ChM_teory2

n

n

2

ci

( f ,

i )

,

)0 ¦

( f

, i

)

i

,

f )0

2

f

2

¦

ci

.

Mi

2

Mi

2

Mi

2

i 0

i 0

Ⱦɥɹ ɮɭɧɤɰɿʀ f (x), ɳɨɛ ɩɨɛɭɞɭɜɚɬɢ ȿɇɋɄɇ ɩɨɤɥɚɞɟɦɨ H

L2,D (a,b) , ɜ

ɹɤɨɦɭ ɫɤɚɥɹɪɧɢɣ ɞɨɛɭɬɨɤ ɜɢɛɟɪɟɦɨ ɧɚɫɬɭɩɧɢɦ ɱɢɧɨɦ

(u,v)

³b u(x)v(x)dD(x)

(ɿɧɬɟɝɪɚɥ ɋɬɿɥɶɬ‘ɽɫɚ),

ɞɟ				a
	(x) - ɡɪɨɫɬɚɸɱɚ ɮɭɧɤɰɿɹ. Ɇɨɠɥɢɜɿ ɜɢɩɚɞɤɢ:							³b U(x)u(x)v(x)dx;
1.	D(x) C1[a,b], ɬɨɞɿ			c(x)	(x)	0 ɬɚ (u,v)
	x				x			a	d x d xk , k 1,
2.		– ɮɭɧɤɰɿɹ ɫɬɪɢɛɤɿɜ,				(xk	0), ɞɟ xk 1
							n
ɜɜɟɫɬɢ		k	(xk 0)	(xk	0), ɬɨ (u,v)		¦ Uiu(xi )v(xi ).
							i 1
	ɉɟɪɲɢɣ ɜɢɛɿɪ			(x)	ɜɢɤɨɪɢɫɬɨɜɭɽɬɶɫɹ			ɩɪɢ	ɚɩɪɨɤɫɢɦɚɰɿʀ

ɧɟɩɟɪɟɪɜɧɨɝɨ ɚɪɝɭɦɟɧɬɭ, ɚ ɞɪɭɝɢɣ – ɞɥɹ ɬɚɛɥɢɱɧɢɯ ɮɭɧɤɰɿɣ.

N . əɤɳɨ

ɮɭɧɤɰɿɣ

8.5. ɋɢɫɬɟɦɢ ɨɪɬɨɝɨɧɚɥɶɧɢɯ ɮɭɧɤɰɿɣ [ȻɀɄ, 19-102], [ɅɆɋ, 388-382]

əɤ ɜɢɛɢɪɚɬɢ ɨɪɬɨɧɨɪɦɚɥɶɧɭ ɚɛɨ ɨɪɬɨɝɨɧɚɥɶɧɭ ɫɢɫɬɟɦɭ ɮɭɧɤɰɿɣ {Mi}i 0?

Ɋɨɡɝɥɹɧɟɦɨ ɞɟɹɤɿ ɡ ɧɚɣɛɿɥɶɲ ɜɠɢɜɚɧɢɯ ɬɚɤɢɯ ɫɢɫɬɟɦ.

10. əɤɳɨ H

L2 (

1,1);

1 (ɜɚɝɨɜɢɣ ɦɧɨɠɧɢɤ), ɬɨ i (x)

Li (x) - ɫɢɫɬɟɦɚ

ɛɚɝɚɬɨɱɥɟɧɿɜ Ʌɟɠɚɧɞɪɚ, ɹɤɿ ɦɚɸɬɶ ɜɢɝɥɹɞ

Ln (x)

1

d n

(x2

1)n .

2n n! dxn

ȼɢɤɨɪɢɫɬɨɜɭɸɬɶ ɬɚɤɨɠ ɪɟɤɭɪɟɧɬɧɿ ɮɨɪɦɭɥɢ

(n

1)Ln 1 (x) (2n 1)xLn (x) nLn 1 (x) ,

ɞɨ ɹɤɢɯ ɞɨɞɚɽɦɨ ɭɦɨɜɢ

L0 (x) 1,L1(x)

x .

ɐɟ ɨɪɬɨɝɨɧɚɥɶɧɚ ɫɢɫɬɟɦɚ ɜ ɬɨɦɭ ɫɟɧɫɿ, ɳɨ

(Li , Lk )

³11 Li (x)Lk (x)dx

Gik

Li (x)

2 ,

Li (x)

2

2i2 1

f , Li

2i 1

f , Li .

ɿ ɬɨɦɭ ci

2

Li

2

Ɂɚɭɜɚɠɟɧɧɹ əɤɳɨ ɩɨɬɪɿɛɧɨ ɩɨɛɭɞɭɜɚɬɢ ɧɚɛɥɢɠɟɧɧɹ ɧɚ ɞɨɜɿɥɶɧɨɦɭ ɩɪɨɦɿɠɤɭ

(a,b) , ɬɨ ɛɚɠɚɧɨ ɩɟɪɟɣɬɢ						ɞɨ ɩɪɨɦɿɠɤɭ ( 1,1) , ɬɨɛɬɨ		ɩɨ f (x) ɧɚ [a,b]
		(t) ɡ t [				ɡɚɦɿɧɨɸ x At B, t		ɬɚ ɞɥɹ ɩɨɛɭɞɨɜɢ
ɩɨɛɭɞɭɜɚɬɢ f			1,1]				x
ɛɚɝɚɬɨɱɥɟɧɚ ɇɋɄɇ ɞɥɹ						ɜɢɤɨɪɢɫɬɚɬɢ ɛɚɝɚɬɨɱɥɟɧɢ Ʌɟɠɚɧɞɪɚ Li (t) .
			f (t)

Ɇɨɠɧɚ ɪɨɛɢɬɢ ɧɚɜɩɚɤɢ - ɫɢɫɬɟɦɭ ɛɚɝɚɬɨɱɥɟɧɿɜ ɩɟɪɟɜɟɫɬɢ ɡ [a,b] ɧɚ [ 1,1], ɚɥɟ ɰɟ ɜɢɦɚɝɚɽ ɛɿɥɶɲɟ ɨɛɱɢɫɥɟɧɶ ɿ ɩɪɨɰɟɫ ɩɨɛɭɞɨɜɢ ȿɇɋɄɇ ɫɤɥɚɞɧɿɲɟ.

71

Ɋɟɤɭɪɟɧɬɧɚ ɮɨɪɦɭɥɚ:

20.	əɤɳɨ	H L2,U ( 1,1),	U(x)	1	ɫɤɚɥɹɪɧɢɣ			ɞɨɛɭɬɨɤ
				,
			1	x2
(u,v)	1 u(x)v(x) dx (ɰɟ ɧɟɜɥɚɫɧɿ ɿɧɬɟɝɪɚɥɢ ɞɪɭɝɨɝɨ ɪɨɞɭ), ɬɨ					i	(x)	T (x), ɞɟ
	³1 1	x2						i
^Ti (x)` ɫɢɫɬɟɦɚ ɨɪɬɨɝɨɧɚɥɶɧɢɯ ɛɚɝɚɬɨɱɥɟɧɿɜ ɑɟɛɢɲɨɜɚ 1-ɝɨ ɪɨɞɭ,								ɹɤɿ ɦɚɸɬɶ

ɜɢɝɥɹɞ

Tn (x) cos(narccos(x)).

Ɋɟɤɭɪɟɧɬɧɚ ɮɨɪɦɭɥɚ ɞɥɹ ɰɢɯ ɛɚɝɚɬɨɱɥɟɧɿɜ:

Tn 1 (x) 2xTn

(x) Tn 1 (x), T0

1 T1

x,

2

­

°

,

n

0

®

Tn

°

¯ 2

,n

1,2...

U(x)

(1 x) .

30. H

ɝɿɥɶɛɟɪɬɿɜ ɩɪɨɫɬɿɪ ɡ ɜɚɝɨɜɢɦ

ɦɧɨɠɧɢɤɨɦ

(1

x)

ɋɢɫɬɟɦɚ

Mi (x)

Pn(D,E) (x)- ɛɚɝɚɬɨɱɥɟɧɿɜ

əɤɨɛɿ,

,

1

( ,

–

ɱɢɫɥɨɜɿ

ɩɚɪɚɦɟɬɪɢ)

ɨɪɬɨɝɨɧɚɥɶɧɚ

ɜ

ɫɟɧɫɿ

ɫɤɚɥɹɪɧɨɝɨ

ɞɨɛɭɬɤɭ

(u,v)

³1

1 x D 1

x E u(x)v(x)dx . ɐɹ ɫɢɫɬɟɦɚ ɽ ɭɡɚɝɚɥɶɧɟɧɧɹɦ ɜɢɩɚɞɤɿɜ 10 ɬɚ

20.

ɞɟ

1

Ⱦɢɮɟɪɟɧɰɿɚɥɶɧɚ ɮɨɪɦɭɥɚ ɞɥɹ ɛɚɝɚɬɨɱɥɟɧɿɜ:

Pn(D,E) (x)

(

1)n

(1 x) D (1

x) E

d n

>(1 x)n D (1 x)n E @;

n

n

2

n!

dx

2(n 1)(n D E 1)(2n D E)Pn(D1,E) (x)

E2 ]Pn(D,E) (x)

(2n D E 1)[(2n D E)(2n D E 2)x D2

2(n D)(n E)(2n D E 2)Pn(D1,E) (x) , n

1,2,...

P(D,E)

1,

P(D,E)

0,

P(D,E)

2

2D E 1 *(D n 1)*(E n 1)

,

0

1

n

n!(D E 2n 1)*(D E n 1)

ɬɚ

*(z)

³0f e t t z 1dt, *(z 1)

z*(z),

Ʉɨɥɢ D

E

0 : Pn(0,0) (x)

Ln (x) , ɚ ɞɥɹ D

E

1

:

2

L2,U[0,f),

U(x)

xDe x ,

D 1.

40. H

ɐɶɨɦɭ

ɜɿɞɩɨɜɿɞɚɽ

ɫɢɫɬɟɦɚ

ɛɚɝɚɬɨɱɥɟɧɿɜ

Ʌɚɝɟɪɪɚ

Mi (x)

ɞɢɮɟɪɟɧɰɿɚɥɶɧɨɸ ɮɨɪɦɭɥɨɸ:

1 n x D ex

>xD ne

LDn

(x)

d n

n

dx

ɚɛɨ ɜ ɪɟɤɭɪɟɧɬɧɿɣ ɮɨɪɦɿ

*(n 1)				n!
P (	1	,	1	) (x) T (x) .
	2		2
n				n

ɜɚɝɨɜɨɦɭ ɦɧɨɠɧɢɤɭ

LDi (x) , ɹɤɿ ɡɚɞɚɸɬɶɫɹ

x @,

72

(2n D 1 x)LDn

(n D)LDn 1 ,

(n 1)LDn 1

ɞɟ

LD0

1, LD1

0 ɬɚ ɡ ɧɨɪɦɨɸ

LDn

2

n!*(D n

1) .

L2,D ( f,f),

U(x)

e x2

50.

H

. ɋɢɫɬɟɦɭ ɨɪɬɨɝɨɧɚɥɶɧɢɯ ɮɭɧɤɰɿɣ ɜɢɛɢɪɚɽɦɨ

ɹɤ

ɫɢɫɬɟɦɭ

ɛɚɝɚɬɨɱɥɟɧɿɜ

ȿɪɦɿɬɚ

i (x)

Hi (x) ,

ɹɤɿ

ɡɚɞɚɸɬɶɫɹ

ɞɢɮɟɪɟɧɰɿɚɥɶɧɨɸ ɮɨɪɦɭɥɨɸ:

n

x

2 d n

x2

Hn (x) ( 1)

e

(e

)

ɚɛɨ ɜ ɪɟɤɭɪɟɧɬɧɿɣ ɮɨɪɦɿ

dxn

Hn 1

2xHn

2nHn 1 ,

ɞɟ

2

H0

1, H 1

0 ɬɚ Hn

2n n! .

60.

H

L2 (0,2

),

(x)

1

f (x)

f (x

2 ).

f (x) –

2 -

ɩɟɪɿɨɞɢɱɧɿ

ɮɭɧɤɰɿʀ. Ɂɚ ɫɢɫɬɟɦɭ ɨɪɬɨɝɨɧɚɥɶɧɢɯ ɮɭɧɤɰɿɣ ɜɢɛɢɪɚɽɦɨ ɬɪɢɝɨɧɨɦɟɬɪɢɱɧɭ

ɫɢɫɬɟɦɭ M0

1 ,

M2k 1 (x)

1 cos(kx),

M2k (x)

1 sin(kx),

ɞɟ

2

Mm

2 1.

ȿɥɟɦɟɧɬ ɧɚɣɤɪɚɳɨɝɨ ɫɟɪɟɞɧɶɨɤɜɚɞɪɚɬɢɱɧɨɝɨ ɧɚɛɥɢɠɟɧɧɹ ɩɪɟɞɫɬɚɜɥɹɽ

ɫɨɛɨɸ ɬɪɢɝɨɧɨɦɟɬɪɢɱɧɢɣ ɛɚɝɚɬɨɱɥɟɧ

a0

n

)0 (x)

Tn (x)

¦(ak cos(kx) bk sin(kx)) ,

2

k

1

ɮɨɪɦɭɥɢ ɞɥɹ ɨɛɱɢɫɥɟɧɧɹ ɰɢɯ ɤɨɟɮɿɰɿɽɧɬɿɜ ɧɚɜɟɞɟɧɿ ɜ ɧɚɫɬɭɩɧɨɦɭ ɩɭɧɤɬɿ.

70. əɤɳɨ ɩɨɬɪɿɛɧɨ ɚɩɪɨɤɫɢɦɭɜɚɬɢ ɬɚɛɥɢɱɧɭ ɮɭɧɤɰɿɸ, ɬɨ

N

N1 1

i¦0uivi

H l2, xi

i, i 0,N, (u,v)

ɿ ɡɚ ɫɢɫɬɟɦɭ ɨɪɬɨɝɨɧɚɥɶɧɢɯ ɮɭɧɤɰɿɣ ɜɢɛɢɪɚɽɦɨ ɧɚɫɬɭɩɧɭ ɫɢɫɬɟɦɭ

ɛɚɝɚɬɨɱɥɟɧɿɜ

Mk (x)

pk(N ) (x),

(m d

– ɫɢɫɬɟɦɭ

ɛɚɝɚɬɨɱɥɟɧɿɜ

k

0,m

N)

ɑɟɛɢɲɨɜɚ ɞɢɫɤɪɟɬɧɨɝɨ ɚɪɝɭɦɟɧɬɭ, ɹɤɿ ɡɚɞɚɽɬɶɫɹ ɮɨɪɦɭɥɨɸ

(N)

k

( 1) j ckjckj j

( j)

pk

(x)

¦

x

,

j

N

( j)

j 0

x( j) x(x 1)...(x

1) -

ɮɚɤɬɨɪɿɚɥɶɧɢɣ

ɛɚɝɚɬɨɱɥɟɧ;

Ckj

-

ɱɢɫɥɨ

ɫɩɨɥɭɤ.

Ɋɟɤɭɪɟɧɬɧɚ ɮɨɪɦɭɥɚ

§ N

·

m(N m 1)

(m 1)(N

m)

(N)

(N )

(N)

(N )

(N )

pm 1

¨

x¸ pm

pm 1

,

p0

1,

p

0.

2(2m

1)

2

2(2m 1)

1

©

¹

73

ɇɚɩɪɢɤɥɚɞ

p1(N ) 1

2x

, p2(N )

1

6x

6x2

.

N

N

N(N

1)

ɍ ɜɢɩɚɞɤɭ, ɹɤɳɨ ɡɚɞɚɧɿ

ɜɭɡɥɢ ti

t0 ih, i

, ɬɨ ɪɨɛɢɦɨ ɡɚɦɿɧɭ

0, N

x

ti t0

i.

i

h

8.6. ɋɟɪɟɞɧɶɨɤɜɚɞɪɚɬɢɱɧɟ ɧɚɛɥɢɠɟɧɧɹ ɩɟɪɿɨɞɢɱɧɢɯ ɮɭɧɤɰɿɣ [ɅɆɋ,60-61], [ȻɀɄ,166-182]

ɇɟɯɚɣ ɦɚɽɦɨ ɩɟɪɿɨɞɢɱɧɭ ɮɭɧɤɰɿɸ f (x) ɧɟɩɟɪɟɪɜɧɨɝɨ ɚɪɝɭɦɟɧɬɭ, ɡ
ɩɟɪɿɨɞɨɦ T 2 , ɬɨɛɬɨ f (x 2 )	f (x). ȼ ɩɪɨɫɬɨɪɿ H2 L2[0,2 ]
ɜɢɡɧɚɱɟɧɢɣ ɫɤɚɥɹɪɧɢɣ ɞɨɛɭɬɨɤ	2³u(x)v(x)dx.
(u,v)
	0

ȼ ɹɤɨɫɬɿ ɫɢɫɬɟɦɢ ɥɿɧɿɣɧɨ-ɧɟɡɚɥɟɠɧɢɯ ɮɭɧɤɰɿɣ {Mi } ɜɢɛɟɪɟɦɨ ɬɪɢɝɨ- ɧɨɦɟɬɪɢɱɧɭ ɫɢɫɬɟɦɭ ɮɭɧɤɰɿɣ:

	M0 (x) 1; M2k 1(x)	coskx; 2k (x)			sinkx, k 1, 2,	;
	ɹɤɚ ɽ ɩɨɜɧɨɸ ɧɨɪɦɨɜɚɧɨɸ ɫɢɫɬɟɦɨɸ ɜ L2[0,2				] .
	Ȼɭɞɟɦɨ ɲɭɤɚɬɢ )(x) ɭ ɜɢɝɥɹɞɿ ɬɪɢɝɨɧɨɦɟɬɪɢɱɧɨɝɨ ɛɚɝɚɬɨɱɥɟɧɚ
			a0	n
	)(x) Tn (x)			¦(ak cos kx bk sin kx) .		(1)
		2
				k 1

Ɂɚ ɬɟɨɪɿɽɸ ɧɚɣɤɪɚɳɨɝɨ ɫɟɪɟɞɧɶɨɤɜɚɞɪɚɬɢɱɧɨɝɨ ɧɚɛɥɢɠɟɧɧɹ ɤɨɟɮɿɰɿɽɧɬɢ ɨɛɱɢɫɥɸɽɦɨ ɡɚ ɮɨɪɦɭɥɚɦɢ:

­

1

2

° a0 ( f ,M0)

³ f (x)dx,

2

°

0

°

1

2

®ak

( f ,Mkc )

³ f (x)coskxdx,

(2)

°

0

°

2

³

° b

( f ,

s )

1

f (x)sinkxdx.

° k

Mk

ȼɿɞɯɢɥɟɧɧɹ:

¯

0

n

2

f

2

§

2

¦

2

bk

2

·

' ( f )

¨2 a0

(ak

)¸ .

©

k 1

¹

	Ɍɟɩɟɪ ɧɟɯɚɣ ɮɭɧɤɰɿɹ f (x) ɡɚɞɚɧɚ ɬɚɛɥɢɱɧɨ:
							fi f (xi ), i 1, N .			H L2 ( ) ɞɥɹ
Ɍɪɢɝɨɧɨɦɟɬɪɢɱɧɚ ɫɢɫɬɟɦɚ M0 (x), M2k 1 (x), M2k (x) ɨɪɬɨɝɨɧɚɥɶɧɚ ɜ
Z	­						½
	®xi	i		,i	1, N¾ ɜ ɫɟɧɫɿ ɫɤɚɥɹɪɧɨɝɨ ɞɨɛɭɬɤɭ
	¯		N		¿

74

									N
	(u,v)			1					¦uivi , ui	u(xi ) .
Ɍɨɞɿ							N i 1
	­
				1					N
	°	a0						i¦1
						N
	°								fi ,
	°	2			N
	°®ak				i¦1 fi coskxi ,					(3)
			N
	°	2			N
	°				i 1
	¯		N ¦ fi sinkxi.
	°bk

ɐɟ ɮɨɪɦɭɥɢ Ȼɟɫɫɟɥɹ . ȼ ɮɨɪɦɭɥɿ (1)

(x) Tn (x)

( ɬɨɛɬɨ ɛɚɝɚɬɨɱɥɟɧ ɬɨɣ ɠɟ ),

ɚɥɟ ɤɨɟɮɿɰɿɽɧɬɢ ɜɢɡɧɚɱɚɽɦɨ ɡɚ ɮɨɪɦɭɥɨɸ (3).

2n 1 . ɥɟ ɹɤɳɨ

Ɂɚɭɜɚɠɟɧɧɹ əɤ

ɩɪɚɜɢɥɨ ɤɿɥɶɤɿɫɬɶ

ɞɚɧɢɯ ɡɧɚɱɟɧɶ N

N 2n 1, ɬɨ n

N 1

ɿ N -ɧɟɩɚɪɧɟ. ɉɪɢ ɰɶɨɦɭ TN 1 (x)– ȻɇɋɄɇ ɿ ɡɜɿɞɫɢ

2

2

'2 ( f )

f (x)

TN 1 (x)

2

1

N ª f (xi )

TN 1 (xi )º2

o inf

N

¦«

»

ak ,bk

2

2

i 1 «

»

¬

¼

Ɉɫɤɿɥɶɤɢ ɧɚɣɦɟɧɲɟ ɡɧɚɱɟɧɧɹ ɜɿɞɯɢɥɟɧɧɹ '2 ( f )

0 ,

ɬɨ ɬɪɢɝɨɧɨɦɟɬɪɢɱɧɢɣ

ɛɚɝɚɬɨɱɥɟɧ ɧɚɣɤɪɚɳɨɝɨ ɫɟɪɟɞɧɶɨɤɜɚɞɪɚɬɢɱɧɨɝɨ ɧɚɛɥɢɠɟɧɧɹ ɫɩɿɜɩɚɞɚɽ ɡ ɿɧɬɟɪɩɨɥɹɰɿɣɧɢɦ ɬɪɢɝɨɧɨɦɟɬɪɢɱɧɢɦ ɛɚɝɚɬɨɱɥɟɧɨɦ ɿ

TN 1 (xi ) f (xi ).

2

Ⱦɥɹ ɜɢɡɧɚɱɟɧɧɹ ɤɨɟɮɿɰɿɽɧɬɿɜ ai ,bi ɡɚ ɮɨɪɦɭɥɨɸ Ȼɟɫɫɟɥɹ (3) ɧɟɨɛɯɿɞɧɚ

ɤɿɥɶɤɿɫɬɶ ɨɩɟɪɚɰɿɣ Q

O(N 2 ). ȱɫɧɭɸɬɶ ɚɥɝɨɪɢɬɦɢ, ɹɤɿ ɞɨɡɜɨɥɹɸɬɶ ɨɛɱɢɫɥɢɬɢ

ɡɚ

Q

O(N log N) ɨɩɟɪɚɰɿɣ.

ɐɟ ɬɚɤ ɡɜɚɧɢɣ ɚɥɝɨɪɢɬɦ ɲɜɢɞɤɨɝɨ ɩɟɪɟɬɜɨɪɟɧɧɹ

Ɏɭɪ’ɽ. əɤɳɨ ɜ (3) ɿɫɧɭɽ ɝɪɭɩɚ ɞɨɞɚɧɤɿɜ , ɹɤɿ ɪɿɜɧɿ ɦɿɠ ɫɨɛɨɸ, ɬɨɛɬɨ ɱɢɫɥɨ N

ɦɨɠɧɚ

ɩɪɟɞɫɬɚɜɢɬɢ

ɹɤ

N p1 p2 , ɬɨ

ɦɨɠɧɚ ɬɚɤ ɜɢɛɪɚɬɢ

ɫɿɬɤɭ, ɳɨ

Q

O(N max(p , p

2

)). əɤɳɨ ɠ N

nm , ɬɨ Q

O(Nm)

O(N log

2

N) .

1

8.7. Ɇɟɬɨɞ ɧɚɣɦɟɧɲɢɯ ɤɜɚɞɪɚɬɿɜ (ɆɇɄ) [ɅɆɋ 61-65], [ȼ,88-93]

ɇɟɯɚɣ ɜ ɪɟɡɭɥɶɬɚɬɿ ɜɢɦɿɪɸɜɚɧɶ ɮɭɧɤɰɿʀ

f (x) ɦɚɽɦɨ ɬɚɛɥɢɰɸ ɡɧɚɱɟɧɶ:

, xi [a,b] .

yi

f (xi ), i

1, N

(1)

Ɂɚ ɞɚɧɢɦɢ

ɰɿɽʀ ɬɚɛɥɢɰɿ

ɬɪɟɛɚ

ɩɨɛɭɞɭɜɚɬɢ

ɚɧɚɥɿɬɢɱɧɭ ɮɨɪɦɭɥɭ

)(x;a0 ,a2 , ,an ) ɬɚɤɭ, ɳɨ

)(xi ;a0 ,a2 ,

,an ) yi , i

1, N .

(2)

ȼɢɤɨɧɭɜɚɬɢ ɰɟ ɿɧɬɟɪɩɨɥɸɜɚɧɧɹɦ ɬɨɛɬɨ ɡɚɞɚɜɚɬɢ

)(xi ;a0 ,a2 ,

,an )

yi , i

1, N

(3)

75

ɧɟɪɚɰɿɨɧɚɥɶɧɨ, ɛɨ N n )ɿ ɫɢɫɬɟɦɚ ɩɟɪɟɜɢɡɧɚɱɟɧɚ; ʀʀ ɪɨɡɜ’ɹɡɤɢ ɹɤ ɩɪɚɜɢɥɨ ɧɟ ɿɫɧɭɸɬɶ. ȼɢɝɥɹɞ ɮɭɧɤɰɿʀ (x;a0 ,a2 , ,an ) ɿ ɱɢɫɥɨ ɩɚɪɚɦɟɬɪɿɜ ai ɭ ɞɟɹɤɢɯ

ɜɢɩɚɞɤɚɯ ɜɿɞɨɦɿ. ȼ ɿɧɲɢɯ ɜɢɩɚɞɤɚɯ ɜɨɧɢ ɜɢɡɧɚɱɚɸɬɶɫɹ ɡɚ ɝɪɚɮɿɤɨɦ, ɩɨɛɭɞɨɜɚɧɢɦ ɡɚ ɜɿɞɨɦɢɦɢ ɡɧɚɱɟɧɧɹɦɢ f (xi ) ɬɚɤ, ɳɨɛ ɡɚɥɟɠɧɿɫɬɶ (2) ɛɭɥɚ

ɞɨɫɢɬɶ ɩɪɨɫɬɨɸ ɿ ɞɨɛɪɟ ɜɿɞɨɛɪɚɠɚɥɚ ɪɟɡɭɥɶɬɚɬɢ ɫɩɨɫɬɟɪɟɠɟɧɶ. ɥɟ ɬɚɤɿ

ɦɿɪɤɭɜɚɧɧɹ ɧɟ ɞɚɸɬɶ ɡɦɨɝɭ ɩɨɛɭɞɭɜɚɬɢ ɽɞɢɧɢɣ ɟɥɟɦɟɧɬ ɬɚ ɣ ɳɟ ɧɚɣɤɪɚɳɨɝɨ ɧɚɛɥɢɠɟɧɧɹ.

Ɍɨɦɭ ɜɢɡɧɚɱɚɸɬɶ ɩɚɪɚɦɟɬɪɢ a0 , ,an ɬɚɤ, ɳɨɛ ɭ ɞɟɹɤɨɦɭ ɪɨɡɭɦɿɧɧɿ ɜɫɿ

ɪɿɜɧɹɧɧɹ ɫɢɫɬɟɦɢ (2) ɨɞɧɨɱɚɫɧɨ ɡɚɞɨɜɨɥɶɧɹɥɢɫɶ ɡ ɧɚɣɦɟɧɲɨɸ ɩɨɯɢɛɤɨɸ, ɧɚɩɪɢɤɥɚɞ, ɳɨɛ ɜɢɤɨɧɭɜɚɥɨɫɹ

			N	)(xi ;a0 , ,an )]2 o min .
	I(a0 ,	,an )	¦[yi				(4)
			i 1
Ɍɚɤɢɣ ɦɟɬɨɞ ɪɨɡɜ’ɹɡɚɧɧɹ ɫɢɫɬɟɦɢ (2) ɿ ɧɚɡɢɜɚɸɬɶ ɦɟɬɨɞɨɦ ɧɚɣɦɟɧɲɢɯ
ɤɜɚɞɪɚɬɿɜ,	ɨɫɤɿɥɶɤɢ	ɦɿɧɿɦɿɡɭɽɬɶɫɹ			ɫɭɦɚ	ɤɜɚɞɪɚɬɿɜ	ɜɿɞɯɢɥɟɧɧɹ
)(x;a0 ,a2 ,	,an ) ɜɿɞ ɡɧɚɱɟɧɶ		f (xi ) .

Ⱦɥɹ ɪɟɚɥɿɡɚɰɿʀ ɦɿɧɿɦɭɦɭ ɧɟɨɛɯɿɞɧɨ ɬɚ ɞɨɫɬɚɬɧɶɨ ɜɢɤɨɧɚɧɧɹ ɭɦɨɜ:

					I
əɤɳɨ )(xi ;a0 ,					wai	0, i	0,n.
	,an ) ɥɿɧɿɣɧɨ ɡɚɥɟɠɢɬɶ ɜɿɞ ɩɚɪɚɦɟɬɪɿɜ a0, ,an , ɬɨɛɬɨ
			)(xi ;a0 ,				n
						,an )	¦akMk (x),
ɬɨ ɡ (3) ɦɚɽɦɨ ɋɅ	Ɋ						k 0
			n
			¦akMk (xi ) yi , i						1, N	,
			k 0
ɹɤɭ ɧɚɡɢɜɚɸɬɶ ɫɢɫɬɟɦɨɸ ɭɦɨɜɧɢɯ ɪɿɜɧɹɧɶ. ɉɨɡɧɚɱɢɜɲɢ
C (Mk (xi ))ik					, a& (a0 ,		7 , y& (y1, , yN )				7 ,
		1,N				,an )
		0,n
ɦɚɽɦɨ ɦɚɬɪɢɱɧɢɣ ɡɚɩɢɫ ɋɅ Ɋ (7)
					Ca	y

(5)

(6)

(7)

(8)

ɉɨɦɧɨɠɢɜɲɢ ɫɢɫɬɟɦɭ ɭɦɨɜɧɢɯ ɪɿɜɧɹɧɶ (8) ɡɥɿɜɚ ɧɚ ɬɪɚɧɫɩɨɧɨɜɚɧɭ ɞɨ C ɦɚɬɪɢɰɸ CT ɨɬɪɢɦɚɽɦɨ ɫɢɫɬɟɦɭ ɧɨɪɦɚɥɶɧɢɯ ɪɿɜɧɹɧɶ

	C	T &	C	T	&	gik @in,k 0 ,				(9)
		Ca			y
G	A C7C , dimG				n 1, G				·n
N 7	N		N		xi Mj xi ,	C	T &	§ N
gik ¦cij cjk	¦cjicjk		¦Mk				y	¨¦cik	yi ¸	,
j 1	j 1		j 1					© i 1	¹k 0
ɡ ɹɤɨʀ ɜɥɚɫɧɨ ɿ ɨɛɱɢɫɥɸɸɬɶ ɧɟɜɿɞɨɦɿ ɤɨɟɮɿɰɿɽɧɬɢ.
ɉɨɤɚɠɟɦɨ, ɳɨ ɆɇɄ ɽ ɦɟɬɨɞɨɦ ɡɧɚɯɨɞɠɟɧɧɹ ȿɇɋɄɇ, ɹɤɳɨ ɜɢɡɧɚɱɢɬɢ
ɫɤɚɥɹɪɧɢɣ ɞɨɛɭɬɨɤ		u,v	¦u xi v xi .
			N

i 1

76

ɉɨɫɬɚɜɢɦɨ ɡɚɞɚɱɭ ɡɧɚɯɨɞɠɟɧɧɹ ȿɇɋɄɇ:

N

'( f ,))

f

)

2

f

), f

)

¦ yi

) xi ,a& 2 o inf .

i 1

Ɂɚ ɬɟɨɪɿɽɸ ɫɟɪɟɞɧɶɨɤɜɚɞɪɚɬɢɱɧɨɝɨ ɧɚɛɥɢɠɟɧɧɹ ɞɥɹ ɰɶɨɝɨ ɧɟɨɛɯɿɞɧɨ, ɳɨɛ

ɤɨɟɮɿɰɿɽɧɬɢ a0 ,

,an

ɡɧɚɯɨɞɢɥɢɫɹ ɡ ɫɢɫɬɟɦɢ:

n

Mk ,M j Mk , f

¦ak

, k

,

0,n

j 0

ɚ ɰɟ ɫɩɿɜɩɚɞɚɽ ɡ (9).

əɤɳɨ

ɜɿɞɨɦɚ ɿɧɮɨɪɦɚɰɿɹ

ɩɪɨ

ɨɛɱɢɫɥɸɜɚɥɶɧɭ

ɩɨɯɢɛɤɭ

ɞɥɹ

ɡɧɚɱɟɧɶ

f (xi )

xi

f

yi

i ,

ɬɨ ɜɢɛɢɪɚɸɬɶ ɬɚɤɢɣ ɫɤɚɥɹɪɧɢɣ ɞɨɛɭɬɨɤ u,v

N

12 .

¦ Uiu xi v xi , ɞɟ Ui

) x,a0,...,an

i

1

Hi

ɇɟɯɚɣ

ɬɟɩɟɪ

-

ɧɟɥɿɧɿɣɧɚ

ɮɭɧɤɰɿɹ

ɩɚɪɚɦɟɬɪɿɜ

a a0, ,an , ɧɚɩɪɢɤɥɚɞ: )

a0ea1x

a2ea3x ...,

ɚɛɨ

)

a0 cos a1x

a2 sin a3 x ... .

ɋɤɥɚɞɟɦɨ ɮɭɧɤɰɿɨɧɚɥ:

1 Ui >yi

) x,a& @2

S a0 ,...,an

o inf&

(10)

¦

a

i

1

Ɉɫɤɿɥɶɤɢ ɬɟɩɟɪ ) x,a0 ,...,an ɧɟɥɿɧɿɣɧɚ, ɬɨ ɡɚɫɬɨɫɭɽɦɨ ɦɟɬɨɞ ɥɿɧɟɚɪɢɡɚɰɿʀ.

ɇɟɯɚɣ ɜɿɞɨɦɿ ɧɚɛɥɢɠɟɧɿ ɡɧɚɱɟɧɧɹ a&0

a00 ,..., an0

. Ɋɨɡɤɥɚɞɟɦɨ

)(x,a) ɜ

ɨɤɨɥɿ a&0 . Ɍɨɞɿ ɨɬɪɢɦɚɽɦɨ ɥɿɧɿɣɧɟ ɧɚɛɥɢɠɟɧɧɹ ɞɨ

(x,a):

n

) x,a& ) x,a&0 ¦k 0

wak

x,a&0 ak

ak0 .

əɤɳɨ ɜɜɟɫɬɢ ɩɨɡɧɚɱɟɧɧɹ

) x,a&0 , cik

)cak

z& a& a&0 yi*

xi ,

0 ,

yi

a

ɬɨ ɨɬɪɢɦɚɽɦɨ ɫɢɫɬɟɦɭ ɭɦɨɜɧɢɯ ɪɿɜɧɹɧɶ ɜɿɞɧɨɫɧɨ ɩɨɩɪɚɜɨɤ ɞɨ a&0 :

		&			*
			y						(11)
	Cz
	Ɂɚɦɿɧɢɦɨ ʀʀ ɧɚ ɫɢɫɬɟɦɭ ɧɨɪɦɚɥɶɧɢɯ ɪɿɜɧɹɧɶ
		T	&			T		*
	C				C		y		(12)
		Cz
	Ɂɧɚɣɲɨɜɲɢ z&, ɨɛɱɢɫɥɸɽɦɨ ɧɚɫɬɭɩɧɟ ɧɚɛɥɢɠɟɧɧɹ: a&1 a&0								z&. ɐɟɣ ɩɪɨɰɟɫ
	ɦɨɠɧɚ ɩɪɨɞɨɜɠɭɜɚɬɢ: ɧɚ ɤɨɠɧɿɣ				ɿɬɟɪɚɰɿʀ ɡɧɚɯɨɞɢɦɨ				z&m ,m 0,1,... ɿ
	ɭɬɨɱɧɸɽɦɨ ɧɚɛɥɢɠɟɧɧɹ ɞɨ a&: a&m		a&m 1 z&m 1 .

ɍɦɨɜɚ ɩɪɢɩɢɧɟɧɧɹ ɿɬɟɪɚɰɿɣ

77

	z&m	¨§ ¦			zkm 2 ¸·	2	H.
						1
				n	¹
				© k 0
ȼɚɠɥɢɜɢɦ ɽ ɜɢɛɿɪ ɩɨɱɚɬɤɨɜɨɝɨ ɧɚɛɥɢɠɟɧɧɹ a&0 . Ɂ ɫɢɫɬɟɦɢ ɭɦɨɜɧɢɯ
ɪɿɜɧɹɧɶ (ɧɟɥɿɧɿɣɧɨʀ) ɜɢɛɟɪɟɦɨ ɞɟɹɤɿ					n 1.	Ɋɨɡɜ‘ɹɡɨɤ ɰɿɽʀ ɫɢɫɬɟɦɢ ɿ ɞɚɫɬɶ
ɩɨɱɚɬɤɨɜɟ ɧɚɛɥɢɠɟɧɧɹ.

Ⱦɥɹ ɞɟɹɤɢɯ ɩɪɨɫɬɢɯ ɧɟɥɿɧɿɣɧɢɯ ɡɚɥɟɠɧɨɫɬɟɣ ɜɿɞ ɧɟɜɟɥɢɤɨʀ ɤɿɥɶɤɨɫɬɿ ɩɚɪɚɦɟɬɪɿɜ ɡɚɞɚɱɭ ɦɨɠɧɚ ɥɿɚɧɟɪɢɡɭɜɚɬɢ ɚɧɚɥɿɬɢɱɧɨ. ɇɚɩɪɢɤɥɚɞ, ɪɨɡɝɥɹɧɟɦɨ

ɧɚɛɥɢɠɟɧɧɹ ɞɚɧɢɯ ɚɥɨɦɟɬɪɢɱɧɢɦ ɡɚɤɨɧɨɦ
yi	f xi	, ) x, A,D			AxD .
ɋɢɫɬɟɦɚ ɭɦɨɜɧɢɯ ɪɿɜɧɹɧɶ ɦɚɽ ɜɢɝɥɹɞ:
) xi		AxiD yi ,i				.
					1, N
ɉɪɨɥɨɝɚɪɢɮɦɭɽɦɨ ʀʀ:
\ xi ln) xi ln A Dln xi
					ln yi,i 1,N
ȼɜɟɞɟɦɨ a ln A . Ɍɟɩɟɪ ɮɭɧɤɰɿɹ		x,a,	ɥɿɧɿɣɧɚ. ɋɢɫɬɟɦɚ ɭɦɨɜɧɢɯ ɪɿɜɧɹɧɶ
ɜɿɞɧɨɫɧɨ ɩɚɪɚɦɟɬɪɿɜ a ɬɚ	ɦɚɽ ɜɢɝɥɹɞ.		, b ln yi i11 ,
Cz& b , z		a,
		¨		¸
		§ 1	ln x1	·
	C	¨		¸
		¨ .................		¸ .
		©1	ln xN	¹

Ɂɚɩɢɲɟɦɨ ɫɢɫɬɟɦɭ ɧɨɪɦɚɥɶɧɢɯ ɪɿɜɧɹɧɶ ɞɥɹ ɦɟɬɨɞɭ ɧɚɣɦɟɧɲɢɯ ɤɜɚɞɪɚɬɿɜ

§

N

T &

T

·

¨

C Cz

C b ,

i 1

¸

¨

N

¦ln xi

¸

G C

T

C

¨

N

N

2

¸

, C

T

b

¨

¦ln xi

¦ ln xi

¸

© i 1

i 1

¹

Ɋɨɡɜ’ɹɡɚɜɲɢ ɫɢɫɬɟɦɭ (13), ɡɧɚɯɨɞɢɦɨ ɿ

, ɬɚ A

§		N		·	(13)
¨		i 1		¸
¨	¦ln yi			¸
¨	N			¸ .
© i 1				¹
¨	¦ln xi		ln yi ¸
exp a		.

8.8. Ɂɝɥɚɞɠɭɸɱɿ ɫɩɥɚɣɧɢ [Ɇɚɪɱɭɤ Ƚ. ɂ. Ɇɟɬɨɞɵ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ

	ɦɚɬɟɦɚɬɢɤɢ, ɫ. 184–181]	~	fi Hi , ɬɨ ɡɚɫɬɨɫɨɜɭɸɬɶ
	əɤɳɨ ɡɧɚɱɟɧɧɹ ɜ ɬɨɱɤɚɯ xi ɧɟɬɨɱɧɨ
		fi

ɡɝɥɚɞɠɭɜɚɧɧɹ. Ⱦɥɹ ɰɶɨɝɨ ɬɪɟɛɚ ɩɨɛɭɞɭɜɚɬɢ ɧɨɜɭ ɬɚɛɥɢɰɸ ɿɡ ɡɝɥɚɞɠɟɧɢɦɢ

ɡɧɚɱɟɧɧɹɦɢ
		fi .
ɇɚɜɟɞɟɦɨ ɞɟɹɤɿ ɩɪɨɫɬɿ ɮɨɪɦɭɥɢ ɡɝɥɚɞɠɭɜɚɧɧɹ:
					1	~	~	~
m 1:	fi					>fi 1	fi	fi	1	@ , N 3.
					3
					1	~		~
	fi					>fi 2	... fi		2	@ , N 5.
					5

78

1

ª

~

~

º

fi

«

2

2

» ,

N - ɩɚɪɧɟ.

« f

N ... f

N

N

¬

i

i

»¼

1

~

~

~

~

~

m 3:

fi

3

5

>

3 fi 2 12 fi 1

17 fi

12 fi 1

3 fi 2 @, N 5.

Ȳɯ ɨɬɪɢɦɭɽɦɨ ɜ ɬɚɤɢɣ ɫɩɨɫɿɛ:

ɞɨ

~

ɛɭɞɭɽɦɨ

fi ɡɚɫɬɨɫɨɜɭɽɦɨ ɚɩɪɨɤɫɢɦɚɰɿɸ,

ɛɚɝɚɬɨɱɥɟɧ ɇɋɄɇ

Qm x

¦ck pkN x ,

m

k 0

ɞɟ pkN –ɫɢɫɬɟɦɚ

ɛɚɝɚɬɨɱɥɟɧɿɜ

ɑɟɛɢɲɨɜɚ ɞɢɫɤɪɟɬɧɨɝɨ ɚɪɝɭɦɟɧɬɭ.

Ȼɟɪɟɦɨ

ɡɧɚɱɟɧɧɹ

xi ,

fi

Qm

ɹɤɿ ɩɪɢɜɨɞɹɬɶ ɞɨ ɧɚɜɟɞɟɧɢɯ ɜɢɳɟ ɮɨɪɦɭɥ.

ɥɟ ɰɿ ɮɨɪɦɭɥɢ ɧɟ ɞɚɸɬɶ ɝɚɪɚɧɬɿɸ, ɳɨ ɜ ɪɟɡɭɥɶɬɚɬɿ ɦɢ ɨɬɪɢɦɚɽɦɨ

ɮɭɧɤɰɿɸ, ɹɤɚ ɡɚɞɨɜɨɥɶɧɹɽ ɭɦɨɜɿ:

Hi

fi

fi

Ɂɝɥɚɞɠɭɸɱɿ ɫɩɥɚɣɧɢ ɞɚɸɬɶ ɦɨɠɥɢɜɿɫɬɶ ɩɨɛɭɞɭɜɚɬɢ ɧɚɛɥɢɠɟɧɧɹ ɡ ɡɚɞɚɧɨɸ ɬɨɱɧɿɫɬɸ. ɇɚɝɚɞɚɽɦɨ ɞɟɹɤɿ ɜɿɞɨɦɨɫɬɿ ɩɪɨ ɫɩɥɚɣɧɢ. əɜɧɢɣ ɜɢɝɥɹɞ

ɤɭɛɿɱɧɨɝɨ ɫɩɥɚɣɧɚ:

1

mi

1

¨ fi

¸

s x mi

i

i

1

i 1 i

i

x

x

3

x x

3

§

m

h

2 · x

x

©

¹

6hi

6hi

6

hi

(1)

§

m h 2 · x

x >xi 1, xi @, hi

x

©

¹

s xi

¨ fi

i i

¸

h

i 1 ,

xi

xi 1.

6

i

Ɍɭɬ

, ɚ mi

xi ɡɚɞɨɜɨɥɶɧɹɸɬɶ ɫɢɫɬɟɦɭ:

fi

,i

0,n

s

­

hi

mi 1

hi hi 1

mi

hi 1

mi 1

fi 1

fi

fi

fi 1

®

1

i

i

°

3

6

h

h

,

i

1,n 1,

(2)

° 6

¯m0 mn

0.

ȼ ɦɚɬɪɢɱɧɿɣ ɮɨɪɦɿ ɰɹ ɫɢɫɬɟɦɚ ɦɚɽ ɜɢɝɥɹɞ:

&

Hf .

(3)

Am

Ɍɭɬ

&

m1

,...,mn 1 7 ,

f0 ,..., fn 7 ,

§ h1 h2

m

f

§

·

·

h2

·

1

§ 1

1

1

¨

¸

¨ h1

¨ h1 h2

¸

h2

¸

3

6

¨

0

¸

¨

0 ....¸

6

3

© ...

¹

¨

¸

¨

¸

A

¨

h2

h2 h3

...¸n

1,

H

¨ ...

... ...

¸n

1.

¨

¸

¨

¸

¨

0

...

...

¸

¨ 0

...

... ...

¸

©

¹

©

¹

n 1

n

1

79

Ʉɭɛɿɱɧɢɣ ɿɧɬɟɪɩɨɥɹɰɿɣɧɢɣ ɫɩɥɚɣɧ ɦɿɧɿɦɿɡɭɽ ɮɭɧɤɰɿɨɧɚɥ:

) s inf ) u U

)(u)

b³

ucc 2dx :

a

u x :u xi

, u x W22 a,b

(5)

fi ,i

0,n

u U

u xi @ .

ȼɜɟɞɟɦɨ ɮɭɧɤɰɿɨɧɚɥ

) u

¦ Ui

fi

)1 u

n

~

2

i

0

Ɂɝɥɚɞɠɭɸɱɢɦ ɫɩɥɚɣɧɨɦ ɧɚɡɜɟɦɨ ɮɭɧɤɰɿɸ g , ɹɤɚ ɽ ɪɨɡɜ‘ɹɡɤɨɦ ɡɚɞɚɱɿ:

)1 g

inf2

1 u

,

(5)

u W2

a.b

ɉɟɪɲɢɣ ɞɨɞɚɧɨɤ ɜ )1 u ɞɚɽ ɦɿɧɿɦɭɦ „ɡɝɢɧɭ”, ɞɪɭɝɢɣ – ɫɟɪɟɞɧɶɨɤɜɚɞɪɚɬɢɱɧɟ

ɧɚɛɥɢɠɟɧɧɹ ɞɨ ɡɧɚɱɟɧɶ

~

. ɉɨɤɚɠɟɦɨ, ɳɨ g

ɽ ɫɩɥɚɣɧɨɦ.

fi

s xi

ɇɟɯɚɣ

ɿɫɧɭɽ ɮɭɧɤɰɿɹ g(x).

ɉɨɛɭɞɭɽɦɨ

ɤɭɛɿɱɧɢɣ

ɫɩɥɚɣɧ

ɬɚɤɢɣ,

ɳɨ

g xi . Ɂ ɬɨɝɨ, ɳɨ g(x)

ɽ ɪɨɡɜ’ɹɡɤɨɦ ɡɚɞɚɱɿ (5), ɦɚɽɦɨ

>

@

)

1

s

t )

1

g

³

scc

dx

¦ U >

f

xi

xi

@

t ³

gcc

dx

¦ U

fi g

xi

b

2

~

2

b

2

~

2

Ɂɜɿɞɫɢ

a

s t

g

.

a

i

Ɍɚɤ ɹɤ ɤɭɛɿɱɧɢɣ ɿɧɬɟɪɩɨɥɹɰɿɣɧɢɣ ɫɩɥɚɣɧ s(x) ɦɿɧɿɦɿɡɭɽ ɮɭɧɤɰɿɨɧɚɥ (4), ɬɨ

) s d ) g . Ɍɨɦɭ ) s

) g . Ɂɜɿɞɤɢ s

g .

ɉɨɡɧɚɱɢɦɨ

Pi

g xi , i

0,n

(6)

əɤɳɨ ɛ ɦɢ ɡɧɚɥɢ

i

, ɬɨ ɞɥɹ ɩɨɛɭɞɨɜɢ g ɞɨɫɬɚɬɧɶɨ ɛɭɥɨ ɛ ɪɨɡɜ‘ɹɡɚɬɢ ɫɢɫɬɟɦɭ

ɉɿɞɫɬɚɜɢɦɨ (1) ɬɚ (6) ɜ )1 g :

Am

H

(7)

¸

dx ¦Ui fi

Pi

inf )1 u . (8)

)1 g

¦

¨mi 1

mi

2

n

xi

§

xi

x

x xi 1 ·

n

~

2

i 1 xi³1 ©

¹

hi

h

i 0

ɉɿɫɥɹ ɩɟɪɟɬɜɨɪɟɧɶ ɦɚɽɦɨ

mi

¸

dx

¦Ui fi

Pi

)1

g

¦

¨mi 1

n

xi

§

xi

x

x xi 1 ·

2

n

~

i 1 xi³1

2

©

¹

fi

hi

mi

hi

i 0

Pi

¦mi ¨mi 1

hi 1

mi 1 ¸

¦Ui

n

§

hi

hi

hi 1

·

n

~

2

©

¹

i

1

6

fi

3

6

i 0

Am,m

¦Ui

Pi

&

&

n

~

2

i 0

Ɂɚɞɚɱɚ 28 ɉɨɤɚɡɚɬɢ, ɳɨ ɞɥɹ ɤɭɛɿɱɧɨɝɨ ɡɝɥɚɞɠɭɸɱɨɝɨ ɫɩɥɚɣɧɭ g ɦɚɽ ɦɿɫɰɟ ɮɨɪɦɭɥɚ:

80