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y0 y1

;

y1x0 y0 x1

x0 x1

ɉɨɞɫɬɚɜɥɹɹ ɤɨɷɮɮɢɰɢɟɧɬɵ ɜ P

(x) ɩɨɥɭɱɢɦ

x x1

x x0

P (x)

;

ɬ.ɟ.

ɩɨɥɢɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧ ɜ ɜɢɞɟ

0 x

1 x

ɫɭɦɦɵ ɞɜɭɯ ɥɢɧɟɣɧɵɯ ɮɭɧɤɰɢɣ ɧɟɡɚɜɢɫɹɳɢɯ ɨɬ ɨɪɞɢɧɚɬ ɭɦɧɨ ɠɟɧɧɵɯ ɧɚ ɨɪɞɢɧɚɬɵ ɢ ɨɛɥɚɞɚɸɳɢɯ ɫɜɨɣɫɬɜɨɦ

P (x ) y 1 y 0, P (x ) y 0 y 1.
1	0	0	1	1	1	0	1

ȼ ɷɬɨɦ ɫɨɫɬɨɢɬ ɢɞɟɹ ɩɨɫɬɪɨɟɧɢɹ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɨɝɨ ɩɨɥɢɧɨɦɚ Ʌɚɝɪɚɧɠɚ Ⱦɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ n ɡɚɩɢɲɟɦ ɢɧɬɟɪɩɨɥɹɰɢ ɨɧɧɵɣ ɩɨɥɢɧɨɦ ɜ ɜɢɞɟ

Pn (x)

y0 L0 (x) y1L1(x) ... yn Ln (x),

ɝɞɟ Li

ɩɨɥɢɧɨɦɵ ɫɬɟɩɟɧɢ ɧɟ ɜɵɲɟ n, ɧɟ ɡɚɜɢɫɹɳɢɟ ɨɬ ɨɪɞɢɧɚɬ ɢ

ɨɛɥɚɞɚɸɳɢɟ

ɫɥɟɞɭɸɳɢɦ

ɫɜɨɣɫɬɜɨɦ

Li (xi )

1, Li (x j )

0 .

ɂɡ ɪɚɜɟɧɫɬɜɚ Li (x j )

0 ɫɥɟɞɭɟɬ ɱɬɨ Li ɢɦɟɟɬ n ɤɨɪɧɟɣ ɪɚɫɫɦɚɬ

ɪɢɜɚɸɬɫɹ ɨɞɧɨɤɪɚɬɧɵɟ ɤɨɪɧɢ

L (x)

(x x0 )(x x0 )...(x xi 1)(x xi 1)...(x xn ) ,

ɝɞɟ Ni — ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɬɨɪɵɣ ɧɚɯɨɞɢɬɫɹ ɢɡ ɭɫɥɨɜɢɹ Li (xi )

ȼ ɪɟɡɭɥɶɬɚɬɟ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɩɨɥɢɧɨɦ Ʌɚɝɪɚɧɠɚ ɢɦɟɟɬ ɜɢɞ

P (x)

¦ y

(x x0 )(x x1)...(x xi 1)(x xi 1)...(x xn )

. (2.2.1)

i 0

x )(x

x )...(x

)(x x

)...(x

)

i 1

Ⱦɨɫɬɨɢɧɫɬɜɚ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɨɝɨ ɩɨɥɢɧɨɦɚ Ʌɚɝɪɚɧɠɚ ɹɜɥɹɟɬɫɹ ɩɪɨɫɬɨɬɚ ɤɨɧɫɬɪɭɤɰɢɢ ɉɪɢ ɡɚɞɚɧɧɨɦ ɧɚɛɨɪɟ ɚɛɫɰɢɫɫ ɭɡɥɨɜɵɯ ɬɨ

ɱɟɤ ɢ ɜɵɛɪɚɧɧɨɣ ɪɚɫɱɟɬɧɨɣ ɬɨɱɤɟ x* ɭɩɪɨɳɚɟɬɫɹ ɜɵɱɢɫɥɟɧɢɹ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɨɪɞɢɧɚɬ yi ɇɟɞɨɫɬɚɬɨɤ — ɞɨɛɚɜɥɟɧɢɟ n+1)-ɨɝɨ ɭɡɥɚ

(xn 1, yn 1) ɬɪɟɛɭɟɬ ɩɟɪɟɪɚɫɱɟɬɚ ɜɫɟɯ ɫɥɚɝɚɟɦɵɯ

ɉɨɝɪɟɲɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɹ ɩɭɫɬɶ f (x) — ɮɭɧɤɰɢɹ, ɞɢɮɮɟɪɟɧ ɰɢɪɭɟɦɚɹ n+1 ɪɚɡ, ɢ Pn (x) — ɩɪɢɛɥɢɠɚɸɳɢɣ ɟɺ ɢɧɬɟɪɩɨɥɹɰɢɨɧ ɧɵɣ ɩɨɥɢɧɨɦ

f (x) P

(x)

M n 1

(x x

)(x x )

(x x

)

(n !)!

ɝɞɟ M n 1 max

f (n 1) (x)

, x >a,b@.

ɂɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɩɨɥɢɧɨɦ Ʌɚɝɪɚɧɠɚ ɩɪɢ ɥɢɧɟɣɧɵɯ ɩɪɟɨɛ ɪɚɡɨɜɚɧɢɹɯ x = at + b (t — ɧɨɜɚɹ ɩɟɪɟɦɟɧɧɚɹ ɫɨɯɪɚɧɹɟɬ ɫɜɨɣ ɜɢɞ

ɂɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɩɨɥɢɧɨɦ ɇɶɸɬɨɧɚ

ɉɭɫɬɶ n ɬɨɝɞɚ P0 (x) y0 ɟɫɥɢ n = ɬɨ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɨ

ɥɢɧɨɦɚ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ P (x)	y	y1	y0	(x x ) ɬ ɟ ɩɨ

1	0	x1	0
			x0

ɜɟɞɟɧɢɟ ɩɪɢɛɥɢɠɚɸɳɟɣ ɮɭɧɤɰɢɢ ɫ ɞɨɛɚɜɥɟɧɢɟɦ ɭɡɥɨɜ ɭɬɨɱɧɹɟɬ ɫɹ ɜɛɥɢɡɢ ɬɨɱɤɢ ɯ0 Ʉɨɧɫɬɪɭɤɰɢɹ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɨɝɨ ɩɨɥɢɧɨɦɚ ɇɶɸɬɨɧɚ ɬɚɤɨɜɚ

Pn (x) a0 a1(x x0 ) a2 (x x0 )(x x1) a3 (x x0 )(x x1)(x x2 )

an (x x0 )(x x1) (x xn 1).

Ɋɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɪɚɜɧɨɦɟɪɧɚɹ ɫɟɬɤɚ ɬ ɟ xi x0 ih .

Ⱦɥɹ ɞɚɥɶɧɟɣɲɟɝɨ ɚɧɚɥɢɡɚ ɜɜɨɞɢɬɫɹ ɩɨɧɹɬɢɟ ɤɨɧɟɱɧɨɣ ɪɚɡɧɨ ɫɬɢ Ʉɨɧɟɱɧɨɣ ɪɚɡɧɨɫɬɶɸ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɧɚɡɵɜɚɟɬɫɹ ɜɟɥɢɱɢɧɚ

'y(x) y(x h) y(x), x [x0 , xn ].

Ʉɨɧɟɱɧɚɹ ɪɚɡɧɨɫɬɶ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɩɟɪɜɨɣ

'2 y(x) 'y(x h) 'y(x) y(x 2h) 2 y(x h) y(x)

ɢ ɬ ɞ , ɤɨɧɟɱɧɚɹ ɪɚɡɧɨɫɬɶ i-ɝɨ ɩɨɪɹɞɤɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɱɟɪɟɡ ɪɟɤɭɪ ɪɟɧɬɧɨɟ ɫɨɨɬɧɨɲɟɧɢɟ

'(i 1) y(x)	'(i) y(x h) '(i) y(x)
ɢ ɡɚɜɢɫɢɬ ɨɬ ɡɧɚɱɟɧɢɣ y ɜ i + 1)-ɣ ɬɨɱɤɟ
ȼɵɪɚɠɟɧɢɟ ɜɢɞɚ x[n]	x(x h)(x 2h) (x (n 1)h) ɧɚɡɵɜɚɟɬ

ɫɹ ɨɛɨɛɳɟɧɧɵɦ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ȿɝɨ ɩɟɪɜɚɹ ɤɨɧɟɱɧɚɹ ɪɚɡɧɨɫɬɶ ɪɚɜɧɚ

'x[n] (x h)[n] x[n] (x h)x(x h)...

...(x (n 2)h) x(x h)(x 2h)...(x (n 1)h) nhx[n 1] . (2.3.1)

Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɤɨɧɟɱɧɵɯ ɪɚɡɧɨɫɬɟɣ ɞɥɹ ɨɛɨɛ ɳɟɧɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ ɜɵɫɲɢɯ ɩɨɪɹɞɤɨɜ

ɉɨɞɫɬɚɜɥɹɹ x0 ɜ Pn (x) ɩɨɥɭɱɢɦ a0 Pn (x0 ) y0 Ⱦɚɥɟɟ ɨɩɪɟ ɞɟɥɢɦ ɤɨɧɟɱɧɭɸ ɪɚɡɧɨɫɬɶ ɜ ɬɨɱɤɟ x0 ɂɡ ɫɜɨɣɫɬɜɚ ɩɨɥɭɱɢɦ


'Pn (x) \|x x0	{a1h 2a2h(x x0 ) ...
... nanh(x x0 )(x x1)		(x xn 2 )} \|x x0			a1h 'y0.
Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ ɱɬɨ a			'y0	Ɍɨɱɧɨ ɬɚɤ ɠɟ ɢɡ ɫɥɟɞɭɟɬ
Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ ɱɬɨ a				Ɍɨɱɧɨ ɬɚɤ ɠɟ ɢɡ ɫɥɟɞɭɟɬ
	1		h
			h

ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɤɨɧɟɱɧɨɣ ɪɚɡɧɨɫɬɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɜ ɬɨɱɤɟ x0 :

'2 P (x) \|	x x	{2a h2	3 2a h2		(x x )
n	x x	2		3	0
	0
n(n 1)a h2 (x x )(x x )					(x x	)} \|	x	x	2a h2	'2 y .
	n		0	1	n 3		x	x	2	0
								0
Ɉɛɳɚɹ ɮɨɪɦɭɥɚ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ a								'(i) y
Ɉɛɳɚɹ ɮɨɪɦɭɥɚ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ a									0 .
						i		i!hi
								i!hi

ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɦ ɩɟɪɜɵɣ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɩɨɥɢɧɨɦ ɇɶɸɬɨɧɚ

				'y				'(2) y
P	(x)	y		0 (x x		)		0	(x x	)(x	x ) ...
P	(x)	y		0 (x x		)			(x x	)(x	x ) ...
n		0		hi	0			2!h2	0		1
				hi				2!h2			(2.3.2)
			'(n) y								(2.3.2)
		...	'(n) y				)(x x )		(x x		).
		...		0 (x x			)(x x )		(x x		).
				n!hn		0		1		n 1
				n!hn

ɉɨɫɬɪɨɟɧɧɵɣ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɩɨɥɢɧɨɦ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɭɡɥɨɜɵɟ ɬɨɱɤɢ

ȼɬɨɪɨɣ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɩɨɥɢɧɨɦ ɇɶɸɬɨɧɚ ɩɨɡɜɨɥɹɟɬ ɧɚ ɱɚɬɶ ɢɧɬɟɪɩɨɥɹɰɢɸ ɫ ɬɨɱɤɢ xn ɬ ɟ ɭɥɭɱɲɢɬɶ ɬɨɱɧɨɫɬɶ ɩɪɢɛɥɢɠɟ

ɧɢɹ ɧɚ ɩɪɚɜɨɣ ɝɪɚɧɢɰɟ ɢɧɬɟɪɜɚɥɚ ɢɧɬɟɪɩɨɥɹɰɢɢ

Pn (x) a0 a1(x xn ) a2 (x xn 1)(x xn )

a3 (x xn 2 )(x xn 1)(x xn ) an (x x1)(x xn 1) (x xn ).

ɂɡ ɫɬɪɭɤɬɭɪɵ ɩɨɥɢɧɨɦɚ ɫɥɟɞɭɟɬ ɱɬɨ a0 yn .

'Pn (xn 1) \|x x	'yn 1	{ha1	2ha2	(x xn ) } \|x x ;
n				n

a	'yn 1	; a	2	'2 yn 2	ɢ ɬɚɤ ɞɚɥɟɟ Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ

1	h 1!			h2 2!

P	(x)	y	n	'yn 1	(x x	)	'2 yn 2		(x x	)(x x	) ...
P	(x)	y		'yn 1	(x x	)			(x x	)(x x	) ...
n				h	n		h2	2!	n	n 1	(2.3.3)
				h			h2	2!
					'n y
				...	'n y			(x x ).
				...	0 (x x )			(x x ).
					hnn!		n		1
					hnn!

ɉɪɢ ɪɚɫɱɺɬɚɯ ɢ ɚɥɝɨɪɢɬɦɢɡɚɰɢɢ ɜɵɱɢɫɥɟɧɢɹ ɢɧɬɟɪɩɨɥɹɰɢɨɧ ɧɨɝɨ ɩɨɥɢɧɨɦɚ ɩɪɢɦɟɧɹɟɬɫɹ ɬɚɛɥɢɰɚ ɤɨɧɟɱɧɵɯ ɪɚɡɧɨɫɬɟɣ

Ɍɚɛɥɢɰɚ

ʋ	x	y	'y	'2 y	'3 y	…
0	x0	y0	'y0	'2 y	'3 y	…
				0	0
1	x1	y1	'y1	'2 y	…
				1
2	x2	y2	'y2	…
3	x3	y3	…
…	…	…

Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ -ɝɨ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɨɝɨ ɩɨɥɢɧɨɦɚ ɇɶɸɬɨɧɚ ɧɟɨɛɯɨɞɢɦɚ -ɹ ɫɬɪɨɤɚ ɬɚɛɥ Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ -ɝɨ ɢɧɬɟɪɩɨɥɹ ɰɢɨɧɧɨɝɨ ɩɨɥɢɧɨɦɚ ɇɶɸɬɨɧɚ ɧɟɨɛɯɨɞɢɦɚ ɩɨɛɨɱɧɚɹ ɞɢɚɝɨɧɚɥɶ ɬɚɛɥɢɰɵ Ɉɛɵɱɧɨ ɩɪɢ ɦɚɲɢɧɧɵɯ ɪɚɫɱɺɬɚɯ ɦɚɫɫɢɜ ɨɪɞɢɧɚɬ ɭɡɥɨ ɜɵɯ ɬɨɱɟɤ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɜ ɦɚɫɫɢɜ ɤɨɷɮɮɢɰɢɟɧ ɬɨɜ ai ɬɚɤ, ɱɬɨ ɨɧɢ ɡɚɩɨɦɢɧɚɸɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɷɥɟɦɟɧɬɚɯ

ɦɚɫɫɢɜɚ

ɉɪɢɦɟɪɵ ɢ ɡɚɞɚɧɢɹ ɞɥɹ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɧɹɬɢɣ

ɉɪɢɦɟɪ. Ⱦɚɧɚ ɬɚɛɥɢɰɚ ɭɡɥɨɜ ɉɨɫɬɪɨɢɬɶ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɩɨ ɥɢɧɨɦ Ʌɚɝɪɚɧɠɚ ɢ ɩɪɨɜɟɫɬɢ ɩɪɨɜɟɪɤɭ ɬɚɛɥ

Ɍɚɛɥɢɰɚ

N	0	1	2	3

X	0	0,5	1	1,5
Y	1	2	3	1

ȼ ɜɵɪɚɠɟɧɢɟ ɞɥɹ n=3
P (x)		y	(x x1)(x x2 )(x x3 )							y	(x x0 )(x x2 )(x x3 )
P (x)		y								y
3		0 (x x )(x x )(x x )								1 (x x )(x x )(x x )
			0	1	0	2	0	3				1	0	1	2	1	3
y2	(x x0 )(x x1)(x x3 )							y3		(x x0 )(x x1)(x x2 )
y2	(x x )(x x )(x x )							y3	(x x )(x x )(x x )
2		0	2	1	2	3				3	0	3	1	3	2

ɧɟɨɛɯɨɞɢɦɨ ɩɨɞɫɬɚɜɢɬɶ ɞɚɧɧɵɟ ɢɡ ɬɚɛɥ :

P (x)

1(x 0,5)(x 1)(x 1,5) 2

(x 0)(x 1)(x 1,5)

(0 0,5)(0 1)(0 1,5)

(0,5 0)(0,5 1)(0,5 1,5)

(x 0)(x 0,5)(x 1,5)

(x 0)(x 0,5)(x 1)

(1 0)(1 0,5)(1 1,5)

(1,5 0)(1,5 0,5)(1,5 1)

ɉɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨɥɭɱɢɦ P (x) 4x3 6x2

ɉɪɨɜɟɪɤɚ:

(x0 )

P3 (0)

1 { y0 , P3 (x1)

P3 (0,5)

4 / 8 6 / 4 1

2 { y1,

(x2 )

P3 (1)

4 6 1

3 { y2 ,

(x3 )

P3 (1,5)

4 27 / 8 6 9 / 4 1

1 { y3.

ɉɪɢɦɟɪ ɉɨɫɬɪɨɢɬɶ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɟ ɩɨɥɢɧɨɦɵ ɇɶɸɬɨɧɚ ɩɨ ɩɪɟɞɵɞɭɳɟɣ ɬɚɛɥɢɰɟ ɭɡɥɨɜɵɯ ɬɨɱɟɤ

ɉɟɪɜɵɣ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɩɨɥɢ

ʋ	x		y	'y		'2 y	'3y		ɧɨɦ ɇɶɸɬɨɧɚ
0	0	1		1	0		-3		P (x)		y 'y		(x x )
1	0,5	2		1		-3			P (x)		y 'y		(x x )
1	0,5	2		1		-3		3			0	h	0
2	1	3		-2						'2 y		h
2	1	3		-2
3	1,5		1
										0	(x x0 )(x x1)
										2h2

'3 y0 (x x0 )(x x1)(x x2 ); 6h3

P3 (x) 1 2x 0	3	x(x 0,5)(x 1)	1 2x
	6(0,5)3

4(x3 1,5x2 0,5) 4x3 6x2 1.

ȼɬɨɪɨɣ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɩɨɥɢɧɨɦ ɇɶɸɬɨɧɚ

		'y	2		'2 y			'3 y
P (x)	y		2	(x x )	1 (x x )(x x )			0 (x x )(x x )(x x );
P (x)	y			(x x )	1 (x x )(x x )			0 (x x )(x x )(x x );
3	3	h	3		2h2	3	2	6h3	3	2	1
		h			2h2			6h3

			P (x)	1 2	x 1,5			3	(x 1,5)(x 1)
			P (x)	1 2					(x 1,5)(x 1)
		3				0,5		2(0,5)2
						0,5		2(0,5)2
		3		(x 1,5)(x 1)(x 0,5) 4x3 6x2 1.
		6(0,5)3		(x 1,5)(x 1)(x 0,5) 4x3 6x2 1.
		6(0,5)3
ȼɚɪɢɚɧɬɵ ɡɚɞɚɸɬɫɹ ɩɨ ɧɨɦɟɪɚɦ ɫɬɨɥɛɰɨɜ ɬɚɛɥ ɢ ɜ ɜɢɞɟ
ɞɪɨɛɟɣ	N y		ɧɚɩɪɢɦɟɪ			9	ɨɡɧɚɱɚɟɬ ɱɬɨ ɞɥɹ ɭɡɥɨɜɵɯ ɬɨɱɟɤ ɩɨ ɯ ɢ
ɞɪɨɛɟɣ	Nx		ɧɚɩɪɢɦɟɪ			2	ɨɡɧɚɱɚɟɬ ɱɬɨ ɞɥɹ ɭɡɥɨɜɵɯ ɬɨɱɟɤ ɩɨ ɯ ɢ
	Nx					2

ɭ ɜɵɛɢɪɚɸɬɫɹ ɞɟɜɹɬɵɣ ɢ ɜɬɨɪɨɣ ɜɚɪɢɚɧɬɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ Ʉɚɠ ɞɵɣ ɫɬɭɞɟɧɬ ɞɨɥɠɟɧ ɩɨɥɭɱɢɬɶ ɬɪɢ ɬɚɤɢɯ ɞɪɨɛɢ ɞɥɹ ɪɚɫɱɟɬɚ ɢɧ ɬɟɪɩɨɥɹɰɢɨɧɧɨɝɨ ɩɨɥɢɧɨɦɚ Ʌɚɝɪɚɧɠɚ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ ɢɧɬɟɪ ɩɨɥɹɰɢɨɧɧɨɝɨ ɩɨɥɢɧɨɦɚ ɇɶɸɬɨɧɚ Ɋɟɡɭɥɶɬɚɬ ɧɟɨɛɯɨɞɢɦɨ ɩɪɟɞɫɬɚ

ɜɢɬɶ ɜ ɜɢɞɟ P3 (x) a0 a1x a2 x2 a3 x3 ɝɞɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɚ

ɜɢɥɶɧɵɟ ɢɥɢ ɧɟɩɪɚɜɢɥɶɧɵɟ ɞɪɨɛɢ ɧɟ ɞɟɫɹɬɢɱɧɵɟ. ɉɪɨɜɟɪɤɚ ɩɪɨ ɢɡɜɨɞɢɬɫɹ ɩɨɞɫɬɚɧɨɜɤɨɣ ɭɡɥɨɜɵɯ ɬɨɱɟɤ

Ɍɚɛɥɢɰɚ

	ȼɚɪɢɚɧɬɵ Nx
n	1	2	3
0	0	-0,5	-1
1	0,5	0	-0,5
2	1	0,5	0
3	1,5	1	0,5

Ɍɚɛɥɢɰɚ

n								ȼɚɪɢɚɧɬɵ N y
n	1	2	3	4	5	6	7		8	9	10	11	12	13	14	15	16
	1	2	3	4	5	6	7		8	9	10	11	12	13	14	15	16
0	-1	2	1	-1	2	1	-1		2	1	1	-1	0	2	2	1	3
1	2	-1	2	0	1	1	2		-2	2	-1	-1	-1	0	1	-2	0
2	1	1	-1	-1	-1	-1	-2		-1	-1	2	2	2	-1	0	-1	-1
3	1	0	1	2	2	2	1		1	-1	1	-2	1	2	-2	-1	2

ɑɂɋɅȿɇɇɕȿ ɆȿɌɈȾɕ ɊȿɒȿɇɂɃ ɌɊȺɇɋɐȿɇȾȿɇɌɇɕɏ ɂ ȺɅȽȿȻɊȺɂɑȿɋɄɂɏ ɍɊȺȼɇȿɇɂɃ

Ɉɛɳɢɣ ɜɢɞ ɭɪɚɜɧɟɧɢɹ f (x) 0 Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ ɬ ɟ ɧɚɣɬɢ ɟɝɨ ɤɨɪɟɧɶ ɨɡɧɚɱɚɟɬ ɨɩɪɟɞɟɥɢɬɶ x* ɬɚɤɨɟ ɱɬɨ f (x* ) { 0 .

ȼɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɬɨɱɧɨɟ ɡɧɚɱɟɧɢɟ x* ɧɚɣɬɢ ɧɟɜɨɡɦɨɠɧɨ ɩɨ ɷɬɨɦɭ ɢɫɩɨɥɶɡɭɸɬɫɹ ɩɪɢɛɥɢɠɟɧɧɵɟ ɦɟɬɨɞɵ ɤɨɝɞɚ ɡɧɚɱɟɧɢɟ ɤɨɪɧɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɶɸ H Ƚɟɨɦɟɬɪɢɱɟɫɤɢ ɤɨɪɟɧɶ – ɷɬɨ ɩɟɪɟɫɟɱɟɧɢɟ ɝɪɚɮɢɤɨɦ ɮɭɧɤɰɢɢ f (x) ɨɫɢ x .

Ɂɚɞɚɱɚ ɞɟɥɢɬɫɹ ɧɚ ɷɬɚɩɚ 1. Ʌɨɤɚɥɢɡɚɰɢɹ ɤɨɪɧɹ – ɬ ɟ ɧɚɯɨɠɞɟɧɢɟ ɢɧɬɟɪɜɚɥɚ ɧɚ ɤɨɬɨɪɨɦ

ɢɡɨɥɢɪɨɜɚɧ ɟɞɢɧɫɬɜɟɧɧɵɣ ɧɭɠɧɵɣ ɧɚɦ ɤɨɪɟɧɶ ȼɵɛɨɪ ɢɧɬɟɪɜɚɥɚ ɩɪɨɢɡɜɨɞɢɬɫɹ ɩɭɬɟɦ ɚɧɚɥɢɡɚ ɡɧɚɤɚ f (x) ɜ ɪɹɞɟ ɩɪɨɛɧɵɯ ɬɨɱɟɤ ɗɬɨɬ ɩɪɨɰɟɫɫ ɜ ɨɛɳɟɦ ɜɢɞɟ ɧɟ ɚɥɝɨɪɢɬɦɢɡɢɪɭɟɬɫɹ

2. ɍɬɨɱɧɟɧɢɟ ɩɨɥɨɠɟɧɢɹ ɤɨɪɧɹ ɧɚ ɢɧɬɟɪɜɚɥɟ ɢɡɨɥɹɰɢɢ. ɋɜɨɣɫɬɜɚ ɮɭɧɤɰɢɢ ɧɚ ɢɧɬɟɪɜɚɥɟ ɢɡɨɥɹɰɢɢ [a, b]:

2.1.f (x) ɧɟɩɪɟɪɵɜɧɚ ɧɚ >a, b];

2.2.f (x) ɦɨɧɨɬɨɧɧɚ ɧɚ >a, b@ ɬ ɟ f c(x) ! 0 ɢɥɢ f c(x) 0 ɱɬɨ

ɨɛɭɫɥɚɜɥɢɜɚɟɬ ɟɞɢɧɫɬɜɟɧɧɨɫɬɶ ɤɨɪɧɹ;

2.3. f (x) ɦɟɧɹɟɬ ɡɧɚɤ ɧɚ >a, b], f (a) f (b) 0 ɬ ɟ ɤɨɪɟɧɶ ɫɭɳɟ

ɫɬɜɭɟɬ;

2.4. f (x) ɧɟ ɢɦɟɟɬ ɬɨɱɟɤ ɩɟɪɟɝɢɛɚ ɬ ɟ f cc(x) ! 0 ɢɥɢ f cc(x) 0. ɉɨɫɥɟɞɧɢɟ ɭɫɥɨɜɢɹ ɧɟ ɹɜɥɹɸɬɫɹ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɨɛɹɡɚɬɟɥɶɧɵ

ɦɢ ɧɨ ɞɥɹ ɫɯɨɞɢɦɨɫɬɢ ɧɟɤɨɬɨɪɵɯ ɦɟɬɨɞɨɜ ɨɧɢ ɧɟɨɛɯɨɞɢɦɵ Ɍɚɤ ɟɫɥɢ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɤɨɪɟɧɶ ɜ ɬɨɱɤɟ ɫɜɨɟɝɨ ɥɨɤɚɥɶɧɨɝɨ ɦɢɧɢɦɭɦɚ ɭɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ ɨɞɧɚɤɨ ɨɧɨ ɧɟɨɛɯɨɞɢɦɨ ɞɥɹ ɫɯɨɞɢ ɦɨɫɬɢ ɦɟɬɨɞɨɜ ɞɢɯɨɬɨɦɢɢ ɯɨɪɞ ɢ ɫɟɤɭɳɢɯ Ⱦɥɹ ɫɯɨɞɢɦɨɫɬɢ ɦɟɬɨ ɞɚ ɫɟɤɭɳɢɯ ɬɚɤɠɟ ɧɟɨɛɯɨɞɢɦɨ ɜɵɩɨɥɧɟɧɢɟ ɭɫɥɨɜɢɹ

ɇɚɯɨɠɞɟɧɢɟ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɤɨɪɧɹ – ɷɬɨ ɢɬɟɪɚɰɢɨɧ ɧɵɣ ɩɪɨɰɟɫɫ ɤɨɝɞɚ ɩɨ ɩɪɟɞɵɞɭɳɟɦɭ ɩɪɟɞɵɞɭɳɢɦ ɡɧɚɱɟɧɢɹɦ ɤɨɪɧɹ ɧɚɯɨɞɢɬɫɹ ɫɥɟɞɭɸɳɟɟ ɩɪɢɛɥɢɠɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɂɬɟɪɚɰɢɨɧ ɧɵɣ ɩɪɨɰɟɫɫ ɩɪɟɤɪɚɳɚɟɬɫɹ ɤɨɝɞɚ ɞɨɫɬɢɝɚɟɬɫɹ ɡɚɞɚɧɧɚɹ ɬɨɱɧɨɫɬɶ

f (xn H) f (xn H) 0 .

(3.1)

Ⱦɥɹ ɷɬɨɝɨ ɧɟɨɛɯɨɞɢɦɨ ɱɬɨɛɵ ɩɪɨɰɟɫɫ ɢɬɟɪɚɰɢɣ ɫɯɨɞɢɥɫɹ Ɋɚɫ ɫɦɨɬɪɢɦ ɧɟɫɤɨɥɶɤɨ ɢɬɟɪɚɰɢɨɧɧɵɯ ɩɪɨɰɟɞɭɪ

Ɇɟɬɨɞ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɞɥɹ ɪɟɲɟɧɢɹ ɧɟɥɢɧɟɣɧɵɯ ɢ ɬɪɚɧɫɰɟɧɞɟɧɬɧɵɯ ɭɪɚɜɧɟɧɢɣ

ɍɪɚɜɧɟɧɢɟ f (x)

0 ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɤ ɜɢɞɭ

I(x) ,

(3.1.1)

ɢ ɟɫɥɢ ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ

(3.1.2)

(x)

ɬɨ ɢɬɟɪɚɰɢɨɧɧɵɣ ɩɪɨɰɟɫɫ

(3.1.3)

xk 1 I(xk )

ɫɯɨɞɢɬɫɹ

ɬɨɱɧɨɦɭ

ɡɧɚɱɟɧɢɸ

Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ

xk 1 xk

I(xk ) I(xk 1) ɢɡ ɬɟɨɪɟɦɵ ɨ ɫɪɟɞɧɟɦ ɫɥɟɞɭɟɬ ɨɰɟɧɤɚ

xk 1 xk

d M1

xk xk 1

ɬ ɟ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɬɨɱɤɚɦɢ ɩɨɫɥɟ

ɞɨɜɚɬɟɥɶɧɨɫɬɢ ɭɦɟɧɶɲɚɟɬɫɹ ɟɫɥɢ M1 1 — ( M1 max Ic = q –

ɡɧɚɦɟɧɚɬɟɥɶ ɫɯɨɞɢɦɨɫɬɢ ɉɨ ɬɟɨɪɟɦɟ ɨ ɧɟɩɨɞɜɢɠɧɨɣ ɬɨɱɤɟ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɭɳɟɫɬɜɭɟɬ ɩɪɟɞɟɥ — ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɇɚɱɚɥɶɧɚɹ ɬɨɱɤɚ x0 — ɥɸɛɚɹ ɬɨɱɤɚ ɢɧɬɟɪɜɚɥɚ ɥɨɤɚɥɢɡɚɰɢɢ ɤɨɪɧɹ Ɂɧɚɦɟɧɚ

ɬɟɥɶ ɫɯɨɞɢɦɨɫɬɢ ɡɚɜɢɫɢɬ ɨɬ ɜɢɞɚ I(x) ɍɪɚɜɧɟɧɢɟ f (x) 0 ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɨɛɪɚɡɨɜɚɧɨ ɤ ɢɬɟɪɚɰɢɨɧɧɨɦɭ ɜɢɞɭ ɦɧɨɠɟɫɬɜɨɦ ɪɚɡɥɢɱɧɵɯ ɫɩɨɫɨɛɨɜ – ɦɨɞɢɮɢɤɚɰɢɣ ɨɞɧɨɲɚɝɨɜɨɝɨ ɫɬɚɰɢɨɧɚɪɧɨ ɝɨ ɦɟɬɨɞɚ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɫɦ ɬɚɤɠɟ ɜɵɛɨɪɨɦ ɤɨɬɨɪɵɯ ɦɨɠɧɨ ɞɨɛɢɬɶɫɹ ɦɢɧɢɦɭɦɚ ɡɧɚɦɟɧɚɬɟɥɹ ɫɯɨɞɢɦɨɫɬɢ ɇɚɩɪɢɦɟɪ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɷɤɜɢɜɚɥɟɧɬɧɨ ɫɥɟɞɭɸɳɟɦɭ

x x Of (x) Ⱦɨɫɬɚɬɨɱɧɨɟ ɭɫɥɨɜɢɟ ɫɯɨɞɢɦɨɫɬɢ ɜɵɩɨɥɧɹ

ɟɬɫɹ ɟɫɥɢ O	1	c			max		c	, x [a,b] .
	1	c					c
		, f (x) ! 0, ɝɞɟ M1					f (x)
		M1
		M1
		Ɇɟɬɨɞ ɯɨɪɞ ɢ ɫɟɤɭɳɢɯ
ɇɚ ɢɧɬɟɪɜɚɥɟ [a,b]			ɡɚɦɟɧɢɦ f (x)		ɥɢɧɟɣɧɵɦ ɢɧɬɟɪɩɨɥɹɰɢɨɧ
ɧɵɦ ɩɨɥɢɧɨɦɨɦ ɩɪɨɯɨɞɹɳɟɦ ɱɟɪɟɡ ɬɨɱɤɢ (a, f (a) ɢ (b, f (b)) :
		P (x)	f (a)	f (b) f (a)		(x a) .
		P (x)	f (a)			(x a) .
		1		b a
				b a

ȼ ɤɚɱɟɫɬɜɟ ɩɟɪɜɨɝɨ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɤɨɪɧɹ ɜɵɛɟɪɟɦ

ɤɨɪɟɧɶ ɩɨɥɢɧɨɦɚ P1(x) 0 ɬɨɝɞɚ

x a	f (a)(b a)	.	(3.2.1)

1	f (b) f (a)

Ⱦɚɥɟɟ ɟɫɥɢ ɩɨɜɟɞɟɧɢɟ f cc(x) ɧɟɢɡɜɟɫɬɧɨ ɬɨ ɜɵɛɢɪɚɸɬ ɢɧɬɟɪɜɚɥ

ɧɚ ɤɨɬɨɪɨɦ f (x) ɦɟɧɹɟɬ ɡɧɚɤ [a; x1] ɢɥɢ [x1, a] ɢ ɧɚ ɧɟɦ ɫɬɪɨɹɬ ɧɨɜɭɸ ɯɨɪɞɭ ɬ ɟ ɜ ɮɨɪɦɭɥɭ ɩɨɞɫɬɚɜɥɹɟɦ ɧɨɜɵɟ ɝɪɚɧɢɰɵ ɢɧɬɟɪ ɜɚɥɚ ɢ ɬ ɞ ɞɨ ɞɨɫɬɢɠɟɧɢɹ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɢ

ȿɫɥɢ f (x)	ɧɟ ɢɦɟɟɬ ɬɨɱɤɢ ɩɟɪɟɝɢɛɚ ɧɚ [a;b] ɬɨ ɨɞɢɧ ɢɡ ɤɨɧɰɨɜ
ɦɧɨɠɟɫɬɜɚ ɯɨɪɞ ɧɟɩɨɞɜɢɠɟɧ ɍɫɥɨɜɢɟ ɧɟɩɨɞɜɢɠɧɨɣ ɬɨɱɤɢ
			a,ɟɫɥɢf a f			cc
		ɫ	a,ɟɫɥɢf a f			a ²		(3.2.2)
		ɫ	®			cc		(3.2.2)
			¯b,ɟɫɥɢf b f			b ²
cc
Ⱥɧɚɥɢɡ f	(x) ɩɨɡɜɨɥɹɟɬ ɨɩɪɟɞɟɥɢɬɶ ɧɟɩɨɞɜɢɠɧɭɸ ɬɨɱɤɭ c ɢ ɞɥɹ
ɧɚɯɨɠɞɟɧɢɹ xn 1 ɢɫɩɨɥɶɡɨɜɚɬɶ ɢɬɟɪɚɰɢɨɧɧɭɸ ɮɨɪɦɭɥɭ
	xn 1	I(xn )		xn	f (xn )		c xn ,	(3.2.3)
	xn 1	I(xn )		xn	f (c) f (xn )		c xn ,	(3.2.3)
					f (c) f (xn )

ɩɪɢɱɟɦ x0 c .

ɉɪɢ ɨɬɫɭɬɫɬɜɢɢ ɬɨɱɤɢ ɩɟɪɟɝɢɛɚ ɜ ɨɛɥɚɫɬɢ ɥɨɤɚɥɢɡɚɰɢɢ ɤɨɪɧɹ ɛɨɥɟɟ ɷɮɮɟɤɬɢɜɧɵɦ ɹɜɥɹɟɬɫɹ ɞɜɭɯɲɚɝɨɜɵɣ ɦɟɬɨɞ ɫɟɤɭɳɢɯ ɜ ɤɨ ɬɨɪɨɦ ɩɨɫɥɟɞɭɸɳɟɟ ɩɪɢɛɥɢɠɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɤɨɪɧɹ ɧɚɯɨɞɢɬɫɹ ɩɨ ɞɜɭɦ ɩɪɟɞɵɞɭɳɢɦ ɑɟɪɟɡ ɩɟɪɜɵɟ ɞɜɟ ɬɨɱɤɢ ɩɪɨɜɨɞɢɬɫɹ ɫɟɤɭɳɚɹ ɩɟɪɟɫɟɱɟɧɢɟ ɤɨɬɨɪɨɣ ɫ ɨɫɶɸ ɚɛɫɰɢɫɫ ɞɚɟɬ ɫɥɟɞɭɸɳɟɟ ɩɪɢɛɥɢɠɟɧ ɧɨɟ ɡɧɚɱɟɧɢɟ ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɪɢɯɨɞɢɦ ɤ ɢɬɟɪɚɰɢɨɧɧɨɣ ɮɨɪɦɭɥɟ

xn 1 xn	f (xn )	xn xn 1	.	(3.2.4)
	f (xn ) f (xn 1)

Ⱥɧɚɥɨɝɢɱɧɚɹ ɮɨɪɦɭɥɚ ɩɨɥɭɱɚɟɬɫɹ ɟɫɥɢ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɮɨɪɦɭɥɵ ɦɟɬɨɞɚ ɇɶɸɬɨɧɚ ɜɦɟɫɬɨ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɮɭɧɤɰɢɢ ɩɨɞɫɬɚɜɢɬɶ ɟɺ ɤɨɧɟɱɧɨɪɚɡɧɨɫɬɧɭɸ ɚɩɩɪɨɤɫɢɦɚɰɢɸ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɜ ɬɨɱɤɟ xn .

Ɇɟɬɨɞ ɤɚɫɚɬɟɥɶɧɵɯ (ɦɟɬɨɞ ɇɶɸɬɨɧɚ

ȼ ɷɬɨɦ ɦɟɬɨɞɟ ɜ ɤɚɱɟɫɬɜɟ x0 ɜɵɛɢɪɚɟɬɫɹ ɨɞɧɚ ɢɡ ɝɪɚɧɢɰ ɢɧɬɟɪ ɜɚɥɚ [a,b] ɢ ɢɡ ɷɬɨɣ ɬɨɱɤɢ ɫɬɪɨɢɬɫɹ ɤɚɫɚɬɟɥɶɧɚɹ ȼ ɤɚɱɟɫɬɜɟ ɩɪɢ ɛɥɢɠɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɤɨɪɧɹ x1 ɩɪɢɧɢɦɚɟɬɫɹ ɬɨɱɤɚ ɩɟɪɟɫɟɱɟɧɢɹ ɤɚɫɚɬɟɥɶɧɨɣ ɫ ɨɫɶɸ ɚɛɫɰɢɫɫ ɂɡ ɬɨɱɤɢ (x1, f (x1) ɩɪɨɜɨɞɢɬɫɹ ɧɨɜɚɹ

ɤɚɫɚɬɟɥɶɧɚɹ ɢ ɬ ɞ ɞɨ ɞɨɫɬɢɠɟɧɢɹ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɢ ɍɪɚɜɧɟɧɢɟ ɤɚɫɚɬɟɥɶɧɨɣ ɜ ɬɨɱɤɟ xn ɢɦɟɟɬ ɜɢɞ

yk (x) f (xn ) f c(xn ) (x xn ) , yk (xn 1) 0 ,

ɨɬɫɸɞɚ ɫɥɟɞɭɟɬ ɢɬɟɪɚɰɢɨɧɧɵɣ ɩɪɨɰɟɫɫ

xn 1 I(xn ) xn	f (xn )	f c(xn )	.	(3.3.1)

ȼɵɪɚɠɟɧɢɟ ɞɥɹ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɢ x0 ɫɨɜɩɚɞɚɟɬ ɫ

Ɇɟɬɨɞ ɇɶɸɬɨɧɚ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɦɨɞɢɮɢɤɚɰɢɟɣ ɦɟɬɨɞɚ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɩɪɢ I(x) x f (x) f c(x) ɍɫɥɨɜɢɹ ɫɯɨɞɢɦɨɫɬɢ ɦɟɬɨɞɚ ɫɥɟɞɭɸɬ ɢɡ ɚ ɢɦɟɧɧɨ ɞɥɹ ɜɫɟɯ x ɢɡ ɨɛɥɚɫɬɢ ɥɨɤɚ ɥɢɡɚɰɢɢ ɤɨɪɧɹ ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ


q	f (x) f		''(x)	< 1.	(3.3.2)
q	( f	'(x))2		< 1.	(3.3.2)
	( f	'(x))2

ɂɡ ɫɥɟɞɭɟɬ ɱɬɨ ɱɟɦ ɦɟɧɶɲɟ ɨɛɥɚɫɬɶ ɥɨɤɚɥɢɡɚɰɢɢ ɤɨɪɧɹ ɬɟɦ ɦɟɧɶɲɟ ɡɧɚɦɟɧɚɬɟɥɶ q ɫɯɨɞɢɦɨɫɬɢ ɦɟɬɨɞɚ ɇɶɸɬɨɧɚ ɢ ɜ ɩɪɟɞɟɥɟ

q o 0 ɩɪɢ x o x Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨɣ ɨɛɥɚɫɬɢ ɥɨɤɚɥɢɡɚɰɢɢ ɤɨɪɧɹ ɫɯɨɞɢɦɨɫɬɶ ɦɟɬɨɞɚ ɇɶɸɬɨɧɚ ɛɟɡɭɫɥɨɜɧɚɹ

3.4. ɋɤɨɪɨɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɢɬɟɪɚɰɢɨɧɧɵɯ ɦɟɬɨɞɨɜ

ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɹ 'x x x*, 'x x x Ⱦɥɹ ɨɰɟɧɤɢ

n n n n n 1

ɫɤɨɪɨɫɬɢ ɫɯɨɞɢɦɨɫɬɢ ɧɟɨɛɯɨɞɢɦɨ ɨɩɪɟɞɟɥɢɬɶ ɡɚɜɢɫɢɦɨɫɬɶ ɦɟɠɞɭ

'x	ɢ 'x .
n 1	n

ȿɫɥɢ ɜ ɩɪɨɰɟɫɫɟ ɢɬɟɪɚɰɢɣ ɧɚɱɢɧɚɹ ɫ ɧɟɤɨɬɨɪɨɝɨ n ɜɵɩɨɥɧɹɟɬɫɹ

n 1 d q ɝɞɟ r, q const , (3.4.1)

'xn r

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