,,,
1.
2.
2.1
2.2
3. -
4.
5.
. . , . , . , . . . , 300 . . , 6 . n-
a0xn+a1xn-1++an-1x+an=0, a0¹0
n³5 , , , , .
cos(x)=0
. , .
, " ", , . , .
f(x) =0,
- f(x), , ..
() .
.
1.
:
.
, , (, ). , F(X) [A;B].
, F(X), - X0, .
.
.
1.
, [-0.4,0].
x=f(x),
.
:
.
2.
.
.
3
x= (x),
1 .
2 .
3 .
2.
2.1
f(x)=0 (2.1)
X[a, b]. (2.1) :
x=φ(x). (2.2)
, . :
x=g(x) f(x) + x ≡ φ(x),
g(x) - , [a,b].
x(0) - - x ( x(0)=(a+b)/2). :
x(k+1)=φ(x(k)), k=0, 1, 2, ... (2.3)
x(0).
: 1 {x(k)} φ , x=φ(x)
:
. (2.4)
x(k+1)=φ(x(k)) (2.4), φ (2.4)
.
x*=φ(x*). , x* - (2.2), .. X=x*.
{x(k)}. :
2.1: ( ) x=φ(x) [a,b] :
1) φ(x) C1[a,b];
2) φ(x) [a,b] " x [a,b];
3) q > 0: | φ '(x) | ≤ q < 1 x [a,b]. T {x(k)}, x(k+1) = φ(x(k)), k=0, 1, ... x(0) [a,b].
: {x(k)}: x(k) = φ(x(k-1)) x(k+1) = φ(x(k)) T 2) x(k) x(k+1) [a,b], :
x (k+1) - x (k) = φ(x (k)) - φ(x (k-1)) = φ '(c k )(x (k) - x (k-1)),
c k (x (k-1), x (k)).
:
| x (k+1) - x (k) | = | φ '(c k ) | | x (k) - x (k-1) | ≤ q | x (k) - x (k-1)| ≤
≤ q ( q | x (k-1) - x (k-2) | ) = q 2 | x (k-1) - x (k-2) | ≤ ... ≤ q k | x (1) - x (0) |. (2.5)
S∞ = x (0) + ( x (1) - x (0) ) + ... + ( x (k+1) - x (k) ) + ... . (2.6)
, ,
Sk = x (0) + ( x (1) - x (0) ) + ... + ( x (k) - x (k-1) ).
,
Sk = x (k)). (2.7)
, {x(k)}.
p (2.6) ( x(0))
q 0 | x (1) - x (0) | + q 1 |x (1) - x (0)| + ... + |x (1) - x (0)| + ..., (2.8)
( q < 1). (2.5) (2.6) (2.8) ( (2.8) (2.6). (2.6) . T {x(0)}.
,
|X - x (k+1)|
.
X - x(k+1) = X - Sk+1 = S∞ - Sk+1 = (x(k+2) - (k+1) ) + (x(k+3) - x(k+2) ) + ... .
|X - x(k+1)| ≤ |x(k+2) - (k+1) | + |x(k+3) - x(k+2) | + ... ≤ qk+1 |x(1) - x(0) | + qk+2 |x(1) - x(0) | + ... = qk+1|x(1) - x(0) | / (1-q).
|X - x(k+1)| ≤ qk+1|x(1) - x(0) | / (1-q). (2.9)
x(0) x(k), x(1) - x(k+1) ( ) , qk+1 ≤ q :
|X - x(k+1)| ≤ qk+1|x(k+1) - x(k) | / (1-q) ≤ q|x(k+1) - x(k) | / (1-q).
, :
|X - x(k+1)| ≤ q|x(k+1) - x(k) | / (1-q). (2.10)
. x=φ(x) , ε,
|X - x(k+1)| ≤ ε.
(2.10) , ε ,
|x(k+1)-x(k)| ≤ (1-q)/q. (2.11)
, x=φ(x) , ε(1-q)/q.
2.2: q
.
2.2
. , - :
3.
- x .
4.
. 0, . . , :
5.
3. -
- 6, 7.
:
FN, F ;
X, START ;
E, PRECISION ;
N, COUNT_ITER .
6 SIMPLE_ITER
7
4.
SIMPLE_ITER.txt
;,
(DEFUN SIMPLE_ITER (N E X FN)
(COND
((AND (<= N 0) (> (ABS (- (FUNCALL FN X) X)) (* E (FUNCALL FN X)))) X)
(T (SIMPLE_ITER (- N 1) E (FUNCALL FN X) FN))
)
)
;
(LOAD "D:\\FUNCTION.TXT")
;
(SETQ START (/ (- (CADR INTERVAL) (CAR INTERVAL)) 2))
;
(SETQ ROOT (SIMPLE_ITER COUNT_ITER PRECISION START (FUNCTION F)))
;
(SETQ OUTPUT_STREAM (OPEN "D:\\ROOT.TXT" :DIRECTION :OUTPUT))
;
(PRINT 'ROOT OUTPUT_STREAM)
(PRINT ROOT OUTPUT_STREAM)
;
(TERPRI OUTPUT_STREAM)
(CLOSE OUTPUT_STREAM)
FUNCTION.txt ( 1)
;
(DEFUN F (X)
(/ (+ (- (* X X) (* 5 (COS X))) 3.25) 3)
)
;
(SETQ COUNT_ITER 100)
;,
(SETQ INTERVAL '(-0.4 0))
;
(SETQ PRECISION 0.0001)
FUNCTION.txt ( 2)
;
(DEFUN F (X)
(- (* X X) (COS X))
)
;
(SETQ COUNT_ITER 60)
;,
(SETQ INTERVAL '(1 1.5))
;
(SETQ PRECISION 0.0001)
5.
1.
8
9
2.
10
11
, , . , , .
. , .. . .
1. , .. [] / .., ... .: , 2007. 708 .
2. , .. : . [] / .., 3- .:-, 2006. C. 412.
3. , .. . [ ] / .. . .: , 2001. . 504.
4. [ ] : http://solidbase.karelia.ru/edu/meth_calc/files/12.shtm
5. , .. . [] / .., ... .: , 2006. C. 346.
6. , .. [] / .., .., ... : , 2002. 160 .
7. , .. Lisp. [ ] / .., .. . .: , 2003. . 79.
8. . [] / ., .. .: , 1990. 460 .
1. 2. 2.1 2.2 3. - 4.
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