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ߠ

,
- ,
.
.



, n .
( ),
,
, . .
(
1 ) ( 1 ).

xi i-
yi

, (
, ,
.. ).

, xi - yi i-
, .
, ,
.

xik i- ,
k- ,
k.

1
堠 ࠠ 頠
.
堠 򠠠

. ( i )
( i )
1 2 k n
.
( å ik )


1 11 12
1k 1n å 1k 1 1

2 21
22 2k 2n å 2k 2 2



i i1 xi2 xik xin å xik yi xi

n xn1 xn2 xnk xnn å xnk yn xn

.
å i1 å xi2 å xik å xin
k-



, , 1
:
1
- ( 11 + 12 + + 1n
) = 1
2 - ( 21 + 22 + + 2n
) = 2 ( 1 )
. . .
. . . . . . . . . . . . . . . . . . . . . .
xn - ( xn1 + xn2 + + xnn ) = yn

,

.

( ik , yi .. ) ,
, , ,
. ( 1 )
, .

y1 , y2 , , yn ,
, :
_
= (
1 , 2 , , yn ) , ( 2 )
x1 , x2 , , xn ,
-
:
_
x = ( x1 , x2 , , xn ). ( 3 )


( 1 ). ,
,
- , ..
k , n2 xik , xk.

. aik :
xik
aik = ( i , k = 1 , 2 , , n ).
xk

aik

. i-
, k-
, k-
. , aik
, , , ..,

xik xik

= = aik = const
( 4 )
xk xk

xik = aikxk , ( 5 )
.. i- k-
, , ,
xk. ( 5 )
.

aik ( 4 ),
,

a11 a12
a1k a1n
a21 a22
a2k a2n
A= .
ai1 ai2
aik ain
an1 an2
ank ann

. , aik .
>0 .


,
.1

xik = aik = xk ( 1 ), :
x1 - ( a11x1 + a12x2
+ + a1nxn ) = y1
x2 - ( a21x1
+ a22x2 + + a2nxn ) = y2 ( 6 )

xn - ( an1x1
+ an2x2 + + annxn ) = yn ,
- ,
.1

( 6 ) ,
:

_ _ _
ŷ
-
= ,

_ _
( -
)
= , ( 6'
)

n-

1-a11 -a12 -a1n
E - A= -a21 1-a22
-a2n

-an1 -an2 1-ann

( 6 ) 2n ( xi yi ). ,
n ,
( 6 ) n - .

= (
y1 , y2 , , yn )
- = ( 1 , 2 , n ).

,
:




.2
񠠠 堠
 頠

򠠠 򠠠
񠠠 1 2

0.2 0.4

1 100 160 260 240 500
0.55 0.1

2 275 40 315 85 400

򠠠 575
k- 375 200


575

,
.2

:

100
160
275 40
11
= = 0.2 ; 12 = = 0.4 ; 21 = = 0.55 ; 22
= = 0.1

500
400 500 400

.2 .

( 6 ), .2
1
- 0.21 - 0.42 = 1
2
- 0.551 - 0.12 = 2

1 2 1 2,

..
,
, 1=240 2=85, 1=500
2=400, 1=480 2=170, 1=1000
2=800 ..


.

.

( 6 ).

, ,
>0 >0, ..
-, .
( 6' ) .

, .
,
,


0.9 0.8 0.1 -0.8
( 6' )
=
, - =

0.6 0.9 -0.6 0.1

0.1 -0.8 1 1
-0.6 0.1 2 2
0.11
- 0.82 = 1 ( a )
-0.61
+ 0.12 = 2


,
-0.51
- 0.72 = 1 + 2,

1 2, 1>0
2>0 ( 1=2=0 1=2=0
).

( ( 6 ) )
( ( 6 ) ).

, ,
.
. >0, ( - )>0,
.. ( 6' ) x>0, >0, >0

.

, ( - ) .

,
( - )' = ',
- '
'
, '>0. , (
6'
) x>0. , ( 6'
) ( - ) .

( - )-1 S = || sik+ ||,
( 6''
)
_ _
= SӠ ( 7 )


S = ( E - A )-1,
- .

( 7 ) :
x1 = S11y1 + S12y2
+ + S1nyn
x2 = S21y1
+ S22y2 + + S2nyn ( 8 )

xn = Sn1y1
+ Sn2y2 + + Snnyn


.

Sik S.

1- , ..

1
_ 0
1
= :

0

( 7 ),
1 S11
_ 0 S21 _
= S : = : = S1
0 Sn1 0

_ 1
젠 2 = 0 , 젠

:
0
0 S12
_ 1 S22 _
= S :
= : = S2
0 Sn2

, ,
k- ,


0 S1k
_ : S2k _
= S 1
= : = Sk , ( 9 )
: Snk
0
.. k- S.

( 9 ) :
k-
, 1- 1=S1k, 2- 2=S2k .., i-
xi=Sik , , n-
xn=Snk .

k- ,
S1k, S2k, , Sik,
, Snk,
1-, 2- .., n-
k- .
a1k, a2k,
, aik, , ank k- ,
, k-
. , , .
i-
k-
aik, k-
, 1- ( a1k ), 2- (a2k ) ..
, i-
( ai1, ai2, ..).
.2
2-
( k=2 ) 1-
. .2 ,
2- ( 2=1 ) : 1- a12=0.4 2- a22=0.1.

. 2=100.
1- 1=0.4100=40 ? , , .. ,
1- ( 11=0.2 ),
: 1=40+0.240=48. , ..
1- 1'=48
.. . .2 2-
1- 2-
, 2=100.
1- . 
ꠠ 頠
, 1=0 蠠 2=1 ( .2 ):
0.81
- 0.42 = 0
-0.551
+ 0.92 = 1

, 1=0.8 2=1.5. ,
2- , 1-
1=0.8.
S12. , 12=0.4
1-
2- , 2- (
), S12
1-
( 12 ), ,
( 1- ) ,
2- .
S12-a12=0.8-0.4=0.4


, 12=0.4 2=1,

.
,
Sik

i-
k- , ( aik
), ( Sik
- aik )
.

, Sik > aik.

k k-
,
( 8 ):
x1 = S1kyk, x2
= S2kyk,
, xn = Snkyk ,
:
_
_
x = Skyk ( 10 )
,
, -
_
1
=
: ,
k-
xk,
n
, ( 10
) Sk ,
..
_ _
xk = Sk1y1 + Sk2y2
+ + Sknyn = Sky , ( 11 )
- ( 7 ) S .

,

S, ( 7 )
( 11 )
.

, - D = ( D1, D2, , Dn )
D = ( D1, D2, , Dn )
:

_ _
D = SD , ( 12 )

.2.
:
0.2 0.4

=

0.55 0.1
,
1
-0.2 -0.4 0.8 -0.4
-
=
=

-0.55 1 -0.1 -0.55 0.9

0.8 -0.4
D [ E - A ] = = 0.5
-0.55 0.9
( - )*.
:
0.9 0.4
( -
)* = ,
0.55 0.8
,
,
:
1 0.9 0.4 1.8 0.8
S = (
- )-1 = =
0.5 0.55 0.8 1.1 1.6

, 1- 2- ,
1- , S11=0.8 S21=1.5.
11=0.2 21=0.55, ,
1.8-0.2=1.6 1.1-0.55=0.55.

, 1- 2-
2- S12=0.8 S22=1.5,
򠠠 0.8-0.4=0.4
1.6-0.1=1.5.

480 1-
170 2- .
= 1 ( 7 ):

2
_ _ 1.8 0.8 480 1000
= S = =
1.1 1.6 170 800 .



..

.1, , xik, ,
.. .
n+1-, n+2- ..
.

k- xn+1,k, xn+2,k
( k = 1, 2, , n ). aik,

xn+1,k

an+1,k = ,

xk
xn+2,k

an+2,k = ,


xk
, k-
. ( ..
),
:
a11 a12 a1k
a1n
a21 a22 a2k
a2n

' = ai1 ai2 aik
ain

an1 an2
ank ann
an+1,1 an+1,2
an+1,k
an+1,n
an+2,1
an+2,2 an+2,k
an+2,n
-
( ).
, ,
.
, , , 1-
, ..
_ 1
= 0

:

0 .



S11
_
_ S21
x = S1 = :
Sn1
Sn+1,1. , an+1,k
k- S11, S12, , S1n,
,
1- an+1,1S11,
2- an+1,2S21 .., n- an+1,nSn1. ,
1- , :

_ _
Sn+1,1 = an+1,1S11 + an+1,2S21
+ + an+1,nSn1 = an+1S1 ,
.. ( n+1 )-
',
an+1, 1- S.

,
k-
, :
_ _
Sn+1,k = an+1Sk ( 13 )
.
,

:
_ _
Sn+2,k = an+2Sk ( 14 )
S , Sn+1,k Sn+2,k,
:
S11 S12 S1k
S1n
S21 S22 S2k
S2n .


S' = Si1 Si2 Sik
Sin
( 15 )
Sn1 Sn2
Snk Snn
Sn+1,1
Sn+1,2 Sn+1,k Sn+1,n
Sn+2,1 Sn+2,2 Sn+2,k
Sn+2,n




( S ),
xn+1, xn+2 .., .

,
xn+1 = Sn+1,1y1
+ Sn+1,2y2 + + Sn+1,nyn , ( 16 )
xn+2 = Sn+2,1y1
+ Sn+2,2y2 + + Sn+2,nyn ,
.. ,
,
S'
.

, ( 7 ) ( 16 ),
:

x1
x2
_ : _
x = xn = S'Ӡ ( 17 )
xn+1
xn+2


, .2,
xn+1,k ( . - )
xn+2,k ( . . ), .3

aik, :

0.2 0.4
'
= 0.55 0.1

0.5 0.2

1.5 2.0

3
頠 堠 
頠 頠

򠠠 򠠠 ꠠ


1 2

1 100 160 260 240
500

2 275 40
315 85 400



250 80 330

- 750 800 1550


S = ( E - A )-1
.

( 13 ) ( Sn+1,k=S3,k ):
_ _
S31 = a3S1
= 0.5
1.8 + 0.2
1.1 = 1.12 ;
_
_
S32 = a3S2
= 0.5
0.8 + 0.2
1.6 = 0.72
Sn+2,k
= S4,k:
_ _
S41 = a4S1
= 1.5
1.8 + 2.0
1.1 = 4.9 ;
_ _
S42 = a4S2
= 1.5
0.8 + 2.0
1.6 = 4.4 .
, S'
:

1.8 0.8
S' = 1.1 1.6

1.12 0.72

4.9 4.4

蠠 ࠠ
頠 䠠 젠
=
240 ,
( 16 ) xn+1
85
xn+2, xn+1 = x3 = 1,12 240 + 0.72
85 = 268.8 + 61.2 = 330 . .-. xn+2 = xn
= 4.9
240 + 4.4
85 = 1176 + 374 = 1550 .., .3.

.3,

( 250 80 750 800 ),
: 1- 268.8 2-
61.2; 1176 374.

, ,
( 17
).
,
=
480 .

170

_ 1
1.8 0.8 1000

= 2 =
1.1 1.6 480
= 800
3 1.12 0.72 170 600
4 4.9 4.4
3100

, 1- 2- : 1=1000
2=800, 3=660 .
.-. 4=3100 ..

, ,
.

.




, -
, 1 .-.




I II III
I 1.4 2.4 0.8
a4 5
II 0.6
1.6 a5 12
III 2.0 1.8
2.2 a6 2
10
20 20 7 12


:
) ,
;
) ,
;
) ,
;
) ( . )
;
)
.

:
) I ,
1- , ..
_
_ 235
4
= ( 1.4; 2.4; 0.8 ) 186 = 1088
397

II ..

:

1.4 2.4 0.8 235
1088 I

0 0.6 1.6 186 = 746 II

2.0 1.8 2.2 397
1678

0.1 0.2 0.2 1409
-.
) I
1- ( 1=1 ) 1.4S11 + 2.4S21 + 0.8S31. ,
,
:

I
II III
1.4 2.4
0.8 1.04 0.21
0.02 1.97 2.92
1.36 I

0 0.6 1.6 0.21 1.05 0.13
= 0.17 0.84
2.09 II

2.0 1.8 2.2 0.03 0.13 1.26 2.53
2.60 5.23
10 20
20 15.2 24.8
28.0


, , 1=1 1.97
I, 0.17 II, 2.53 15.2
.-.
) , ..

. :
I II III
I 330 440 318
II 0 111 635

 470 335
873

䠠 2350 3720
7940
)
( 5; 12; 2; 1.2 ) :
330 440 318
0 111 635 I II III
( 5;
12; 2; 1.2 ) 470 335
873 = ( 5410; 8666; 20484 )
2350 3720
7940
) ,
, ,
, .., :
1.97 2.92 1.36
0.17 0.84 2.09 I II III
( 5;
12; 2; 1.2 ) 2.53 2.60
5.23 = ( 35.3; 59.6; 75.7
)
15.2 24.8 28.0

, I, II III
: 35.3 ., 59.6 ., 75.7 .


. . . . . . . . . . . . . . .

      ©2010