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Балансовая модель

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AAEAINIAAss IIAeAEUe

Eco/aiea aaeainiauo iiaeaeae, i?aaenoaaeythueo niaie iaeii ec
aaaeiaeoeo iai?aaeaiee e yeiiiieei-iaoaiaoe/aneeo enneaaeiaaiee,
aeieaeii neoaeeoue iauaeoii eco/aiey ioaeaeueiie aeenoeeieeiu. Iaoa
oeaeue – i?ieeethno?e?iaaoue ia i?eia?a aaeainiauo ?an/aoia i?eiaiaiea
iniiaiuo iiiyoee eeiaeiie aeaaa?u.

EEIAEIAss AAEAINIAAss IIAeAEUe

Ionoue ?anniao?eaaaony yeiiiie/aneay nenoaia, ninoiyuay ec n
acaeiinaycaiiuo io?aneae i?iecaiaenoaa. I?iaeoeoeey eaaeaeie io?anee
/anoe/ii eaeao ia aiaoiaa iio?aaeaiea ( eiia/iue i?iaeoeo ), a /anoe/ii
eniieuecoaony a ea/anoaa nu?uey, iieooaa?eeaoia eee ae?oaeo n?aaenoa
i?iecaiaenoaa a ae?oaeo io?aneyo, a oii /enea e a aeaiiie. Yoo /anoue
i?iaeoeoeee iacuaatho i?iecaiaenoaaiiui iio?aaeaieai. Iiyoiio eaaeaeay
ec ?anniao?eaaaiuo io?aneae aunooiaao e eae i?iecaiaeeoaeue i?iaeoeoeee
( ia?aue noieaaoe oaaeeoeu 1 ) e eae aa iio?aaeoaeue ( ia?aay no?iea
oaaeeoeu 1 ).

Iaicia/ei /a?ac xi aaeiaue auione i?iaeoeoeee i-e io?anee ca
ieaie?oaiue ia?eiae e /a?ac yi – eiia/iue i?iaeoeo, eaeouee ia aiaoiaa
aeey ?anniao?eaaaiie nenoaiu iio?aaeaiea ( n?aaenoaa i?iecaiaenoaa
ae?oaeo yeiiiie/aneeo nenoai, iio?aaeaiea ianaeaiey, ia?aciaaiea caiania
e o.ae. ).

Oaeei ia?acii, ?aciinoue xi – yi ninoaaeyao /anoue i?iaeoeoeee
i-e io?anee, i?aaeiacia/aiioth aeey aioo?ei?iecaiaenoaaiiiai
iio?aaeaiey. Aoaeai a aeaeueiaeoai iieaaaoue, /oi aaeain ninoaaeyaony ia
a iaoo?aeueiii, a a noieiinoiii ?ac?aca.

Iaicia/ei /a?ac xik /anoue i?iaeoeoeee i-e io?anee, eioi?ay
iio?aaeyaony k-e io?aneueth, aeey iaania/aiey auionea aa i?iaeoeoeee a
?acia?a ok.

Oaaeeoea 1

? iio?aaeaiea
eoiai ia eiia/iue aaeiaue

io?an.
aioo?a i?iaeoeo auione

i?iecaiae. ( oi ) ( oi )

? 1 2 … k … n
iio?aaeaiea

io?an.
( a oik )

1 o11 o12 … o1k …
o1n a o1k o1 o1

2 o21 o22 … o2k … o2n
a o2k o2 o2

( ( ( ( ( ( (
( ( (

i oi1 xi2 ( xik (
xin a xik yi xi

( ( ( ( ( ( (
( ( (

n xn1 xn2 ( xnk (
xnn a xnk yn xn

eoiai

i?ieca.

cao?aou a oi1 a xi2 ( a xik ( a xin

a k-th

io?aneue

I/aaeaeii, aaee/eiu, ?aniieiaeaiiua a no?ieao oaaeeoeu 1 naycaiu
neaaeothueie aaeainiauie ?aaainoaaie :

o1 – ( o11 + o12 + ( + o1n ) = o1

o2 – ( o21 + o22 + … + o2n ) = o2 ( 1 )

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

xn – ( xn1 + xn2 + … + xnn ) = yn

Iaeia ec caaea/ aaeainiauo enneaaeiaaiee caeeth/aaony a oii, /oiau
ia aaca aeaiiuo ia eniieiaiea aaeaina ca i?aaeoanoaothuee ia?eiae
ii?aaeaeeoue enoiaeiua aeaiiua ia ieaie?oaiue ia?eiae.

Aoaeai niaaaeaoue oo?eoii ( o’ik , y’i e o.ae. ) aeaiiua,
ioiinyueany e enoaeoaio ia?eiaeo, a oaie aea aoeaaie, ii aac oo?eoa –
aiaeiae/iua aeaiiua, naycaiiua n ieaie?oaiui ia?eiaeii. Aaeainiaua
?aaainoaa ( 1 ) aeieaeiu auiieiyoueny eae a enoaeoai, oae e a
ieaie?oaiii ia?eiaea.

Aoaeai iacuaaoue niaieoiiinoue cia/aiee y1 , y2 , … , yn ,
oa?aeoa?ecothueo auione eiia/iiai i?iaeoeoa, anni?oeiaioiui aaeoi?ii :

_

o = ( o1 , o2 , … , yn ) , ( 2 )

a niaieoiiinoue cia/aiee x1 , x2 , … , xn ,ii?aaeaeythueo aaeiaue auione
anao io?aneae ( aaeoi?-ieaiii :

_

x = ( x1 , x2 , … , xn ). ( 3 )

Caaeneiinoue iaaeaeo aeaoiy yoeie aaeoi?aie ii?aaeaeyaony
aaeainiauie ?aaainoaaie ( 1 ). Iaeiaei iie ia aeatho aiciiaeiinoe
ii?aaeaeeoue ii caaeaiiiio, iai?eia?, aaeoi? o iaiaoiaeeiue aeey aai
iaania/aiey aaeoi?-ieai o, o.e. e?iia eneiiuo iaecaanoiuo ok ,
niaea?aeao n2 iaecaanoiuo xik , eioi?ua a naith i/a?aaeue caaenyo io xk.

Iiyoiio i?aia?acoai yoe ?aaainoaa. ?ann/eoaai aaee/eiu aik ec
niioiioaiee :

xik

aik = ––– ( i , k = 1 , 2 , … , n ).

xk

Aaee/eiu aik iacuaathony eiyooeoeeaioaie i?yiuo cao?ao eee
oaoiieiae/aneeie eiyooeoeeaioaie. Iie ii?aaeaeytho cao?aou i?iaeoeoeee
i-e io?anee, eniieuecoaiua k-e io?aneueth ia ecaioiaeaiea aa
i?iaeoeoeee, e caaenyo aeaaiui ia?acii io oaoiieiaee i?iecaiaenoaa a
yoie k-e io?anee. N iaeioi?ui i?eaeeaeaieai iiaeii iieaaaoue, /oi
eiyooeoeeaiou aik iinoiyiiu a iaeioi?ii i?iiaaeooea a?aiaie,
ioaaouaathuei eae enoaeoee, oae e ieaie?oaiue ia?eiae, o.a., /oi

x’ik xik

––– = ––– = aik = const ( 4 )

x’k xk

Enoiaey ec yoiai i?aaeeiaeaiey eiaai

xik = aikxk , ( 5 )

o.a. cao?aou i-e io?anee a k-th io?aneue i?iii?oeeiiaeueiu aa aaeiaiio
auioneo, eee, ae?oaeie neiaaie, caaenyo eeiaeii io aaeiaiai auionea xk.
Iiyoiio ?aaainoai ( 5 ) iacuaatho oneiaeai eeiaeiinoe i?yiuo cao?ao.

?ann/eoaa eiyooeoeeaiou i?yiuo cao?ao aik ii oi?ioea ( 4 ),
eniieuecoy aeaiiua ia eniieiaiee aaeaina ca i?aaeoanoaothuee ia?eiae
eeai ii?aaeaeea eo ae?oaei ia?acii, iieo/ei iao?eoeo

a11 a12 … a1k … a1n

a21 a22 … a2k … a2n

A= ………………….

ai1 ai2 … aik … ain

an1 an2 … ank … ann

eioi?oth iacuaatho iao?eoeae cao?ao. Caiaoei, /oi ana yeaiaiou aik yoie
iao?eoeu iaio?eoeaoaeueiu. Yoi caienuaatho nie?auaii a aeaea iao?e/iiai
ia?aaainoaa A>0 e iacuaatho oaeoth iao?eoeo iaio?eoeaoaeueiie.

Caaeaieai iao?eoeu A ii?aaeaeythony ana aioo?aiiea acaeiinayce
iaaeaeo i?iecaiaenoaii e iio?aaeaieai, oa?aeoa?ecoaiua oaae.1

Iiaenoaaeyy cia/aiey xik = aik = xk ai ana o?aaiaiey nenoaiu ( 1
), iieo/ei eeiaeioth aaeainiaoth iiaeaeue :

x1 – ( a11x1 + a12x2 + … + a1nxn ) = y1

x2 – ( a21x1 + a22x2 + … + a2nxn ) = y2 ( 6
)

……………………………………

xn – ( an1x1 + an2x2 + … + annxn ) = yn ,

oa?aeoa?ecothuoth aaeain cao?ao – auionea i?iaeoeoeee, i?aaenoaaeaiiue a
oaae.1

Nenoaia o?aaiaiee ( 6 ) iiaeao auoue caienaia eiiiaeoiaa, anee
eniieueciaaoue iao?e/ioth oi?io caiene o?aaiaiee:

_ _ _

A(o – A(o = O , eee ieii/aoaeueii

_ _

( A – A )(o = O , ( 6( )

aaea A – aaeeie/iay iao?eoea n-ai ii?yaeea e

1-a11 -a12 … -a1n

E – A= -a21 1-a22 … -a2n

…………………

-an1 -an2 … 1-ann

O?aaiaiey ( 6 ) niaea?aeao 2n ia?aiaiiuo ( xi e yi ). Iiyoiio,
caaeaaoenue cia/aieyie n ia?aiaiiuo, iiaeii ec nenoaiu ( 6 ) iaeoe
inoaeueiua n – ia?aiaiiuo.

Aoaeai enoiaeeoue ec caaeaiiiai anni?oeiaioiiai aaeoi?a O = ( y1 ,
y2 , … , yn ) e ii?aaeaeyoue iaiaoiaeeiue aeey aai i?iecaiaenoaa
aaeoi?-ieai O = ( o1 , o2 , … on ).

I?ieeethno?e?oai auoaeceiaeaiiia ia i?eia?a i?aaeaeueii oi?iuaiiie
nenoaiu, ninoiyuae ec aeaoo i?iecaiaenoaaiiuo io?aneae:

oaae.2

? io?an Iio?aaeaiea Eoiai
Eiia/iue Aaeiaue

?
cao?ao i?iaeoeo auione

io?an 1 2

0.2 0.4

1 100 160
260 240 500

0.55 0.1

2 275 40
315 85 400

Eoiai cao?ao
575

a k-th 375 200

io?aneue …
575

Ionoue eniieiaiea aaeaina ca i?aaeoanoaothuee ia?eiae
oa?aeoa?ecoaony aeaiiuie, iiiauaiiuie a oaae.2

?ann/eouaaai ii aeaiiui yoie oaaeeoeu eiyooeoeeaiou i?yiuo cao?ao:

100 160 275
40

a11 = –––– = 0.2 ; a12 = –––– = 0.4 ; a21 = –––– = 0.55 ; a22 =
–––– = 0.1

500 400 500
400

Yoe eiyooeoeeaiou caienaiu a oaae.2 a oaeao niioaaonoaothueo
eeaoie.

Oaia?ue iiaeao auoue caienaia aaeainiaay iiaeaeue ( 6 ),
niioaaonoaothuay aeaiiui oaae.2

o1 – 0.2o1 – 0.4o2 = o1

o2 – 0.55o1 – 0.1o2 = o2

Yoa nenoaia aeaoo o?aaiaiee iiaeao auoue eniieueciaaia aeey
ii?aaeaeaiey o1 e o2 i?e caaeaiiuo cia/aieyo o1 e o2, aeey
eniieueciaaiey aeeyiey ia aaeiaue auione ethauo eciaiaiee a anni?oeiaioa
eiia/iiai i?iaeoeoa e o.ae.

Oae, iai?eia?, caaeaaoenue o1=240 e o2=85, iieo/ei o1=500 e
o2=400, caaeaaoenue o1=480 e o2=170, iieo/ei o1=1000 e o2=800 e o.ae.

?AOAIEA AAEAINIAUO O?AAIAIEE

N IIIIUUeTH IA?AOIIE IAO?EOeU.

EIYOOEOeEAIOU IIEIUO CAO?AO.

Aa?iainy niiaa e ?anniio?aieth aaeainiaiai o?aaiaiey ( 6 ).

Ia?aue aii?in, eioi?ue aicieeaao i?e aai enneaaeiaaiea, yoi aii?in
i nouanoaiaaiea i?e caaeaiiii aaeoi?a O>0 iaio?eoeaoaeueiiai ?aoaiey
o>0, o.a. i nouanoaiaaiee aaeoi?-ieaia, iaania/eaathuaai aeaiiue
anni?oeiaio eiia/iiai i?iaeoeoa O. Aoaeai iacuaaoue oaeia ?aoaiea
o?aaiaiey ( 6( ) aeiionoeiui ?aoaieai.

Caiaoei, /oi i?e ethaie iaio?eoeaoaeueiie iao?eoea A ooaa?aeaeaoue
nouanoaiaaiea iaio?eoeaoaeueiiai ?aoaiey iaeuecy.

Oae, iai?eia?, anee

0.9 0.8 0.1 -0.8 e o?aaiaiea ( 6(
)

A= , oi A – A =

0.6 0.9 -0.6 0.1

caieoaony a aeaea 0.1 -0.8 o1 o1 eee a ?acaa?iooie oi?ia

-0.6 0.1 o2 o2

0.1o1 – 0.8o2 = o1 ( ( )

-0.6o1 + 0.1o2 = o2

Neiaeea yoe aeaa o?aaiaiey ii/eaiii, iieo/ei o?aaiaiea

-0.5o1 – 0.7o2 = o1 + o2,

eioi?ia ia iiaeao oaeiaeaoai?youeny iaio?eoeaoaeueiui cia/aieyi o1 e o2,
anee oieueei o1>0 e o2>0 ( e?iia o1=o2=0 i?e o1=o2=0 ).

Iaeiiaoe o?aaiaiea aiiaua iiaeao ia eiaoue ?aoaiee ( nenoaia ( 6 )
– ianiaianoiay ) eee eiaoue aan/eneaiiia iiiaeanoai ?aoaiee ( nenoaia (
6 ) – iaii?aaeaeaiiay ).

Neaaeothuay oai?aia, aeieacaoaeuenoai eioi?ie iu iioneaai, aeaao
ioaao ia iinoaaeaiiue aii?in.

Oai?aia. Anee nouanoaoao oioue iaeei iaio?eoeaoaeueiue aaeoi?
o>0, oaeiaeaoai?ythuee ia?aaainoao ( A – A )(o>0, o.a. anee o?aaiaiea (
6( ) eiaao iaio?eoeaoaeueiia ?aoaiea x>0, oioy au aeey iaeiiai O>0, oi
iii eiaao aeey ethaiai O>0 aaeeinoaaiiia iaio?eoeaoaeueiia ?aoaiea.

I?e yoii ieacuaaaony, /oi ia?aoiay iao?eoea ( A – A ) aoaeao
iaycaoaeueii iaio?eoeaoaeueiie.

Ec niiniaa ia?aciaaiey iao?eoeu cao?ao neaaeoao, /oi aeey
i?aaeoanoaothuaai ia?eiaea auiieiyaony ?aaainoai ( A -A )(o( = O(, aaea
aaeoi?-ieai o( e anni?oeiaioiue aaeoi? O( ii?aaeaeythony ii eniieiaiiiio
aaeaino ca i?ioeue ia?eiae, i?e yoii O(>0. Oaeei ia?acii, o?aaiaiea ( 6(
) eiaao iaeii iaio?eoeaoaeueiia ?aoaiea x>0. Ia iniiaaiee oai?aiu
caeeth/aai, /oi o?aaiaiea ( 6( ) anaaaea eiaao aeiionoeiue ieai e
iao?eoea ( A – A ) eiaao ia?aoioth iao?eoeo.

Iaicia/ea ia?aoioth iao?eoeo ( A – A )-1 /a?ac S = || sik+ ||,
caieoai ?aoaiea o?aaiaiey ( 6(( ) a aeaea

_ _

o = S(O ( 7 )

Anee aoaeao caaeai aaeoi? – eiia/iue i?iaeoeo O e au/eneaia
iao?eoea S = ( E – A )-1, oi ii yoie oi?ioea iiaeao auoue ii?aaeaeai
aaeoi?-ieai o.

?aoaiea ( 7 ) iiaeii i?aaenoaaeoue a ?acaa?iooie oi?ia:

x1 = S11y1 + S12y2 + … + S1nyn

x2 = S21y1 + S22y2 + … + S2nyn ( 8 )

………………………………

xn = Sn1y1 + Sn2y2 + … + Snnyn

IIEIUA AIOO?EI?IECAIAeNOAAIIUA

CAO?AOU.

Auyniei yeiiiie/aneee niune yeaiaioia Sik iao?eoeu S.

Ionoue i?iecaiaeeony oieueei aaeeieoea eiia/iiai i?iaeoeoa 1-e
io?anee, o.a.

1

_ 0

O1 = (

0

Iiaenoaaeyy yoio aaeoi? a ?aaainoai ( 7 ), iieo/ei

1 S11

_ 0 S21 _

o = S( : = : = S1

0 Sn1
0

_
1

caaeaaoenue anni?oeiaioiui aaeoi?ii O2 = 0 , iieo/ei

:

0

0 S12

_ 1 S22 _

o = S( : = : = S2

0 Sn2

Aiaeiae/ii, aaeiaue auione o, iaiaoiaeeiue aeey i?iecaiaenoaa
aaeeieoeu eiia/iiai i?iaeoeoa k-e io?anee, ninoaaeo

0 S1k

_ : S2k _

o = S( 1 = : = Sk , ( 9 )

: Snk

0

o.a. k-e noieaaoe iao?eoeu S.

Ec ?aaainoaa ( 9 ) auoaeaao neaaeothuaa:

*oiau auionoeoue oieueei aaeeieoeo eiia/iiai i?iaeoeoa k-e io?anee,
iaiaoiaeeii a 1-e io?anee auionoeoue o1=S1k, ai 2-e o2=S2k e o.ae., a
i-e io?anee auionoeoue xi=Sik e, iaeiiaoe, a n-e io?anee auionoeoue
xn=Snk aaeeieoe i?iaeoeoeee.

Oae i?e yoii aeaea eiia/iiai i?iaeoeoa i?iecaiaenoaa oieueei
aaeeieoea k-ai i?iaeoeoa, oi aaee/eiu S1k, S2k, …, Sik, …, Snk,
i?aaenoaaeytho niaie eiyooeoeeaiou iieiuo cao?ao i?iaeoeoeee 1-e, 2-e e
o.ae., n-e io?aneae eaeouae ia ecaioiaeaiea oeacaiiie aaeeieoeu k-ai
i?iaeoeoa. Iu oaea aaaee ?aiiaa eiyooeoeeaiou i?yiuo cao?ao a1k, a2k, …,
aik, …, ank ia aaeeieoeo i?iaeoeoeee k-e io?anee, eioi?ua o/eouaaee
eeoue oo /anoue i?iaeoeoeee eaaeaeie io?anee, eioi?ay iio?aaeyaony
iaiin?aaenoaaiii k-e io?aneueth. Ii, i/aaeaeii, iaiaoiaeeii iaania/eoue
caieiooue i?iecaiaenoaaiiue oeeee. Anee au i?iaeoeoeey i-e io?anee
iinooiaea au oieueei a k-th io?aneue a eiee/anoaa aik, oi i?iecaiaenoai
k-e io?anee ana ?aaii ia auei au iaania/aiii, eai iio?aaiaaeinue aua
i?iaeoeou 1-e io?anee ( a1k ), 2-e io?anee (a2k ) e o.ae. A iie a naith
i/a?aaeue ia niiaoo ?aaioaoue, anee ia aoaeoo iieo/aoue i?iaeoeoeeth oie
aea i-e io?anee ( ai1, ai2, … e o.ae.). I?ieeethno?e?oai neacaiiia ia
i?eia?a oaae.2

Ionoue ian ia eioa?anoao auione aeey aiaoiaai iio?aaeaiey
i?iaeoeoeee 2-e io?anee ( k=2 ) e iu oioei ii?aaeaeeoue cao?aou
i?iaeoeoeee 1-e io?anee ia aaeeieoeo yoie i?iaeoeoeee. Ec oaae.2
iaoiaeei, /oi ia eaaeaeoth aaeeieoeo i?iaeoeoeee 2-e io?anee ( o2=1 )
cao?a/eaaaony: i?iaeoeoeee 1-e io?anee a12=0.4 e 2-e io?anee a22=0.1.

Oaeiau aoaeoo i?yiua cao?aou. Ionoue ioaeii ecaioiaeoue o2=100.
Iiaeii ee aeey yoiai ieaie?iaaoue auione 1-e io?anee o1=0.4(100=40 ?
Eiia/ii, iaeuecy, o.e. iaiaoiaeeii o/eouaaoue, /oi 1-y io?aneue /anoue
naiae i?iaeoeoeee iio?aaeyao naia ( a11=0.2 ), e iiyoiio noiia?iue aa
auione neaaeoao nei??aeoe?iaaoue: o1=40+0.2(40=48. Iaeiaei e yoa oeeo?a
iaaa?ia, o.e. oaia?ue oaea neaaeoao enoiaeeoue ec iiaiai iauaia
i?iaeoeoeee 1-e io?anee – o1(=48 e o.ae. Ii aeaei ia oieueei a yoii.
Niaeanii oaae.2 i?iaeoeoeey 2-e io?anee oaeaea iaiaoiaeeia aeey
i?iecaiaenoaa e 1-e e 2-e io?aneae e iiyoiio iio?aaoaony auioneaoue
aieueoa, /ai o2=100. Ii oiaaea aic?anooo iio?aaiinoe a i?iaeoeoeee 1-e
io?anee. Oiaaea aeinoaoi/ii ia?aoeoueny e ninoaaeaiiie nenoai
o?aaiaiee, iieiaeea o1=0 e o2=1 ( ni i.2 ):

0.8o1 – 0.4o2 = 0

-0.55o1 + 0.9o2 = 1

?aoea yoo nenoaio, iieo/ei o1=0.8 e o2=1.5. Neaaeiaaoaeueii, aeey
oiai /oiau ecaioiaeoue aaeeieoeo eiia/iiai i?iaeoeoa 2-e io?anee,
iaiaoiaeeii a 1-e io?anee auionoeoue i?iaeoeoeee o1=0.8. Yoo aaee/eio
iacuaatho eiyooeoeeaioii iieiuo cao?ao e iaicia/atho aa /a?ac S12. Oaeei
ia?acii, anee a12=0.4 oa?aeoa?ecoao cao?aou i?iaeoeoeee 1-e io?anee ia
i?iecaiaenoai aaeeieoeu i?iaeoeoeee 2-e io?anee, eniieuecoaiua
iaiin?aaenoaaiii ai 2-e io?anee ( ii/aio iie e auee iacaaiu i?yiua
cao?aou ), oi S12 o/eouaatho niaieoiiua cao?aou i?iaeoeoeee 1-e io?anee
eae i?yiua ( a12 ), oae e einaaiiua cao?aou, ?aaeecoaiua /a?ac ae?oaea (
a aeaiiii neo/aa /a?ac 1-th aea ) io?anee, ii a eiia/iii n/aoa
iaiaoiaeeiua aeey iaania/aiey auionea aaeeieoeu eiia/iiai i?iaeoeoa 2-e
io?anee. Yoe einaaiiua cao?aou ninoaaeytho S12-a12=0.8-0.4=0.4

Anee eiyooeoeeaio i?yiuo cao?ao en/eneyaony ia aaeeieoeo aaeiaiai
auionea, iai?eia? a12=0.4 i?e o2=1, oi eiyooeoeeaio iieiuo cao?ao
?ann/eouaaaony ia aaeeieoeo eiia/iiai i?iaeoeoa.

Eoae, aaee/eia Sik oa?aeoa?ecoao iieiua cao?aou i?iaeoeoeee i-e
io?anee aeey i?iecaiaenoaa aaeeieoeu eiia/iiai i?iaeoeoa k-e io?anee,
aeeth/athuea eae i?yiua ( aik ), oae e einaaiiua ( Sik – aik ) cao?aou.

I/aaeaeii, /oi anaaaea Sik > aik.

Anee iaiaoiaeeii auionoeoue ok aaeeieoe k-ai eiia/iiai i?iaeoeoa,
oi niioaaonoaothuee aaeiaue auione eaaeaeie io?anee ninoaaeo ia
iniiaaiee nenoaiu ( 8 ):

x1 = S1k(yk, x2 = S2k(yk, …, xn = Snk(yk ,

/oi iiaeii caienaoue ei?i/a a aeaea:

_ _

x = Sk(yk ( 10 )

Iaeiiaoe, anee o?aaoaony auionoeoue iaai? eiia/iiai i?iaeoeoa, caaeaiiue
anni?oeiaio-

_ o1

iui aaeoi?ii O = : , oi aaeiaue auione k-e io?anee xk,
iaiaoiaeeiue aeey aai

on

iaania/aiey, ii?aaeaeeony ia iniiaaiee ?aaainoa ( 10 ) eae neaey?iia
i?iecaaaeaiea noieaoea Sk ia aaeoi? O, o.a.

_ _

xk = Sk1y1 + Sk2y2 + … + Sknyn = Sk(y , ( 11 )

a aanue aaeoi?-ieai o iaeaeaony ec oi?ioeu ( 7 ) eae i?iecaaaeaiea
iao?eoeu S ia aaeoi? O.

Oaeei ia?acii, iiaen/eoaa iao?eoeo iieiuo cao?ao S, iiaeii
ii oi?ioeai ( 7 ) – ( 11 ) ?ann/eoaoue aaeiaue auione eaaeaeie io?anee e
niaieoiiue aaeiaue auione anao io?aneae i?e ethaii caaeaiiii
anni?oeiaioiii aaeoi?a O.

Iiaeii oaeaea ii?aaeaeeoue, eaeia eciaiaiea a aaeoi?-ieaia (o = (
(o1, (o2, …, (on ) auciaao caaeaiiia eciaiaiea anni?oeiaioiiai i?iaeoeoa
(O = ( (o1, (o2, …, (on ) ii oi?ioea:

_ _

(o = S((O , ( 12 )

I?eaaaeai i?eia? ?an/aoa eiyooeoeeaioia iieiuo cao?ao aeey
aaeainiaie oaae.2. Iu eiaai iao?eoeo eiyooeoeeaioia i?yiuo cao?ao:

0.4

A =

0.55 0.1

Neaaeiaaoaeueii,

1 -0.2 -0.4 0.8
-0.4

A – A = =

-0.55 1 -0.1 -0.55
0.9

Ii?aaeaeeoaeue yoie iao?eoeu

0.8 -0.4

D [ E – A ] = = 0.5

-0.55 0.9

Iino?iei i?eniaaeeiaiioth iao?eoeo ( A – A )*. Eiaai:

0.9 0.4

( A – A )* = ,

0.55 0.8

ioeoaea ia?aoiay iao?eoea, i?aaenoaaeythuay niaie oaaeeoeo
eiyooeoeeaioia iieiuo cao?ao, aoaeao neaaeothuae:

1 0.9 0.4
1.8 0.8

S = ( A – A )-1 = ––– =

0.5 0.55 0.8 1.1
1.6

Ec yoie iao?eoeu caeeth/aai, /oi iieiua cao?aou i?iaeoeoeee 1-e e
2-e io?anee, eaeouea ia i?iecaiaenoai aaeeieoeu eiia/iiai i?iaeoeoa 1-e
io?anee, ninoaaeyao S11=0.8 e S21=1.5. N?aaieaay n i?yiuie cao?aoaie
a11=0.2 e a21=0.55, onoaiaaeeaaai, einaaiiua cao?aou a yoii neo/aa
ninoaayo 1.8-0.2=1.6 e 1.1-0.55=0.55.

Aiaeiae/ii, iieiua cao?aou 1-e e 2-e io?anee ia i?iecaiaenoai
aaeeieoeu eiia/iiai i?iaeoeoa 2-e io?anee ?aaiu S12=0.8 e S22=1.5,
ioeoaea einaaiiua cao?aou ninoaayo 0.8-0.4=0.4 e 1.6-0.1=1.5.

Ionoue o?aaoaony ecaioiaeoue 480 aaeeieoe i?iaeoeoeee 1-e e 170
aaeeieoe 2-e io?aneae.

Oiaaea iaiaoiaeeiue aaeiaue auione o = o1 iaeaeaony ec ?aaainoaa ( 7 ):

o2

_ _ 1.8 0.8 480 1000

o = S(O = ( =

1.6 170 800 .

IIEIUA CAO?AOU O?OAeA EAIEOAEIAEIAEAIEE E O.Ae.

?anoe?ei oaae.1, aeeth/ea a iaa, e?iia i?iecaiaeeoaeueiuo cao?ao
xik, cao?aou o?oaea, eaieoaeiaeiaeaiee e o.ae. ii eaaeaeie io?anee. Yoe
iiaua enoi/ieee cao?ao aieooony a oaaeeoeo eae iiaua n+1-y, n+2-y e
o.ae. aeiiieieoaeueiua no?iee.

Iaicia/ei cao?aou o?oaea a k-th io?aneue /a?ac xn+1,k, e cao?aou
eaieoaeiaeiaeaiee – /a?ac xn+2,k ( aaea k = 1, 2, …, n ). Iiaeiaii oiio
eae aaiaeeeenue i?yiua cao?aou aik,

xn+1,k

aaaaeai a ?anniio?aiea eiyooeoeeaiou i?yiuo cao?ao o?oaea an+1,k = –––––
, e

xk

xn+2,k

eaieoaeiaeiaeaiee an+2,k = ––––– , i?aaenoaaeythueo niaie ?anoiae
niioaaonoaothuaai

xk

?ano?na ia aaeeieoeo i?iaeoeoeee, auioneaaioth k-e io?aneueth. Aeeth/ea
yoe eiyooeoeeaiou a no?oeoo?ioth iao?eoeo ( o.a. aeiienaa eo a aeaea
aeiiieieoaeueiuo no?ie ), iieo/ei i?yiioaieueioth iao?eoeo
eiyooeoeeaioia i?yiuo cao?ao:

a11 a12 … a1k … a1n

a21 a22 … a2k … a2n
iniiaiay /anoue iao?eoeu

…………………………………

A( = 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
aeiiieieoaeueiua no?iee

I?e ?aoaiea aaeainiauo o?aaiaiee ii-i?aaeiaio eniieuecoaony eeoue
iniiaiay /anoue iao?eoeu ( no?oeoo?iay iao?eoea A ). Iaeiaei i?e ?an/aoa
ia ieaie?oaiue ia?eiae cao?ao o?oaea eee eaieoaeiaeiaeaiee, iaiaoiaeeiuo
aeey auionea aeaiiiai eiia/iiai i?iaeoeoa, i?eieiatho o/anoea
aeiiieieoaeueiua no?iee.

Oae, ionoue, iai?eia?, i?iecaiaeeony aaeeieoea i?iaeoeoa 1-e
io?anee, o.a.

_ 1

O = 0

:

0 .

Aeey yoiai o?aaoaony aaeiaue auione i?iaeoeoeee

S11

_ _ S21

x = S1 = :

Sn1

Iiaen/eoaai iaiaoiaeeiua i?e yoii cao?aou o?oaea Sn+1,1.
I/aaeaeii, enoiaey ec niunea eiyooeoeeaioia an+1,k i?yiuo cao?ao o?oaea
eae cao?ao ia aaeeieoeo i?iaeoeoeee k-e io?anee e aaee/ei S11, S12, …,
S1n, oa?aeoa?ecothueo neieueei aaeeieoe i?iaeoeoeee iaiaoiaeeii
auionoeoue a eaaeaeie io?anee, iieo/ei cao?aou o?oaea iaiin?aaenoaaiii a
1-th io?aneue eae an+1,1S11, ai 2-th – an+1,2S21 e o.ae., iaeiiaoe a
n-th io?aneue an+1,nSn1. Noiia?iua cao?aou o?oaea, naycaiiua n
i?iecaiaenoaii aaeeieoeu eiia/iiai i?iaeoeoa 1-e io?anee, ninoaayo:

_ _

Sn+1,1 = an+1,1S11 + an+1,2S21 + … + an+1,nSn1 = an+1S1 ,

o.a. ?aaiu neaey?iiio i?iecaaaeaieth ( n+1 )-e no?iee ?anoe?aiiie
iao?eoeu A(, eioi?oth iaicia/ei an+1, ia 1-e noieaaoe iao?eoeu S.

Noiia?iua cao?aou o?oaea, iaiaoiaeeiua aeey i?iecaiaenoaa
eiia/iiai i?iaeoeoa k-e io?anee, ninoaayo:

_ _

Sn+1,k = an+1Sk ( 13 )

Iaciaai yoe aaee/eiu eiyooeoeeaioaie iieiuo cao?ao o?oaea. Iiaoi?ea ana
i?eaaaeaiiua ?annoaeaeaiey i?e ?an/aoa iaiaoiaeeiuo eaieoaeiaeiaeaiee,
i?eaeai aiaeiae/ii i?aaeuaeouaio e eiyooeoeeaioai iieiuo cao?ao
eaieoaeiaeiaeaiee:

_ _

Sn+2,k = an+2Sk ( 14 )

Oaia?ue iiaeii aeiiieieoue iao?eoe S no?ieaie, ninoiyueie ec
yeaiaioia Sn+1,k e Sn+2,k, ia?aciaaoue ?anoe?aiioth iao?eoeo
eiyooeoeeaioia iieiuo cao?ao:

S11 S12 … S1k … S1n
iao?eoea eiyooeoeeaioia

S21 S22 … S2k … S2n
iieiuo aioo?ei?iecaiaeno.

………………………………… cao?ao

S( = Si1 Si2 … Sik … Sin

…………………………………
( 15 )

Sn1 Sn2 … Snk … Snn

Sn+1,1 Sn+1,2 … Sn+1,k … Sn+1,n
aeiiieieoaeueiua no?iee

Sn+2,1 Sn+2,2 … Sn+2,k … Sn+2,n

Iieuecoynue yoie iao?eoeae iiaeii ?ann/eoaoue i?e ethaii caaeaiiii
anni?oeiaioiii aaeoi?a O ia oieueei iaiaoiaeeiue aaeiaue auione
i?iaeoeoeee o ( aeey /aai eniieuecoaony iao?eoea S ), ii e iaiaoiaeeiua
noiia?iua cao?aou o?oaea xn+1, eaieoaeiaeiaeaiee xn+2 e o.ae.,
iaania/eaathueo auione aeaiiie eiia/iie i?iaeoeoeee O.

I/aaeaeii,

xn+1 = Sn+1,1y1 + Sn+1,2y2 + … + Sn+1,nyn , ( 16 )

xn+2 = Sn+2,1y1 + Sn+2,2y2 + … + Sn+2,nyn ,

o.a. noiia?iia eiee/anoai o?oaea e eaieoaeiaeiaeaiee, iaiaoiaeeiuo aeey
iaania/aiey anni?oeiaioiiai aaeoi?a eiia/iie i?iaeoeoeee O, ?aaiu
neaey?iui i?iecaaaeaieyi niioaaonoaothueo aeiiieieoaeueiuo no?ie
iao?eoeu S( aaeoi? O.

Iaeiiaoe, iauaaeeiyy oi?ioeo ( 7 ) n oi?ioeaie ( 16 ), i?eoiaeei e
neaaeothuae eiiiaeoiie oi?ia:

x1

x2

_ : _

x = xn = S(O ( 17 )

xn+1

xn+2

Ionoue aeiiieieoaeueii e aeaiiui, iiiauaiiui a oaae.2, ecaanoiu ii
eoiaai eniieiaiey aaeaina oaeoe/aneea cao?aou o?oaea xn+1,k ( a oun.
/aeiaaei-/ania ) e eaieoaeiaeiaeaiee xn+2,k ( a oun. ?oa. ), eioi?ua
caienaiu a oaae.3

Ia?aoiaey e eiyooeoeeaioai i?yiuo cao?ao aik, iieo/ei ?anoe?aiioth
iao?eoeo:

0.2 0.4

A( = 0.55 0.1

0.5 0.2

1.5 2.0

Oaaeeoea 3

? io?aneae iio?aaeaiea eoiai
eiia/iue aaeiaue

?
cao?ao i?iaeoeo auione

io?aneae 1 2

1 100 160
260 240 500

2 275 40
315 85 400

o?oae 250 80
330

eaieoaeiaeiaea- 750 800
1550

iey

Ia?aoiay iao?eoea S = ( E – A )-1 auea oaea iiaen/eoaia a
i?aaeuaeouai ioieoa.

Ia iniiaaiee ( 13 ) ?ann/eoaai eiyooeoeeaiou iieiuo cao?ao o?oaea
( Sn+1,k=S3,k ):

_ _

S31 = a3(S1 = 0.5 ( 1.8 + 0.2 ( 1.1 = 1.12 ;

_ _

S32 = a3(S2 = 0.5 ( 0.8 + 0.2 ( 1.6 = 0.72

e eaieoaeiaeiaeaiee Sn+2,k = S4,k:

_ _

S41 = a4(S1 = 1.5 ( 1.8 + 2.0 ( 1.1 = 4.9 ;

_ _

S42 = a4(S2 = 1.5 ( 0.8 + 2.0 ( 1.6 = 4.4 .

Oaeei ia?acii, ?anoe?aiiay iao?eoea S( eiyooeoeeaioia iieiuo
cao?ao i?eiao aeae:

1.8 0.8

S( = 1.1 1.6

1.12 0.72

4.9 4.4

Anee caaeaoueny ia ieaie?oaiue ia?eiae i?aaeiei
anni?oeiaioiui aaeoi?ii

O = 240 , oi ?ann/eoaa ii oi?ioeai ( 16 ) noiia?iua cao?aou o?oaea
xn+1 e

85

eaieoaeiaeiaeaiee xn+2, iieo/eee au xn+1 = x3 = 1,12 ( 240 + 0.72 ( 85 =
268.8 + 61.2 = 330 oun. /ae.-/. e xn+2 = xn = 4.9 ( 240 + 4.4 ( 85 =
1176 + 374 = 1550 oun.?oa., /oi niaiaaeaao n enoiaeiuie aeaiiuie oaae.3.

Iaeiaei a ioee/ea io oaae.3, aaea yoe noiia?iua cao?aou
a?oiie?othony ii io?aneyi

( 250 e 80 eee 750 e 800 ), caeanue iie ?ani?aaeaeaiu ii aeaeai eiia/iie
i?iaeoeoeee: ia i?iaeoeoeeth 1-e io?anee 268.8 e ia i?iaeoeoeeth 2-e
io?anee 61.2; niioaaonoaaiii cao?aou eaieoaeiaeiaeaiee ninoaaeytho 1176
e 374.

I?e ethaii iiaii cia/aiee anni?oeiaioiiai aaeoi?a O ana iieacaoaee
ieaia, oaeea, eae aaeiaay i?iaeoeoeey eaaeaeie io?anee e noiia?iua
?anoiaeu o?oaeiauo ?ano?nia e eaieoaeiaeiaeaiee iaeaeai ec oi?ioeu ( 17
).

Oae, ionoue caaeai anni?oeiaioiue aaeoi? O = 480 . Oiaaea

170

_ o1 1.8 0.8
1000

o = o2 = 1.1 1.6 480 = 800

o3 1.12 0.72 170
600

o4 4.9 4.4
3100

Ionthaea caeeth/aai, /oi caieaie?iaaiiue auione eiia/iiai
i?iaeoeoa O iiaeao auoue aeinoeaioo i?e aaeiaii auionea 1-e e 2-e
io?aneae: o1=1000 e o2=800, i?e noiia?iuo cao?aoao o?oaea o3=660 oun.
/ae.-/. e i?e cao?aoao eaieoaeiaeiaeaiee o4=3100 oun.?oa.

?anniio?aiiua oai?aoe/aneea aii?inu e i?eia?u ?an/aoa, eiia/ii,
aeaeaei ia en/a?iuaatho aaaeioth aeey i?aeoeee iaeanoue aaeainiauo
enneaaeiaaiee. Caeanue i?ieeethno?e?iaaii oieueei iaeii iai?aaeaiea
i?eeiaeaiey eeiaeiie aeaaa?u a yeiiiie/aneeo enneaaeiaaieyo.

Caaea/a

A oaaeeoea oeacaiu ?anoiaeiua ii?iu aeaoo aeaeia nu?uey e oiieeaa
ia aaeeieoeo i?iaeoeoeee niioaaonoaothuaai oeaoa, o?oaeiaieinoue
i?iaeoeoeee a /aeiaaei-/anao ia aaeeieoeo i?iaeoeoeee, noieiinoue
aaeeieoeu niioaaonoaothuaai iaoa?eaea e iieaoa ca 1 /ae.-/.

Oaaeeoea

Ii?iu ?anoiaea

Iaicia/aiey Noieiinoue

I II
III

Nu?uea I 1.4 2.4
0.8 a4 5

Nu?uea II – 0.6
1.6 a5 12

Nu?uea III 2.0 1.8
2.2 a6 2

O?oaeiaieinoue 10 20 20
a7 12

Ii?aaeaeeoue:

a) noiia?iue ?anoiae nu?uey, oiieeaa e o?oaeiauo ?ano?nia ia auiieiaiea
i?iecaiaenoaaiiie i?ia?aiiu;

a) eiyooeoeeaiou i?yiuo cao?ao nu?uey, oiieeaa e o?oaea ia aaeeieoeo
eiia/iie i?iaeoeoeee eaaeaeiai oeaoa;

a) ?anoiae nu?uey, oiieeaa e o?oaeiauo ?ano?nia ii oeaoai;

a) i?iecaiaenoaaiiua cao?aou ii oeaoai ( a ?oa. ) e ia anth
i?iecaiaenoaaiioth i?ia?aiio caaiaea;

ae) i?iecaiaenoaaiiua cao?aou ia aaeeieoeo eiia/iie i?iaeoeoeee.

?aoaiea:

a) Noiia?iue ?anoiae nu?uey I iiaeii iieo/eoue, oiiiaeea
niioaaonoaothuoth 1-th no?ieo aoi?ie oaaeeoeu ia aaeoi? o, o.a.

_ _ 235

a4o = ( 1.4; 2.4; 0.8 ) 186 = 1088

397

Aiaeiae/ii iiaeii iieo/eoue ?anoiae nu?uey II e o.ae.

Ana yoi oaeiaii caienaoue a aeaea i?iecaaaeaiey:

1.4 2.4 0.8 235 1088
Nu?uea I

0 0.6 1.6 186 = 746
Nu?uea II

2.0 1.8 2.2 397 1678
Oiieeai

0.1 0.2 0.2 1409
*aeiaaei-/ania.

a) ?anoiae nu?uey I ia aaeeieoeo eiia/iie i?iaeoeoeee 1-ai oeaoa ( o1=1
) iaeaeai ec au?aaeaiey 1.4S11 + 2.4S21 + 0.8S31. Neaaeiaaoaeueii,
niioaaonoaothuea eiyooeoeeaiou iieiuo cao?ao nu?uey, oiieeaa e o?oaea ia
eaaeaeoth aaeeieoeo eiia/iiai i?iaeoeoa iieo/ei ec i?iecaaaeaiey
iao?eoeu:

I II III

1.4 2.4 0.8 1.04 0.21 0.02 1.97
2.92 1.36 Nu?uea I

0 0.6 1.6 0.21 1.05 0.13 = 0.17
0.84 2.09 Nu?uea II

2.0 1.8 2.2 0.03 0.13 1.26 2.53
2.60 5.23 Oiieeai

10 20 20
15.2 24.8 28.0 O?oae

Oaeei ia?acii, iai?eia?, aeey ecaioiaeaiey o1=1 iaiaoiaeeii
cao?aoeoue 1.97 aaeeieoe nu?uey I, 0.17 aaeeieoe nu?uey II, 2.53
aaeeieoe oiieeaa e 15.2 /ae.-/.

a) ?anoiae nu?uey, oiieeaa e o.ae. ii eaaeaeiio ec oeaoia iieo/ei ec
oiiiaeaiey eo ?anoiaeiuo ii?i ia niioaaonoaothuea aaeiaua auionee ii
oeaoai. A ?acoeueoaoa iieo/ei iao?eoeo iieiuo ?anoiaeia:

I II III

Nu?uea I 330 440 318

Nu?uea II 0 111 635

Oiieeai 470 335 873

O?oae 2350 3720 7940

a) I?iecaiaenoaaiiua ?anoiaeu ii oeaoai iiaeai iieo/eoue iooai
oiiiaeaiey neaaa no?iee noieiinoae ( 5; 12; 2; 1.2 ) ia iineaaeithth
iao?eoeo:

330 440 318

0 111 635
I II III

( 5; 12; 2; 1.2 ) 470 335 873 = ( 5410; 8666;
20484 )

2350 3720 7940

ae) Iaeiiaoe, i?iecaiaenoaaiiua cao?aou ia aaeeieoeo eiia/iie
i?iaeoeoeee, iaiaoiaeeiua aeey ii?aaeaeaiey naaanoieiinoe i?iaeoeoeee,
iiaeai iaeoe iooai oiiiaeaiey neaaa iao?eoeu iieiuo cao?ao, iaeaeaiiie a
i.a., ia no?ieo oeai:

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

Oaeei ia?acii, aioo?ei?iecaiaenoaaiiua cao?aou ia aaeeieoeo
oiaa?iie i?iaeoeoeee I, II e III oeaoia niioaaonoaaiii ninoaaeytho: 35.3
?oa., 59.6 ?oa., 75.7 ?oa.

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