, , . . , f (x), v [a,b], , f (x) .
x 0 [a,b] f (x):
y = f (x 0)+ f (x 0) (x - x 0).
y =0, x 1 O x:
: y = f (x) x = x 1 x 2 :
, { x n},
(6) |
, : | x n+1 - x n| < e. x * x *= x n+1. - . 3.
y=f(x) x = xn, (.9).
. 9. ( )
, (.10), (.11).
. 10. . 11.
( ). f(x) [a,b],
| f (x)|³ m >0, | f (x)|£ M, x Î[a,b].
, x = v (1) [a,b], .. a< v <b. , d: 0<d£min(v -a,b- v), [ v -d, v +d]Ì[a,b] (6) v.
, , x 0 [a,b], :
f (x 0) f (x)>0. | (7) |
0=a x 0=b , (7).
, | f' (x)| , .. y = f (x) .
. ( 10) .
|
|
2.4.
- x 0 (1), . (1)
x =j(x). | (8) |
x 0 j(x) { xn }:
xn +1=j(xn), n =0,1,2, | (9) |
, | xn +1 xn | < e, x * x *= xn +1.
(9) ( ). , j(x) [a,b]. , (. [5], 15, .29)
j(x 2)- j(x 1)=j(x)(x 2- x 1), xÎ[ x 1, x 2]Ì[a,b].
, (8) (8)
( ). x = v (8) j(x) [ v -d, v +d], d>0, j(x),
|j(x)|£ q <1. | (10) |
x 0 [ v -d, v +d] { xn } (9) x = v, (8) [ v -d, v +d].
. , j x y =j(x). (10) , j : x 1 x 2 , y 1=j(x 1) y 2=j(x 2). v j, : v =j(v). (9), , { xn } v.
, (9) , [a,b], v , (10). . (10), |j(x)| 1, , (9) , . |j(x)| , , , . x 0 , . - . 4. , (10) , . , , (10) , (9) .
, (1) (8), (10), . : f (x)=0 x = x - kf (x). j(x)= x - kf (x) j(x) =
=1-k f (x). (10), | j(x) | < 1, |1-k f (x) | < 1, 0<k< , . k |1-kM|=|1-km|, , . k=2/(M+m).
|
|
. 12.
) ) )
. 12.
(, ) , () (| j(x) | > 1).