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View of A modelling by rational approximations

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$IÌtLrÌ

l|',AhÍrrt,rstì

ì\xðllittlrlülì trl. t,tr

l.nlì()1ìilì

Dlì

r,ltptnoxIMA'i'I{,N

'i't¡r¡tr P,ilr ltio

lÍ191r, pp.55_(ig

A iTIODELLING BY RATIONAL APPROÀ,IAXIONS

I'AYO[- OLIOCt IOL¿\]'Y (lh'at islava)

1. tt't'ItoI)L{;:t'I()i[ t\

A

scientist who lìas compiled tables

of

tlat,a $/ishes

to

lecluce

thcm to a mole

conr¡enient a,nc[ conì]jlehensil¡le

folrn. rre

accornplisli;;

tlrilfi;

rep_r'esenting

tlte tlata

I'n

fuuctional form, Hc

selects

a

clais

of functiolis

ancl chooses fronr

this

class tìre onc

that

bcst f its his clai,¿r,.

,,*,..,,1,t]

1 :orìrtltj

presL

"fllulcl\'^t,epr.cscnl

(,on_

tinuous

functions.

class

to

l:e

the lational

f Lrnctions, zr,nd

the¡' ¿r

effi_

9-lg1-t,

torm of ap¡l'oximation. :\ ratiônai fulction ne) : p(|lg(t)

is -cne

wtrìclì can

bc'clalu¿rtecl as

thc

cluotient

of tu'o

pol-vnomial.s.'ü

eft)+0,

t,hen

a

single clir.isìon

lielcls the iìnal

r,csu1t.

_ Äctuall¡',

a r,athet' differ,cnt algot'ithnt clepenrÌine on

the timc

rcc¡uir.ecl for,-the mu r'ations

in par,ticulal computel bcing

usctl.

det'ir.ctl b¡. 1,.'*,r.*tn¡rnirrg

(t) jlrto

¿r,

còntilne

y€Íll's sorìì.e

theoÌctical

aspecl,s

of

continuetl

fractions

har-e been ¿isc¡s- 1gcl

i1

nìan)- pal)ers, T.c.

[2], t3l. rf

the clivision anc-[lnultiplication up"rà

tiolrs

tahe abou.t

thc

sauro 1,irne

in thc

conrputcr',

then

Lhdr,e

is a

tlec:-iclerl

adr-:rntagc

in

usìrrg

a

continuecl-fi'trct,icln cxpansion.

'l.h-¡r puÌpose

oj this

papgr. is

to

dcrnonitr¿ltc ¿-r, rnellrod

fol evalll,ing tlte

coeTficients

of

functioìs-11(t) :

(1.1)

å,f-ld

(r.2)

r(/) :

A

t1-

r+...

t* ,_nriË;

n-lreLe^ (,q

-

I

)

clenotcs

the

stage

of the pol¡'nomial qe) antl Ktl j :

: Ir2, ...,

e

-- 2, alo

prescr,ibcrl rrahtcs

ol functions I{¡:

KjØ;þ;_L),

(2)

l-¡G P. Chochoìat

L.G-3)

'¡l-3

i'

f",;"-1 *

I-llr Ii onal Appl'oxiutati orrs

ancl

(1.3) ¡

).,

R(1,)

:

ÌÌ,(t, Lþ,, Lþ,, . .

.,

l,ro_r)

=

'fhe fir.st

elifferences

from

valrres zr,;. $,e catì express

in the

form

x

À1

t

(tr,,

I

'1r,,,)

.rlto- r,

(tt'¡ |

Tr,¡,1

1':112,

At (2,1)

â1,r+r

-

à¡,

: :

l¡to) At,;

¿+I

az,i,

.l-| Áo_r,

rvhere

.Ðr- :

(I.Ít),

Llrit .., Lfl),j :

1, 2)

...,

e

_ 2, ale pÌesclibed

va_

Ines,

l-ff\ :Lt! +

/,,,,,,

,,,_1,"2,...,10t- 1, tir;rrO.

lVe llol

fol,rrrulalo

the

c'ol,ó

of i

(1'4)

ør,¡s-r

:

?",,"-r(f",0*_,

¡

&s,t T*rt)

for

s

: l,

2r. . .¡ (l

- 1

¿nd unknoìr,ns

a,r,e orerïrents

of the

vector.

?o: þnr, .'.'.,:rr'rl_i;'" ''

r'ì'¿r,)'

whe.e

To.,n-, Norv, suppose tha1,

holrls

Evirteruí for,r.: !.

Z. rár,

..":, riieü

i,9 _.

2,a.ndi, :

I

,2, ...rl;,.

rç,i!''j')'2i,' .=..,,h;,_.¿:'_,.r),

(2.1)

to

(2.8)

giv'

(2.4) ?t,rtn+1

- ? ht I'l¡-1 \ -

r 'r+P

:

,ì, (g

1('':

"¡')

d''"

"F :

7 ,

2, '

.

.,'li,

(ii p

=

Nox,c 0,

then rhar'tz.jl

(2.4) liecorn

lmliie_'

es (2.2t\.

iil,iålm*ing

resurr :

+I

As.i'a

t l',['_r',

* l,is

4

' nt.tq_7 or',

in

shot.t,

(2.2)

rn'hele the r,el*tion

(2.3) dt+t,t:

3r,r+t

-

âr,,

: t

hr

't",

tt;,

t

'l'2,rt

tt,rn,

I ?rj,

dr,¡, j 'Lb,i,.dr,ít

(1.5)

is jusl,ifiecl

for.

arr<l

for

b

:

2) S)

Ítr,rr-, : f

i.:1

(2.5)

'çvher;e

(2.8)

Q'l'tI¿,it: Cn(Ur,;,), d : 0, 1, . .. ,k, _ 7, s

:

1,

2, ...,

Q

- Ir.

r.vhere 2I:,

{

h,_,

_X,

#",.j" _,.4,,r,

. . .,

ll - l

rve tlenote

l,t,i¡t

l'rr|_1rt.

Untler the assnulltiol thal; /rrr 1

holcls k,1

:11,o6, fol j :2, ...

. .

.t

e

- 7, nr¡ : Ilit

â,rct tr,;,_r,

,.

z, B;_.-.,

i_ ,i iJie flre

'alues

of

ü,¡o,

io: \r2r.

.

., lo,

r,r,.,

.-

tl

. '

i{.r:ö f#åi:iì ,#É#l*åîi"",ffi

tion¿ll

l,esults ar,

2.'I"tItì

tt,{tNl rr-Ulit}ìtìICAL 0ONS.fItUCI.ION

'f'hus

the

suitable

appr"oxi

some

form of

(1.1), (1.2), (1.3), rve

gct

a s¡'stonr,of 7co

eclu

m (1.4), (1.5). Iror,one,

rlctermine the vsctol I'o uirde^r,

bhàt

q':'Clãr:'*jl"î,jiJijï

\\,e

û"kt) :

e(7

-

e|2)(I-y)i2y

Ì-Xence, insl,eacl

af

(2.1) .ive have

É, (þi

c"tu,,,,))

. l¡t

(11'8)

F,r"o,- r'(rt,îtp)' -

?¡,r,¡,)

: e,

,,

- Ir 1, ...,

À.0 _- r;,

_

1,

where the par,arneters n¿,,. ale

the

mentioned (mnltipliels,.

y)

l2y

(2.7)

Èr,,+.n+t

-

zt,t+p

- il,,¡¡

J?

.:1, 2, ...rlty

(3)

5B P. (lhocholal"ú Á

_Ilsi.ng

ïelåtions (2.2\

antl (2.7)

in

(2.8)

for

câ,ch i1, dr

:1r 2, ...,1c'

$.e obt&in irnmediatoly

the

ecluations

rt

is uo'w cleal

that

t hc acculact. unctel rvhich \re carl courptrte r,oobs

'ttr1,'11 'tt 1,2t . ' ' , 'Ht,t,

of tlrc

f ir's1. polr:norrrial ctlrrat

ion rì,

1z.r

depcnd;

e'qscntiallv or-i

tlte

âccut'atìy untteî"rnlhich \r,c càn evalutc

ilre

coefficienl,s

J, of

1¡¡1, ¡rul.t.rrontial.

\,\.esce,Iz,¡,,i.t:1,2,.-..,/.,ras¡lr.esultoTrelation(2.8) (for,r._1) is s-ell tlefinctl. : I,2, ...,l;:,

for. (1,1)

Tr-

c,ur' alc-- ,spccial' c,alculated.

cäsc

\\,as q

: 2 that

,r'r"a.rs ,,?,i,7 i1

_

ficl

.r'lri'e'Jr.\'r, rr

.cratioirs

(r.2)

antr (2.J),

i.c. iI

rve

[alrc

Ç

- s,

iL

i'ollorr,s

llral

1lrõ

cocfficicrl

nt¡,,.orì t.," r,orrrlirrl"il'try

rni,,ii,g Ilre

uonlincar, equzr,tion

5

(2.L7)

rrr,1

:

l{atîonal ,\¡rpr.oxirn atì ons

59

(2.e)

¡7

C"(u,,.,,)

:0

Now, usi's (2.6),'"--..1,';;

¿;

i; *i:'*t

(2.10)

þ:o

fJorrl,-r:s,

for r': Ir2, ...,

/ro

-

/i:r

-

1, u'here

J,

depenti.s

linearl;r oft

?t,6,rr tl!.r,

...

"'.;lrtr,¡

ancl tt,,¿, e (0t

1), it - I,zt ...,

/i;r. \'\¡e kuo'lv -{rom algebra

that

the coefficients

J, of

each polynomial equation

in

(2.10) can bc expressed

in telrns of the

loots.

In

praci.ice,

it is

possible

to

f

ind

explesnions

in the

forrn

(2.11)

frt,,,

:

Qo(u,,r, ,t6,zt . .

11,,t,\t

l':1,2) ...,,

l:o

- /i, - 1; l::7r2, ..., hr. Using equality (2.5) for

't{¡,¡,r'í1

: Ir2, ...rltr,

and appl¡'ing 1,lte

lrlopelty of

roots

of the

polyno-

urials

(2.10), (2.11) l'¡ecoures

(2.I2)

'trtt,r

:

'Y¡,(tr,r,1, 1trt,2, . .. , ?úr,¿,),

't'- 1rz, ...,lto - l't - 1; b:Ir2, ...,

/ir, rvhere t{1,;, 2ü'e

loots of

the

f

ilst

pol"1'nomial ecluation

in

(2.10). 'I.he roots of

tiris

polynornial ctluation can

lre

explessecl

irr telms t¡f lhe

coefficients

Jr,, i.c.

flo,t,trt,t¡ ...trùh,,t.

Tnlning to our

eclnations (2.12) we have

(2.13) flu,,:,

}u(no,r,tvt-,t-,

...,,ìtt,,t),

y:a¡21 ...:

/,:n

-

h,

- 1i It: I,2, ...,1ír.

Substitutìng

(2.13)

iir

(2.8) 'n'e obtain a,

linear

olelcletci'rniucrl sys- tern

(2.74)

r':l,2,...rlúo-

/i:1

-1, containing /,,r unlinorvn

pzrr':lmeter.s ht,t, hz,t, . . .t 1trh,,t.

Such

overclelelrninecl systern can be solvetl

in the

I;1 s€tìs€.

Rackground

material

on

Lt minimization

can be

fountl in

lJalrotlale ancl

Eobelts

[11.

\\¡e

plopose

to

solve s)'stem Q.AÐ

in

the leas1, sttruares sense

(2.r5) n

Áo -,fr,- /:1

1

2h(ù

{

m¡,))

(fr' -

(l.,r

f

lr¿,) (fr,,

+

lt{o,

!

nt¡,|)

(1, ,(1t,.

I

mr,\

I lf.)

((¿i,i

+

2/¿(o)X¿r,,

-f

2h(o\

j-

,tt-tu\

* ÍLl

to

he t,lì

,T

scnsc ltrrrlcr,

lhc

assrrnr¡rl irnt

lhat

i,, is

ccriirrq lcsull

s

n'c

ol¡tli lr llrc

lL¡r¡rror

folrn of

(1.1) antt (1.2).

Sinc attcntion to

t,hc appr.

/J(t) rvliti ).

Thelcfor." .uo ,rr'o'rt

same_appr

bcfole.

llhls

is, at the

possiblc 1

¿it,i,:

'f'rjr(lr,¿,, !t'r,

I,r\, 4 : l ,2, ...r

l;:r.

ll. l\lUllliltl{,rf

t.

ìiXt'ìÌtìt¡iru{.tì

k1

rtt,LÌ

i- L

,rur-n,,

þ-1

1r-t

(r{ r -

(lt ,r -l- l(o)

+ ne")(\¡ t

h1

),

t u,-r,r Zr,¡,(21,r1 ?tr-rrt

.,.tzr,,

t-0,

It) :0,

þ:o

,{r

T

,rn,-0,, Z,,p(4,r, ø7,t+tt,' . , 31,r +1ae)

þ:o

\\rc trv 1o

cletctmine pat'arnetet's tt¿,r,

b:I,2, ...,1,:r, florn tite

sysl'ern

(2.16) oD

=-= o.

0ntr,t

Oue

rlay

oJrt:riu

ilre

r 1i1i ttt o, ot' gnc t¡lt.,t' strl. t¡ll e

¿rs

lo

l'a vuur. 1 Irc ¡ìr'or,r,ss of

s('(¡uotìcc

of

/,n tncltsril.r,ltrr,ltts 1,,.. r

t'lor,k irtrlical cd 1 ¡t¡,c¡ 1r,¡,,, ¿rrrtl u

"'åi,n of

thc

functic¡rr -R(l) g'ir,ör Lrr.relatiol

llsitrg illc lcsulls

r>i'(9.

I0) I'ol i,, - 2,ll

u.c ¡arL ol¡1.¿¡ilr

fr.o¡r

(2.j())

tltc llc-rl

ior'rrrull,qr

i.e.

,ir,,¡,,

ii '.

I

,2i..., i.,

^r,"

,óot- .,i {be si,;ì,ì;;;É

2

(4)

60 lr. Chõcholaty fi

7

(Ìr;o

-

h,

- 1)

equaf,ions :

a")

hL:

2

(3.1) (1

-

rh,,)u1,,

+

(1

f

B+rr,,. _,ttz,,) M,.,

l

Jn,r,,

_

Ð

b) /rr:B

(nr,,

-

tur,,)u1,,

+ (f +

nr,,

_

'n,r.,

I nr,,)

tfi,,

! (3'2) + (1 -

itn,",,

-l

6tt.",,.

|

2rtr,,)n,.,,

# 6ru,r.,:

ç

lYlter,e

r :

7., Zt .

.., h, _

/i;,

_

1.

lìational A¡rp;'oxinr alions

Olvitrg 1o Llre t.eason

of llrc

sirn¡rlicity,

out.

<,h-nir¡ue r¡,ill lte pl,e$ekì_

lerl

rnials

fol jll íin: (J.l), (Z.ta)'bet,ornes'"

15 arrtl,/,',,-_ 2.

Applyi,ig

Ð

iir"

"-

plopo

of

loolñ

of

the

poìyrro_

(3.3)

?r.¡,r+r

t't'rr

=:

2",'-l

t4.,,

-

3

2't;;+;*. + (i r

'iù2'''¡t

-

-wÌrere

1':1,

2)

..., 11,

ol, exacilt.

'l'l' I't -

2n,r.,

-l

+tr,,

f

6

?'ablc Z

(3.4)

ot1.r

:

Br'tr.,

I

on,r,,

|

È) 1ì"(1). Ã"(/)

Ðnr,,

I

år.n,zt

l-

G ' I cxact - ,¡ì.(/) I / cxact - -Rr(r)¡

where

the'coefficients

B, ,

,/ ale

given

in

Tlalrle 1.

'l-uble I

1

2 .10

20 30

0.66666?

0.47ti:l9l 0. ít750c{) o. l5757tt 0 . t)87ri:t f) fi .06082t)

o. Ic0475_ 1íì

0 . 209507_ 1:i 0.229261 _73 0.104ii61-1ir 0.10t401_ i7 0.706125_ tf)

BAÐ81ïGÌtIJ

t).455i9J_14 0. B32tr(ì7_

t).16ir8,t0-i4 t).102ii60_ irì t). :to3:t56_ 13 Q.725122_2{)

1

2 3 5 (;

7

ll

70 7t 12

1

2

-2-1 _13

_.1(l--Jl -r;1-26 -97

o0

1 -:l 0 -rj - 1 -Ir 8 -24

- I ìf

8 - -.tJ1(ì -.1 ¡ -h.l

48

-tì0

- 21 -3:l -

fJO

-

120

-

99 -14:)

001010

:.1 1 (i _1 1. o

j :J l0 -t cl ,

325û01 'J4 1027216

20 15 2rJ 5 3 10

9712321-r

3l¡ 28 /¡í¡ 7 4 10 2.j

44 3cî 5ir 20 j 5 2S

18 ll-r 22 g 7 i2

r]:i 55 7u :ili 2åì 45

i7 drj û 1 4.,j 3(i t15

Iìratn,pl,e 2

+

0.1

l(t,; -

?r

/¿(o) -'o.J"',sorr,å

. F{elc ï.c consicle

,

al,

i.he 1õ erJì] efuuction

0.5/(tr,¿,-

1)*

r csnlts t,u,

¡r,fii' .ltijr"til

t;',

- 2,

rr,,

:

5, Tu|,!c J

t t,(t) clir(:[

- l?¿(1) -11tì

4 5 10 20

:JO

0.:ì5()000 0 .21t)(;67 0 .1ãu:i33 0.06uoãrì 0.031fJ71 t) .020¿,ì13

Now, \\'e

complete_onl' (]xpres-rion

b¡'

\\,orkiruJ for,rvartrs

in

(?.g)

(subsúitutio^

of

(8..4))

ard

obt¿r,in a Ii'e¡r,r, oi'elcleterminecl s],stem

nr,rl-,[2r,.

_l_

(I _

C)zr,,r,

+ ((] _ ]i,);y,rz*

Fcr.,*r]

*

(3.õ)

-l-

rlr'l-LLør,, + (l:{ _

B)z.,,*t

+ (})

__ ß))î.1,r+z

I Ezr,,*"]I + l-Jãr,

1- @

- D)2r,,*r+ (D _

G)21,,*21_

{ir;,,*rl :

g,

?'

:

1, 21 . ..1X2.

f).111:lij-9_i4 o.15(ì4iJO-- 14 0.595{)J0_15 o.24(i114_16 ().13269ii_ I 7 o.203859_10

i'

-

_.llhesc

lesults âïe

typi¡¿¡l r¡f oar ap¡,licrt

tr a rrliel.r.

of

ii;i'il,;i,i;i,,

å3(Í) rlclrcrrrls

o" lJic rlclu,,u,iiiìiiír,,"à

lt,¡., io

:

1r

2,

. .

.,

/,,0.. pr.ior,

to tt,o-.¡,äil,nr of

u.

j1

?)

,:

nse

iu conrputin¡

¿r best apploxirnal,ic¡n is

to

pr'cfrr. r'a1ior¿tr

to poiyno',iui;;;o;ir'atio',

or

possiirry

orre.

fo*nrs

(5)

62 P. Chocholatú

spline 'of the

K"....

...r

1{o-ror

L|),t L\tl,

. .

., Ltil

¡oo.

NEVUII Ð'ÀNÂI,YSE NUMERIQUE tì,I'rr}r.'TËIEONTD I}D T,,ÄPPROXJTTA?:ION Tc¡¡rei¿lT No, ¡;': l!Ìs4, ¡p. 6Û_6Ð

[ìEtìE]ìUNC]ts EEST APPROXII\{ATiON IN SPACES OF BOUNDED

VECTOR-VALUED SESUENCES

1. Barrotlale, I., lìobclts, F.D.tri.' An inrprouccl alqori|hm for tlisr.rel.e L, lÌttt'lr appro.tit¡1liu1, SIAIII J. Nnrncr. Ânal., t0 (1973), S39-S48.

2. -llovsl-atl, lì. ll., (,bnlinned fructiotr tails antl irrrdiotttili!¡1, l'hc Ììockr¡ ll,totnt.ail J. IIaÛr.

r0, 4 (t989), I035_I041.

3. Jactrlrsell, L., \\raarleìatrd, I I., .{?¡ etirJ,tlplot¡(' ¡tro¡tt:rl¡¡ for !uiIs oI lintiÍ periottic cotttinttctl

f rttcliotts,'l'hc ]ìoriiy ì{onntain J. l\fath. ¿0, I (1 l}gl)), l5J _ j63.

4. Jotrcs, \\r. ì1., I'holn, \V. .1., Co¡tlitttttt! fractiotts, nnaltjiie llteor¡¡ ltttl rt¡t¡:licnlions,ììrrcyclg- pcclia of :\'{¿rLlrclral-ics anrl its ÂppìicaLion. (Arìclir-on-\Vcslci,, lr.caclinu, J\.lassrrclrrrsctts,

1 $80 ).

5' Waarlclantl , Fl., I'octLI Properlies of con!inueil frut:liotrs, l,cct¡lrc Notes ill lTlath., 12117 (1fìS7)

239-25(i.

Ç. CoBZil$

(Cluj -- \la¡roca)

1. tNrROt)ufj.lÌ01\I

Iìcceiveel 1 \/II 1992

Ï-¡e

x

be ¡¡' norrnecr space

and y

fì,

_rìon-void subset

of -y. For

n e

x

pnt ct(r, y) : inf {ll;, _ i¡¡, y . r¡-_

rh¿

¿ista"cã'tr.åìo, to y,

anct ter

(1) Py(u):

{y

e

y :lla _

Ull

: rt(r, y)},

n z

is a surr,space

of 'y a.d y a'on-r,oicr

bo.naecr subset of

r

then

tlre

chebvsrrev'aclius

ot r'ì"itììì'J.rri.r to z is ã;ìil;iiry

(2)

racl(Y,

Z) :inf s¡p lly

-

:ez

),ey

the problem of bes1, tr,pplorirnation

3il;il""u

sequerlces rry-nrn,rrunis

ii.

";Jr¡:

¿-(¿l ilre

-Ftarracl. spacc of

1r t

. . .

j,

equ

Ípped wiûÌr

f he

(3)

ll ø ll

:

sup {ll r(rr,) }l : ?¿ € ry:,r,

lor

n e I*(-E).

D c ¡t u rlrn r.: rt t of Nr urr riic l I _{ rr a1¡7.s í.s

¿¡ttl OplíntizttIit¡n Ct¡nt:.¡tíus U i1 i Ders i t u I ) t, ¡t ! i sl t t utt

Mlyttsli(t tktliutt ifiS- 842 1r¡ Brnlisluu¡t

Sl¡nuli io

Referințe

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