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Киевский национальный университет им. Т. Шевченко

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[НЕСОРТИРОВАННОЕ]

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ChM_teory2

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A 1K&

A 1Cx&

A 1Cz&

A 1

z& d A

C x

Ɉɰɿɧɤɚ ɩɨɯɢɛɤɢ

A 1

G(x) d

G(A)¸ d

A 1

condA

G(A)¸.

condAG(A)

ɇɚɫɥɿɞɨɤ əɤɳɨ C

0, ɬɨ G(x&) d condAG(b)

ȼɥɚɫɬɢɜɨɫɬɿ

condA:

10. condA t 1;

20. condA t

max

i (A)

;

min

i (A)

30. cond(AB) d condA condB;

40. AT A 1

condA

Ⱦɪɭɝɚ ɜɥɚɫɬɢɜɿɫɬɶ ɦɚɽ ɦɿɫɰɟ ɨɫɤɿɥɶɤɢ ɞɨɜɿɥɶɧɚ ɧɨɪɦɚ ɦɚɬɪɢɰɿ ɧɟ ɦɟɧɲɟ ʀʀ

ɧɚɣɛɿɥɶɲɨɝɨ ɡɚ ɦɨɞɭɥɟɦ ɜɥɚɫɧɨɝɨ ɡɧɚɱɟɧɧɹ. Ɂɧɚɱɢɬɶ

t max

. Ɉɫɤɿɥɶɤɢ

ɜɥɚɫɧɿ ɡɧɚɱɟɧɧɹ ɦɚɬɪɢɰɶ A 1 ɬɚ A ɜɡɚɽɦɧɨ ɨɛɟɪɧɟɧɿ, ɬɨ

A 1

t max

min

əɤɳɨ condA

1, ɬɨ ɫɢɫɬɟɦɚ ɧɚɡɢɜɚɽɬɶɫɹ ɩɨɝɚɧɨ ɨɛɭɦɨɜɥɟɧɨɸ.

Ɉɰɿɧɤɚ ɜɩɥɢɜɭ ɩɨɯɢɛɨɤ ɡɚɨɤɪɭɝɥɟɧɧɹ ɩɪɢ ɨɛɱɢɫɥɟɧɧɿ ɪɨɡɜ‘ɹɡɤɭ ɋɅ Ɋ

ɬɚɤɚ (Ⱦɠ. ɍɿɥɤɿɧɫɨɧ): G(A)

O(nE t ) ,

G(b)

O(E t ),

ɞɟ

- ɪɨɡɪɹɞɧɿɫɬɶ

ȿɈɆ, t

- ɤɿɥɶɤɿɫɬɶ ɪɨɡɪɹɞɿɜ, ɳɨ ɜɿɞɜɨɞɢɬɶɫɹ ɩɿɞ ɦɚɧɬɢɫɭ ɱɢɫɥɚ. Ɂ ɨɰɿɧɤɢ (3)

ɜɢɬɿɤɚɽ:

G(x&)

condAuO(nE t ). ȼɢɫɧɨɜɨɤ: ɧɚɣɩɪɨɫɬɿɲɢɣ ɫɩɨɫɿɛ ɩɿɞɜɢɳɢɬɢ

ɬɨɱɧɿɫɬɶ ɨɛɱɢɫɥɟɧɧɹ ɪɨɡɜ‘ɹɡɤɭ ɩɨɝɚɧɨ ɨɛɭɦɨɜɥɟɧɨʀ ɋɅ Ɋ – ɡɛɿɥɶɲɢɬɢ ɪɨɡɪɹɞɧɿɫɬɶ ȿɈɆ ɩɪɢ ɨɛɱɢɫɥɟɧɧɹɯ. ȱɧɲɿ ɫɩɨɫɨɛɢ ɩɨɜ‘ɹɡɚɧɿ ɡ ɪɨɡɝɥɹɞɨɦ ɰɿɽʀ

ɋɅ Ɋ ɹɤ ɧɟɤɨɪɟɤɬɧɨʀ ɡɚɞɚɱɿ ɿɡ				ɡɚɫɬɨɫɭɜɚɧɧɹɦ ɜɿɞɩɨɜɿɞɧɢɯ ɦɟɬɨɞɿɜ ʀʀ
ɪɨɡɜ‘ɹɡɚɧɧɹ (ɞɢɜ. [ȻɀɄ, ɫ. 306]).
ɉɪɢɤɥɚɞ ɩɨɝɚɧɨ ɨɛɭɦɨɜɥɟɧɨʀ ɫɢɫɬɟɦɢ – ɫɢɫɬɟɦɢ ɡ ɦɚɬɪɢɰɟɸ Ƚɿɥɶɛɟɪɬɚ
§	1		·n		9
Hn ¨			¸	, ɧɚɩɪɢɤɥɚɞ condH8 10		.

© i j		1¹i, j 1

4. ȱɬɟɪɚɰɿɣɧɿ ɦɟɬɨɞɢ ɞɥɹ ɫɢɫɬɟɦ ɪɿɜɧɹɧɶ

4.1. ȱɬɟɪɚɰɿɣɧɿ ɦɟɬɨɞɢ ɪɨɡɜ’ɹɡɚɧɧɹ ɋɅ Ɋ [ɋȽ, 82-86], [ȻɀɄ, 269-270, 287-

290]

ɋɢɫɬɟɦɭ

ɡɜɨɞɢɦɨ ɞɨ ɜɢɝɥɹɞɭ

Ax&

(1)

Bx&

(2)

Ȼɭɞɶ ɹɤɚ ɫɢɫɬɟɦɚ

C Ax&

(3)

ɦɚɽ ɜɢɝɥɹɞ (2) ɿ ɩɪɢ detC

0 ɟɤɜɿɜɚɥɟɧɬɧɚ ɫɢɫɬɟɦɿ (1). ɇɚɩɪɢɤɥɚɞ, ɞɥɹ

Ax&

(3c)

4.1.1. Ɇɟɬɨɞ ɩɪɨɫɬɨʀ ɿɬɟɪɚɰɿʀ (ɦ.ɩ.ɿ.)

ɐɟɣ ɦɟɬɨɞ ɡɚɫɬɨɫɨɜɭɽɬɶɫɹ ɞɨ ɪɿɜɧɹɧɧɹ (2)

x&0

x&k

Bx&k

f ,

(4)

–ɩɨɱɚɬɤɨɜɟ ɧɚɛɥɢɠɟɧɧɹ ɡɚɞɚɧɨ. ȱɬɟɪɚɰɿɣɧɢɣ ɩɪɨɰɟɫ ɡɛɿɝɚɽɬɶɫɹ, ɬɨɛɬɨ

x&k

o 0, k o f , ɹɤɳɨ

(5)

ɉɪɢ ɰɶɨɦɭ ɦɚɽ ɦɿɫɰɟ ɨɰɿɧɤɚ

x&n

x&1

x&0

(6)

4.1.2. Ɇɟɬɨɞ əɤɨɛɿ

i . Ɂɜɟɞɟɦɨ ɫɢɫɬɟɦɭ (1) ɞɨ ɜɢɝɥɹɞɭ

ɉɪɢɩɭɫɬɢɦɨ aii

¦ aij

¦ aij xj bi ,i 1,n .

j i 1

j 1 aii

aii

ȱɬɟɪɚɰɿɣɧɢɣ ɩɪɨɰɟɫ ɡɚɩɢɲɟɦɨ ɭ ɜɢɝɥɹɞɿ

xik 1

aij

xkj

aij

xkj

k 0,1....,i 1,n

(7)

j 1 aii

j i

1 aii

aii

ȱɬɟɪɚɰɿɣɧɢɣ ɩɪɨɰɟɫ ɡɛɿɝɚɽɬɶɫɹ ɞɨ ɪɨɡɜ’ɹɡɤɭ, ɹɤɳɨ ɜɢɤɨɧɭɽɬɶɫɹ ɭɦɨɜɚ

, i

aij

aii

1,n

ɐɟ ɭɦɨɜɚ ɞɿɚɝɨɧɚɥɶɧɨʀ ɩɟɪɟɜɚɝɢ ɦɚɬɪɢɰɿ A. əɤɳɨ ɠ

n	d q aii , i 1,n,0 d q 1,
¦ aij	d q aii , i 1,n,0 d q 1,	(8)

j 1 i j

ɬɨ ɦɚɽ ɦɿɫɰɟ ɨɰɿɧɤɚ ɬɨɱɧɨɫɬɿ:

x&n x&

x&1 x&0

1 q

4.1.3. Ɇɟɬɨɞ Ɂɟɣɞɟɥɹ

ȼ ɤɨɦɩɨɧɟɧɬɧɨɦɭ ɜɢɝɥɹɞɿ ɿɬɟɪɚɰɿɣɧɢɣ ɦɟɬɨɞ Ɂɟɣɞɟɥɹ ɡɚɩɢɫɭɽɬɶɫɹ ɬɚɤ:

	i 1				n		aij		bi
xkj 1	¦	aij	xkj	1	¦			xkj		(9)
									aii
	j 1 aii				j i	1 aii

ɇɚ ɜɿɞɦɿɧɭ ɜɿɞ ɦɟɬɨɞɭ əɤɨɛɿ ɧɚ k-ɦɭ-ɤɪɨɰɿ ɩɨɩɟɪɟɞɧɿ ɤɨɦɩɨɧɟɧɬɢ ɪɨɡɜ‘ɹɡɤɭ ɛɟɪɭɬɶɫɹ ɡ k+1- ɨʀ ɿɬɟɪɚɰɿʀ.

Ⱦɨɫɬɚɬɧɹ ɭɦɨɜɚ ɡɛɿɠɧɨɫɬɿ ɦɟɬɨɞɭ Ɂɟɣɞɟɥɹ - AT A 0.

4.1.4.Ɇɚɬɪɢɱɧɚ ɿɧɬɟɪɩɪɟɬɚɰɿɹ ɦɟɬɨɞɿɜ əɤɨɛɿ ɿ Ɂɟɣɞɟɥɹ

ɉɨɞɚɦɨ ɦɚɬɪɢɰɸ A ɭ ɜɢɝɥɹɞɿ

A A1 D A2 ,

ɞɟ A1 - ɧɢɠɧɿɣ ɬɪɢɤɭɬɧɢɤ ɦɚɬɪɢɰɿ Ⱥ, A2 - ɜɟɪɯɧɿɣ ɬɪɢɤɭɬɧɢɤ ɦɚɬɪɢɰɿ Ⱥ, D - ʀʀ ɞɿɚɝɨɧɚɥɶ. Ɍɨɞɿ ɫɢɫɬɟɦɭ (1) ɡɚɩɢɲɟɦɨ ɭ ɜɢɝɥɹɞɿ

ɚɛɨ		Dx& A1x& A2 x& b
	x& D 1A1x& D 1A2 x& D 1b

Ɇɚɬɪɢɱɧɢɣ ɡɚɩɢɫ ɦɟɬɨɞɭ əɤɨɛɿ:				A2 )x&k D 1b ;
	x&k 1	D 1(A1
ɦɟɬɨɞɭ Ɂɟɣɞɟɥɹ:			D 1b ɚɛɨ		(D A1)x&k 1	A2 x&k b .
x&k 1	D 1A1x&k 1 D 1A2 x&k
ɇɟɨɛɯɿɞɧɚ ɿ ɞɨɫɬɚɬɧɹ ɭɦɨɜɚ ɡɛɿɠɧɨɫɬɿ ɦɟɬɨɞɭ əɤɨɛɿ:
- ɜɫɿ ɤɨɪɟɧɿ ɪɿɜɧɹɧɧɹ det D			A1	A2	0 ɩɨ ɦɨɞɭɥɸ ɛɿɥɶɲɟ 1.
ɇɟɨɛɯɿɞɧɚ ɿ ɞɨɫɬɚɬɧɹ ɭɦɨɜɚ ɡɛɿɠɧɨɫɬɿ ɦɟɬɨɞɭ ɦɟɬɨɞɚ Ɂɟɣɞɟɥɹ:
- ɜɫɿ ɤɨɪɟɧɿ ɪɿɜɧɹɧɧɹ det A1			D	A2	0 ɩɨ ɦɨɞɭɥɸ ɛɿɥɶɲɟ 1.

4.1.5.Ɉɞɧɨɤɪɨɤɨɜɿ ( ɞɜɨɲɚɪɨɜɿ ) ɿɬɟɪɚɰɿɣɧɿ ɦɟɬɨɞɢ

Ʉɚɧɨɧɿɱɧɨɸ ɮɨɪɦɨɸ ɨɞɧɨɤɪɨɤɨɜɨɝɨ ɿɬɟɪɚɰɿɣɧɨɝɨ ɦɟɬɨɞɭ ɪɨɡɜ’ɹɡɤɭ

ɋɅ Ɋ ɽ ɣɨɝɨ ɡɚɩɢɫ ɭ ɜɢɝɥɹɞɿ

	k	1	k
	x	x		k
Bk	x	x		Ax b	(10)
Bk		Wk 1		Ax b	(10)
		Wk 1

Ɍɭɬ {Bk }- ɩɨɫɥɿɞɨɜɧɿɫɬɶ ɦɚɬɪɢɰɶ (ɩɟɪɟ ɨɛɭɦɨɜɥɸɸɱɿ ɦɚɬɪɢɰɿ), ɳɨ ɡɚɞɚɸɬɶ ɿɬɟɪɚɰɿɣɧɢɣ ɦɟɬɨɞ ɧɚ ɤɨɠɧɨɦɭ ɤɪɨɰɿ; { k 1}- ɿɬɟɪɚɰɿɣɧɿ ɩɚɪɚɦɟɬɪɢ.

əɤɳɨ Bk	E , ɬɨ ɿɬɟɪɚɰɿɣɧɢɣ ɩɪɨɰɟɫ ɧɚɡɢɜɚɽɬɶɫɹ ɹɜɧɢɦ
	xk 1 xk	Wk 1 Axk b .
əɤɳɨ Bk
əɤɳɨ Bk	E, ɬɨ ɿɬɟɪɚɰɿɣɧɢɣ ɩɪɨɰɟɫ ɧɚɡɢɜɚɽɬɶɫɹ ɧɟɹɜɧɢɦ
	B x&k 1 Fk .
		&

ȼ ɰɶɨɦɭ ɜɢɩɚɞɤɭ ɧɚ ɤɨɠɧɿɣ ɿɬɟɪɚɰɿʀ ɧɟɨɛɯɿɞɧɨ ɪɨɡɜ‘ɹɡɭɜɚɬɢ ɋɅ							Ɋ.
əɤɳɨ	k 1	, Bk B, ɬɨ	ɿɬɟɪɚɰɿɣɧɢɣ		ɩɪɨɰɟɫ		ɧɚɡɢɜɚɽɬɶɫɹ
ɫɬɚɰɿɨɧɚɪɧɢɦ; ɿɧɚɤɲɟ – ɧɟɫɬɚɰɿɨɧɚɪɧɢɦ.
Ɇɟɬɨɞɚɦ, ɳɨ ɪɨɡɝɥɹɧɭɬɿ ɜɢɳɟ ɜɿɞɩɨɜɿɞɚɸɬɶ:
ɦɟɬɨɞɭ ɩɪɨɫɬɨʀ ɿɬɟɪɚɰɿʀ (3c):			Bk	E,	k 1	;
ɦɟɬɨɞɭ əɤɨɛɿ:			Bk	D,	k 1	1;
ɦɟɬɨɞɭ Ɂɟɣɞɟɥɹ:			Bk	D	A1,	k 1	1.

4.1.6. Ɂɛɿɠɧɨɫɬɿ ɫɬɚɰɿɨɧɚɪɧɢɯ ɿɬɟɪɚɰɿɣɧɢɯ ɩɪɨɰɟɫɿɜ ɭ ɜɢɩɚɞɤɭ ɫɢɦɟɬɪɢɱɧɢɯ

ɦɚɬɪɢɰɶ

Ɋɨɡɝɥɹɧɟɦɨ

ɜɢɩɚɞɨɤ ɫɢɦɟɬɪɢɱɧɢɯ

ɦɚɬɪɢɰɶ

ɫɬɚɰɿɨɧɚɪɧɢɣ

ɿɬɟɪɚɰɿɣɧɢɣ ɩɪɨɰɟɫ Bk

E ,

k 1

ɇɟɯɚɣ ɞɥɹ Ⱥ ɫɩɪɚɜɟɞɥɢɜɿ ɧɟɪɿɜɧɨɫɬɿ

2 E ,

(11)

W W0

Ɍɨɞɿ

ɩɪɢ

ɜɢɛɨɪɿ

J1 J2

ɿɬɟɪɚɰɿɣɧɢɣ

ɩɪɨɰɟɫ

ɡɛɿɝɚɽɬɶɫɹ.

ɇɚɣɛɿɥɶɲ

ɬɨɱɧɢɦ

ɡɧɚɱɟɧɧɹɦ

1 ,

ɩɪɢ

ɹɤɢɯ

ɜɢɤɨɧɭɸɬɶɫɹ

ɨɛɦɟɠɟɧɧɹ

(11) ɽ

min

i A ,

2 max

i A .

Ɍɨɞɿ

J2 J1

1 [

(Ɂɚɭɜɚɠɢɦɨ, ɳɨ ɚɧɚɥɨɝɿɱɧɨ ɨɛɱɢɫɥɸɽɬɶɫɹ

q ɿ

ɞɥɹ ɦɟɬɨɞɭ

ɪɟɥɚɤɫɚɰɿʀ

ɪɨɡɜ‘ɹɡɚɧɧɹ

ɧɟɥɿɧɿɣɧɢɯ

ɪɿɜɧɹɧɶ,

ɞɟ

min

f (x)

max

f (x)

) ɿ ɫɩɪɚɜɟɞɥɢɜɚ ɨɰɿɧɤɚ

x&n

x&0

əɜɧɢɣ ɦɟɬɨɞ ɡ ɛɚɝɚɬɶɦɚ ɩɚɪɚɦɟɬɪɚɦɢ

E, ^

k `: min q

o min,

ɹɤɿ ɨɛɱɢɫɥɸɸɬɶɫɹ ɡɚ ɞɨɩɨɦɨɝɨɸ ɧɭɥɿɜ ɛɚɝɚɬɨɱɥɟɧɚ ɑɟɛɢɲɨɜɚ, ɧɚɡɢɜɚɸɬɶɫɹ ɿɬɟɪɚɰɿɣɧɢɦ ɦɟɬɨɞɨɦ ɡ ɱɟɛɢɲɟɜɫɶɤɢɦ ɧɚɛɨɪɨɦ ɩɚɪɚɦɟɬɪɿɜ.

4.1.7. Ɇɟɬɨɞ ɜɟɪɯɧɶɨʀ ɪɟɥɚɤɫɚɰɿʀ

ɍɡɚɝɚɥɶɧɟɧɧɹɦ

ɦɟɬɨɞɭ

Ɂɟɣɞɟɥɹ

ɦɟɬɨɞ

ɜɟɪɯɧɶɨʀ

ɪɟɥɚɤɫɚɰɿʀ:

D ZA1 x

Z x

Ax&k

b ,

ɞɟ D - ɞɿɚɝɨɧɚɥɶɧɚ ɦɚɬɪɢɰɹ ɡ ɟɥɟɦɟɧɬɚɦɢ aii

ɩɨ ɞɿɚɝɨɧɚɥɿ.

ɡɚɞɚɧɢɣ

ɱɢɫɥɨɜɢɣ ɩɚɪɚɦɟɬɪ.

Ɍɟɩɟɪ

A1,

əɤɳɨ

ɬɨ ɦɟɬɨɞ

ɜɟɪɯɧɶɨʀ

ɪɟɥɚɤɫɚɰɿʀ ɡɛɿɝɚɽɬɶɫɹ ɩɪɢ ɭɦɨɜɿ 0

2 .

ɉɚɪɚɦɟɬɪ

ɩɿɞɛɢɪɚɽɬɶɫɹ

ɟɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɡ ɭɦɨɜɢ ɦɿɧɿɦɚɥɶɧɨʀ ɤɿɥɶɤɨɫɬɿ ɿɬɟɪɚɰɿɣ.

4.1.8.Ɇɟɬɨɞɢ ɜɚɪɿɚɰɿɣɧɨɝɨ ɬɢɩɭ

Ⱦɨ ɰɢɯ ɦɟɬɨɞɿɜ ɜɿɞɧɨɫɹɬɶɫɹ: ɦɟɬɨɞ ɦɿɧɿɦɚɥɶɧɢɯ ɧɟɜ’ɹɡɨɤ, ɦɟɬɨɞ

ɦɿɧɿɦɚɥɶɧɢɯ ɩɨɩɪɚɜɨɤ, ɦɟɬɨɞ ɧɚɣɲɜɢɞɲɨɝɨ ɫɩɭɫɤɭ, ɦɟɬɨɞ ɫɩɪɹɠɟɧɢɯ ɝɪɚɞɿɽɧɬɿɜ. ȼɨɧɢ ɞɨɡɜɨɥɹɸɬɶ ɨɛɱɢɫɥɸɜɚɬɢ ɧɚɛɥɢɠɟɧɧɹ ɛɟɡ ɜɢɤɨɪɢɫɬɚɧɧɹ

ɚɩɪɿɨɪɧɨʀ ɿɧɮɨɪɦɚɰɿʀ ɩɪɨ

2 ɜ (11).

ɇɟɯɚɣ B E .

Ⱦɥɹ ɦɟɬɨɞɭ ɦɿɧɿɦɚɥɶɧɢɯ ɧɟɜ’ɹɡɨɤ ɩɚɪɚɦɟɬɪɢ k 1

ɨɛɱɢɫɥɸɸɬɶɫɹ ɡ ɭɦɨɜɢ

2Wk 1 r&k , Ar&k Wk2 1`

2 o min .

Ɍɨɦɭ

r&k

Ar&k

Ar&k ,r&k

ɞɟ r&k

Ax&k b- ɧɟɜ’ɹɡɤɚ.

ɍɦɨɜɚ ɞɥɹ ɡɚɜɟɪɲɟɧɧɹ ɿɬɟɪɚɰɿɣɧɨɝɨ ɩɪɨɰɟɫɭ:

r&n

ɒɜɢɞɤɿɫɬɶ ɡɛɿɠɧɨɫɬɿ ɰɶɨɝɨ ɦɟɬɨɞɭ ɫɩɿɜɩɚɞɚɽ ɿɡ ɲɜɢɞɤɿɫɬɸ ɦɟɬɨɞɭ ɩɪɨɫɬɨʀ

ɿɬɟɪɚɰɿʀ ɡ ɨɞɧɢɦ ɨɩɬɢɦɚɥɶɧɢɦ ɩɚɪɚɦɟɬɪɨɦ W0	J1	2
ɿɬɟɪɚɰɿʀ ɡ ɨɞɧɢɦ ɨɩɬɢɦɚɥɶɧɢɦ ɩɚɪɚɦɟɬɪɨɦ W0	J1	J2 .
ɧɚɥɨɝɿɱɧɨ ɛɭɞɭɸɬɶɫɹ ɦɟɬɨɞɢ ɡ B	E .	Ɇɚɬɪɢɰɹ B ɧɚɡɢɜɚɽɬɶɫɹ

ɩɟɪɟɨɛɭɦɨɜɥɸɜɚɱɟɦ ɿ ɞɨɡɜɨɥɹɽ ɩɿɞɜɢɳɢɬɢ ɲɜɢɞɤɿɫɬɶ ɡɛɿɠɧɨɫɬɿ ɿɬɟɪɚɰɿɣɧɨɝɨ ɩɪɨɰɟɫɭ. Ƀɨɝɨ ɜɢɛɢɪɚɸɬɶ ɡ ɭɦɨɜ ɚ) ɥɟɝɤɨ ɪɨɡɜ’ɹɡɭɜɚɬɢ ɋɅ Ɋ Bx&k F&k (ɞɿɚɝɨɧɚɥɶɧɢɣ, ɬɪɢɤɭɬɧɿɣ, ɞɨɛɭɬɨɤ ɬɪɢɤɭɬɧɿɯ ɬɚ ɿɧɲɟ); ɛ) ɡɦɟɧɲɟɧɧɹ ɱɢɫɥɚ ɨɛɭɦɨɜɥɟɧɨɫɬɿ ɦɚɬɪɢɰɿ B 1 A ɭ ɩɨɪɿɜɧɹɧɧɿ ɡ A .

4.2. Ɇɟɬɨɞɢ ɪɨɡɜ’ɹɡɚɧɧɹ ɧɟɥɿɧɿɣɧɢɯ ɫɢɫɬɟɦ [ɋȽ, 209-211]

Ɋɨɡɝɥɹɧɟɦɨ ɫɢɫɬɟɦɭ ɪɿɜɧɹɧɶ:
° f1(x1,	, xn )	0
®
°	, xn )	0
¯ fn (x1,	, xn )	0
ɉɟɪɟɩɢɲɟɦɨ ʀʀ ɭ ɜɟɤɬɨɪɧɨɦɭ ɜɢɝɥɹɞɿ :
f (x&) 0		(1)

4.2.1.Ɇɟɬɨɞ ɩɪɨɫɬɨʀ ɿɬɟɪɚɰɿʀ

ȼ ɰɶɨɦɭ ɦɟɬɨɞɿ ɪɿɜɧɹɧɧɹ (1) ɡɜɨɞɢɬɶɫɹ ɞɨ ɟɤɜɿɜɚɥɟɧɬɧɨɝɨ ɜɢɝɥɹɞɭ

x& )(x&)	(2)
ȱɬɟɪɚɰɿɣɧɢɣ ɩɪɨɰɟɫ ɩɪɟɞɫɬɚɜɢɦɨ ɭ ɜɢɝɥɹɞɿ:
&	(3)
x&k 1 )(x&k ),	(3)

ɩɨɱɚɬɤɨɜɟ ɧɚɛɥɢɠɟɧɧɹ)x&0 – ɡɚɞɚɧɨ.

ɇɟɯɚɣ ɨɩɟɪɚɬɨɪ ɜɢɡɧɚɱɟɧɢɣ ɧɚ ɦɧɨɠɢɧɿ H . Ɂɚ ɬɟɨɪɟɦɨɸ ɩɪɨ ɫɬɢɫɤɭɸɱɿ ɜɿɞɨɛɪɚɠɟɧɧɹ ɿɬɟɪɚɰɿɣɧɢɣ ɩɪɨɰɟɫ (3) ɫɯɨɞɢɬɶɫɹ, ɹɤɳɨ

ɜɢɤɨɧɭɽɬɶɫɹ ɭɦɨɜɚ

)(y&)

d q

, 0 q 1,

(4)

)(x&)

ɚɛɨ

(x&)

§ wMi

·n

(5)

)c(x)

. Ⱦɥɹ ɩɨɯɢɛɤɢ ɫɩɪɚɜɟɞɥɢɜɚ ɨɰɿɧɤɚ

¹i, j 1

x&m x&

x&1

x&0

ɑɚɫɬɢɧɧɢɦ ɜɢɩɚɞɤɨɦ ɦɟɬɨɞɭ ɩɪɨɫɬɨʀ ɿɬɟɪɚɰɿʀ ɽ ɦɟɬɨɞ ɪɟɥɚɤɫɚɰɿʀ ɞɥɹ ɪɿɜɧɹɧɧɹ

(1)

x&k 1 x&k WF&(x&k ) ,

ɞɟ W		.

	Fc2x&

4.2.2.Ɇɟɬɨɞ ɇɶɸɬɨɧɚ

Ɋɨɡɝɥɹɧɟɦɨ ɪɿɜɧɹɧɧɹ

F(x&)

ɉɪɟɞɫɬɚɜɢɦɨ ɣɨɝɨ ɭ ɜɢɝɥɹɞɿ

x&k ) 0,

F(x&k ) Fc([k )(x&

(6)

x&k

(x&k

x&k ), 0

F(x)

·n

ɞɟ [

Ɍɭɬ

i ¸

ɦɚɬɪɢɰɹ əɤɨɛɿ

ɞɥɹ F(x&):. Ɇɨɠɟɦɨ ɧɚɛɥɢɠɟɧɨ ɜɜɚɠɚɬɢ

j ¹i, j 1

x&k .

Ɍɨɞɿ ɡ (6) ɦɚɬɢɦɟɦɨ:

x&k ) 0

F(x&k ) Fc(x&k )(x&k 1

(7)

ȱɬɟɪɚɰɿɣɧɢɣ ɩɪɨɰɟɫ ɩɪɟɞɫɬɚɜɢɦɨ ɭ ɜɢɝɥɹɞɿ:

x&k 1

x&k

Fc(x&k )@

1 F(x&k ) .

(8)

Ⱦɥɹ ɪɟɚɥɿɡɚɰɿʀ ɦɟɬɨɞɭ ɇɶɸɬɨɧɚ ɩɨɬɪɿɛɧɨ, ɳɨɛ ɿɫɧɭɜɚɥɚ ɨɛɟɪɧɟɧɚ ɦɚɬɪɢɰɹ

)

Fc(x

Ɇɨɠɧɚ ɧɟ ɲɭɤɚɬɢ ɨɛɟɪɧɟɧɭ ɦɚɬɪɢɰɸ, ɚ ɪɨɡɜ’ɹɡɭɜɚɬɢ ɧɚ ɤɨɠɧɿɣ ɿɬɟɪɚɰɿʀ

ɋɅ Ɋ

)

A z

F(x

0,1,2,...

(9)

°x

ɞɟ x&0 —ɡɚɞɚɧɨ, ɚ ɦɚɬɪɢɰɹ Ak

Fc(x&k ).

Ɇɟɬɨɞ ɦɚɽ ɤɜɚɞɪɚɬɢɱɧɭ ɡɛɿɠɧɿɫɬɶ, ɹɤɳɨ ɞɨɛɪɟ ɜɢɛɪɚɧɨ ɩɨɱɚɬɤɨɜɟ

ɧɚɛɥɢɠɟɧɧɹ. ɋɤɥɚɞɧɿɫɬɶ ɦɟɬɨɞɭ (ɩɪɢ ɭɦɨɜɿ ɜɢɤɨɪɢɫɬɚɧɧɹ ɦɟɬɨɞɭ Ƚɚɭɫɫɚ

ɪɨɡɜ‘ɹɡɚɧɧɹ ɋɅ Ɋ (9)) ɧɚ ɤɨɠɧɿɣ ɿɬɟɪɚɰɿʀ

n3 O(n2 ),

ɞɟ n – ɪɨɡɦɿɪɧɿɫɬɶ ɫɢɫɬɟɦɢ (1).

4.2.3. Ɇɨɞɢɮɿɤɨɜɚɧɢɣ ɦɟɬɨɞ ɇɶɸɬɨɧɚ

ȱɬɟɪɚɰɿɣɧɢɣ ɩɪɨɰɟɫ ɦɚɽ ɜɢɝɥɹɞ : @

x&k 1 x&k Fc(x&0 ) 1 F(x&k ).

Ɍɟɩɟɪ ɨɛɟɪɧɟɧɚ ɦɚɬɪɢɰɹ ɨɛɱɢɫɥɸɽɬɶɫɹ ɬɿɥɶɤɢ ɧɚ ɧɭɥɶɨɜɿɣ ɿɬɟɪɚɰɿʀ. ɇɚ ɿɧɲɢɯ

ɧɚɛɥɢɠɟɧɧɹ

ɡɜɨɞɢɬɶɫɹ ɞɨ ɦɧɨɠɟɧɧɹ

ɦɚɬɪɢɰɿ

ɨɛɱɢɫɥɟɧɧɹ ɧɨɜɨɝɨ

ɧɚ ɜɟɤɬɨɪ

& &k

Fc x

F(x

) ɬɚ ɞɨɞɚɜɚɧɧɹ ɞɨ x

Ɂɚɩɢɲɟɦɨ ɦɟɬɨɞ ɭ ɜɢɝɥɹɞɿ ɫɢɫɬɟɦɢ ɥɿɧɿɣɧɢɯ ɪɿɜɧɹɧɶ (ɚɧɚɥɨɝ (9))

° 0

)

A z

F (x

® &k

&k .

(10)

°x

Ɉɫɤɿɥɶɤɢ ɦɚɬɢɰɹ A0 ɪɨɡɤɥɚɞɚɽɬɶɫɹ ɧɚ ɬɪɢɤɭɬɧɿ (ɚɛɨ ɨɛɟɪɬɚɽɬɶɫɹ) ɨɞɢɧ ɪɚɡ, ɬɨ

ɫɤɥɚɞɧɿɫɬɶ ɰɶɨɝɨ ɦɟɬɨɞɭ ɧɚ ɨɞɧɿɣ ɿɬɟɪɚɰɿʀ (ɨɤɪɿɦ ɧɭɥɶɨɜɨʀ) Qn O(n2 ) .

ɥɟ ɰɟɣ

ɦɟɬɨɞ ɦɚɽ ɥɿɧɿɣɧɭ ɲɜɢɞɤɿɫɬɶ ɡɛɿɠɧɨɫɬɿ.

Ɇɨɠɥɢɜɟ ɰɢɤɥɿɱɧɟ ɡɚɫɬɨɫɭɜɚɧɧɹ ɦɨɞɢɮɿɤɨɜɚɧɨɝɨ ɦɟɬɨɞɭ ɇɶɸɬɨɧɚ,

ɬɨɛɬɨ ɤɨɥɢ ɨɛɟɪɧɟɧɭ ɦɚɬɪɢɰɸ ɩɨɯɿɞɧɢɯ ɲɭɤɚɽɦɨ ɬɚ ɨɛɟɪɬɚɽɦɨ ɱɟɪɟɡ ɩɟɜɧɟ ɱɢɫɥɨ ɤɪɨɤɿɜ ɿɬɟɪɚɰɿɣɧɨɝɨ ɩɪɨɰɟɫɭ.

Ɂɚɞɚɱɚ 9 ɉɨɛɭɞɭɜɚɬɢ ɚɧɚɥɨɝ ɦɟɬɨɞɭ ɫɿɱɧɢɯ ɞɥɹ ɫɢɫɬɟɦ ɧɟɥɿɧɿɣɧɢɯ ɪɿɜɧɹɧɶ.

5. Ⱥɥɝɟɛɪɚʀɱɧɚ ɩɪɨɛɥɟɦɚ ɜɥɚɫɧɢɯ ɡɧɚɱɟɧɶ.

ɇɟɯɚɣ ɡɚɞɚɧɨ ɦɚɬɪɢɰɸ A: (n n).		Ɍɨɞɿ	ɡɚɞɚɱɚ ɧɚ	ɜɥɚɫɧɿ ɡɧɚɱɟɧɧɹ
ɫɬɚɜɢɬɶɫɹ ɬɚɤ: ɡɧɚɣɬɢ ɱɢɫɥɨ	ɬɚ ɜɟɤɬɨɪ x	0, ɳɨ ɡɚɞɨɜɨɥɶɧɹɸɬɶ ɪɿɜɧɹɧɧɸ
	Ax	x .		(1)
ɧɚɡɢɜɚɽɬɶɫɹ ɜɥɚɫɧɢɦ ɡɧɚɱɟɧɧɹɦ A, ɚ x - ɜɥɚɫɧɢɦ ɜɟɤɬɨɪɨɦ. Ɂ (1)
det(A E)	Pn ( ) ( 1)n	n an	n 1 ... a0	0.

Ɍɭɬ Pn ( ) – ɯɚɪɚɤɬɟɪɢɫɬɢɱɧɢɣ ɛɚɝɚɬɨɱɥɟɧ.

Ⱦɥɹ ɪɨɡɜ’ɹɡɚɧɧɹ ɛɚɝɚɬɶɨɯ ɡɚɞɚɱ ɦɟɯɚɧɿɤɢ, ɮɿɡɢɤɢ, ɯɿɦɿʀ ɩɨɬɪɿɛɧɟ

ɡɧɚɯɨɞɠɟɧɧɹ ɜɫɿɯ ɜɥɚɫɧɢɯ ɡɧɚɱɟɧɶ i , i					, ɚ ɿɧɨɞɿ ɣ ɜɫɿɯ ɜɥɚɫɧɢɯ ɜɟɤɬɨɪɿɜ
				1,n
x&	, ɳɨ ɜɿɞɩɨɜɿɞɚɸɬɶ	i	. ɐɸ ɡɚɞɚɱɭ ɧɚɡɢɜɚɸɬɶ ɩɨɜɧɨɸ ɩɪɨɛɥɟɦɨɸ ɜɥɚɫɧɢɯ
i

ɡɧɚɱɟɧɶ.

ȼ ɛɚɝɚɬɶɨɯ ɜɢɩɚɞɤɚɯ ɩɨɬɪɿɛɧɨ ɡɧɚɣɬɢ ɥɢɲɟ ɦɚɤɫɢɦɚɥɶɧɟ ɚɛɨ ɦɿɧɿɦɚɥɶɧɟ ɡɚ ɦɨɞɭɥɟɦ ɜɥɚɫɧɟ ɡɧɚɱɟɧɧɹ ɦɚɬɪɢɰɿ. ɉɪɢ ɞɨɫɥɿɞɠɟɧɧɿ ɫɬɿɣɤɨɫɬɿ

ɤɨɥɢɜɚɥɶɧɢɯ ɩɪɨɰɟɫɿɜ ɿɧɨɞɿ ɩɨɬɪɿɛɧɨ ɡɧɚɣɬɢ ɞɜɚ ɦɚɤɫɢɦɚɥɶɧɢɯ ɡɚ ɦɨɞɭɥɟɦ ɜɥɚɫɧɢɯ ɡɧɚɱɟɧɧɹ ɦɚɬɪɢɰɿ.

Ɉɫɬɚɧɧɿ ɞɜɿ ɡɚɞɚɱɿ (2)-(4) ɧɚɡɢɜɚɸɬɶ ɱɚɫɬɤɨɜɢɦɢ ɩɪɨɛɥɟɦɚɦɢ ɜɥɚɫɧɢɯ ɡɧɚɱɟɧɶ.

5.1. ɋɬɟɩɟɧɟɜɢɣ ɦɟɬɨɞ [ȻɀɄ, 309-314], [ɄȻɆ, 149-157]
1) Ɂɧɚɯɨɞɠɟɧɧɹ max : 1	max	2 t	3 t ....

ɇɟɯɚɣ x&0 - ɡɚɞɚɧɢɣ ɜɟɤɬɨɪ, ɛɭɞɟɦɨ ɩɨɫɥɿɞɨɜɧɨ ɨɛɱɢɫɥɸɜɚɬɢ ɜɟɤɬɨɪɢ

Ɍɨɞɿ x&k Ak x&0 . ɇɟɯɚɣ {e&

x&k

Ax&k ,k

0,1....

- ɫɢɫɬɟɦɚ ɜɥɚɫɧɢɯ ɜɟɤɬɨɪɿɜ. ɉɪɟɞɫɬɚɜɢɦɨ

ɜɢɝɥɹɞɿ:

x&0

¦cie&i .

Ɉɫɤɿɥɶɤɢ Ae&i

ie&i , ɬɨ x&k

ik e&i

. ɉɪɢ ɜɟɥɢɤɢɯ k x&k c1 1k e&1. Ɍɨɦɭ

¦ci

O¨

¸ .

(k)

xmk 1

k ·

Ɂɧɚɱɢɬɶ P1(k) o 1 .

kof

(2) x&0 ɭ

əɤɳɨ

ɜɟɤɬɨɪɿɜ

ɦɚɬɪɢɰɹ

ɫɢɦɟɬɪɢɱɧɚ,

ɬɨ

ɿɫɧɭɽ

ɨɪɬɨɧɨɪɦɨɜɚɧɚ ɫɢɫɬɟɦɚ

(e&

,e&

)

. Ɍɨɦɭ

,¦cj

¦ci2

&k 1

k &

i2k 1

)

¨¦ci i

jej ¸

(k)

¦ci

¦ci2

i2k

(xk ,xk )

ei ,¦cj

jej

2 2k 1

j§

2k ·

...

c1 1

O¨

...

kof

ɐɟ ɨɡɧɚɱɚɽ ɡɛɿɠɧɿɫɬɶ ɞɨ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɡɚ ɦɨɞɭɥɟɦ ɜɥɚɫɧɨɝɨ ɡɧɚɱɟɧɧɹ ɡ ɤɜɚɞɪɚɬɢɱɧɨɸ ɲɜɢɞɤɿɫɬɸ.

əɤɳɨ 1 1, ɬɨ ɩɪɢ ɩɪɨɜɟɞɟɧɧɿ ɿɬɟɪɚɰɿɣ ɜɿɞɛɭɜɚɽɬɶɫɹ ɡɪɿɫɬ ɤɨɦɩɨɧɟɧɬ

ɜɟɤɬɨɪɚ x&k , ɳɨ ɩɪɢɜɨɞɢɬɶ ɞɨ ”ɩɟɪɟɩɨɜɧɟɧɧɹ” (overflow). əɤɳɨ ɠ 1 1, ɬɨ ɰɟ ɩɪɢɜɨɞɢɬɶ ɞɨ ɡɦɟɧɲɟɧɧɹ ɤɨɦɩɨɧɟɧɬ (underflow). ɉɨɡɛɭɬɢɫɹ ɧɟɝɚɬɢɜɭ

ɬɚɤɨɝɨ ɹɜɢɳɚ ɦɨɠɧɚ ɧɨɪɦɭɸɱɢ ɜɟɤɬɨɪɢ x&k

ɧɚ ɤɨɠɧɿɣ ɿɬɟɪɚɰɿʀ.

ɥɝɨɪɢɬɦ ɫɬɟɩɟɧɟɜɨɝɨ ɦɟɬɨɞɭ ɡɧɚɯɨɞɠɟɧɧɹ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɡɚ ɦɨɞɭɥɟɦ

ɜɥɚɫɧɨɝɨ ɡɧɚɱɟɧɧɹ ɡ ɬɨɱɧɿɫɬɸ

ɜɢɝɥɹɞɚɽ ɬɚɤ:

1. x&0 o e&0

;

x&k 1

2. x&k 1

Ax&k ,P1(k)

(x&k 1,e&k )

,e&k 1

, k 0,1...;

x&k 1

P1(k 1)

P1(k)

t H goto2;

P1(k 1).

4. 1

Ɂɚ ɰɢɦ ɚɥɝɨɪɢɬɦɨɦ ɞɥɹ ɫɢɦɟɬɪɢɱɧɨʀ ɦɚɬɪɢɰɿ ȺɌ

Ⱥ ɲɜɢɞɤɿɫɬɶ ɩɪɹɦɭɜɚɧɧɹ

P1(k) ɞɨ max - ɤɜɚɞɪɚɬɢɱɧɚ.

t .... ɇɟɯɚɣ

1 , e&1

Ɂɧɚɯɨɞɠɟɧɧɹ

2 :

ɜɿɞɨɦɿ.

Ɂɚɞɚɱɚ 10 Ⱦɨɜɟɫɬɢ, ɳɨ ɹɤɳɨ

, ɬɨ

P2(k)

xmk 1

1xmk

2 , ɞɟ

x&k 1

Ax&k

xk 1

kof

xmk – m-ɬɚ ɤɨɦɩɨɧɟɧɬɚ

x&k .

2 , e&2

Ɂɚɞɚɱɚ 11 ɉɨɛɭɞɭɜɚɬɢ ɚɥɝɨɪɢɬɦ ɨɛɱɢɫɥɟɧɧɹ

ɜɢɤɨɪɢɫɬɨɜɭɸɱɢ

ɧɨɪɦɭɜɚɧɧɹ ɜɟɤɬɨɪɿɜ ɬɚ ɫɤɚɥɹɪɧɿ ɞɨɛɭɬɤɢ ɞɥɹ ɨɛɱɢɫɥɟɧɧɹ

P2(k) .

Ɂɧɚɯɨɞɠɟɧɧɹ ɦɿɧɿɦɚɥɶɧɨɝɨ ɜɥɚɫɧɨɝɨ ɱɢɫɥɚ

min (A)

min

i (A)

ɉɪɢɩɭɫɬɢɦɨ

ɳɨ

i (A)

ɬɚ

ɜɿɞɨɦɟ

max . Ɋɨɡɝɥɹɧɟɦɨ

ɦɚɬɪɢɰɸ

max E

A. Ɇɚɽɦɨ

A ,

max

Ɍɨɦɭ max i (B)

max

min

i (A). Ɂɜɿɞɫɢ

min A

max

B .

VE A,V

əɤɳɨ

i :

ɿ (Ⱥ)

0, ɬɨ ɛɭɞɭɽɦɨ ɦɚɬɪɢɰɸ

ɿ ɞɥɹ ɧɟʀ

ɩɨɩɟɪɟɞɧɿɣ ɪɨɡɝɥɹɞ ɞɚɽ ɧɟɨɛɯɿɞɧɢɣ ɪɟɡɭɥɶɬɚɬ.

Ɂɚɦɿɫɬɶ

max

ɦɚɬɪɢɰɿ B

ɦɨɠɧɚ ɜɢɤɨɪɢɫɬɨɜɭɜɚɬɢ

ɓɟ ɨɞɢɧ ɫɩɨɫɿɛ ɨɛɱɢɫɥɟɧɧɹ ɦɿɧɿɦɚɥɶɧɨɝɨ ɜɥɚɫɧɨɝɨ ɡɧɚɱɟɧɧɹ ɩɨɥɹɝɚɽ ɜ

ɜɢɤɨɪɢɫɬɚɧɧɹ ɨɛɟɪɧɟɧɢɯ ɿɬɟɪɚɰɿɣ:

Ax&k 1

x&k ,k

0,1...

(3)

ɥɟ ɰɟɣ ɦɟɬɨɞ ɜɢɦɚɝɚɽ ɛɿɥɶɲɨʀ ɤɿɥɶɤɨɫɬɿ ɚɪɢɮɦɟɬɢɱɧɢɯ ɨɩɟɪɚɰɿɣ: ɫɤɥɚɞɧɿɫɬɶ ɦɟɬɨɞɭ ɧɚ ɨɫɧɨɜɿ ɮɨɪɦɭɥɢ (2) Q O(n2 ), ɚ ɧɚ ɨɫɧɨɜɿ (3) - Q O(n3 ) , ɨɫɤɿɥɶɤɢ ɬɪɟɛɚ ɪɨɡɜ‘ɹɡɭɜɚɬɢ ɋɅ Ɋ, ɚɥɟ ɡɛɿɝɚɽɬɶɫɹ ɦɟɬɨɞ (3) ɲɜɢɞɱɟ.

5.2. ȱɬɟɪɚɰɿɣɧɢɣ ɦɟɬɨɞ ɨɛɟɪɬɚɧɧɹ [ɄȻɆ, 157-161] ɐɟ ɦɟɬɨɞ ɪɨɡɜ’ɹɡɚɧɧɹ ɩɨɜɧɨʀ ɩɪɨɛɥɟɦɢ ɜɥɚɫɧɢɯ ɡɧɚɱɟɧɶ ɞɥɹ

ɫɢɦɟɬɪɢɱɧɢɯ ɦɚɬɪɢɰɶ AT A. ȱɫɧɭɽ ɦɚɬɪɢɰɹ U, ɳɨ ɩɪɢɜɨɞɢɬɶ ɦɚɬɪɢɰɸ Ⱥ ɞɨ ɞɿɚɝɨɧɚɥɶɧɨɝɨ ɜɢɞɭ:

A U U T ,

/ (1)

ɞɟ - ɞɿɚɝɨɧɚɥɶɧɚ ɦɚɬɪɢɰɹ, ɩɨ ɞɿɚɝɨɧɚɥɿ ɹɤɨʀ ɫɬɨɹɬɶ ɜɥɚɫɧɿ ɡɧɚɱɟɧɧɹ i ; U -

ɭɧɿɬɚɪɧɚ ɦɚɬɪɢɰɹ, ɬɨɛɬɨ: U 1 UT . Ɂ (1) ɦɚɽɦɨ

U T AU

(2)

G 1, i j.

ɇɟɯɚɣ

- ɦɚɬɪɢɰɹ, ɬɚɤɚ ɳɨ

U AU ɿ

ij i? j

Ɍɨɞɿ ɞɿɚɝɨɧɚɥɶɧɿ ɟɥɟɦɟɧɬɢ ɦɚɥɨ ɜɿɞɪɿɡɧɹɸɬɶɫɹ ɜɿɞ ɜɥɚɫɧɢɯ ɡɧɚɱɟɧɶ

i (A)

H H(G).

ȼɜɟɞɟɦɨ t(A)

¦(aij )2 .

Ɂ ɦɚɥɨɫɬɿ ɜɟɥɢɱɢɧɢ

t(A) ɜɢɬɿɤɚɽ,

ɳɨ ɞɿɚɝɨɧɚɥɶɧɿ

i, j 1

ɟɥɟɦɟɧɬɢ ɦɚɥɿ. ɉɨ A

ɡɚ ɞɨɩɨɦɨɝɨɸ ɦɚɬɪɢɰɶ ɨɛɟɪɬɚɧɧɹ Uk

cosM

sinM

¨0

0¸ m i

sinM

cosM

¨0

0¸m j

1¹

n i

n j

Ak `

ɳɨ ɩɨɜɟɪɬɚɸɬɶ ɫɢɫɬɟɦɭ ɜɟɤɬɨɪɿɜ ɧɚ ɤɭɬ

, ɩɨɛɭɞɭɽɦɨ ɩɨɫɥɿɞɨɜɧɿɫɬɶ

ɬɚɤɭ, ɳɨ Ak

o /

ɩɪɢ k o f .

Ɂɚɞɚɱɚ 12 ɉɨɤɚɡɚɬɢ , ɳɨ ɦɚɬɪɢɰɹ ɨɛɟɪɬɚɧɧɹ Uk

ɽ ɭɧɿɬɚɪɧɨɸ: Uk-1=UkT.

ɉɨɫɥɿɞɨɜɧɨ ɛɭɞɭɽɦɨ:

Ak 1

UkT AkUk ,

t Ak .

(3)

ɉɪɨɰɟɫ (3) ɧɚɡɢɜɚɽɬɶɫɹ ɦɨɧɨɬɨɧɧɢɦ, ɹɤɳɨ: t Ak 1

Ɂɚɞɚɱɚ 13 Ⱦɨɜɟɫɬɢ, ɳɨ ɞɥɹ ɦɚɬɪɢɰɿ (3) ɜɢɤɨɧɭɽɬɶɫɹ :

aij(k 1)

aij(k) cos2M

a(jjk)

aii(k)

sin 2M.

(4)

ɉɨɤɚɡɚɬɢ, ɳɨ t Ak 1

t(Ak )

2 aij(k) @2 , ɹɤɳɨ ɜɢɛɢɪɚɬɢ

ɡ ɭɦɨɜɢ aij(k 1)

Ɂɜɿɞɫɢ M

2a(k)

, ɞɟ

aij(k)

aml(k)

. Ɍɨɞɿ

arctan p(k)

p(k)

max

(k)

m,l

m l

t Ak o 0,k o f . ɑɢɦ ɛɿɥɶɲɟ n ɬɢɦ ɛɿɥɶɲɟ ɿɬɟɪɚɰɿɣ ɧɟɨɛɯɿɞɧɨ ɞɥɹ ɡɜɟɞɟɧɧɹ

A ɞɨ / .

əɤɳɨ ɦɚɬɪɢɰɹ ɧɟɫɢɦɟɬɪɢɱɧɚ, ɬɨ ɡɚɫɬɨɫɨɜɭɸɬɶ QR, QL ɦɟɬɨɞɢ.

6. ȱɧɬɟɪɩɨɥɸɜɚɧɧɹ ɮɭɧɤɰɿɣ

6.1 ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɿ ɿɧɬɟɪɩɨɥɸɜɚɧɧɹ [ɋȽ, 151-155]

ɇɟɯɚɣ ɮɭɧɤɰɿɹ

f (x) C[a,b]

ɡɚɞɚɧɚ ɫɜɨʀɦɢ ɡɧɚɱɟɧɧɹɦɢ

yi f xi ,xi

[a,b], i

, ɩɪɢɱɨɦɭ ɩɪɢ

xj ɞɥɹ i

0,n

j .

Ɏɭɧɤɰɿɹ

) x

ɧɚɡɢɜɚɽɬɶɫɹ

ɿɧɬɟɪɩɨɥɸɸɱɨɸ ɞɥɹ

f (x)

ɧɚ ɫɿɬɰɿ {xi}in

0 ,

ɹɤɳɨ ) xi

yi , i

0,n.

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