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(x)
(x)
(5.4)
2
5 3x
2
2
3x
(1) = 1,5
(2) = 6
(x) < 1,
(5.6) 3 25 x 3 )25(3
2
x
(1) = 0,32
(2) = 0,667
1< (x) < 0,
(5.7)
xx
152 23
110
xx
(1) = 9
(2) = 1
(x) 1,
5.1,
(x) (5.6),
.
5.1.3
, f(x) = 0 -
(5.2) x = (x),
, . 5.1.
(x), .
ba, . (5.2) - ,
,,,1' baxqx (5.10)
q 0 1. , -
. : f(x) = 0,
ba, , ? , , , f '(x) [a, b]. -
,
xfxx .
x = (x) f(x) = 0 -
0. , ,
(x) (5.10). ,
xfx '1'
(5.10) ,
.1'1 qxfq (5.11)
-
41
, ,
baxMxfm ,,'0 11
(, m1 1 -
f (x) [, b]).
.1'11 11 mxfM , (5.11) ,
,1,1 11 qmqM
;2
11 mM ,
11
11
mM
mMq
, q -
10 q - m1 > 0 1 > 0. f '(x) -
[a, b], -
, m1 1. ,
(x), , -
.)(')('
2xf
bfafxx
(5.12)
(5.12) -
0523 xx [1, 2].
f(x) 23' 2 xxf , m1 = f '(1) = 5 M1 = f (2) = 14. (5.12),
(x):
.5219
252
145
2 33
xxxxxxx
, 0523 xx
.519
2 3 xxxx (5.13)
,
(5.13). (x)
.5219
2 3 xxxx
'(x) [1, 2]: '(1) = 0.474;
'(2) = 0.474. 1)( ,
() ..
-
42
5.2
5.2.1
. . 5.3 - -
.
5.3 -
() ++
:
//
float bis(float a, float b, float eps)
{ float xr=(a+b)/2;
while (fabs(b-a)>=eps)
{ if (f(a)*f(xr)
-
43
++
float f (float )
{ return x + sin(x) 1; }
- bis
[1, 2] eps -
k = bis (1, 2, eps);
k.
k, , -
f(k) 0.
5.2.2
. 5.4 -
.
5.4 -
r > ?
f(a) f"(a) > 0 ?
x0=b x0=a
)()( 000 xfxfxxr
xr
x0=xr
r=|x0 xr|
: f(x), f '(x), f''(x), a, b,
-
44
++
:
//
newton(float a, float b, float eps, float &xr)
{ float x0, r;
if (f(a)*f2(a)>0) x0=a;
else x0=b;
do
{ xr=x0-f(x0)/f1(x0);
r=fabs(x0-xr);
x0=xr; }
while (r>=eps);
}
. : a
, b , eps -
. -
xr. f(x),
() ++ -
. ,
x + sin(x) 1= 0
++
// ,
float a, b;
// f(x)=0
float f (float )
{ return x + sin(x) 1; }
// f '(x)
float f1 (float )
{ return 1+ cos(x); }
// f ''(x)
float f2 (float )
{ return sin(x); }
// ( )
. . . . . . . . . . . . . . . . . . . . . . .
//
. . . . . . . . . . . . . . . . . . . . . . .
newton()
[1, 2] eps :
newton(1, 2, eps, k);
k. -
k, , f(k) 0.
20
bax
-
45
5.2.3
. 5.5 - -
.
5.5 -
++
:
//
iter(float a, float b, float eps, float &xr)
{ float x0,r;
x0=(a+b)/2;
do { xr=f_iter(x0);
r=fabs(x0-xr);
x0=xr;}
while (r>=eps);
}
. : a
, b , eps -
. -
xr. f(x),
)()()(
2)( xf
bfafxx
- ++
iter(). ,
x + sin(x) 1= 0
)( 0xxr
x0 = (a+b)/2
x0 = xr
r = |x0 xr|
r > ?
:
(x), a, b,
xr
20
bax
-
46
++
float a, b; // ,
// f(x)=0
float f (float )
{ return x + sin(x) 1; }
// f '(x)
float f1 (float )
{ return 1+ cos(x); }
// (x)
float f_iter (float )
{ return x 2/(f1(a)+f1(b))*f(x);}
// ( ) . . . . . . . . . . . . . . . . . . . . . . .
// . . . . . . . . . . . . . . . . . . . . . . .
iter() -
, .
5.2.4
. 5.6 - -
.
5.6 -
f(a)>0 ?
x0 = b x0 = a
f(a)>0 ?
x0 = xr
r = |x0 xr|
axafxf
xfxxr
0
0
0
0
r > ?
00
0
0 xbxfbf
xfxxr
:
f(x), a, b,
xr
20
bax
-
47
++
:
//
float xord(float a, float b, float eps)
{ float x0, r, xr;
if (f(a)> 0) x0=b; else x0=a;
do
{ if (f(a) > 0) xr=x0-f(x0)/(f(x0)- f(a))*(x0-a);
else xr=x0-f(x0)/(f(b)-f(x0))*(b-x0);
r=fabs(x0-xr);
x0=xr; }
while (r>=eps);
return xr;
}
. : a
, b , eps -
. f(x)
- ++ -
- . xord() -
, .
-
48
6
6.1
, m n -
:
.
...
............
...
...
21
22221
11211
mnmm
n
n
aaa
aaa
aaa
A
mn. aij
.
, (m = 1), -
; , (n = 1), -
. m = n, , n
.
mn
d
a
a
a
...00
............
0...0
0...0
22
11
A
. aii
, , aii =1, -
E:
E =
1...00
............
0...10
0...01
.
, , -
0.
, ,
:
.
...
............
...
...
21
22212
12111
mnnn
m
m
aaa
aaa
aaa
A
-
49
, -
:
=
.
, = , -
=
1.
:
() ;
() - 1;
1. , ,
( ) , . ,
T =
nn
n
n
t
tt
ttt
...00
............
...0
...
222
12111
,
tij = 0 (j < i) . ,
tij = 0 (j > i), .
,
,
. -
. -
. -
, 1/n. -
, ,
. -
.
6.2
.
:
p A=
n
i
iia1
. (6.1)
. -
aij, i, j = 1, 2, , n (n ). -
n! nnkkk
k aaa ...)1( 221 1 , -
n , -
. k 2
2 n - . , -
1, 2, 3 : 123, 132, 213, 231, 312, 321.
-
50
k1, k2, , kn 1, 2, 3, , n... |A|
det A. , -
det A = a11a22a33+a12a23a31+ a13a21a32 a31a22a13 a32a23a11
a33a21a12.
, , . ij -
aij , -
i- j- .
aij ij ,
Aij = (1)i+j
|Mij|.
k- , -
, - k - k
.
. k = r(A),
k, ,
k + 1 .
. 1
-
, , -
:
1
= 1
= .
:
1
=
nnnn
n
n
AAAA
AAA
AAA
1121
22212
12111
............
...
...
,
= det A , ij -
aij (i, j = 1, 2, , n) ...
, ,
det A 0.
. = [x1, x2, , xn] -
|| x ||,
|| x || > 0 x 0,
|| Cx || = |||| x || - ,
|| x + y|| || x || +|| y || ,
y , y = [y 1, y 2, , y n].
-
51
:
|| x ||m = max | xi |, i = 1, 2, , n (m- ),
|| x ||l =
n
i
ix1
(l- ),
|| x || =
n
i
ix1
2 ( ).
, -
.
||||, -
, :
|| ||>0, ;
|| C || = | ||| || - ;
|| + || || ||+|| ||, ,
, ,
|| || || || || ||.
.
, ,
|| || || || || ||.
:
n
jij
ima
1
maxA (m- ); (6.2)
n
iij
jla
1
maxA (l- ); (6.3)
n
i
n
j
ijEa
1 1
2
A ( ). (6.4)
. =
987
654
321
.
:
|| A ||m = max (1+2+3, 4+5+6, 7+8+9) = max (6, 15, 24) = 24;
|| A ||l = max (1+4+7, 2+5+8, 3+6+9) = max (12, 15, 18) =
18;
|| A ||E = .9,16285987654321222222222
-
52
. -
0det EA ,
( )
.
...
............
...
...
21
22221
11211
nnnn
n
n
aaa
aaa
aaa
EA
( )
n
() = (1)
( + 1
1 + 2
2 + + 1 + ) (65)
n , ,
. i (i = 1, 2, , n)
p() = 0 . -
:
p() = (1)n ( 1) ( 2) ( n).
, i, -
v = [v1, v2, , vn], -
v = i v, (6.6)
a11v1 + a12v2 + + a1nvn = i v1;
a21v1 + a22v2 + + a2nvn = i v2;
. . . . . . . . . . . . . . . . . . . . . . . . . .
an1v1 + an2v2 + + annvn = i vn. . . . . . . . . . . . . . . . .
. . . . . . . . . .
6.3
-
, -
, , -
.
, (6.5), -
, . -
.
-
53
-
. , -
(6.5). .
.
310
121
013
A .
(A E) :
.12198310
121
01323
EA
01219823
1 = 1, 2 = 3, 3 = 4 .
(6.6)
v iv = 0 ( i)v = 0.
i ( )
-
v ,
( i ) = 0. (6.7)
(6.6) v -
i. -
1 = 1. -
(6.6) :
2v1 v2 = 0;
v1 + v2 v3 = 0;
v2 + 2v3 = 0.
.
v(1)
=
1
2
1
.
,
2 = 3, (6.6)
v2 = 0;
v1 v2 v3 = 0;
v2 = 0.
-
54
v(2)
=
1
0
1
.
,
3 = 4, (6.6)
v1 v2 = 0,
v1 2 v2 v3 = 0,
v2 v3 = 0.
v(3)
=
1
1
1
.
,
(n > 3) -
.
-
. , ,
n ,
, :
D = diag (1, 2, , n) =
,...00
............
0...,0
0...0,
2
1
n
= 1
, | | 0.
.
:
;
v , -
, , y = 1
v .
V1
A V = D = diag( ),...,, 21 n , (6.8)
V = [v(1)
v(2)
v(n)
] , .
-
55
,
310
121
013
A
V = [v(1)
v(2)
v(n)
] :
111
102
111
V .
V1
333.0333.0333.0
5.005.0
167.0333.0167.01
V .
(6.8) :
V1
A V =
333.0333.0333.0
5.005.0
167.0333.0167.0
310
121
013
111
102
111
=
400
030
001
.
, ,
1 = 1, 2 = 3, 3 = 4:
V1
A V = D = diag(1, 3, 4).
-
, , .
,
-
. , ,
.
, 20
, ,
D, -
[9]. , -
. , -
, , ,
.
( n
) ,
-
56
-
nn
nnn
ac
bac
ac
bac
ba
1
112
32
221
11
0...000
...000
.....................
000...0
000...
000...0
A ,
ai ; bi ci , -
(bi , ci
).
.
. -
. , ,
, , -
.
,
v, .
v0 =
1
...
1
1
.
-
yk = A v
k, k+1 = max{ y
k }, v
k+1 =
1
1
k y
k, (6.9)
k ; ck+1 yk
( ). {ck} {vk} -
-
v.
, ,
10264
7172
5110
A .
-
57
v0 =
1
1
1
.
(6.9),
:
y0= Av
0 =
10264
7172
5110
1
1
1
=
12
8
6
,
1 = max{6, 8, 12}=12, v1 =
1
1
y
0 =
12
1
12
8
6
=
1212
128
126
=
1
67.0
5.0
.
y1= Av
1 =
10264
7172
5110
1
67.0
5.0
=
333.5
333.3
333.2
,
2 = 5.333, v2 =
2
1
y
1 =
333.5
1
333.5
333.3
333.2
=
1
625.0
438.0
.
:
y2 = Av
2 =
10264
7172
5110
1
625.0
438.0
=
5.4
75.2
875.1
,
3 = 4.5, v3 =
2
1
y
2 =
5.4
1
5.4
75.2
875.1
=
1
611.0
417.0
.
, ,
k vkp, ,
v:
k = 4, vk v =
1
6,0
4,0
, k .
( k = 8)
0.01.
-
58
7
7.1
.
, . 7.1.
R1, R2, R3 1 2
() . .
, , -
, . ,
Ii Ri
( = 1, 2, , n). ,
. 7.1, :
I1 + I2 I3 = 0;
R1 I1 + R2 I2 = E1 + E2;
R2 I2 + R3 I3 = E2.
, -
I1, I2, I3. Ii.
() . -
:
a11x1 + a12x2 + a13x3 + + a1n xn = b1;
a21x1 + a22x2 + a23x3 + + a2n xn = b2;
(7.1)
an1x1 + an2x2 + an3x3 + + ann xn = bn ,
x , ( -
); aj ; b
I1
E1
R2 I2 R1
I3 R3
7.1
E2
-
59
; n ; i, j = 1, , n.
:
= , (7.2)
; ;
:
,
...
...........
...
...
21
22121
11211
nnnn
n
n
aaa
aaa
aaa
A
,...
2
1
nx
x
x
....
2
1
nb
b
b
,
(det A 0), (7.1) (7.2)
( x1, x2, , xn) .
-
-
. (
) -
.
-
. -
-
, -
. -
, -
. -
, .
, -
. , , -
( 100 . -
75 ).
7.2
7.2.1
, , -
(7.1) .
(7.2),
1
(. . 6.2):
= 1 . (7.3)
-
60
.
31 2 = 5;
21+ 2+ 3 = 0 ; (7.4)
21 2+ 3 =15.
= (7.2):
.
15
0
5
412
112
013
3
2
1
x
x
x
det A = 5 0,
.
= 1 (7.3):
.
3
1
2
15
0
5
2.02.00
6.04.22
2.08.01
3
2
1
x
x
x
, (7.4): 1 = 2; 2 = 1; 3 = 3.
n > 4
1
, (7.3)
( ).
7.2.2
(7.3)
:
AA
~
det
1 ,
det A
iix
(7.5)
A~
, -
ij:
,
...
............
...
...
~
21
22212
12111
nnnn
n
n
AAA
AAA
AAA
A
det A (. 6.2); i ,
- - :
.
......
.....................
......
......
1,1,1
21,221,221
11,111,111
nninninn
nii
nii
i
aabaa
aabaa
aabaa
(7.6)
-
61
, (7.1) n
(n+1)- n.
. (7.4) .
(7.6) :
;10
4115
110
015
1
;5
4152
102
053
2 .15
1512
012
513
1
, , det A = 5 0, -
;25
10
det
11
Ax ;1
5
5
det
22
Ax .3
5
15
det
33
Ax
(7.4) -
.
7.2.3
. ( -
) 200 .
-
.
.
,
.
(n 1) .
1- . x1 -
i = 2, 3, , n. , a11 0. -
1- .
qi1 = ai1/a11 (i = 2, 3, , n),
1- . , -
, , n- ,
q21, q31, , qn1, ... x1
, .
a11x1 + a12x2 + a13x3 + + a1nxn = b1;
a22(1)
x2 + a23(1)
x3 + + a2n(1)
xn = b2(1)
;
a32(1)
x2 + a33(1)
x3 + + a3n(1)
xn = b3(1)
;
. . . . . . . . . . . . . . .
an2(1)
x2 + an3(1)
x3 + + ann(1)
xn = bn(1)
...,
-
62
aij(1)
bij(1)
aij(1)
= aij qi1a1j ; bi(1)
= bi qi1b1.
2- . x2 -
i = 3, 4, , n, ... a22(1)
0, a22(1)
,
( ) 2- . 2- -
(1)(1)
2 222 aaq
ii (i = 3, 4, , n)
, , , n-
, q32, q42, , qm2, ... .
a11x1+ a12x2 + a13x3 + + a1nxn = b1;
a22(1)
x2 + a23(1)
x3 + + a2n(1)
= b2(1)
;
a33(2)
x3 + + a3n(2)
xn = b3(2)
;
. . . . . . . . . . . . . . . . . . .
an3(2)
x3 + + ann(2)
xn = bn(2)
.
aij(2)
bij(2)
:
aij(2)
= aij(1)
qi2a2j(1)
, bi(2)
= bi(1)
qi2b2(1)
.
. k- .
k- . , () k-
akk(k1)
, k-
1)()1( kkik kkikaaq (i = k + 1, , n)
(k + 1)-, , n- -
k-e , qk+1,k, qk+2,k, , qnk.
(n 1)-
a11x1 + a12x2 + a13x3 + + a1nxn = b1;
a22(1)
x2 + a23(1)
x3 + + a2n(1)
xn = b2(1)
;
a33(2)
x3 + + a3n(2)
xn = b3(2)
; (7.7)
. . . . . . . . . . . . . . . . . . .
ann(n1)
xn = bn(n1)
.
A(n1)
.
.
. xn.
xn , xn1.
, xn1, xn2, , x1. -
xn = bn(n1)
/ ann(n1)
;
xk = (bn(k1)
ak,k+1(k1)
xk+1 akn(k1)
xn) / akk(k1)
-
63
,)1(1
)1(,
)1(
k
kk
n
kii
kik
kn
ka
xab
x k = n 1, n 2, , 1. (7.8)
, ,
akk(k1)
. ,
, -
. , , -
, .
. -
, ,
. ,
.
, -
. [9],
n3/3 . ,
/ 15 , -
10- 0.005 , 100 -
5 , 1000 5000 .
. (7.4) .
. 1- . a11 = 3 0
1- . 1- (qi1 = ai1/a11, i = 2, 3):
3
221 q ,
3
231 q .
,
q21, q31. -
3x1 x2 = 5;
3
1x2 + x3 = 3
10;
3
1 x2 + 4x3 = 3
35.
2- . x2 i = 3.
a22(1)
=3
1 0 2- . 2-
q32 = .13
1
3
1
-
64
, q32.
3x1 x2 = 5;
;3
10
3
132 xx
5x3 = 15.
.
. (5x3 = 15)
.35
153
x3 = 3
,3
10
3
132 xx
.13
13
3
10
3
1
3
1032
x
(3x1 x2 = 5) 2 = 1
x1 = (5 + x2) / 3 = (5+ 1) / 3 = 2.
(1 = 2, 2 = 1, 3 = 3) -
(7.4) , -
.
7.3
7.3.1
-
A x = B,
x = D x + C (7.9)
,0...
.................................................
...0
...0
2211
2221212
1121211
nnnnn
nn
nn
cxxdxdx
cxdxxdx
cxdxdxx
.0 , , ijii
ii
ii
ijij aji
a
bc
a
ad
-
65
(7.9) -
, (k+1)- -
:
x(k+1)
= D x(k)
+ C . (7.10)
x(0)
= . -
,
)(
)(1)(
k
kk
x
xx,
; )(1)( kk xx ,
)(kx
.
. :
5001002
;600102006
;20026100
321
321
321
xxx
xxx
xxx
(7.11)
0.001.
(7.9). -
1 (100), 2 (200), -
3 (100) :
.502.001.0
;305.003.0
;202.006.0
321
321
321
xxx
xxx
xxx
1, 2, 3
:
502.001.0
;305.003.0
;202.006.0
213
312
321
xxx
xxx
xxx
5
3
2
002.001.0
05.0003.0
02.006.00
3
2
1
3
2
1
x
x
x
x
x
x
. (7.12)
5
3
2)0(
3
2
1
x
x
x
,
-
66
(7.12)
92.4
19.3
92.1
5
3
2
5
3
2
002.001.0
05.0003.0
02.006.00)1(
3
2
1
x
x
x
.
-
(7.12)
917.4
1884.3
907.1
5
3
2
92.4
19.3
92.1
002.001.0
05.0003.0
02.006.00)2(
3
2
1
x
x
x
.
,
917162.4
18864.3
907036.1
5
3
2
917.4
1884.3
907.1
002.001.0
05.0003.0
02,006.00)3(
3
2
1
x
x
x
.
,
,
0.001:
.001.00000473.01627455.6
0002917.0
(2)
)2()3(
(k)
(k)1)(k
x
xx
x
xx
, (3)
= {1.907036; 3.18864; 4.917162}
(7.11).
, -
, -
.
-
: (7.9),
, D ,
1D . (7.13)
(6.2) (6.4), -
:
); -( 1max1
mdn
jij
im
D
-
67
); -( 1max1
ldn
iij
jl
D
1 1 1
2
n
i
n
jijE
dD ( ).
,
,
002.001.0
05.0003.0
02.006.00
D
(7.12),
: ||D||m = 0.08, ||D||l = 0.08 ||D|| = 0.088882.
, (7.13).
(7.4)
31 2 = 5;
21 + 2 + 3 = 0;
21 2 + 3 =15
(7.9),
.4
15
4
1
2
1
2
;3
5
3
1
213
;312
21
xxx
xxx
xx
D C
.
415
03
5
;
04
1
2
1102
03
10
D
D:
||D||m = 3, ||D||l =2.5 ||D|| = 2.32886.
,
(7.13).
-
68
7.3.2
-
. ,
(k + 1)- xi (k + 1)-
x1, x2, , xi 1.
: .,...,,)0()0(
2
)0(
1 nxxx
, , (k)- )(knx , -
(k+1)-
....
. . . . . . . . . . . . . . . . . . . . .
;...
;...
)()1(22
)1(11
)1(
2)(
2)(
323)1(
121)1(
2
1)(
1)(
313)(
212)1(
1
nk
nnnk
nk
nk
n
knn
kkk
knn
kkk
cxdxdxdx
cxdxdxdx
cxdxdxdx
(7.14)
, , -
( (7.13)).
-, , .
.
141022
;13102
;1210
321
321
321
xxx
xxx
xxx
(7.15)
0.001.
(7.9):
.4.12.02.0
;3.11.02.0
;2.11.01.0
213
312
321
xxx
xxx
xxx
(7.16)
0,0,0 )0(3)0(
2
)0(
1 xxx
(7.16):
.2.12.101.001.0)1(
1 x
-
69
, (7.16)
1 ,2.1)1(
1 x 3
:0)0(3x
.06.13.101.02.12.0)1(2 x
, (7.16) 1 -
,2.1)1(
1 x 2 :06.1)1(
2 x
.948.04.106.12.02.12.0)1(3 x
,
(7.15):
999098.04.100536.12.09992.02.0
00536.13.1948.01.09992.02.0
9992.02.1948.01.006.11.0
)2(
3
)2(
2
)2(
1
x
x
x
(7.14) -
:
.000053.14.10001801.12.0999555.02.0
;0001801.13.1999098.01.0999555.02.0
;999555.02.1999098.01.000536.11.0
)3(3
)3(2
)3(1
x
x
x
:
.0000048.14.19999993.02.0999976.02.0
;9999993.03.1000053.11.0999976.02.0
;999976.02.1000053.11.00001801.11.0
)4(3
)4(2
)4(1
x
x
x
, -
, -
0.001:
.001.0000266.0731928.1
000461.0
(3)
)3((4)
(k)
(k)1)(k
x
xx
x
xx
-
70
, (7.15) 0.001
= {1.000; 1.000; 1.000}.
7.3.3
, ,
, . -
. -
, , -
.
,
.
. -
, ,
-
. ,
. -
.
-
71
8
8.1
(. 7)
. -
. n n
x1, x2, ..., xn
:
.0)..., ,,(
. . . . . . . . . .
;0)..., ,,(
;0).,.. ,,(
21
212
211
nn
n
n
xxxf
xxxf
xxxf
(8.1)
f1, f2, , fn - ,
.
,
= {x1, x2, , xn},
(8.1). , , -
.
.
,
2 x
2 x
y 5
x + 1 = 0;
x + 3 lg x y2 = 0; (8.2)
(8.1):
.0lg3),(
;0152),(2
211212
121
2
1211
xxxxxf
xxxxxxf
(8.3)
f1 f2 . -
(8.2) :
;152 2
x
xxy
xxy lg3 .
, -
. . 8.1 -
(1.5; 1.5) (3.5; 2.2), -
. , (8.2) -
, -
, .
-
72
8.1
(8.2)
,
.
. ,
.
.
,
, , .
8.2
-
(8.1)
).,...,,(
................................
);,...,,(
);,...,,(
21
2122
2111
nnn
n
n
xxxGx
xxxGx
xxxGx
(8.4)
y
x
x
xxy
152 2
xxy lg3
-
73
(0)
:
(0)
=
)0(
)0(
2
)0(
1
...
nx
x
x
(8.4). -
, .
(8.4). ,
(k+1)-
).,...,,(
................................
);,...,,(
);,...,,(
)()(
2
)(
1
)1(
)()(
2
)(
12
)1(
2
)()(
2
)(
11
)1(
1
k
n
kk
n
k
n
k
n
kkk
k
n
kkk
xxxGx
xxxGx
xxxGx
(8.5)
,
,
(k)
(k)1)(k
x
xx (8.6)
; (k)1)(k xx ,
(k)x
.
. -
(8.2) 0.0001. (8.4),
:
).,(lg3
);,(2
1)5(
212112
211
21
1
xxGxxx
xxGxx
x
(8.7)
(0)
(. . 8.1) .2.2 ,5.3)0(
2
)0(
1 xx -
G1 G2
:
.265436.25.3lg35.3lg3
;478505.32
1)52.2(5.3
2
1)5(
)0(
1
)0(
1
)1(
2
)0(
2
)0(
1)1(
1
xxx
xxx
-
74
-
(k+1)- (8.5):
.lg3
;2
1)5(
)(
1
)(
1
)1(
2
)(
2
)(
1)1(
1
kkk
kk
k
xxx
xxx
(8.2) -
. 8.1.
8.1
(8.2)
k )(
1
k
)(
2
k
0 3.5 2.2
1 3.478505 2.265436
2 3.483738 2.258912
3 3.484834 2.260503
4 3.485804 2.260836
5 3.486391 2.261131
6 3.486771 2.261309
7 3.487013 2.261424
(8.6):
3-
;001.0000465.01520012.4
0019325.0
(2)
)2()3(
x
xx
7-
.0001.00000645.01558503.4
0002682.0
(6)
)6()7(
x
xx
, , 3- ,
(8.4) 0,001; , 7-
, 0.0001,
1 = 3.487013; 2 = 2.261424.
-
75
(8.4)
, G'() -
,
||G(x)|| < 1, (8.8)
G() , :
G(x) =
n
nnn
n
n
x
G
x
G
x
G
x
G
x
G
x
G
x
G
x
G
x
G
...
............
...
...
21
2
2
2
1
2
1
2
1
1
1
.
, (8.7), -
(8.2), .
0lg32
4343,031
2
1)5(4
2
1)5(4
5
)('
11
1
21
1
21
2
2
2
1
2
2
1
1
1
xx
x
xx
x
xx
x
x
G
x
G
x
G
x
G
xG.
(1 = 3.5 0.1; 2 = 2.2 0.1)
.0 ;42.0 ;27.0 ;54.02
2
1
2
2
1
1
1
x
G
x
G
x
G
x
G
G'() :
.735.0(x) ';81.0(x) ';96.0(x) ' Elm
GGG
, G'() ,
(8.7)
(||G(x)|| < 1, 8.8).
8.3
,
, -
-
76
. , (k+1)-
(x1) k- ,
x1 (k+1)- , -
k- :
). ..., , , ,(
. . . . . . . . . . . . . . . . . . .
); ..., , , ,(
); ..., , , ,(
); ..,. , , ,(
)()1(
3
)1(
2
)1(
1
)1(
)()(
3
)1(
2
)1(
13
)1(
3
)()(
3
)(
2
)1(
12
)1(
2
)()(
3
)(
2
)(
11
)1(
1
k
n
kkk
n
k
n
k
n
kkkk
k
n
kkkk
k
n
kkkk
xxxxGx
xxxxGx
xxxxGx
xxxxGx
(8.9)
k .
-
(8.6) (8.6).
.
(8.2) 0.0001. , -
, (8.7):
).,(lg3
);,(2
1)5(
212112
211
21
1
xxGxxx
xxGxx
x
(0)
(8.2) .2.2 ,5.3)0(
2
)0(
1 xx -
G1 G2 (8.9) -
:
.258912.2478505.3lg3478505.3lg3
;478505.32
1)52.2(5.3
2
1)5(
)1(
1
)1(
1
)1(
2
)0(
2
)0(
1)1(
1
xxx
xxx
.
-
(8.9) (k+1)- :
)1(
1
)1(
1
)1(
2
)(
2
)(
1)1(
1
lg3
2
1)5(
kkk
kk
k
xxx
xxx
-
77
(8.2) -
. 8.2.
8.2
(8.2)
k )(
1
k
)(
2
k
0 3.5 2.2
1 3.478505 2.258912
2 3.482109 2.260008
3 3.484260 2.260662
4 3.485544 2.261052
5 3.486310 2.261284
6 3.486767 2.261423
7 3.487039 2.261506
(8.6):
2-
;001.0000908.01476118.4
0037664.0
(1)
)1()2(
x
xx
9-
.0001.00000244.0156346.4
0001014.0
)8(
)8()9(
x
xx
, , 2- ,
(8.2) 0.001; , 9-
, 0.0001,
1 = 3.487299; 2 = 2.261585.
,
, (||G(x)|| < 1,
(8.8)).
8.4
f1, f2, , fn -
,
.
-
78
x = x0
).()(!
1...)()(
!2
1)()(
!1
1)()(
0
)(
00
2
0000xfxx
nxfxxxfxxxfxf nn
f1, f2, , fn (8.1) -
( )
(0)
= { )0()0(2)0(
1 ..,. ,, nxxx }:
), ..., ,,()(...) ..., ,,()(
) ..., ,,()() ..., ,,() ..., ,,(
)0()0(2
)0()0()0()0(2
)0(
2
)0(22
)0()0(2
)0(
1
)0(1
)0()0(2
)0(21
11
111
nin
nnni
ninini
xxxfx
xxxxxfx
xx
xxxfx
xxxxxfxxxf
= 1, 2, , n.
:
)( )0()0( ii xxxi i- ,
fi i- ,
) ..., ,,( )0()0(2)0('
1 nij
ij xxxfx
F
fi xj.
n
xj:
;...
...................................................
;...
;...
2211
22222121
11212111
nnnnnn
nn
nn
fxFxFxF
fxFxFxF
fxFxFxF
(8.10)
nnnnnn
n
n
f
f
f
x
x
x
FFF
FFF
FFF
......
...
............
...
...
2
1
2
1
''
2
'
1
'
2
'
22
'
21
'
1
'
12
'
11
.
(F')(x) = (f),
(F') ,
.
-
79
( det(F') 0) -
x= (F')1
(f).
, :
(1)
= (0)
+ x(0)
.
,
(k+1)
= (k)
+ x(k)
(8.11)
(k+1)
= (k)
F1
((k)
) f((k)
); (8.12)
F1
(x(k)
) F' (k)
={)()(
2)(
1 ..., ,,k
nkk xxx }, k = 0, 1, 2,
-
(8.6).
,
:
1 (0)
= { )0()0(2)0(
1 ..., ,, nxxx };
2 (F') '
jiF (k) (k );
3 (8.10) -
x(k)
= (F')1
(x(k)
) f(x(k)
);
4 (k)
x(k)
(8.11) (k+1) )
= (k)
+ x(k)
;
5 -
(8.6): , k -
. 2, .
, , -
, ,
(. . 3.1).
.
(8.3) 0.0001:
.0lg3),(
;0152),(
2
1212
121
2
211
21
1
xxxxxf
xxxxxxf
-
80
(0)
(. . 8.1) .2.2 ,5.3)0(
2
)0(
1 xx -
f1 f2
f(x) = .292204.0
300000.0
lg3
1522
211
121
2
1
xxx
xxxx
F' =
2
1
121
24343.03
1
54
xx
xxx
.
:
F'((0)
) =
4.4372.1
5.38.6; det(F'(
(0))) = 25.12 0.
, F'((0)
) .
F1
(x(0)
) =
2707.00546.0
1393.01751.0.
(8.12)
x(1)
= x(0)
F1
(x(0)
) f(x(0)
) =
= .2627187.0
3488164.0
292204.0
300000.0
2707.00546.0
1393.01751.0
2.2
5.3
. -
. 8.3.
8.3
(8.3)
k
)(
1
k
)(
2
k f1(
)(k) f2(
)(k)
0 3.5 2.2 0.3 0.2922041
1 3.488164 2.262718 0.001022 0.003941
2 3.487443 2.261629 2.5404107
1.211106
3 3.4874428 2.2616286 1.77631014
8.8991013
4 3.4874428 2.2616286 0
0
-
81
x(3)
,
1012
, (8.3)
1 = 3.4874428; 2 = 2.2616286.
3-
(8.6):
.0001.01046602.9156588.4
0000004.0 8)2(
)2()3(
x
xx
-
: , , , . -
, -
:
1 F'((0)
)
F1
, ,
|| F1
((0)
)|| A;
2 f(x)
, :
|| F1
((0)
) f(x(0)
)|| ;
3
,)(
1
2
Cxx
xfn
k kj
i
i, j = 1, 2,, n; x
(0);
4 ,
2nABC 1.
(8.3):
1
F'=
2
1
121
24343.03
1
54
xx
xxx
x(0)
:
F1
(x(0)
)=
2707.00546.0
1393.01751.0.
|| F1
((0)
)|| = 0.325366 < 0.33, A = 0.33;
-
82
2
f(x) x(0)
:
F1
((0)
) f(x(0)
) =
062718.0
011836.0
292204.0
300000.0
2707.00546.0
1393.01751.0,
|| F1
((0)
) f(x(0)
)|| = 0.063826 < 0.1,
= 0.1;
3 -
x(0)
:
F'' =
2003029.1
0114
2
1x, F''(
(0)) = .
20010636.0
0114
F''((0)
)
n
k kj
i
xx
xf
1
2 )(, i, j = 1, 2;
x (0):
210636.0
15
5, = 5;
4 , , n = 2
2 n A B C = 2 2 0.33 0.1 5 = 0.66 1.
, (8.3)
.
-
83
,
1
:
- Mathcad; - -
;
- ; - ; - ; - ; - .
:
- Mathad; - Mathad ++; - ++
Mathad.
- Mathad; - Mathad; - Mathad.
-
1 , , : f(x) = 0
[, b] :
1 f(x) [, b];
2 f(x) [, b];
3 f(x) [, b] 0;
4 f(x) [, b] 1;
5 f(x) [, b] 1.
2 , , : F(x) = 0 -
[, b] :
1 f(x) [, b];
2 f(x) [, b];
3 f(x) [, b];
4 f(x) [, b];
5 f(x) 1 [, b];
6 f(x) [, b].
-
84
3 , , :
)('
)(
1
11
n
nnn
xf
xfxx
:
1 ;
2 ;
3 ;
4 .
4 , , : :
1 ;
2 ;
3 ;
4 ;
5 .
5 , , :
1
10 ))(2)()((
2
n
kkn xfxfxf
hI
:
1 ;
2 ;
3 .
6 , , :
:
1 ;
2 ;
3 .
7 , , : , :
1 ;
2 ;
3 ;
4 .
-
85
8 , , : :
1 ;
2 ;
3 ;
4 ;
5 ;
6 .
9 , , : :
1 , ;
2 , f '(x) 0;
3 , f(x) -
.
10 , , : ()
, :
1 0;
2 0;
3 0;
4 .
11 , , : :
1 , 0;
2 , ;
3 .
12 , , : :
1 ;
2 ;
3 .
13 , , : :
1 ,
;
2 , ;
3 .
-
86
14 , , : :
1 ;
2 ;
2 ;
2 ;
3 .
15 , , : :
1 ;
2 ;
3 ;
4 ;
5 ;
6 .
16 , , : :
1 ;
2 ;
3 ;
4 ;
5 .
17 :
1 x(k)
;
2 ;
3 (0)
= { )0()0(2)0(
1,...,,
nxxx };
4
;
5 (k)
x(k)
(k+1)
.
18 , , : :
1 F'((0))
F1
, ;
2 0;
3
f(x) , ;
-
87
4
;)(
1
2
Cxx
xfn
k kj
i
5 , 2nABC 1.
19 : ,
. . . . . . . . . . .
20 , , : :
1 ;
2 ;
3 ;
4 .
21 , , : :
1 p() = (1) n
(n + 1
n1 + 2
n2 + +
n1+
n);
2 a11v1 + a12v2 + + a1nvn = iv1;
3 p() = (1) n
( 1) ( 2) ( n).
22 :
, i, -
. . . . . . . . . . .
23 , , : :
1 ;
2 ;
3 .
24 , , : Mathad ,
2 3 0.1,
1 x := a, a + h .. b a := 2 b := 3 h := 0.1
2 a := 2 b := 3 h := 0.1 x := a, a+h .. b
3 x := 2, 2+h .. 3 h := 0.1
4 x := 2, 1.9 .. 3
5 x := 2, 0.1 .. 3
-
88
25 , , : Mathad sin
2x
3, = 1, :
1 y := sin 2 x
3
2 x := 1 y := (sin(x 3))
2
3 y := sin(x3)
2 x := 1
4 y(x) := sin2 x
3
5 y(x) := (sin(x3))
2 x := 1 y(x) =
26 , , :
Mathad ,sin 2
ax
ab a:=2 b:=3 x:=4, -
:
1
xa
baxy
2sin:)( a := 2 b := 3 x := 4 y(x) =
2 ax
abxy
2sin:)( a := 2 b := 3 x := 4
3 ax
abxy
2sin:)( a := 2 b := 3 x := 5
4 a := 2 b := 3 x := 5 ax
abxy
)sin(:)(
2
y(x) =
-
89
1
,
1
..
