$IÌtLrÌ
l|',AhÍrrt,rstìì\xðllittlrlülì trl. t,tr
l.nlì()1ìilìDlì
r,ltptnoxIMA'i'I{,N'i't¡r¡tr P,ilr ltio
l¡
lÍ191r, pp.55_(igA 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 tablesof
tlat,a $/ishesto
leclucethcm to a mole
conr¡enient a,nc[ conì]jlehensil¡lefolrn. rre
accornplisli;;tlrilfi;
rep_r'esenting
tlte tlata
I'nfuuctional form, Hc
selectsa
claisof functiolis
ancl chooses fronrthis
class tìre oncthat
bcst f its his clai,¿r,.,,*,..,,1,t]
1 :orìrtltj
presL"fllulcl\'^t,epr.cscnl
(,on_tinuous
functions.
classto
l:ethe lational
f Lrnctions, zr,ndthe¡' ¿r
effi_9-lg1-t,
torm of ap¡l'oximation. :\ ratiônai fulction ne) : p(|lg(t)
is -cnewtrìclì can
bc'clalu¿rtecl asthc
cluotientof tu'o
pol-vnomial.s.'üeft)+0,
t,hen
a
single clir.isìonlielcls the iìnal
r,csu1t._ Äctuall¡',
a r,athet' differ,cnt algot'ithnt clepenrÌine onthe timc
rcc¡uir.ecl for,-the mu r'ationsin par,ticulal computel bcing
usctl.det'ir.ctl b¡. 1,.'*,r.*tn¡rnirrg
(t) jlrto
¿r,còntilne
y€Íll's sorìì.e
theoÌctical
aspecl,sof
continuetlfractions
har-e been ¿isc¡s- 1gcli1
nìan)- pal)ers, T.c.[2], t3l. rf
the clivision anc-[lnultiplication up"ràtiolrs
tahe abou.tthc
sauro 1,irnein thc
conrputcr',then
Lhdr,eis a
tlec:-iclerladr-:rntagc
in
usìrrga
continuecl-fi'trct,icln cxpansion.'l.h-¡r puÌpose
oj this
papgr. isto
dcrnonitr¿ltc ¿-r, rnellrodfol evalll,ing tlte
coeTficientsof
functioìs-11(t) :(1.1)
å,f-ld
(r.2)
r(/) :
A
t1- IÇ
r+...
t* ,_nriË;
n-lreLe^ (,q
-
I)
clenotcsthe
stageof the pol¡'nomial qe) antl Ktl j :
: Ir2, ...,
e-- 2, alo
prescr,ibcrl rrahtcsol functions I{¡:
KjØ;þ;_L),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
elifferencesfrom
valrres zr,;. $,e catì expressin the
formx
À1t
(tr,,I
'1r,,,)
.rlto- r,
(tt'¡ |
Tr,¡,11':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,rrrulalothe
c'ol,óof i
(1'4)
ør,¡s-r:
?",,"-r(f",0*_,¡
&s,t T*rt)for
s: l,
2r. . .¡ (l- 1
¿nd unknoìr,nsa,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) liecornlmliie_'
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,rttt,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 tlenotel,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
ofü,¡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
suitableappr"oxi
someform of
(1.1), (1.2), (1.3), rvegct
a s¡'stonr,of 7coeclu
m (1.4), (1.5). Iror,one,rlctermine the vsctol I'o uirde^r,
bhàtq':'Clãr:'*jl"î,jiJijï
\\,eû"kt) :
e(7-
e|2)(I-y)i2y
Ì-Xence, insl,eacl
af
(2.1) .ive haveÉ, (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
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
ecluationsrt
is uo'w clealthat
t hc acculact. unctel rvhich \re carl courptrte r,oobs'ttr1,'11 'tt 1,2t . ' ' , 'Ht,t,
of tlrc
f ir's1. polr:norrrial ctlrration rì,
1z.ro¡
depcnd;e'qscntiallv or-i
tlte
âccut'atìy untteî"rnlhich \r,c càn evalutcilre
coefficienl,sJ, 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,
iLi'ollorr,s
llral
1lrõcocfficicrl
nt¡,,.orì t.," r,orrrlirrl"il'tryrni,,ii,g Ilre
uonlincar, equzr,tion5
(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:ofJorrl,-r:s,
for r': Ir2, ...,
/ro-
/i:r-
1, u'hereJ,
depenti.slinearl;r oft
?t,6,rr tl!.r,...
"'.;lrtr,¡
ancl tt,,¿, e (0t1), it - I,zt ...,
/i;r. \'\¡e kuo'lv -{rom algebrathat
the coefficientsJ, of
each polynomial equationin
(2.10) can bc expressedin telrns of the
loots.In
praci.ice,it is
possibleto
find
explesnionsin the
forrn(2.11)
frt,,,:
Qo(u,,r, ,t6,zt . ..¡
11,,t,\tl':1,2) ...,,
l:o- /i, - 1; l::7r2, ..., hr. Using equality (2.5) for
't{¡,¡,r'í1
: Ir2, ...rltr,
and appl¡'ing 1,ltelrlopelty of
rootsof 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'eloots of
thef
ilst
pol"1'nomial ecluationin
(2.10). 'I.he roots oftiris
polynornial ctluation canlre
explesseclirr telms t¡f lhe
coefficientsJr,, 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 solvetlin the
I;1 s€tìs€.Rackground
material
onLt minimization
can befountl in
lJalrotlale anclEobelts
[11.\\¡e
ploposeto
solve s)'stem Q.AÐin
the leas1, sttruares sense(2.r5) n
Áo -,fr,- /:1)ì
12h(ù
{
m¡,))(fr' -
(l.,rf
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 irntlhat
i,, isccriirrq lcsull
s
n'col¡tli lr llrc
lL¡r¡rrorfolrn of
(1.1) antt (1.2).Sinc attcntion to
t,hc appr./J(t) rvliti ).
Thelcfor." .uo ,rr'o'rtsame_appr
bcfole.llhls
is, at thepossiblc 1
¿it,i,:
'f'rjr(lr,¿,, !t'r,I,r\, 4 : l ,2, ...r
l;:r.ll. l\lUllliltl{,rf
t.
ìiXt'ìÌtìt¡iru{.tìk1
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:riuilre
r 1i1i ttt o, ot' gnc t¡lt.,t' strl. t¡ll e¿rs
lo
l'a vuur. 1 Irc ¡ìr'or,r,ss ofs('(¡uotìcc
of
/,n tncltsril.r,ltrr,ltts 1,,.. rt'lor,k irtrlical cd 1 ¡t¡,c¡ 1r,¡,,, ¿rrrtl u
"'åi,n of
thc
functic¡rr -R(l) g'ir,ör Lrr.relatiolllsitrg illc lcsulls
r>i'(9.I0) I'ol i,, - 2,ll
u.c ¡arL ol¡1.¿¡ilrfr.o¡r
(2.j())tltc llc-rl
ior'rrrull,qri.e.
,ir,,¡,,ii '.
I
,2i..., i.,
^r,"
,óot- .,i {be si,;ì,ì;;;É
2
60 lr. Chõcholaty fi
7
(Ìr;o
-
h,- 1)
equaf,ions :a")
hL:
2(3.1) (1
-
rh,,)u1,,+
(1f
B+rr,,. _,ttz,,) M,.,l
Jn,r,,_
Ðb) /rr:B
(nr,,
-
tur,,)u1,,+ (f +
nr,,_
'n,r.,
I nr,,)
tfi,,! (3'2) + (1 -
itn,",,-l
6tt.",,.|
2rtr,,)n,.,,# 6ru,r.,:
çlYlter,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
rnialsfol jll íin: (J.l), (Z.ta)'bet,ornes'"
15 arrtl,/,',,-_ 2.Applyi,ig
Ðiir"
"-plopo
ofloolñ
ofthe
poìyrro_(3.3)
?r.¡,r+rt't'rr
=:2",'-l
t4.,,-
32'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,ztl-
G ' I cxact - ,¡ì.(/) I / cxact - -Rr(r)¡where
the'coefficients
B, ,,/ ale
givenin
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_ 1ã
t).16ir8,t0-i4 t).102ii60_ irì t). :to3:t56_ 13 Q.725122_2{)
1
2 3 5 (;
7
ll
t¡
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
+
0.1l(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 Jt 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-rionb¡'
\\,orkiruJ for,rvartrsin
(?.g)(subsúitutio^
of
(8..4))ard
obt¿r,in a Ii'e¡r,r, oi'elcleterminecl s],stemnr,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+zI 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'
-
_.llhesclesults âïe
typi¡¿¡l r¡f oar ap¡,licrttr a rrliel.r.
ofii;i'il,;i,i;i,,
å3(Í) rlclrcrrrls
o" lJic rlclu,,u,iiiìiiír,,"à
lt,¡., io
:
1r2,
. ..,
/,,0.. pr.ior,to tt,o-.¡,äil,nr of
u.j1
?),:
nseiu conrputin¡
¿r best apploxirnal,ic¡n isto
pr'cfrr. r'a1ior¿trto poiyno',iui;;;o;ir'atio',
or
possiirryorre.
fo*nrs62 P. Chocholatú
spline 'of the
K"....
...r
1{o-rorL|),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 spaceand y
fì,_rìon-void subset
of -y. For
n ex
pnt ct(r, y) : inf {ll;, _ i¡¡, y . r¡-_
rh¿¿ista"cã'tr.åìo, to y,
anct ter(1) Py(u):
{y
ey :lla _
Ull: rt(r, y)},
n z
is a surr,spaceof 'y a.d y a'on-r,oicr
bo.naecr subset ofr
thentlre
chebvsrrev'acliusot r'ì"itììì'J.rri.r to z is ã;ìil;iiry
(2)
racl(Y,Z) :inf s¡p lly
- z¡
:ez
),eythe problem of bes1, tr,pplorirnation
3il;il""u
sequerlces rry-nrn,rrunisii.
";Jr¡:
¿-(¿l ilre
-Ftarracl. spacc of1r t
z¡
. . .j,
equÍpped 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