ❞✈✐❥❡ t❛↔❦❡ M0 (x0 , f (x0 )) ✐ M1 (x1 , f (x1 ))
♥❛ ❦r✐✈♦❥ ❢✉♥❦❝✐❥❡✳ ◆❡❦❛ ❥❡ ❦♦❞ ♥❛s x0 = b✳ ❙❛❞❛ ❛❦♦ ♣♦✈✉↔❡♠♦ s❥❡❦❛♥t✉ ❦r♦③ f (x0 ) = f (b)
✐ ❦r♦③ f (x1 ) ✐♠❛t ➣❡♠♦ ❦❧❛s✐↔❛♥ ♣r♦❞♦r ♣r❛✈❡ ❦r♦③ ❞✈✐❥❡ t❛↔❦❡ ✐ ❛❦♦ ✉♣♦tr✐❥❡❜✐♠♦ ❢♦r♠✉❧✉
③❛ ♣r♦❞♦r ♣r❛✈❡ ❦r♦③ ❞✈✐❥❡ t❛↔❦❡ ✐♠❛♠♦✿
s : f1 (x) = f (x0 ) +

f (x1 ) − f (x0 )
(x − x0 ).
x1 − x0

❙ ♦❜③✐r♦♠ ❞❛ ❥❡ f1 (x) = 0 ✐♠❛♠♦
f (x0 ) +

f (x1 ) − f (x0 )
(x − x0 ) = 0
x1 − x0

s❛❞❛ ❛❦♦ s✈❡ ♣♦♠♥♦➸✐♠♦ s❛ x1 − x0 ❞❛❧❥❡ ➣❡♠♦ ✐♠❛t✐
(x1 − x0 )f (x0 ) + (f (x1 ) − f (x0 ))(x − x0 ) = 0

s❛ ❥♦➨ ♠❛❧♦ sr❡➒✐✈❛♥❥❛ ♥❛ ❦r❛❥✉ ❞♦❜✐❥❛♠♦
x0 f (x1 ) − x1 f (x0 )
f (x1 ) − f (x0 )

x=

♣r✐ ↔❡♠✉ ❥❡ f (x0 ) 6= f (x1 )✳ ❆❦♦ ♥❛♣r❛✈✐♠♦ ♥✐③ ♦✈❛❦✈✐❤ ❥❡❞♥❛❦♦st✐
x2 =

x0 f (x1 ) − x1 f (x0 )
f (x1 ) − f (x0 )

x3 =

x1 f (x2 ) − x2 f (x1 )
f (x2 ) − f (x1 )

✐ t❛❦♦ ♥❛st❛✈✐♠♦✱ ♥❛ ❦r❛❥✉ ➣❡♠♦ ❞♦❜✐t✐
xn+1 =

xn−1 f (xn ) − xn f (xn−1 )
f (xn ) − f (xn−1 )

♣r✐ ↔❡♠✉ ❥❡ f (xn−1 ) 6= f (xn )✳ ❩❛ ♦✈❛❥ ♥✐③ ✈r✐❥❡❞✐
lim xn = ξ

n→∞

❣❞❥❡ ❥❡ ξ t❛↔♥♦ r❥❡➨❡♥❥❡ ❥❡❞♥❛↔✐♥❡✳ ❇r③✐♥❛ ❦♦♥✈❡r❣❡♥❝✐❥❡ ❥❡

√
1+ 5
.
2

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛

❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✸

❙❧✐❦❛ ✶✿ ▼❡t♦❞❛ s❥❡↔✐❝❡
❑❛♦ ➨t♦ s♠♦ r❡❦❧✐ ♠❡t♦❞❛ s❥❡↔✐❝❡ ❥❡ ♠♦❞✐✜❦❛❝✐❥❛ ◆❡✇t♦♥✲♦✈❡ ♠❡t♦❞❡✳ ❙❛❞❛ ➣❡♠♦
♣♦❦❛③❛t✐ ✐ ③❜♦❣ ↔❡❣❛✳
❩♥❛♠♦ ❞❛ ❥❡ r❡❦✉r③✐✈♥❛ ❢♦r♠✉❧❛ ③❛ ◆❡✇t♦♥✲♦✈✉ ♠❡t♦❞✉
xn+1 = xn −

f (xn )
.
f ′ (xn )

❙❛❞❛ ❛❦♦ ✉③♠❡♠♦ ❞❛ ❥❡
f ′ (xn ) =

lim

xn−1 →xn

✐♠❛♠♦ ❞❛ ❥❡
f ′ (xn ) ≈

f (xn − f (xn−1 ))
xn − xn−1

f (xn ) − f (xn−1 )
.
xn − xn−1

❆❦♦ ♣♦s❧❥❡❞♥❥✉ ❛♣r♦❦s✐♠❛❝✐❥✉ f ′ (xn ) ✉❜❛❝✐♠♦ ✉ ✭✶✮ ❞♦❜✐t ➣❡♠♦
xn+1 = xn −

f (xn )
f (xn )−f (xn−1 )
xn −xn−1

✐ s❛❞❛ ❦❛❞❛ t♦ ♠❛❧♦ sr❡❞✐♠♦ ❞♦❜✐❥❛♠♦
xn+1 =

xn−1 f (xn ) − xn f (xn−1 )
f (xn ) − f (xn−1 )

➨t♦ ❥❡ ✉st✈❛r✐ ❢♦r♠✉❧❛ ③❛ ♠❡t♦❞✉ s❥❡↔✐❝❡ ✐ ♥❛r❛✈♥♦ ❞❛ ♠♦r❛ ✈r✐❥❡❞✐t✐ f (xn ) 6= f (xn−1 )✳
✶✳✷

Pr✐♠❥❡r

▼❡t♦❞♦♠ s❥❡↔✐❝❡ s❛ t❛↔♥♦➨➣✉ ✈❡➣♦♠ ♦❞ ε = 10−4 r✐❥❡➨✐t✐ ❥❡❞♥❛↔✐♥✉ xex − 1 = 0.
❘❥❡➨❡♥❥❡✿

◆❛❝rt❛❥♠♦ ❢✉♥❦❝✐❥✉ f (x) = xex − 1✿

✭✶✮

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛

❇❡❣❛♥♦✈✐➣ ❉❛✈♦r

❙❧✐❦❛ ✷✿ ❋✉♥❦❝✐❥❛

f (x) = xex − 1

♥❛ ✐♥t❡r✈❛❧✉

[0.4, 0.8]

[0.4, 0.8]✳ Pr♦✈❥❡r✐♠♦ s❛❞❛ ❞❛
′
❞❛ ❧✐ ❥❡ f (x) st❛❧♥♦❣ ③♥❛❦❛✿

◆❛ s❧✐❝✐ ✷ ✈✐❞✐♠♦ ❞❛ s❡ r❥❡➨❡♥❥❡ ♥❛❧❛③✐ ✉ ✐♥t❡r✈❛❧✉
♥✉❧❛ ❢✉♥❦❝✐❥❡

f (x)

❥❡❞✐♥st✈❡♥❛ ♥❛ ♦✈♦♠ ✐♥t❡r✈❛❧✉✱ t❥✳

✹

❧✐ ❥❡

f ′ (x) = ex (1 + x)
[0.4, 0.8] ✉✈✐❥❡❦ ♣♦③✐t✐✈♥♦✱ ❞❛❦❧❡ ♥✉❧❛
f (a) · f (b) < 0✿
f (0.4) = −0.40327 < 0,

➨t♦ ❥❡ ♥❛ ✐♥t❡r✈❛❧✉
❞❛ ❧✐ ✈r✐❥❡❞✐

❥❡ ❥❡❞✐♥st✈❡♥❛✳ Pr♦✈❥❡r✐♠♦ s❛❞❛

f (0.8) = 0.78043 > 0
❞❛❦❧❡ ✈r✐❥❡❞✐ ✐ ♦✈❛ ❥ ✉s❧♦✈✳

x1 = 0.6✱

❥❡r ❥❡

❖❞r❡❞✐♠♦ s❛❞❛ t❛↔❦❡

f (0.6) = 0.09327✱
x2 =

x0

✐

x1 ✳

◆❡❦❛ ❥❡

x0 = 0.4✱

❛ ♥❡❦❛ ❥❡

♣❛ ✐ ❞❛❧❥❡ ✈r✐❥❡❞❡ ✉s❧♦✈✐ ♠❡t♦❞❡✳ ■③r❛↔✉♥❛❥♠♦ s❛❞❛

x2 ✿

x0 f (x1 ) − x1 f (x0 )
= 0.56243
f (x1 ) − f (x0 )

x1 f (x2 ) − x2 f (x1 )
= 0.56702
f (x2 ) − f (x1 )
x2 f (x3 ) − x3 f (x2 )
= 0.56714
x4 =
f (x3 ) − f (x2 )
x3 f (x4 ) − x4 f (x3 )
x5 =
= 0.56714
f (x4 ) − f (x3 )
x3 =

✈✐❞✐♠♦ ❞❛ s♠♦ ❞♦❜✐❧✐ ♣♦❦❧❛♣❛♥❥❡ ♥❛ ❞♦ ♣❡t✉ ❞❡❝✐♠❛❧✉✱ ♣❛ ❥❡ ♣r✐❜❧✐➸♥♦ r❥❡➨❡♥❥❡ ❞❛❦❧❡

x¯ = 0.56714✳
✶✳✸

❑♦❞ ✉

▼❛t❤❡♠❛t✐❝❛

✲✐

f [①❴]✿❂x ∗ E ∧ x − 1
P❧♦t[f [x], {x, 0.4, 0.8}]
D[f [x], x]
f [0.4]
f [0.8]
f [0.6]
①✵ = 0.4
①✶ = 0.6
①✷ = (①✵ ∗ f [①✶] − ①✶ ∗ f [①✵])/(f [①✶] − f [①✵])
①✸ = (①✶ ∗ f [①✷] − ①✷ ∗ f [①✶])/(f [①✷] − f [①✶])
①✹ = (①✷ ∗ f [①✸] − ①✸ ∗ f [①✷])/(f [①✸] − f [①✷])
①✺ = (①✸ ∗ f [①✹] − ①✹ ∗ f [①✸])/(f [①✹] − f [①✸])

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛

✶✳✹

❇❡❣❛♥♦✈✐➣ ❉❛✈♦r

✺

❑♦♠❡♥t❛r

❱✐❞✐♠♦ ❞❛ s♠♦ ✉ ③❛❞❛t❦✉ r❡❧❛t✐✈♥♦ ❜r③♦ ❞♦➨❧✐ ❞♦ r❥❡➨❡♥❥❛✱ ➨t♦ ❥❡ ♣♦s❧❥❡❞✐❝❛ ❜r③✐♥❡ ❦♦♥✈❡r✲
❣❡♥❝✐❥❡ ❞❛t❡ ♠❡t♦❞❡✱ ❛❧✐ ✐ tr❡❜❛♠♦ ♣r✐♠✐❥❡t✐t✐ ❞❛ st♦ ✈✐➨❡ s✉③✐♠♦ ♣♦↔❡t♥✐ ✐♥t❡r✈❛❧ t♦ ➣❡♠♦
♣r✐❥❡ ❞♦➣✐ ❞♦ r❥❡➨❡♥❥❛✳

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛

✷

✷✳✶

❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✻

❩❛❞❛t❛❦ ✷✳

◆❡✇t♦♥✲♦✈❛ ♠❡t♦❞❛ ③❛ r❥❡➨❛✈❛♥❥❡ s✐st❡♠❛ ♥❡❧✐♥❡❛r♥✐❤ ❛❧❣❡✲
❜❛rs❦✐❤ ❥❡❞♥❛↔✐♥❛

◆❡❦❛ ❥❡

fi (x1 , x2 , ..., xn ) = 0, (i = 1, 2, ..., n)

✭✷✮

= 0✱ ❣❞❥❡ ❥❡ f (x) = (f1 (x), ..., fn (x))T
(0)
(0)
(0)
✐ x = (x1 , ..., xn )T ✳ ◆❡❦❛ ❥❡ ♣♦↔❡t♥❛ ❛♣r♦❦s✐♠❛❝✐❥❛ ❞❛t❛ s❛ x(0) = (x1 , x2 , ..., xn )✳ ❆❦♦
fi (x1 , x2 , ..., xn ) r❛③✈✐❥❡♠♦ ✉ ❚❛②❧♦r✲♦✈ r❡❞ st❡♣❡♥❛ ✶ ✉ ♦❦♦❧✐♥✐t❛↔❦❡ x(0) ❞♦❜✐t ➣❡♠♦
s✐st❡♠ ♥❡❧✐♥❡❛r♥✐❤ ❛❧❣❡❜❛rs❦✐❤ ❥❡❞♥❛↔✐♥❛✱ ♦❞♥♦s♥♦ f (x)

fi (x) = fi (x(0) ) + ∇fi (x(0) )(x − x(0) ) + R2
♣r✐ ↔❡♠✉ ❥❡
(0)

(0)

∇fi (x )(x − x ) =

n
X
∂fi (x(0) )
j=1

∂xj

(0)

(xj − xj ).

❙❛❞❛ ✉♠❥❡st♦ s✐st❡♠❛ ✭✷✮ r❥❡➨❛✈❛♠♦ s✐st❡♠

f˜i (x) = 0, (i = 1, 2, ..., n),
➨t♦ ♠♦➸❡♠♦ ♥❛♣✐s❛t✐ ✉ ♠❛tr✐↔♥♦♠ ♦❜❧✐❦✉

J (0) s(0) = −f (x(0) ),
❣❞❥❡ s✉

J (0)

 ∂f

1 (x

(0) )

1
 ∂f2∂x
(x(0) )
 ∂x
1
=
 ✳✳
✳


∂f1 (x(0) )
∂x2
∂f2 (x(0) )
∂x2

✳✳
✳

∂fn (x(0) )
∂x2

∂fn (x(0) )
∂x1

s(0)



∂f1 (x(0) )
∂xn
∂f2 (x(0) ) 

∂xn


...
...
✳✳

✳✳
✳

✳

∂fn (x(0) )
∂xn

...
(0) 

x1 − x1
 x − x(0) 
 2
2 
=

✳✳


✳



,


(0)

xn − xn

✐

▼❛tr✐❝✉


f1 (x(0) )
 f2 (x(0) ) 


f (x(0) ) =  ✳✳  .
 ✳ 
fn (x(0) )


J

♥❛③✐✈❛♠♦

❏❛❝♦❜✐ ✲❥❡✈❛

♠❛tr✐❝❛ ✐❧✐

❏❛❝♦❜✐❥❛♥

♥❥❛ t❛❞❛ ❥❡

x(1) = x(0) + s(0) .
❖♣➣❡♥✐t♦✱ ❞♦❜✐❥❛♠♦ ✐t❡r❛t✐✈♥✐ ♣♦st✉♣❛❦

x(k+1) = x(k) + s(k) ,

s✐st❡♠❛✳ ◆♦✈❛ ❛♣r♦❦s✐♠❛❝✐❥❛ r❥❡➨❡✲

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛

❣❞❥❡ ❥❡

s(k)

❇❡❣❛♥♦✈✐➣ ❉❛✈♦r

✼

r❥❡➨❡♥❥❡ s✐st❡♠❛

J (k) s(k) = −f (x(k) ).
Pr✐♠✐❥❡t✐♠♦ s❛❞❛ ❞❛ s❡ ♣r❡t❤♦❞♥✐ ✐t❡r❛t✐✈♥✐ ♣♦st✉♣❛❦ ✉③ ♣r❡t♣♦st❛✈❦✉ r❡❣✉❧❛r♥♦st✐ ❏❛❝♦✲
❜✐❥❛♥❛

J (k)

♠♦➸❡ ♥❛♣✐s❛t✐ ✉ ♦❜❧✐❦✉

x(k+1) = x(k) − (J (k) )−1 f (x(k) ), (k = 0, 1, 2, ..., n).
✷✳✷

Pr✐♠❥❡r

❘✐❥❡➨✐t✐ s✐st❡♠

❣r❡➨❦♦♠

❘❥❡➨❡♥❥❡
x¯

(0)

(

x3 − y 3 = x

◆❡✇t♦♥✲♦✈♦♠ ♠❡t♦❞♦♠✱ ♣r✐ ↔❡♠✉ ❥❡

x3 + y 3 = 3xy
−3
♠❛♥❥♦♠ ♦❞ 10
✳

✿

◆❡❦❛ ❥❡

= (−1, 0.3)

T

F (x, y) = x3 − y 3 − x

✳ Pr✐❦❛➸✐♠♦

G(x, y) = x3 + y 3 − 3xy ✳
s❛❞❛ ➨t❛ s✉ ❦♦❞ ♥❛s f (¯
x(0) ) ✐ J (0) ✳

 3
x − y3 − x
f (¯
x) = 3
x + y 3 − 3xy

♦❞♥♦s♥♦

✐



−0.027
f (¯
x )=
−0.073
(0)

s❛❞❛ ♦❞r❡❞✐♠♦

J (0)

3x2 − 1
−3y 2
J=
3x2 − 3y 3y 2 − 3x
J

(0)

(J (0) )−1
(J

✐ s❛❞❛ ♥❛➒✐♠♦ r❥❡➨❡♥❥❡





♦❞♥♦s♥♦

❖❞r❡❞✐♠♦ s❛❞❛

x0 = −1 ✐ y0 = 1✱

(0) −1

)




2 −0.27
.
=
2.1 3.27




0.4601 0.0380
=
−0.2955 0.2814

x¯(1) = x¯(0) − (J (0) )−1 f (¯
x(0) )



x¯(1) = (−0.9848, 0.3126)T .
❉❛❦❧❡

✷✳✸

❑♦❞ ✉

▼❛t❤❡♠❛t✐❝❛


−0.985
.
x¯ =
0.313


✲✐

f [①❴, ②❴]✿❂x∧ 3 − y ∧ 3 − x
g[①❴, ②❴]✿❂x∧ 3 + y ∧ 3 − 3 ∗ x ∗ y
f [−1, 0.3]
g[−1, 0.3]
❢✵[①❴, ②❴] = {f [x, y], g[x, y]}
❢✵[−1, 0.3]
❥❛❝♦❜[①❴, ②❴] = {{D[f [x, y], x], D[f [x, y], y]}, {D[g[x, y], x], D[g[x, y], y]}}
❥❛❝♦❜[−1, 0.3]
j = ■♥✈❡rs❡[❥❛❝♦❜[x, y]]✴✳{x → −1, y ✲>0.3}
{−1, 0.3} − j.❢✵[−1, 0.3]

s

❯③♠✐♠♦ ❞❛ ❥❡

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛

✷✳✹

❇❡❣❛♥♦✈✐➣ ❉❛✈♦r

✽

❑♦♠❡♥t❛r

❱✐❞✐♠♦ ❞❛ ❥❡ ♦✈❛ ♠❡t♦❞❛ ❞♦st❛ ❜r③❛✱ ↔❛❦ ❥❡ ♥❛❥❜r➸❛ ♦❞ s✈✐❤ ♠❡t♦❞❛ ❦♦ ❥❡ ✐♠❛♠♦ ③❛
r❛↔✉♥❛♥❥❡ s✐st❡♠❛ ♥❡❧✐♥❡❛r♥✐❤ ❛❧❣❡❜❛rs❦✐ ❥❡❞♥❛↔✐♥❛✱ ❛❧✐ ❥♦❥ ❥❡ ♠❛♥❛ ➨t♦ ✐♠❛ ❞♦st❛ r❛↔✉♥❛✱
♠❡➒✉t✐♠ ✉③ ❞❛♥❛➨♥❥✉ r❛↔✉♥❛❧♥✉ t❡❤♥♦❧♦❣✐❥✉ t♦ ✈✐➨❡ ✐ ♥✐❥❡ ♣r♦❜❧❡♠✳

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛
✸

❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✾

❩❛❞❛t❛❦ ✸✳

✸✳✶

▼❡t♦❞❛ ❘✉♥❣❡✲❑✉tt❛ ③❛ r❥❡➨❛✈❛♥❥❡ ❞✐❢❡r❡♥❝✐❥❛❧♥✐❤ ❥❡❞♥❛↔✐♥❛

P♦s♠❛tr❛❥♠♦ ❈❛✉❝❤②✲❡✈ ♣r♦❜❧❡♠

y ′ = f (x, y), y(x0 ) = y0 .
❏❡❞♥❛ ♦❞ ♥❛❥✈❛➸♥✐❥✐ ♠❡t♦❞❛ ③❛ r❥❡➨❛✈❛♥❥❡ ♦✈♦❣ ❈❛✉❝❤②✲❡✈♦❣ ♣r♦❜❧❡♠ ❥❡ ♠❡t♦❞❛ ❘✉♥❣❡✲
✶
❑✉tt❛ ✭❘❑✮ ✳ Pr❡t♣♦st❛✈✐♠♦ ❞❛ ♣♦③♥❛❥❡♠♦ ❛♣r♦❦s✐♠❛❝✐❥✉ yn tr❛➸❡♥❡ ❢✉♥❦❝✐❥❡ x 7−→ y(x)
✉ t❛↔❦✐ xn ✳ ➎❡❧✐♠♦ ♦❞r❡❞✐t✐ (n + 1)✲✈✉ ❛♣r♦❦s✐♠❛❝✐❥✉ yn+1 ✉ t❛↔❦✐ xn + h✳ ❯ t✉ s✈r❤✉ ♥❛
✐♥t❡r✈❛❧✉ (xn , xn + h) ✉ ♥❡❦♦❧✐❦♦ str❛t❡➨❦✐❤ t❛↔❛❦❛ ❛♣r♦❦s✐♠✐r❛t ➣❡♠♦ ✈r✐❥❡❞♥♦st ❢✉♥❦❝✐❥❡
x 7−→ f (x, y(x))✱ t❡ ♣♦♠♦➣✉ ♥❥✐❤ ➨t♦ ❜♦❧❥❡ ❛♣r♦❦s✐♠✐r❛t✐ r❛③❧✐❦✉ yn+1 − yn ✳
◆❛❥❥❡❞♥♦st❛✈♥✐❥✐ ♣r✐♠❥❡r ✐③ ❢❛♠✐❧✐❥❡ ❘❑ ♠❡t♦❞❛ ❥❡ t③✈

❍❡✉♥ ✲♦✈❛

♠❡t♦❞❛

1
yn+1 = yn + (k1 + k2 )
2
❣❞❥❡ ❥❡ k1 = hf (xn , yn )✱ ❛ k2 = hf (xn + h, yn + k1 )✳
Pr✐♠❥❡❞❜❛ ✸✳✶

❑❛❦♦ ❥❡

y(xn+1 ) − y(xn ) =

Z

xn +h
xn

dy
dx =
dx

Z

xn +h

f (x, y(x))dx
xn

❛❦♦ ♣r❡t♣♦st❛✈✐♠♦ ❞❛ ❥❡ f s❛♠♦ ❢✉♥❦❝✐❥❛ ♦❞ x✱ ❍❡✉♥♦✈❛ ♠❡t♦❞❛ ♦❞❣♦✈❛r❛ tr❛♣❡③♥♦♠
♣r❛✈✐❧✉✱ ❦♦❥❛ ✐♠❛ ♣♦❣r❡➨❦✉ r❡❞❛ ✈❡❧✐↔✐♥❡ O(h2 )✳ Pr✐♠✐❥❡t✐♠♦ t❛❦♦➒❡r ❞❛ ❥❡ ③❛ s✈❛❦✉
❛♣r♦❦s✐♠❛❝✐❥✉ yn ♣♦r❡❜♥♦ ❞✈❛ ♣✉t❛ ✐③r❛↔✉♥❛✈❛t✐ ✈r✐❥❡❞♥♦st ❢✉♥❦❝✐❥❡ f ✳
❑❧❛s✐↔♥❛ ❘❑ ♠❡t♦❞❛ ❞❡✜♥✐r❛♥❛ ❥❡ s❛

1
yn+1 = yn + (k1 + 2k2 + 2k3 + k4 )
6
❣❞❥❡ s✉ k1 = hf (xn , yn )✱ k2 = hf (xn + h2 , yn +
hf (xn + h, yn + k3 )✳

k1
)✱
2

k3 = hf (xn + h2 , yn +

k2
)
2

✐ k4 =

Pr✐♠❥❡❞❜❛ ✸✳✷

❆❦♦ ♣r❡t♣♦st❛✈✐♠♦ ❞❛ ❥❡ f s❛♠♦ ❢✉♥❦❝✐❥❛ ♦❞ x✱ ♦♥❞❛ ✐③ ♣r✐♠❥❡❞❜❡ ✸✳✶ ♠♦➸❡♠♦ ♣♦✲
❦❛③❛t✐ ❞❛ ❘❑✲♠❡t♦❞❛ ♦❞❣♦✈❛r❛ ❙✐♠♣s♦♥ ✲♦✈♦❥ ❢♦r♠✉❧✐✱ ✉③ ③❛♠❥❡♥✉ h 7−→ h2 ✳ ❙❥❡t✐♠♦
s❡ ❞❛ ❙✐♠♣s♦♥♦✈❛ ❢♦r♠✉❧❛ ✐♠❛ ♣♦❣r❡➨❦✉ r❡❞❛ ✈❡❧✐↔✐♥❡ O(h2 )✱ ➨t♦ s❡ ♣r❡♥♦s✐ ✐ ♥❛ ❘❑
♠❡t♦❞✉ ✐ ✉ ♦♣➣❡♠ s❧✉↔❛❥✉ ✕ ❦❛❞❛ ❥❡ f ❢✉♥❦❝✐❥❛ ♦❞ x ✐ ♦❞ y ✳ Pr✐♠✐❥❡t✐♠♦ t❛❦♦➒❡r
❞❛ ❥❡ ❦♦❞ ❘❑ ♠❡t♦❞❡ ③❛ s✈❛❦✉ ❛♣r♦❦s✐♠❛❝✐❥✉ yn ♣♦tr❡❜♥♦ ↔❡t✐r✐ ♣✉t❛ ✐③r❛↔✉♥❛✈❛t✐
✈r✐❥❡❞♥♦st ❢✉♥❦❝✐❥❡ f ✳
✶ ■❞❡❥✉

❥❡ ♣r✈✐ ✐③❧♦➸✐♦ ❈✳❘✉♥❣❡ ✉ r❛❞✉ Ü❜❡r ❞✐❡ ♥✉♠❡r✐s❝❤❡ ❆✉✢ös✉♥❣ ✈♦♥ ❉✐✛❡r❡♥t✐❛❧❣❧❡✐❝❤✉♥❣❡♥✱
▼❛t❤❡♠❛t✐s❝❤❡ ❆♥♥❛❧❡♥ ✹✻ ✭✶✽✾✺✮✱ ✶✻✼✕✶✼✽✱ ❛ ❦❛s♥✐❥❡ r❛③✈✐♦ ❲✳❑✉tt❛ ✉ r❛❞✉ ❇❡✐tr❛❣ ③✉r ♥❛❤❡r✉♥❣s✇❡✐s❡♥
■♥t❡❣r❛t✐♦♥ ✈♦♥ ❉✐✛❡r❡♥t✐❛❧❣❧❡✐❝❤✉♥❣❡♥✱ ❩❡✐ts❝❤r✐❢t ❢ür ▼❛t❤❡♠❛t✐❦ ✉♥❞ P❤②s✐❦ ✹✻ ✭✶✾✵✶✮✱ ✹✸✺✕✹✺✸✳

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛

✸✳✷

❇❡❣❛♥♦✈✐➣ ❉❛✈♦r

✶✵

Pr✐♠❥❡r

▼❡t♦❞♦♠ ❘✉♥❣❡ ✲ ❑✉tt❛ r✐❥❡➨✐t✐ ❈❛✉❝❤②✲❡✈ ♣r♦❜❧❡♠

y ′ = x2 − ex + y, y(0) = 1
③❛

x ∈ [0, 0.3]

❘❥❡➨❡♥❥❡
❥❡

∆y =

✿

✐ ♦❝✐❥❡♥✐t✐ ❣r❡➨❦✉✳

Pr✐♠✐❥❡t✐♠♦ ❞❛ ♥❛♠ ❥❡

1
(k
6 1

+ 2k2 + 2k3 + k4 )✳

i
✵

x ∈ [0, 1]✳

❯③♠✐♠♦ s❛❞❛ ❞❛ ♥❛♠ ❥❡

h = 0.15

✐ ♥❡❦❛

❋♦r♠✐r❛❥♠♦ t❛❜❡❧✉

x

y

k = hf (x, y)

∆y

✵

✶

✵

✵

✵✳✵✼✺

✶

✲✵✳✵✸✸✷✸

✲✵✳✵✻✻✹✻

✵✳✵✼✺

✵✳✾✻✻✼✼

✲✵✳✵✶✺✽✷

✲✵✳✵✸✶✻✺

✵✳✶✺

✵✳✾✽✹✶✽

✲✵✳✵✷✸✷✼

✲✵✳✵✷✸✷✼
✲✵✳✵✷✵✷✸

✶

✵✳✶✺

✵✳✾✼✾✼✼

✲✵✳✵✷✸✾✸

✲✵✳✵✷✸✾✸

✵✳✷✷✺

✵✳✾✻✼✽✶

✲✵✳✵✸✺✵✽

✲✵✳✵✼✵✶✼

✵✳✷✷✺

✵✳✾✻✷✷✸

✲✵✳✵✸✺✾✷

✲✵✳✵✼✶✽✹

✵✳✸

✵✳✾✹✸✽✺

✲✵✳✵✹✼✹✵

✲✵✳✵✹✼✹✵
✲✵✳✵✸✺✺✻

❙❛❞❛ ✐③r❛↔✉♥❛ ❥♠♦

y2 ✿
y2 = y1 + ∆y = 0.97977 − 0.03556 = 0.94421.

❖st❛❧♦ ♥❛♠ ❥❡ ❥♦➨ ❞❛ ♣r♦❝✐❥❡♥✐♠♦ ❣r❡➨❦✉✳ ❚♦ ➣❡♠♦ ✉↔✐♥✐t✐ t❛❦♦ ➨t♦ ➣❡♠♦ s✈❡ ✐st♦ ✉r❛❞✐t✐✱
❛❧✐ s❛♠♦ ③❛ ❦♦r❛❦

2h✱

h = 0.3
k1 ✱ k2 ✱ k3 ✱ k 4 ✿

♦❞♥♦s♥♦ t♦ ❜✐ ❦♦❞ ♥❛s s❛❞❛ ❜✐❧♦

❘✉♥❣❡✲♦✈✉ ♦❝❥❡♥✉ ❣r❡➨❦❡✳ ❉❛❦❧❡✱ ✐③r❛↔✉♥❛❥♠♦

✐ ♦♥❞❛ ➣❡♠♦ ❦♦r✐st✐t✐

k1 = 0
k2 = −0.04180
k3 = −0.04807
k4 = −0.09238.
❙❛❞❛ ✐③r❛↔✉♥❛ ❥♠♦

1
y1 = 1 + (0 − 0.04180 − 0.04807 − 0.09238) = 0.96963
6
❖❝❥❡♥✉ ❣r❡➨❦❡ ➣❡♠♦ ✐③r❛↔✉♥❛t✐ ♥❛ s❧❥❡❞❡➣✐ ♥❛↔✐♥

|0.94421 − 0.96963|
1
|yh − y2h |
=
= 0.0017 < 0.005 = 10−2 .
15
15
2
❉❛❦❧❡

y ≈ 0.94.

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛

✸✳✸

❑♦❞ ✉

▼❛t❤❡♠❛t✐❝❛

❇❡❣❛♥♦✈✐➣ ❉❛✈♦r

✶✶

✲✐

f [①❴, ②❴]✿❂x∧ 2 − E ∧ x + y
h = 0.15
❦✶✶ = h ∗ f [0, 1]
❦✶✷ = h ∗ f [0.25, 1]
2 ∗ ❦✶✷
1 − 0.06646/2
❦✶✸ = h ∗ f [0.075, 0.96677]
2 ∗ ❦✶✸
1 − 0.01582
❦✶✹ = h ∗ f [0.15, 0.98418]
1/6 ∗ (0 − 0.06646 − 0.03165 − 0.02327)
1 − 0.02023
❦✷✶ = h ∗ f [0.15, 0.97977]
0.97977 − 0.02393/2
❦✷✷ = h ∗ f [0.225, 0.96781]
2 ∗ ❦✷✷
0.97977 − 0.03508/2
❦✷✸ = h ∗ f [0.225, 0.96223]
2 ∗ ❦✷✸
0.97977 − 0.03592
❦✷✹ = h ∗ f [0.3, 0.94385]
1/6 ∗ (−0.02393 − 0.07017 − 0.07184 − 0.04740)
0.97977 − 0.03556
❦✸✶ = 0.3 ∗ f [0, 1]
❦✸✷ = 0.3 ∗ f [0.15, 1]
❦✸✸ = 0.3 ∗ f [0.15, 1 + ❦✸✷/2]
❦✸✹ = 0.3 ∗ f [0.3, 1 + ❦✸✸]
1 + 1/6 ∗ (0 − 0.04180 − 0.04807 − 0.09238)
(0.96963 − 0.94421)/15
✸✳✹

❑♦♠❡♥t❛r

▼❡t♦❞❛ ❘✉♥❣❡ ✲ ❑✉tt❛ ❥❡ ❥❡❞♥❛ ♦❞ ♥❛❥❜♦❧❥✐❤ ♠❡t♦❞❛ ③❛ r❥❡➨❛✈❛♥❥❡ ❞✐❢❡r❡♥❝✐❥❛❧♥✐❤ ❥❡❞♥❛❞➸❜✐
✉ ♥✉♠❡r✐↔❦♦❥ ♠❡t❡♠❛t✐❝✐✳ ■❛❦♦ ✐♠❛ ♠♥♦❣♦ r❛↔✉♥❛ ✉③ ❞❛♥❛➨♥❥❡ r❛↔✉♥❛r❡ ♣r❡❞st❛✈❧❥❛ ✈r❧♦
❥❛❦ ❛❧❛t ③❛ r❥❡➨❛✈❛♥❥❡ ♦✈♦❣❛ ♣r♦❜❧❡♠❛✳

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛

❇❡❣❛♥♦✈✐➣ ❉❛✈♦r

✶✷

▲✐t❡r❛t✉r❛
❬✶❪ ❘✉❞♦❧❢

❙❝✐t♦✈s❦✐✱

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛✱

❞r✉❣♦ ✐③❞❛♥❥❡✱

❙✈❡✉↔✐❧✐➨t❡

❏✳❏✳

❙tr♦s✲

s♠❛②❡r❛ ✉ ❖s✐❥❡❦✉✱ ❖❞❥❡❧ ③❛ ♠❛t❡♠❛t✐❦✉✱ ❖s✐❥❡❦✱ ✷✵✵✹✳
❬✷❪ ❇✐❧❥❡➨❦❡ s❛ ♣r❡❞❛✈❛♥❥❛ ♣r❡❞♠❡t❛ ◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛ ❦♦❞ ❞r✳ s❝✳ ❊♥❡s❛ ❉✉✈♥❥❛✲
❦♦✈✐➣❛✱ ✈❛♥r✳ ♣r♦❢✳✱ ✷✵✶✼✳