. , , ,

,,,

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

堠 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

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

頠 1 2


1 100 160 260 240 500


2 275 40 315 85 400

䠠 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, 젠

( 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 .

ߠ , - , .

 

 

 

! , , , .
. , :