View of Spline approximation with preservation of moments

ti.\]'t i¡t_r,i.\'_t'tC,\ tìIi\.tr'tì T),-\¡--,,t.\ Siì \,.. \IUtìt()r l.l

tl.t t)Iì .ftIIiORItì l)Iì i,,1\l)pt,ro-\ItI.\i.rôl: '-

L'ANÅT,}'SJì Ì\{,ìì[I.JIITOT]E

ET I,A'T'ÍIIìOXÈil] T}E

T,'AI'TìROXIß,IATION 'I¡¡rne _{19, No}

2,

,tÍì90,

pp. llf ^__lel

SPLINIi API?ROXINIATION \4/ITH PRESERVATI _ON

oF MOl\{iìt{f's

P.ti'I'IìLi tlLr\tìr\

(Clu j-Napoca)

ximating

¿ùn integr;ablc fu_lrc_

ion is considerecl. ,ttre spiine clegree ancl va,riablc _kñots.

11. 'l.he spline apploxirnation

ii""'_ii,ïi'äi¿? åi'üL îîäilïn

^?i

We consider. bhe

namely

first _l,hen

_the

ment functionals

_ancl

,s,,,, be a

pol¡.¡el¡ial

_{spline of degree}

r {.. .{t,,{

1

andthe

correspond._

.,..,1',,)7

rv< ln

_l-

l,

v

: l,

l¿, The

the

lorm

(1.1)

_s,,^(t)

:

p_(t) ^_j_

)ì

"n' n,,,.(t,,_ t),!_t"

11'J¡s¡'1r

.7r,,,(t)

is a polvnoilrial o,

Ua*rn"

,r.'

l'tris foun ís equi,,rl"ni .uliË^ilìc

folton.iug

(L.z)

s,,,,(/) _{=_}

þ,,(t).li

,¡i"",,ij;î ^{e, _}

t),i':.-.,.,

rvhel'c

(r,,,',

-

,ttù[p.Jct,r.,

t,tui.:

nt(t_tt,

_ ll

^.

: :

\nr,._ V_l

^.l

). ^:\lso, for

i;he polynorrrizt,l p,,,(l) orrõ co¡rsid"rl"ìT,u lc-1n,eserr1 ation

þ,,(t) : ^{i,

pirnrisçr

-

^t¡,,

i:0

712 Petru Blaga

,

S¡rli ne a¡rpr-o_xima.tion

113

-!Yith

ancl also I

-1\,

l3r

:

⁾

,,t,.' P',','(r),

'i:0,

m'

Celtainly,

^¿ì, spline Tuncl,ion

in the forrn

(1.1) or' (1.2)

is

co_rnpletely cletsrminated. ^'vrdren

the

coetficients nur.

or

c¿.,u, ?,,¡(ú) ol- _9¿r

and the

hnots f,r,

'

Tho problem v

.: 1,

??,,

ât'c is knol'n. to constrttct a sltline lunction s',*(f)

^so

that

,åP'[#(r""''')],

,

,, 'u-L

-l-

_.¡=-

I

_I

T,

^ou,,

tr.,-0

d$ l,t't!rI (2.1) dúu

(nt,

j- j

_-l-

I _)! î

JflJ(r)dú,

J:0,

j

_lnzl ^-l-

ⁿ .l _rt.

(1.3) f's.,,n(t)df

: trl!)ùt, j :0, N -l

^n,

!

^{n't,, One clenol,es}

where/is

an inl,egrable function on the

inl,elval

[0r

,l]'

ancl -¿Y:?'l_|

fn otliel

ìyor;d.s, bhe splino

function

s,,-(f) must lcproduce all the of the function

/

of order less or eclua,l

to

ltr

-l- n !

^nt,.

"' *r"'

Ìnoments

8¡

--

,)¡("/)

:

^{(rn, -l-. .j}

^-i-

^{1) |}

_{fr( t)dt,} j

^t'rt, !

2. Solution by morncnt funcfiounls. Taking into accounl

^1,hat

and

then

one clefines a, rinoar,fu.nctional

g

_{ott tl.te set}

_of

porynornials _of

flre form t,1-L.p(t), p I

g¡r+,,r,,j,,,-b;^""

tr:- Ú .1

- lm-li ff¡r""1

^clf

^q'*-

^{''+1 ¡¿rr.ii1)}

9(7'n'rt. úr)

: ^g¡, j : oJl-1¡ ¡ *;

using this linea' fu'ctic _nar

_{onc crefines rhe. iJlner plocruct}

for a'y

poly_

nomiars

p

anct

q w*tr p. q-. Ø;,;_;ìî ril;i;ilä,ü''iiìur""

(2.2) _(p,

_lÐ

:

St(tilr+t(7.

-

t),,,F\t(t)r1ft)).

One con¡riclers.(if

.it ^cxists)

^{Lhc rnonic} polynornial

ov :

ô,v(. ;

g\

_oI

*Stråi -"" i'

f i',î 9,?"ål

-ïåf ,ffi

f

ï ^i::ii

ⁱ

^{; il'õî} i,il¿i,lti _i;'

^r

_¡

tã'" t ^r i

i

^ár,::

co"(l)

: r" -l-... and (o¡,

g)

:0, _for ali

_Ç e U,_t.

(r:J) il;::;,,':;,:,:' ,:"#:;fï2,.iî::#

ort'ttos _at ii: ;;i l';Ui"' ,rtrt,

zaro,s^,tlÙ), v

:'L,Ç,

wi,ttt tlte"ruiiìpU",i,íles rv, v

< ,Í") < I.

Moreoaer,

in tltis

oäso

we ha,ae tv:

tlo), v

:

1,

-n.

tlw

cocJficients pt onå, n",""äiä anct

"ü¿irtäa

d,etet.rninatett from,r¿e cond,it.íon

and

using l,ìrc generalizscl

forrnula of the inlegration by parts

rve have

1

!

rra,,,(r)dr

- r:îrt)l,8"

^r,,

[# (r''''n')],,

with the ,Olo"

specified Êr,

go

1,he o1,her,hand, using the

Euler function of the

firs1, hincl

it

^results

that

(t,,

- ^t)T-þtldt :

tu

On f,he basis

of

^l,hese

^relations

we have

l,hat (1.3) is equivalent to

wltore (2.31

9oçt"+r'

pþD -

^S(t,-,.t.

pe)), ?

^{e Øu+,n,}

9oØ):

.fi

p,nt,,,-Ð(r)

+ _X ^'þt

^o,*gn r1tr"¡.

.;:o

_{y?t u_o}

Proof.

_{Using (2.8),} equations (2.1) _can

be

^.u.ritten

in the

form

njlTi'-rnå

^{u' [}

#

^(ü'+r+i¡

^{],:, *}

+ q'¡.--i-tlOf- ú,utl-Pi!'II :

0

ttl?)dt, goltu+r.

_úr)

:

^gr,

j :0, _n{m,

or oquivalentiy

12.4)

j:0, N+.nJ-rn'

9o(t'"+t'

p$)) :

s(yna1.

p(t)), p

€ Øu+nt_*,

714 Petru Blagn

rìr'on

1,he exi.qi,ence a,rìd

ilre

urriclucness

of

1he spline apploxirnation

(],'2) drich

satislies

conditions

(1.3) rvc l'.¿r,çet,lrat tlie,

iineài f¡nction¿ll

-5lo is

l'ell

deÏinecl anrl (1,3) is ecluir.aleirl,

u'ith

(2.4). Lrsiuu the

¡nots

oT 1,he

spline functions

(1.2)

onó

corisicleLs 1trc,

poì¡.rórnjal '

autl b¡, (2.ir.)

ro*(l)

: ÌÌ ^(l -

^¿')"'

(con,, Ø)

- :/!(ttn+r(I-

l)"'ntor,*(l)q(¿))

:

go(tu'tt(1

-

i1),,,-1c0,.,(i)q(l)),

for an¡' polvnornial

_Çe

Ø,,. 'faliing irrto

accoulrt

the definition of

ilrr¿

Junctional

9o it

resu.tls

ttrat

(c,r,i.,

Ç)':

^O.

lti* p.o....,,1r"

";¡;;rlij-;f ^iü;

conclititln'

'ler'a

€gt-.¡n+n,.rirhen*,e

'ave

l(

rnoni ^, rvith

^l? _{o' ,,,,,} z\f)tl ^c,-ro,

. tl'Le

(2

2) q,Ë,i' LîJüïi-åîlx*:

¿f*)1

: I, u:\t

n'.

nt 0

<¿Í¡/)

<...<lt^') ç1, rviilr tho lnull,iplicities r.,, v.:

One

rvlites

succesively

9(t''+t'pQÐ:

^gÌft*+r(7

-

t)ttt'vtr,,,"rr),q(¿))

l- g¡¿,,r,r(ú)) : g(¡,nrt.f(f)).

c)n

the

othr¡r' hancl

one

^conËi Lers

go

_{definecl þ.1, (2.8)}

rvith tu:

-

^ú(n),

^y..- l,

tr,,

:ttttl pr, i .-

O,

rtt,

u-tp.¡ u --.1., ^It

t ! ^0,

^t,,

-1.

^lrnlc_

norvns, tJren

907t"+r '

?(t)) --

S'o(ltn+r(7

-

t¡,,'' 1co,r(/)q(f

)) | I

9o11,"''t

'/(¿)) :

9o1t''+r, r(¿)).

'llhe coefficieni,s _13¡ &ncl ø.,,, are uniquel). cleterminecl e"s

solution of

the li_

near SJ¡si,em

IoTt''+t' li) : g(t"11' tr)t i : ^{0, N} +

^nt.

which

has

a

generalized \randcrlnond.e cletc.r.rnin¿nt.

ins

ss-r'orrartq'¡

,$:'i[ffi1îi-J.,t#-ltl: å:Tr"x

gìene

hc corresitonclinþ.generaliz_

ecl

G

_onsirters

îlat,7;?;;itlõ,ll

rL,ncl

the values "f,,,,(I),

^tt,

: $, ¡,r,

are known.

,,,.^.^^

Ij.lhll^9]'"1?ljzncl

'inLeglation

b¡'

par,ts ior,muta is alrpìioti f o intcgrat

rloln tlìc liglrt sidc in (2.t) if r,,'slllis

thni

Spl iuc ir¡tpr.oxinrr tion

If

one clefines

flrc

rne¡a¡iul.c

c[7,,,,(l)

: L-r]I't

¡rnL',u(t)dt

nt, I

on

f0,1

],

^l,ltcrr

1he't¡latio^s

_(13.t) ârc.tJuir,_Ítlcnt,

to

(3.2)

gn1r,,:-t.

p(t)) :

^g¡¿,,.,r. p(t)),

\\'lrere ₇₎ e Ø,u*,,,,,, _arì{1

go _is

_{defiuerl as}

_{in ilre}

plevicius section :rnr.l

4

175

."Ø) .

j,"','íill'"''

"(l ) !

i-t)

(-

I r,¡( ^¿ )t1 J

À,,(

)ì

^p,

í:o

J

l#(ú"'*'*')],:, t _[ -.rt::(i'"r'i¡I

,, ^/ u--1

-l-

_):1

;:,

_p.-

x

^ou,,

0

l-.r$ r,'''"'tJ

Let

_{( ', '}

)

^tlenol-e the itrneÌ procluct, clcfined _b,y rneans cif

the f.nctional

_{-? :}

(p, q) : ^{g(t'' rrli}

-.- t),,'+1,p(t)q(t)):

î

I

- \

^{r. rr(}

| _

t),,,It1t([)r1Q,)d À,,(¿).

J

(if il exìsts)ilrc

rnonic polynomi:r,t ô¡¡

:

co¡v(. ;

g¡

_otclegree

i'i,t'cspcci

¿o

ilulc'p'odrLet (.,.)

1o

aìl

poll,nonrìals _of 1,h,,n

,,, i.c.

ô,u11¡

: _,i. -i

. .

., ìirí

_iä^1.

_q) : 0, for all Air

anaLogous t'heorem rvil,h thoolcin frorn the prel,iclus sec1ion holtls.

goçt,-rt .

pØ) _ 9,,rt(t,*r.p(t)), ?

^{e Øw+,,.}

e

proof of tjreolein

2.1.

the

solution

of our

pro-

-L,obatto

cluaclraturô or drature.

, "obatto

quadratrue formula

(3'3)

,\u(r)tr

),,,,(L)

:

!,1,r¡r(¿)(0) ¡ n,sr)(t¡l_¡

F ):_'þi

Àg'j)eru.r1f,,rr

+ ^nf,ì,,et;

^ctÀ,,),

-

_í-.o^¿'

(3.1)

f ^(nt+t) (t)ït+I+i

tlt,

:=

0,

.lV

{

^tr,

!

^{n't., ^,,,}

_- ^(-)u

^{{,r)(l), t.}

_-.0t

^ry¿.

6 Spline erpploximzliion 1t7

1n

-where _c1),,,, is

the

a̡ove

specified : l'

^{n'-' âl'e 1'he zel'os}

of the

monic polynomiaf

ô¡r (iÎ ^the

^{i'eai zeros}

ⁱⁿ

^l'he

i,ìr,"*r,itoi¡i,

^-tí.'e rernaindeì

t ^atio*

Rtiinißi dÀ'')

: o'

^{rvìren q e} Ø'v."*z"tt

This quadral,rue fornrula ^is relatecl to t,lre follorving,gener.alizecl ^Garrss-

Christoffel quadratule

^formula

I

(3.4) [ot,loo,,(l) : É

^"Er o{,1)returi?flr¡

* ^nfl)(v; tlo"')'

J v-'l t¿ ⁰

rvhere cl o,,,(ü)o

- r*+t(l-

f)"'+1c17',,,(ü),1'he nocl-ep îç'nl,

' =.1 ^:i'are

the

same as

in

Gauss-I-.,o¡attò

fo"tí"fo,'^íiä'i,tt"

^rveights

^fr'ôm the t*o

quaclrat*re

formulae

are relatecl ^bY

oÍil,

: ""do-'^,,r*,(-

î ^t) [r"*'{r - ')""'']l1rl*,,

, : {rÁ;

¡r

: 0J-- ^{1. llhe}

renrainder

term

satisfies

the relatio' nïr(g i

^do,o)

: 0, when ll

e Ø¡.t+,,-t'

atrt,:ont';lr ^3-2.

l.f tltc

cond'it'ion's

of

'l'heo.t'e'nt'

I'l ^{'. ut'e}

sutisJicd' ^then' me

s1ä'i'i"¡u,icrio,, (i ll

^{whictt, sotaes} ihe proltlem

is

^{git:cn' bu}

Íu:

^Î1"),

^øu,r:

^À(,f;),

':Lr-'tl, P':0' ^t"' -lt

and,

þ¿:

^Bur-r

* \,

^'i'

:

O¡ ^rtu¡

116

ulr'ere ^t\,N), tura (or the

od,

Petru ^Blaga

of

^th,e

¿.+?(¿)d ^À*(¿),

which is

equivalent,

to

[#'*.*',],-,

^{I \- \-} t A /;t ^{l¿f\' 1 ì(N)}"ut"

If, tt^''""J,:,,*' :

v -1 ^P==0

tú

2ß,

'-0

fn'þ1'r-idÀ1?¿(f ₎

, j :0, À. l-

^tt,

)-

^nt,

If

one conìllalcs these relal,ions

to

relations (3,1)

it resultsthe

asseltion

of the

theorem.

Lletnq,rh.If

/

is completelv'monotonic

on [0,1],

^ancl

the multiplities

li¡, v:1,

n,, a,re ocld posit,irre integers

then the

rneasure dÀ,,, ancl also

do.

ãre

positive

lne'Àsr.u'e.

In this

case

the

existence

and the

uniqueness

of the

generalizetl Gauss-Cluistoffel

quadlature

A,te ensut'ed,

and

so the

corrcspontling

Gauss-f:obatto

qlaclrature

_f4,5].

/r. Error of

.spline

l¡r¡rroxirnntior. Fur{her: on

1,he

crror term

t,he

spline apploximation folrnul

l@) :

s,,,n(rr)

I

^e,,,(nl

is

stucliecl.

the euor tern

€,.,,,,(,?)

is

expressed. inrespecb

to

the rem¿¡incler

terms

of

the

genera'1ized- Gauss-Lob¿r,tto quaclrature and Gauss-Cluistoffel quach'ature.

'llrruon,lirtt

4.1. If

tlLc contlí,,ti.otts

of

Thaorent, 3.1, are sati.fiail, th,en,

ot' ^(rrù'!t n e (0,

1) ue

lt,s,ua tl¿,at

e,,,,(n) ₌₌

Il,tI

^(p,,.; rlÀ,n)

:

-Ilfiturlo-; do,,),

uhere ^1ì\1)'),, u,ntl,

Ilï,

^a,ra respecti,aal,t¡ tlt,e ¡'etnrti,nd,ar ternls ^,in, gcneru,li,aod,Gauss- -Lobtttl,o r¡uørllature (3.3) anil qaneral,,[,zarl Gcntss-Ch,ri,stofJet,rlttu,tlra,ture

(3.4),

ull,d,

u

-T:'n,

^{at'e the nodcs}

t¿ o de s' of ^' c orr ^e sP ond'tl ^t' g

G aus s- L ob a't to ^{qu atl,r u-}

-H

^{(p r)}

Gu,ws s- Clt'ristoff eI guad'ratwr e), ßÍ1

i ^:-O,

^{n"¿, and}

çoutt-7sbcúto

ÀtT), u

: l¡

^trt [r

:

0, ru rluadrcrtute.

L,at"o the ueigh,ts in çleneraliø- Proof

.

^{One 1,ahes}

in

^{Gauss-T-,obatto}

quadlature

^formula

g(t)

:

t'n+lp(t), ^{yt e}

Ø**,u,, then nïàØ i

^dÀ,o)

: 0,

and' so

that

,ur+1(l _ _{f¡,,_,,t}

w'ith, IIr,,,*, tl¿e .Hertn'ite, i.tttcrpolat.ittQ ltoh¡n,on't'ial relo,t,iue

to

the fu,ttcl'iou, p, ctnd tlte nocles () and, 1 toi,tlt, tlt,e sftilte n1, -l-

1

m,u,ltiplicity.

Proof

. lly 'I'u'lorts

1ot'ln¿lir

ttc

h¿lr-e p,,(ü)

: (I -- n)\,

^o.,(l)

1

t

)"tt

,Ð.r'[$ ^tt'.'nt'll

^- _]:t

I ^+l",

_{v:l lr:o}^{,t 'r-1}

^ÐÀ!il) [ff tr"'r(t))],:;,*, t

l

/(",) :

)_, ./,,,)(1)(ø

_

_{1),'-l_ (t --} t:)tlJ'tttt ^F1)(¿)i1¿ :

Á-=0 I't _nt,!

/o)1r¡i.u

- ^rl,-i-(o

_J

t ):

1 It, ^t

l,n (

,(t c1

118 Petru Blnga Ot-t

thc other hand it is

hnotr.n

that

8 I _Spline apploximzrtion ₁₁₉

ll'able ir.l-

coLrtains

the elemelts of

^1,hc

cluadratic

spline fu.¡ction n'il,h a sinstre

knot

r, of

ilre murtiplicit¡-

_3,

l-hich'i* -ü"i:tt."

in ilre

form stÐ,r(t)

:2þo

_F

2gr(1 ,) f ^pr(t _

¿),

+

.l-

_oro(rt,

- ^t)i +

^2urr(t,

_

_ú),

*

^2arr(t,

_

_¿)i.

,l,al¡le 5.1

f (t) _{J, I1}

)T

sr',rr(r)

: _X

À:0

p.,(f)tl),,,,(ú) _]

p.,,(ú)c1À,,,(l)

-

--* 1 ¹⁾

l¡!'

I

l,-ì

I

,1"_

dtt'

#u,,,-

^P*X1

^--

^c))'

-Í, ,tf

?,t1)')p(Í)(1{^')).

,, ^t r-7

*)(r)(n -

^I

^{)' -l-}

_v=.1 "1=ì

)i

^7,f,i)

iJ.:0

r

I

I (t

-

^{m)'1, (^')}

ancl so

that,

e, _,n(ã)) =-

+å

llul,

9,,

-

^"(,,

: ]]r,

t"t

arrcl t'helefole

plt)(0)

:0,

anrl

o.4í242(+0) 0.45235(

-

¹⁾

0 46139( 2)

0 .1 E404( i ⁰⁾

0 .18(r{j4 I ⁰⁾

0 .22598( +0) o.4oe:ì7( i ^o)

o .81 824(

i)

0.17034(-1) 0.:ì0:ì2tÌ(

i

^c)

0.15j0lì( |0) 0.83810( 1)

0.28ee7(-3)

-

^{0 .43} 0.30202( -5)⁹³⁰⁽

-

⁶⁾

0 2208s(-2)

-

0.6(js88(. ã)

0.22et)2(-4)

e ¹⁰

1-o

O.497 t-27 0.30427(--5)

o .49ó4óó 0 2:ì:ì:ll( - ^4.)

0.488G41 0.32164(-:))

pÍi)(1)

- e;!)¡, ^(r -

^t:)n,-r;,

2

0 29[Jã - tì D.) rii 0.3100 6(

4(

î(

-1) 3)

-3) c

_,r(r)

1

0

c

cr

- ^$, tr,,ogt{nl |

7l¡ plr,)(rll

.- ^,rl "i]

rtll.,plj,)(îg.)) =:

ir,..0 ulli ^{p- o}

-.

^{ntl,l,( p", 1 cl),,,,).}

If one

clenot,es h,(t)

: 9,Q)

^lI r,,1r( ?,,

) [), ttr"r,

¡,tt;)(0)

.:

/¿(/t(1)

: :0¡

ltt

-:0¡

^tt¿t

rvc can

^g'r'ite

(4.1)

R!,1,,),(tr 1

rl),u)

Rj,''

/ /,g)- - _I

,l o,,,

ì.

\ ¿,,r(l_ ¿),/r "j

illa,'blcs

i.2

^a,nrl õ.8 contain the c.

xirnal-ion Jiuncl

ion ri'ith

t,n,o

tinots I _{: iTiT;}

1,he

mriltipìicity 3

anct

the kirot l, iii _is

_{sin_}

pl¡'. and

f,

has

the rnultiplicjt.l-

3-.

Il

rct,ioir is

'lvritten in the

for.rn

srr,,,r(t) - ^2(tu

)_ 2þ,(7

_

¿)

_i

pr(1

_

¿)' i

--l-

*,o(tl -- f)l -i- a.,,(ú, t),

^..1,

ur,(\ _

_f)g

+

^{u,0(t2 __ t)2,}

anil in thc

seconcl

srr,,,(f) -

²⁹⁰

+

^{2,ßr(1 __}

^¿)

l_ Þr(1

_

_¿), _l

l-

^oro(ü,

- ^t)ì +

^azo(lz

-,l)i +

a"1(t,

_ _t)a i

^ozz(tz

_ _l)i.

fn the

1,hlee t,al¡les

l,e

clcnotect

p.: (go, g,,

(Jz)t,

tlt :

_{sup { lJ(¿)}

_ -f

^,z(¿)

^{i-, o <} ,

" ^tì,,

1ììrd _¿ìllo

in

T¿rirle

¡.r."rve',rãíJtn¿

o" -='çà',"j'o-rr, ø.,r),

n'lrilc in

î¿-¡blcs,J.2

and

ij.B

a.: ^(qß,

s"tt,

etz,

azo)u+rcspeclir)c,iî

"'-:

: (o-to, 'J2g1 7.21, r.r.,)'.

;\11 Dulnelical experiments ltac[ becn e'lfoctuated

in

double

il'ecision on the Iìomanian compulcr COIÌAL

40110.

c*21'

0 . 6ri04rj( -- ¹ ) 0.1229:l( i0) 0 417rì;l( l ⁰⁾

0.50026(,r 0) 0.7800rj(-2)

-

^{() 11ã7o( I 1)}

0.1 80õ4( I ^o)

0 .28:ì 94( - 2) 0.1fr51r-r(-2) 0.09067(+0) 0.28710(- 1)

0.98it68(-2) 0.sl04e( i ⁰⁾ ().17189( ₁₎

o .7544:t( 2)

0.477308 o 20065(-2)

0 .4õ4322 0. r1307(-1)

2 0.4 (;0812 r) .8.1100(- 2)

4

IJui, rve have

tha,t

J?Í;lÌ,(p,,,; cl).,,)

.-

Llii),Ì,( Q,,

- fl,,,,t,,(p"; t);

ri7,.,,,).- (4.2)

=- Itl,\'),(h

;

^{cl 7,,,,).}

Iù'on

(4,1) ancl (a.2)

it

^results

that

e,,u,(ü) -= 7?!'))( o.

;

^{cl o,,,).}

5.

Nnmol'icttÌ

ertuttllles. \Ye

ha,r,e cor-rsiclere¡cl

thc

s¿une exarnples

frorn fil],

^{namely 1,hc} ex1ìorrentinl

furrclion

e-"¿,

c> 0,

anc'l l,r,igonomeilic

f

nrcl,ion

sir, ^1¿-

, ^{t e}

[0,

1]. rn all

^ca,ses quaclrat,io

splirre

appr,oxìmation

is

presented.

2'.

120 Pett.n Iìlaga ₁₀

f1 _Spllnc appl'oxilnatìon

721 T'able s.2

f(r) REFIJIìI]NCES

OP d

(4

I,)t

^'I o.45252(_lo)

0.4223e(-7)

o .45832(-2)

o.12207(-3)

lo sphcricallg sgmntelric tlistributiotts, Numer,

, ^Sp.Iine apprcl,irnations _Io splrcricrill¡ slJnlmelric ,19.y i ð-, G. Y., trIontettt_preseruing spline

õ0(1987),503_518.

,i _a

IoLntnlei dc inlegrare nun.tericít ct luiGauss, 2S- 57.

néral¿s tlc quulrature duIgpe Gctuss_Ctttístoffe l, .alur.e

Iorntulus uitlt multiple Gttttssiun ttod¡:s,

9- ^{1 .lJ.}

qtnctrcdure, r\cta Sci. Nlnilr. (Szcgect), l::(10b0)

e 0.22762(

o.15402(

-3) o.744827

^0.360102 0.14920(_5) 0.1:197

-5

0 0 0

7(t-0) 1) 1)

0.91

-lt)

c" ^B1 852(- O .17Ð24( --2)

0.70192(-4) 0.70727(-4)

0 .3581134 0.7433:-¡2

I 0807(-- o 11377( _4)

a2

0.30327( I ⁰⁾

0 .151 3 7( -F 0) 0 ^. 81 008(- 1)

0.11531(-1) 0.24460(-1) o.?3472(-4)

0 . 14781(- Ít)

0 .3530û5

0 . 73 B¡l9fì 0 .15417(-:t) 0.18397(+0)

0.18255(-1 0) 0 .21075(-1, c)

0 0 0 0,

Iìeccivr:d 10 III.19g0

Uttiue t silg oI CIuj-.Jtrt¡xtnt I;acullg o[ ÌIathernalics

3'100 Clttj-Na¡toca Ilontr'i¡tict 0.344405

0.731303 _0. e7705(-3)

c-21

0 .67768(

-7)

0.13037(+0) 0.30066(+ 0)

0.26s71( Ì-0) 0.93618(+0)

-

0.924:Ì5(-2) 0.s2r02( -2)

0 .32 7560

0.71551r 0.54515(-2)

rct sì ll-

2

f) 50006(+0) 0.22010( +0) 0.72387(-t 0)

-0.10882(-2)o.3s222(-2) . 0.2õ681(_2)

-0,r

^1e?2(-1.1) 0 .:Ji15672

0.702875 0 .3 0658(-2)

'l'tù¡le _5.3

tØ ^DP {t (Iu tr)t ùI

e-lo

o.452,12(_l_0) 0 . 452:J 0/

-

¹⁾

0.458ß4(-2)

0 0

-0

₀

12727(- :ì) 222ß0(-:J)

1 0339(- 5) 7'ì7?3(--5)

o.252266

0 .03634 6 0.14868(-5)

t

.5

0.4093

I

⁰⁾

1) 0 0 -fì0

0.81850(-- ₀

0 250822 6it45B ¹

e 0.16830/-1) _o.11298(

-4)

0 .30112 7(.1- 0) 0.15135( _l-0) 0.81285(-1)

{J )

e2 _0 ) ⁰_(.) ²⁴⁶⁵⁴⁰82917-1 0 ^1.51 50(-i)

0

c-ù

o.18:107(-f ())

0.18242(-t 0) 0.21220(-i 0)

0.09150(-1) 0.13111(-j 0)

-0.22086(-2)

0.23057{) 0.620006

0.83120(-;J 0.943ú2(-:l)

e-2t 0.G7780(-1)

0 . 1?9e0(.+ 0) 0.36572(.1-0)

0.ß1512(+0) 0,0052e(.j-0)

-

^{0. 1 6ü21(- l )}

0 . fi8835(- 2)

O.22625r.t

0.60150åì 0. {650õ(_2)

rl ^t) .5000rì(-1.0)

-

^{0.2831q 2)}

-

⁰ 11e44(-1-1)

0.41J8321 ì-u) sllì

-

0.52386(.j-0)

0.2llirb00

-9.114e1(-

1) 0.5ss65¡i

0.30730( 2) 0.302e2(-2)

)