Tải bản đầy đủ - 74 (trang)
CHƯƠNG II: MỘT VÀI PHƯƠNG PHÁP NỘI SUY TỔNG QUÁT SUY RỘNG ĐỂ GIẢI GẦN ĐÚNG PHUÊONG TRÌNH TOÁN Tư

CHƯƠNG II: MỘT VÀI PHƯƠNG PHÁP NỘI SUY TỔNG QUÁT SUY RỘNG ĐỂ GIẢI GẦN ĐÚNG PHUÊONG TRÌNH TOÁN Tư

Tải bản đầy đủ - 74trang

- 4&



1/^^(x)

con ^'(^Cx),



=



y^(x)



^^(x),...,



-



^^^(x) . . .



^\^x)



^i(x),



(i -



l a h | ha^i x a c ^inh



o,n)



t r o n g khSng



g i a n dang khao o a t , co n i a n g i a t r i cung t h u ^ c voo khong g i a n

do.



Cac he GO C^, Q^ , . . . ,



C^ se du^'c x a c dinh tuy theo t>ng



phitti'ns phojj c\i the*

•Csc phi^o^ng phap dU(^c khao s a t t r o n g chiitmg nay l a dgng

tong q u a t ci'io n g t v a i phuang phap du^c khao s a t t r o n g cni?o'ng I

va n g t s5 k 6 t qu^ cua t a c g i a k h a c du^c coi nhu' nhu'ng tru'»6?ng

hg^ d|ic b i ^ t .



§ ! • ^^IGt v a i k h a i ni.e-n ve t y s a i phan suy ronf; toni^ c u a t

cho t o a n tu" va cou^ thirc ngi suy Iliu^to^n ST^T rpnR*

Tiong phSn nay t a se xay d^'ng k h a i n i ^ n ve t y s a l phan

euy rgng t o n g q u a t cho toan



vu.



Xet ham t n r u t'J'g'ng Ax, ham nay chuyen khong g i a n dinh

chuan X vao khong g i a n djnh chudn X.

Ky h i ^ u c6c khong g i a n cua nhi?ng t o a n tir t\xy6n t i n h X

vao Y l a /fX -> I J .

I^y'ng h0 ham



^Q^^^ '



/1 ^^^ »• • • >



7 k-1 ^^^ ' t r o n g do



^ j ( x ) l a cac Snh x§ b i e n X vao chinh n o .

TJC dem g i a n , t a ky hipu ' i =" / C'[ o» * • •» 1 k - 1 ^ ' I'-^'o'ng

t\^ nhir chutmg I , t a thanh l^p khong g i a n t i c h

S2 =



X.X



:



( t i c h B§ C5c)



3z = X . X . X . , . . . , Ev = . ^ • ^ • • • ^ I t r o n g do k l a

,..

^

k iSn

m£)t so tu' n h i e n .

'^'^"



'•



'-Xii



':VX\











"











'



- 47 PhSn tu cua I^^ ^ ' ^ ^ ^^"^ ^^^^ ^^^

X^ 6 X»



^ ^ » -^o-^' ^^^



^^ " o*"!)* T'.i'O^ng t l / , ph9n t^i cua E^. '^g^c v i e t



db^



Gia sir t o n t a i lagt coan to* tuyen t i n h A(x. , x- ; 'r

•^1

-^o



)



chuyen cac phSn tu' cua kh8ng g i a n Eo vao cac phan t'*'' cua khong

gian ^



X -> I J ' va thoa Juan d i e u ki^m t



(1.1)



A(Xi^^, x^^; f )



U^(^^J = ^^^ " ^^:^^i^-



loon v't A(x^ , Xj » 4 ) dirg'c ggi l a t y s a l phan b^-c nhfit cua

^o

^1

ham tru'u tirg^ng Ax 'rng vo'i h^ han -i dug'c i S y t y l c5c phan ti*?

X. , X.

^o

^1



f^'



X.



Ihay X.



bang mgt ohSn ti^ b S t ky x t



^ J t*? ( 1 . 1 )



ta



suy ra t

(1.2)



Ax



= Ax.



+ A(x, X. \ Cf ) 4 \



^•o



^0



(x).



^^-0



'



C6ng thiJc ( 1 . 2 ) l a cong thu^c n g i suy F i u - t c ^ co phan du

cua ham tti)u t i r g ^ Ax 6ns vo'i ^^ ]

X , X.



» duVc iSy t ^ l cac phan tiJ



*^ X.



Gia sir tBn t j i mgt t o a n tu' song tuySn t i n h

A(x. , X- , X. ; -^ )i

2

-^1-^0



t r o n g do



x.



0



^



X, ( j = oTS), ch\Tycn



cac phSn tu* cua khong g i a n .^- vao cac phSn tu cua khong g i a n

/ " X --^ £'X -^IJ

(1.3)



J



va thoa man d i e u k i ^ n :



A ( x ^ ^ , x^ ;

d



o



.



) - A(x^ , 3^ ; 1^ )



oM



2



-1



^o



=

'^1



(X;



^2



)



.



- 48



To5n tu A(xj^ , Xj^ , x^ ;



^



-



) so ^--g^c ggi l a ty s a i p h e n b ^ c



h a i ciie ham t r i ' u tirg'ng Ax ifeg vol h^

to' X,



tr Xj



/



duVc i S y t^:! c5c p>^.ln



(d = 0 , 2 ) .



i

Toe dyng C8 h a i ve ( I . J ) l e n

(1.4)



W^. (x^. ) t a rVug'c :

' ^0 - 2



AC^^. , X. ; f ) f . (x. ) - A(x. . X. ; T ) Y i (x-:.) =

•^2

^o

"-0 ^2

^1

^o

-"o 'cl

=



A(x,^,, X. , X. ; f )

"2



Yi-iC^^^)



• o



"1



^i



^



(2^ J .

o



fX



Thay x^^ bDjiig mgt phan tu'' b S t ky x ': X va d^i^a vao ( 1 . 1 ) , tu?

( 1 . 4 ) suy !•& ;

( 1 . 5 ) Ax ^ Ax. I- /i(x,. , X. ; V ) ^ ^ (-0 +

^o

^1

^o

^ ^0

+ A(x, X. , X. 5 ^ ) -^ (x) ^ - ( x ) .

^1

-^o

^ -^1

' -^o

G6ng th^o



( 1 . 5 ) l a coiog thil'c ngi crj^'- ITiu-tc;n sviy r j n g



C phSn du cua ham tru*u tu^g^'^ Ax uns vo'i hy han

O

t ^ i cSc phl^ii tiV X, ^ ^ . c



^>



j



duVc I 3 y



^^ " o ' / f ) .



Hoon t o a n tutmg t ^ , t a co t h e dinh n^^la



cho t y s a i



phan bf c k :

( 1 . 6 ) A(x^ , x^

» • • • , X- , XJ ? V ) - A ( x ^ .

,....x. ,x ; O

^k ^ k - 5

^1

^o '

-^k-1

^1 % ^



=



A(x, , x^

^k



, . . . , X, , x^ ; /



^k-1



-1



Lan l i ^ t t a c f?§ng ca h a i ve ( 1 . 6 )

({A



C^^i ) 4^' ( x . ) . . .



w-'.



-o



^



) ti

^ hc-1



len



(r^. ) t a nh^n :



^^ ^'

^k



(1.7) A(x^ , x^

^k



, . . . , X , x^ ;cf)q'.



k-2



-A(x,^



^1



-^o



'



^^'^i^'^--- ^1 ^^1,^ k-2



. . . . . X, , X. ; (f ) ( f ,



-^k - 1



"' I



o



- A(x. , . . . , X. , X



-^k



(X,, ) . . .

k-2



;t^)



^,



Thay x^, beng mgt phan t*? b ? t ky x



o



'fi(^i> =

o



i-:



/ - i . ^••-



k

k



^ 1 <^i^^-



^ X va d^a vao cac cong



th'j'c ( 1 . 1 ) - ( 1 . 4 ) , tir ( 1 . 7 ) ta sioy ra t

( 1 . 8 ) Ax = Ax. + A(x- , X. ; ^ )

^o

1

o



Cf. (x)

o



+ A(x. , X, , X, ; f ) f i

"2



-1



+ . . . + A (x,



*o



zc.



^k-1



H



(x)



-f



^f, (x) -f

-^0



, . . . , - i » ^ i f ^^1,

^1



^0



/ ^ ^ - " ^ ^ ^^^



k-1



-^o



C5ng t^^^c ( 1 . 8 ) l 3 cong tv^u^c n g i suy ITlu-to'n co ph^n di? cua

ham Ax img v6l hf ham



^f



t ^ i cac phan tu' x , x i ^ ^ X



( j - ^TF-T).

I r u - ^ g hgp d$c blC^t, xoSu chgn hg

anh xg tpnh



ti^n)



^±(^0



= x - x^ ( c a c



t h i f^ cac dj-nh nghia ( 1 . 1 ) - ( 1 . S ) se suy r a



cac dinh n g h i a ( 1 . 1 ) - ( 1 , 4 ) cua

Ti?(tog t!/ nhu^ § 1



S 1



chifc^ng I ,



Chu^c;ng I .



co th£ chv?ng minh di^g'c c5c



t y s a l phan tong q u a t t^^uy rgng l e dSi xi^ng thoo cac n 5 c x^.

§2.



VS mgt vai ohi^nix



jh&o nyi suy t;6m: o u a t suy r^nE



p i a l .gan dunr: phL^^n^^ t r i n h d l ) vs B\J' hOl t u cue chun/:*

Gia :ji5 phtrcteg t r i n h (1 ) co nghlgm d\xns l a : ^ ,

t r o n g do :



x*^^^^



- 7^ uOi =



i X : 1 X - :^lj 4
1



;



^



l a h?.ng s6 duvng-^. Cho ti-rc'c



x^ I s xSp : d ban :*Su -^i gSn x * . x^u dyng cac d^nli ngh'ia ve t y

se.i phfxn suy rvng tdng q u a t cWo t o a n t'> da t r i n h bay o-



§ 1,



t a xSy dv*ng mgt s8 phiiwng phap l^v-p sau ddy s

TTgi dun^^ cac PhiTcyn.^ Phap va c6c ^tJA?^ l y y§ t S c ^g hgi tyi.

a/ i'ru'o^ng hg^) k = 1 .

Xu3\; p h S t tu phen t»> x ^ , t a yS.y d'^Jng ph!ln vJ :

(2.1)



x° =



x^ f



MAx^ ,



Xet t y s a l phen bgc hex



(2.2)



ACx, X?, x^; f )

-



A(x, X? ? y )



o


A ( x , x ° , x ^ ; <^ )



f . (x) -



f^(x)



f.,(x) ^



A(x^, x°5 f )



^,^(x).



D'/a vao ( 1 . 1 ) tu' ( 2 . 2 ) suy r e :

(2.5)



Ax = . ^



+ A(x^, x° ; f



) ^^(x)



+ A(x, x^^ x ^ ; f

^ay



li*>



(2.4)



) ^f^(x)



+

^^(x).



:



Ax - Ax^ ^ A(x?, x ^ ^ ) ^ , - ( x ) - A ( : : ^ , x ° ; ^. ) ^; , j ( x ° )

+ A(x?, x ^ ; f ) f ^ ( x ^ ) + A(x, x ^ . x ^ ( H f o^-^> f 1 ^^^)J);'a vao ''»-inh n g h i a ( 1 . 1 ) § 1 chu^o^ng I va ( 1 . 1 ) , tu' ( 2 . 4 )



suy r a :

(2.:^)



AX = Ax° f A ( x ° , r r ^ i f ) f . , ( x , x ° ) ( x - x « )

+ A(x, >f, x ° ; f ) c^^(x)



2h5 nhr-tig s



.i^(r).



v



- 51 (2.6)



i r ^ ^ ( x , x°) - f , , ( x ° , x ° ) 7 (x - x ^ ) =

=



q ' , ( x , >:^, x°)



(x - x ° ) (y. - x°)



Do do :

(2.7)



(fA::, x°) (x - x ° ) = f . ( 4 , x") 0 . - .:°)

^

//)



/



O



0\



^



0>



/



>



ON



+ ^^(^> x^, X ; u^ - X ; vx - : J ^ ; ,

t??ong do



4^1 ^'^o» ^ ' • • • » ^-.^ 1^^ ty oai phan b^c j cua ham



^i(x).

Thay (2.7) vao (2.5) ta '^(^c :

(2.S) Ax = Ax^ f K x ^ , x ^ f ) q^iz:^,

^- A(x^, x ^ ^ )



x^) (x - x^) +



^f^(x, .:^, x ° ) ( : : - x ^ ) ( x - x ^ ) +



+ A(x, x^, x^; f ) ^ ^ ( x )



f,(x).



Xap x i Ax bang da th'^c QQ(X) :

( 2 . 9 ) Ax ^



QQ(X) = Ax^ + A(x^, x^; / )



; ^ ( x ° , x°) (x - x^)



Gia su* ''.^^(x ) - G \*a ton t ? i cac tear, ti? nghich dao

A" (x^, x^; j ) va



•J^Ux?, x ^ ) , tir (2.9) ta suy ra :



x'' = x^ -- ^ A ( x ^ ^ x^; 7 ) ^ ; . ( x ? , x^) 7 "^ Ax^

C?5ng cu5t, nS^-' tr- r^^.t :

r"-*in'i



^-^



-



---"^ +



•T-/ ^•^-^



Tl =



o



-^ '^



t h i ly luCn hoar, toan tu'crig t^' n^-ir t r e n t a as nh^n :

(2.11) Ax = k'jX' + A(x^, x ^ f ) ^\(yXl,

^ A(x^, x ^

-



x^-) (>: - 2;^-) +



f ) f ^ ( x , x^, x^) (x - x^) (x - x^)



- P2 va J

(2.12)



~^^'^'^ ^



-CAi^,



x ^ ^ ) ^ . , ( 4 , :-:'')7"^ '^x^\



n= 0,1,2,...

Tx^ccng hg^ -^^^c b i ^ t , neu



^--f^C^) = (x - :cj) t h i t;> (2.12) ta



se nhg.n qua t r i n h li^p t'/a Aitlccnctephenxen trong £"15_7; ^^eu

^^(x) = (x - x^) va xj = 2 x^' - x ° , trong do x ^ v i x^ l a h a i

phan tu' ban dau lay du gan nhau va r^n gan x' , t h i t ^ (2.12) t a

sc nh'Jn qua t r i n h l$p ci\a



C'^^^^J-



Sy hgi tv Clia qua t r i n h 1-Jp (2.12) du*g'c the hi^^n b -^inh

l y sau dixy :

^nh ly 2 ^ .

Gia BU :

1^/ v5n t ^ i cac toan ti> r^^hich dao irA(x^, x^; ^

rf^(x^, x^)^"^

2^/

uC2n=



va / / A - \ x 3 ^ , x ^ f ) // ^



/ ' f . ( x ) //



^



)7''^,



B, /7ff^ ( 4 , x ^ ^ ) / K ^ -



/ / x * - x^"^//,



t:rx)ng d5 :



1^ • / / x - . ^ V 7 <: (1 + j M L . ) / / x ^ - : . ^ / / J O L^Z/ ,



con I^ l a mgt d?i li^g'ng duxi'ng gio'i ngi di^g'c xac dinh c ^lou

^

k i ^ n t i e p theo.

iP/ / A ( x . , X. ) / / , ^

^•1

^ci ^^n



iJ

'



;



//A(x^^, xi^; f ) / /

L., = nax (7,\, L^); // A(xi^ , x^^,



^i^;^)//^



// fi(xi^, :cj2, XiJ / ^



,^: i" ,

<



Lp,



,^ ^ ^ .



_ 53 4 ° / Cac hang s6 B,iB



, L^ , T'2'-^-2 ^^°^ ^'^^ ^^'^ ^ ^ ^



thi5c :



Khi *6 qua trini- l^p (2.12) ce hyi tv to'i ns^ipni x* cj.a phirc^ng

t r i n h ( 1 ) . T3C -^g '^oi ty ''^uvc cT^nh cia ^ang bSt '^ani;; tt^u'-c :

(2.13)



// X* - x^'-'^ //



^



P^//y^



- x " //^ .



QhiJRg a i s h

De danii' c'-'-'ng luinn duyc rang

TIP dien Iri^^n 1 ° /



y x ^ £ii2(j^i



(i = o , l ) .



cua dinh ly ta suy ro r^.ng toan tv? nghich dao



/f A ( 4 , x^-; (( ) ^f^(x^, x-^) J " ^ tSn t ? i va



// C A ( 4 , x"j ^ ) f ^ ( 4 , x^') 7"- // -• • B ^ .

'=

Djit X = x*",

(2.14)



t'J- (2.11) suy ra :



Ax"= - A ( 4 , x ^ ; p



t|.(x3?~, x-^0 (>:* - x'^') -



- A ( x f . x^^;f ) ^ ^ ( x * , x^.^, X-) (x^-x^') ( . : ^ - 4 )

- A ( x ^ . x^, x ^ ; / )



/;(x*)



'/',(^).



I'T"*? x^ vao ca 1^3i vo (2.12) ta nh§-n :

(2.15) x^^'^- x^ = :c^- x ^ - r . A ( 4 , x ^ ; f ) f ^ ( 4 , ^)



J-^



Ax"



I>\;a vao (2.14) va cac diou kipn 2 ° / , 3 ° / cua dinh l y ,

ti> (2.15) suy ra :

(2.17) // x ^ " ' ' - ^



// ^



B ^ ^ 2 ^1 // ^ - ^''^''•^' ^^ --^ - 4

+ I^ B 3 / / x * - x ^ / / ^ .



// ^



- 54 Thg nh'j'nG. ti" (2.10) suy ra :



4



-x^=



4



- X* > x^ - x-"^ -



x^ - ^



(UAx'^,



= x:^ -- ^

-



|UAx^ - ^v Ax^,



.,- M A(x^, .:^) ( : ^ - x^)



Do do :

(2.18)



/^ x?;^ - X* ^y . 4



(1 > f t l ' i ) // ^ n - ^.± ff



Thay (2.18) vao (2.1?) vn d^a vao gia t h i e t 4 / ctia

dinh. l y , ta suy ro ( 2 . 1 3 ) . ^



l a -Heu p h a i chu'ng minh.



b) TnX^ng hg»p k = J .

?:u5t phat tu' x^, te x^y c^vng cac phSn tii :

(2.19)



x^ = x^ Y ' ' ^ ' ^ ' '



x§ = 2 x ! ? - x ^



I'^P ty s a l p'^^dn suy rgng b^v^c ba

(2.20)



x|:^2.|-xf;.



A(x2, xiz^^^ ^^



A(x?, x | . x ° , x ° ; f ) ^ , ( x ? )



= A(x?, x ° , x , ° ; ^ ) ^ - , ( x 5 ) f , , ( x ° )



y. \ ^j ) :



.1,(4) f . ( x p



=



-



- A ( x ° , xjf. x ^ ; f ) ^ ! i ( x S ) ^ ^ ( a | J .

Dv^a x-ao ( 1 . 1 ) , (1.3) tu' (2.20) ta cuy ra :

( 2 . 2 1 ) Ax| = Ax^ + A(xS, y°i(f)



f , (x°)



+



+ A ( 4 , x°, x ^ f ) f . ( x ? )






+ A(x°. x ° , x ? , x ° ; f ) ^ / ^ ( 4 ) . | 1 , ( x p . 7 1 . ( 4 )

= A ^ . A(xJ, x ° ; f ) ^ ( x p - A(x^, x^; v ) - . ^ ( x ° )



- 55 + A ( x | , 2c°;


Cf.^hX') ^ A(x°, x ° ; f )



f/x^^)



- A ( x ° , x ° ; ^ ) ^ / , ( x ' ^ ) f A ( 4 , x^, x^; f



-



)f^(x?;)f2(^-^



+ A(x;;. x^, .4, x'^f )f^(xp f^(xp V 2^^Dya vao ( I . I ) S1 Oiu'o'ng T v u T l . l ) f> ( 2 . 2 1 ) euy ra s

( 2 . 2 2 ) Axe = Ax^^ + A ( 4 , ^ , ^ ) ^^ ..(zX^, x^) ( ^



- x^)



^



+ A(x|, x^, x°; j ) / : . / . , ( x ? ) y o ( x p + Y ^ ( x ° ) ^ 2 ( ^ ° > - 7 +

^ A ( x | . x | , X?, x ^ f ) . , ^ ^ ( x ° ) ^ - . l ( x p



f2(^3>-



DoV each d^t ( 2 . 1 9 ) , t a co t h e chfn diT^'c a $ t h^ ham ( c b a r ^ hpn

6^j_Cx) = (x - X.;) scio cho :

( 2 . 2 5 ) A(x^, x:?, x ^ ' j ) /fy , ( x ? ) < / 2 ( x 5 ) ^ ^ , ( X ' ^ ) Y ' 2 ( X ° ) 7 = 0

A(x9, 4 ,



x?, x " ; f ) ^ o ( 4 ^ ' / l ( ^ )



V2(xp



=



2

^



A r^o



...0



0



. .0. ^



^



-•'\



/



/_o



I j



o>



2

.



, '



t r o n g cro



Tl



j|



/



0



O^ . _



(xj-x^;:=



/



O



'-^



0\



U ^ - :^:c;'^ ^^5 " ^r



/



O



ON



'•^^^ - ^ ^ ) *



Di^a vao ( 2 . 2 5 ) t v ( 2 . 2 2 ) ta nhdn :

( 2 . 2 4 ) Ax^ - Ax^ + A(:c?, 4 * , ^ ) f , ( x j ,

+ A(x5. 4 ,



X?, x ^ ; p



x^^) (x? - x^) ^



(x^ - x ^ ) ( x ^ - ^ ) ( x ^



-x^)



H^t k h a c , l ^ p t y s a i p^an tong q u a t su^- rgng A(x,X2, 'r^, x ^ ; ^ ) ,

t r o n g d6 X = Xj i- ^ ^ , l y l u ^ n hoan t o a n trc^ng t v n^u' trOn t a

nh&n.



r 56 (2.25)



Ax - Ax° > A ( 4 , X?; f ) f^(x, x°) (x - x^) ^

^ A ( 4 . x°, x°}f ) / : y i ( ^ ) ^ 2 ^ ^ ) ^ f l ^ ^ ° > f 2 ( ^ ° > - 7 ^

+ A(x, x°, X?. x°; f ) ^Q(x)f,,(x) V'2^^^'



ThS nhttog :

(2.26)



T f i ( x . x°) - f^(x^, x ° ) 7 ( x - x ° ) =

= f ^ ( x , x^, x°) (x - x ° ) (x - x p .



L5y (2.26) thoy vao (2.25) ta nh§n t

(2.27) Ax = Ax°+ A(x|, x°j ^ ) ^^Uy



x*^) (x - 3°) +



+ A(3^, x°; J ) ^ ^ ( x , x^, x°) (x - x°) (x - xp *

+ A ( 4 , X?. x ^ f ) r f i ( x ) ^ ' ' 2 ( ^ ) + Y l ( x ° ) y 2 ( x ° ) 7 +

•KA(x, 4 , x°, x ° 5 f )



f o ( x ) f^(x)



f2(x).



D\?a vao (2,24) va (2.27) ta thgy re-ng vol Vx 6iXlst



j



trong d6 :

^^'^tl



= 1 x s / / A ( a ^ , 4 i f ) ^ ^ ( x , x°, x°) ( x - x ° ) I J ^.



+ A ( x | , x ° , x ° » f ) ^ f ^ ( x ) f2(=^)- f . C x ° ) f 2 ( ^ ° ) 7 / - ^

2



2



^//A(x. 4,3c?, x^iy )//rn //x^^//.n

1=0



••

*



i==o



^f.u)//jl

'



ta ee c6 J

(2.28) Ai = Ax*^ + A(3C^, x^; f )



^^"^(x^, x°) (x - x°)



+



+ A(x, 3 | , x^, x ° t f ) ( x - x ° ) ( x - x ^ ) ( x - x ^ ) .



^



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