- . . , .. , , , x2 - sin 5x = 0.
, :
a0xn + a1xn-1 + a2xn-2 +... + an = 0, n³5.
() , [3-5].
f(x) = 0. (30)
:
1) , .. ( ), ;
2) ε .
.
: y=f(x) [a, b] , .. f(a)f(b)<0, .
y=f(x) ( 9).
x* , f(x) [a, b] ( 10).
9
10 f(x)
. .
( ).
.
, , [a, b] , e ( 11). : a - , b - .
11
(. 11). x=a, .. f(a).
d = (a + b) / 2 (31)
f(d).
, .
f(a)×f(d)<0 (32)
a b . (32) , , .. b = d, a = d.
, [a, b] e, ..
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(b-a) < e. (33)
[a, b]:
x* = (a+ b) / 2. (34)
- 12.
12 - -
, y=f(x) [a, b] *=xn ( 13).
13
, (a, f(a)) D (b, f(b)):
.
y=0, ..
.
, :
. (35)
xn , [a, xn] [xn , b] xn - f(xn).
= =xn , ..
f(a)×f(xn)<0, (36)
xn , .. b = xn, a = xn.
,
½ xn+1 - xn ½ ³ e. (37)
-, , 14.
( )
y=f(x) x=x0 , 1 .. ( 15).
x0 [a, b] (.. D), :
f(x0)×f '' (x0) > 0. (38)
14 - -
15
( 15), D, .. x0 = b, :
,
.
, :
xn+1 = xn - f(xn) / f '(xn), (39)
,
½ xn+1 - xn ½ ³ e. (40)
. 10-5 - 10-6 5-6 .
-, , 16.
16 - -
.
, e. , .
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17 - -.
17 - - -