A periodic map for linear barycentric rational trigonometric interpolation

A

periodic

map

for

linear

barycentric

rational

trigonometric

interpolation

Jean-Paul Berrut, Giacomo Elefante

∗

Department of Mathematics, University of Fribourg, Pérolles, Fribourg 1700, Switzerland

Keywords:

Barycentric rational interpolation Trigonometric interpolation Conformal maps

a

b

s

t

r

a

c

t

Considerthesetofequidistantnodesin[0,2π), θk:=k·2π

n , k=0,...,n− 1.

Foranarbitrary2π–periodicfunctionf(θ),thebarycentricformulaforthecorresponding trigonometricinterpolantbetweentheθk’sis

T[f ](θ )= n−1 k=0(−1)kcst( θ−2θk)f(θk) n−1 k=0(−1)kcst( θ−2θk) ,

wherecst(·):=ctg(·)ifthenumber ofnodes nis even,and cst(·):=csc(·)ifn isodd. Baltensperger[3]hasshownthatthecorrespondingbarycentricrationaltrigonometric inter-polantgivenby theright-handsideoftheaboveequationforarbitrarynodes introducedin [9] convergesexponentiallytowardfwhenthenodes aretheimagesoftheθk’sundera

pe-riodicconformalmap.Inthepresentwork,weintroduceasimpleperiodicconformalmap whichaccumulatesnodes inthe neighborhoodofanarbitrarilylocatedfront,as wellas itsextensiontoseveralfronts.Despiteitssimplicity,thismapallowsforaveryaccurate approximationofsmoothperiodicfunctionswithsteepgradients.

1. Introduction

The polynomial interpolating between Chebyshev nodes of the second kindis a very efficient way of approximating smoothfunctionsonafiniteinterval.Itconvergesexponentiallyrapidlyforanalyticfunctions,fasterthanalgebraically (spec-trally)forinfinitelysmoothonesandwithanordercorrespondingtothatofdifferentiabilityforseveraltimesdifferentiable functions.TheMATLAB–basedsoftwareChebfunaswellassome(pseudo)spectralmethodsforpartialdifferentialequations arebasedonit[23,24].

Thebarycentricformulaforitsevaluationreads Pn

(

x

)

= n k=0 (−1) k x−x kF

(

xk

)

n k=0 (−1) k x−x k , (1) ∗ _{Corresponding author.}

E-mail addresses: jean-paul.berrut@unifr.ch (J.-P. Berrut),giacomo.elefante@unifr.ch (G. Elefante).

http://doc.rero.ch

Published in "Applied Mathematics and Computation 371(): 124924, 2020"

which should be cited to refer to this work.

wherethexkaretheinterpolationpoints(nodes),Fistheinterpolatedfunctionandthedoubleprimemeansthatthefirst

andthelasttermofthesumaremultipliedby 1 ₂ .Foradescriptionofthebarycentricrepresentation,thereadermayconsult [13].

In[9],thefirstauthorhassuggestedtheuseof(1)alsowhen theinterpolationpointsxkare nottheChebyshev ones.

TheresultisarationalfunctionwithnopolesonR[9],whichweshalldenotebyrn[f].Baltenspergeretal.[6]haveshown

thatrn[f]retainstheexponentialconvergenceforanalyticfunctions,whenthexk’sareimagesoftheChebyshevnodesunder

aconformalmap.Thismaybeusedtoimprovethestabilityofpseudospectralmethodsintimeevolutionproblems[5]and todiminishthenumberofiterationsinthesolutionofthesystemsofequationsarising frompseudospectralmethodsfor stationaryproblems [10].But itmay alsobe used to concentratethe nodesin thevicinity of fronts inorder to improve accuracy[11],apropertywhichhasledtoseveralapplications,startingwiththesolutionofstationaryproblemswithfronts in[12]andoftimeevolutionproblemsin[22].Theauthorsofthelatterpaperhaveintroducedamapwhichmoves(known) singularitiesofthefunctionawayfromtheirdomain.Thispermittedtheaccuratesolutionofexplosionproblemsin[22]and ofinfiltrationproblemsin[14],forinstance.

Thework[11]makes useoftheBayliss–Turkelmap[7],whichinvolvestwoparametersfordescribingthefront,onefor its location,theother forits strength,andoptimizesthe parametersasfunctionsoff,renderingrn[f] nonlinearinf,with

spectacular results;see [4] fora close alternative. [11] alsopresents a method forextending the map to functions with severalfronts,whichhasbeenappliedrecently[18,20]andwillbeusedinthepresentpaperaswell.

Letting x=cos

θ

,xk=cos

θk

in(1)showsthatthe latteristhecosine polynomialinterpolantofthe 2

π

–periodiceven

functionF(cos

φ

)[8]betweentheequidistantnodes

θ

k:=k·

2

π

n , k=0,...,n− 1. (2)

Foran arbitrary 2

π

–periodicfunction f(

θ

), the corresponding trigonometric interpolantbetweenthe equidistant

θk

from (2)reads[16] T[f ]

(

θ

)

= n−1 k=0

(

−1

)

kcst



_θ−θk 2



f

(

θ

k

)

n−1 k=0

(

−1

)

kcst



_θ−θk 2



, (3)

wherecst

(

·

)

=ctg

(

·

)

ifthenumberofnodesniseven,andcst

(

·

)

=csc

(

·

)

ifnisodd.

AnextensionofChebfuntoperiodicfunctionsusingtrigonometricinterpolationhasrecentlybeenconstructedbyWright andal.[25].

In[9],thefirstauthorhassuggestedtouseT[f](

φ

)asinformula(3)forarbitrarynodes

φk

with

0≤

φ

0 <

φ

1 <· · · <· · · <

φ

n−1 <2

π

(4) andprovedthatthecorrespondinglinearrationaltrigonometricinterpolantdoesnothaveanypoleson[0,2

π

)foranyset ofnodessatisfying(4).

Baltensperger [3]has later shownthat, similarly to the rationalinterpolant (1), (3)converges exponentially towardf whenthenodesaretheimagesofthe

θ

_k’sfrom(2)underaperiodicconformalmap.Baltenspergertestedhisresultsmerely withaperiodicversionoftheTal–Ezermap,whichimprovestheaccuracyonlyforfunctionswitha frontinthecenterof theinterval.

Inhisthesis[21],Teehasconstructedaperiodicmap whichpushes(known)polesoffaway fromthedomain[0,2

π

), butinvolvesspecialfunctions.

Inthepresentwork,weintroduceasimpleperiodicconformalmapwhichaccumulatesnodesintheneighborhoodofan arbitrarilylocatedfront,aswell asitsextensiontoseveralfronts.Despiteitssimplicity,thismapallowsforaveryaccurate approximationofsmoothperiodicfunctionswithsteepgradients.

2. Exponentialconvergenceoftheinterpolant

Asmentionedbefore,Baltensperger[3]wasabletoextendthetheoremfortheexponentialconvergenceofrn[f]tothe

caseoftrigonometricinterpolation.Hisresultreadsasfollows.

Theorem1. Letgbeacomplexvaluedfunctionwithperiod2

π

suchthatg

(

[0,2

π

]

)

=[0,2

π

]andsuchthat

w

(

φ

,

ψ

)

:=cst

((

g

(

φ

)

− g

(

ψ

))

/2

)

cst

(

(

φ

−

ψ

)

/2

)

is boundedand analyticin Sa1× Sa1,where Sa1:=

{

ζ

:

|

Im

ζ |

≤ a1

}

(a1 >0).Letfbea functiontobe approximatedsuchthat

f◦gisacomplex-valuedfunctionwithperiod2

π

.

LetT[f](

φ

):=T[f◦g](

θ

),

φ

=g

(

θ

)

,betherationalfunction(3)interpolatingfbetweenthetransformedpoints

φk

=g

(

θk

)

with

θ

kasin(2).Then,forevery

φ

∈[0,2

π

],wehavethefollowing:

(i) Iff◦ ghassimplejumpdiscontinuitiesinthepthderivative,then

|

f

(

φ

)

− T[f ]

(

φ

)

|

=O

(

n−p

)

;

(ii) Iff◦ gisanalyticinastripSa2:=

{

ζ

:

|

Im

ζ |

≤ a2

}

,wherea1 ≥ a2 ≥ 0and|f(

φ

)|≤ M,then

|

f

(

φ

)

− T[f ]

(

φ

)

|

=O

(

ρ

−n

₎

_{, with}

_ρ

_:=ea2.

Onecanthereforeuseconformalmapstomoveequidistantnodesandstillretainrapidconvergence.

Oneideaistousethemapsforpolynomialinterpolationappropriatelymodifiedinordertomakethemperiodic[3].For instance,themapbyKosloff andTal-Ezer[19]

g

(

x

)

=arcsin

(

α

x

)

/arcsin

(

α

)

, 0 <

α

<1, (5) becomes[3] g

(

θ

)

=2arccos



arcsin

(

α

cos

(

θ

/2

))

arcsin

(

α

)



, 0<

α

<1, (6)

whiletheoneofBaylissandTurkel[7] g

(

x

)

=

β

+ 1

α

tan

(

λ

(

x−

μ

))

(7) becomes g

(

θ

)

=2arccos



β

+1

α

tan

(

λ

(

cos

(

θ

/2

)

−

μ

))



, (8)

withthefollowingparameters

μ

and

λ

:

λ

=

γ

+

η

2 ,

μ

=

γ

−

η

γ

+

η

,

γ

=arctan

(

α

(

1+

β

))

,

η

=arctan

(

α

(

1−

β

))

.

Weshallusethesetwomapsforcomparisoninthenumericalexamples;forothermapsforthenon–periodiccase,see, e.g.[2,18,22].

3. Aperiodicconformalmap

We shall now introduce the conformal maps which constitute the central part of this work. For z∈C, consider the Möbiustransformation[15]

ha

(

z

)

= z+a

1+z¯a, with

|

a

|

<1. (9)

ha mapsthe unit diskonto itself in such a waythat the image ofthe point −ais mapped to 0.Recall that, since ha is

a Möbius transformation,it maps (arcs of) circles to (arcs of) circles. Notice also that the inverse function of ha is h−a . Moreover,ha mapsconnectedsubsetsofCconformallyandleavestheunit circleEinvariant, i.e., mapsitontoitself.The

imageoftheboundarypointeiθ _is

eiψ(θ )_:=ha

₍

_eiθ

₎

₌ eiθ+a

1+eiθ_¯a. (10)

ψ

(

θ

)istheso–calledboundarycorrespondencefunctionofthemaphafromtheunitdiskontoitself[17];itwillbethebasis

ofourchangesofvariable.

Letusconsiderareal;ifwewriteeiθ_{=x+iy,wehavex}2 _+y2 _{=1andwecanwritee}iψ( θ) _as

eiθ_+a 1+eiθ_a= x+iy+a 1+

(

x+iy

)

a= x+2a+a2 x 1+2ax+a2 +i y− a2 _y 1+2ax+a2 =ua

(

x,y

)

+i

v

a

(

x,y

)

.

Thepartialderivativeofua(x,y)withrespecttoais

∂

ua

(

x,y

)

∂

a =

(

x2 − 1

)(

2a2 − 2

)

(

1+2ax+a2

₎

2 =− y2

(

2a2 − 2

)

(

1+2ax+a2

₎

2

andispositivewhen0<a<1,whichmeansthatallpointsofEbut± 1movetotheright.Therefore,manyelementsofthe set{eiθ},where

θ

:=

{

θk

= 2 kπ

n ,k=0,...,n− 1

}

isthesetoftheequidistantnodes(2),willclusteraround1(seeFig.1).

When we consider a complex value ofa such as a=

ρ

eiϕ_,

_ϕ

_{∈(0, 2}

_π

_{), many imagesof {e}iθ_{} will cluster around e}iϕ_,

since thisgeneral caseis the rotation of the previous case by the angle

ϕ

. So, ifwe restrict ourselves to the boundary correspondencefunction,theargumentsofthemappedpointsha(eiθ)willclusteraround

ϕ

.

Fig. 1. Uniform points on the circle and their radii mapped with h a with various values of a = ρ.

3.1. Amapforasinglefront

Ourmapg,then,willbetheboundarycorrespondencefunction,whichweobtainbytakingthelogarithmwithvaluein [0,2

π

)of(10): ga

(

θ

)

=−ilog



eiθ_+a 1+eiθ_¯a



(11)

(sincetheprincipalvalueofthelogarithmliesin

(

−

π

,

π

],wemustchooseanotherbranch).

Ingeneralwehavega(0)=0,inwhichcase,sincethefunctionisperiodic,we considerthefunction ga

(

φ

)

:=ga

(

φ

+

γ

)

foraconstant

γ

suchthat ga

(

0

)

=0.

We can now use these new pointsto interpolate a function with the barycentric trigonometric interpolant and take advantageofTheorem1.Inthepresentwork,wewishtoconstructan exponentiallyconvergentapproximantofafunction fofwhichweknowthatithasafrontat

θ

=

ϕ

.Wedothisbychoosinga

ρ

ina=

ρ

eiϕ_{such thattheinterpolationpoints}

θ

clusteraround

ϕ

andapplyingthebarycentricrationalinterpolant(3)for

φ

:=ga(

θ

)insteadof

θ

.

3.2. Multiplefronts

Incasethefunctionfhasseveralfronts,weneedtoconstructaconformalmapthataccumulatesthepointsaroundallthe locationsofthefronts.As proposedin[12],we considertheinversefunctionsg−1 q withparameters

ρq

,

ϕq

,forq=1,...,J,

onegqforeachfront,andwedefinethemap ¯g−1 as

¯g−1

(

φ

)

=1 J J q=1 g−1 _q

(

φ

)

; (12)

¯gwillcluster thenodes aroundthe variousfronts.Sincethe latterlie atthe (known)points

ϕq

,we wantto choose

ρq

such thataq=

ρq

eiϕq areadequateparametersin ¯g

(

θ

)

.

Wehave 1 J J q=1 −ilog



eiφ_{− a} q 1− eiφ_a¯ k



=

θ

J  k=1



eiφ_{− a} k 1− eiφ_a¯k



=eJiθ_{, (13)} or J  k=1



eiφ_{− a} k



− eJiθ J  k=1



1− eiφ_a¯ k



=0. (14)

Settingz:=eiφ_{,wehavetofindthezeroofapolynomialofdegreelessthanorequaltoJforeveryvalueof}

_φ

_atwhich weneedthatof

θ

,inparticularforeachoftheninterpolationpoints.

However, inthecaseoftwofronts wecancompute theinversefunction analytically.Indeed,letusconsidera=

ρ

1 eiϕ1

andb=

ρ

2 eiϕ2,_{theparametersofthefunctionsg}−1

1 andg−1 2 ,respectively.(13)becomes



eiφ_{− a} 1− eiφ_a



eiφ_{− b} 1− eiφ_b



=e2 iθ or



eiφ_{− a}



_eiφ_{− b}



1+e2 iφ_{ab− e}iφ



_a+b



=e 2 iθ_,

whichleadstotheequation

e2 iφ



_{1− e}2 iθ_¯a¯b



_+eiφ

₍

_e2 iθ



_¯a+¯b

₎

₋

₍

_a+b

₎



_{+ab− e}2 iθ_{=0 (15)} for

φ

.Settingz:=eiφ_{,thisturnsinto}

z2



1−e2 iθ_¯a¯b



_+z



_e2 iθ

₍

_¯a+¯b

₎

₋

₍

_a+b

₎



_{+ab− e}2 iθ

=z2



e−iθ− eiθ_¯a¯b



_+z



_eiθ

₍

_¯a+¯b

₎

_{− e}−iθ

₍

_a+b

₎



_+e−iθ_{ab− e}iθ

=z2



e−iθ− eiθ_¯a¯b



_{− 2izIm}



_e−iθ

₍

_a+b

₎



_+e−iθ_{ab− e}iθ_=0.

Thereduceddiscriminantofthisseconddegreeequationforzis



4 =



iIm



e−iθ

(

a+b

)



2 +



e−iθ− eiθ_¯a¯b



_eiθ_{− e}−iθ_ab



=



eiθ_{− e}−iθ_ab



2

− Im



e−iθ

(

a+b

)



2 andthesolutionsare

z1 ,2

(

θ

)

= iIm



e−iθ

₍

_a+b

₎



_±

₍

 4

)

1 /2 e−iθ_{− e}iθ_¯a¯b . But z1

(

θ

+

π

)

=−i Im



e−iθ

₍

_a+b

₎



₊

₍

 4

)

1 /2

−e−iθ_+eiθ_¯a¯b =

iIm



e−iθ

₍

_a+b

₎



₋

₍

 4

)

1 /2

e−iθ_{− e}iθ_¯a¯b =z2

(

θ

)

,

sothatthesolutionisuniqueuptoatranslation;translatingittohavetheimageof0in0,wehave

φ

=−ilog



z1

(

θ

)



. (16)

Onecould alsoobtainanalyticalmapsforthecasesofthreeandfourfronts,butthisrequirestheuseofmoreinvolved formulas such as that of Cardano for the third degree equation and even more complicatedones for the fourthdegree equation;moreover,astableimplementationofsuchformulasrequirescare.

4. Numericalexamples

4.1.Onefrontfunction

WehavefirstconsideredafunctionsimilartothatofExample1in[11],butmodifieditinordertomakeitperiodic, f1

(

θ

)

=e1 /(sin (θ+ π )+1 .5)+cos

(

4

(

sin

(

θ

+

π

)

+0.5

))

+erf

(

δ

(

sin

(

θ

+

π

)

+1

))

erf

(

δ

)

,

Fig. 2. Left: an example with the function f 1 ; Right: various point sets.

Table 1

Error with the function f 1 with = 10 4 .

n Equi KT BT BRT

10 5.7201e-01 5.7295e-01 (0.05) 3.7049e-01 (3.05) 4.5603e-01 (0.04) 20 4.3166e-01 4.3332e-01 (0.05) 5.0872e-01 (11.30) 3.5469e-02 (0.40) 40 6.0408e-02 5.0592e-02 (0.99) 6.0059e-02 (2.05) 2.0856e-03 (0.46) 80 4.9557e-03 5.0356e-03 (0.05) 1.3034e-02 (2.05) 5.8500e-07 (0.51) 160 4.5164e-05 4.2683e-05 (0.86) 3.1541e-03 (2.05) 1.9540e-14 (0.54) 320 1.1680e-10 1.1921e-10 (0.47) 7.8051e-04 (2.05) 1.7764e-14 (0.62)

Table 2

Error with the function f 1 with = 10 6 .

n Equi KT BT BRT

15 9.6326e-01 5.9939e-01 (0.92) 4.8577e-01 (3.60) 4.6128e-01 (0.52) 30 9.9971e-01 5.1234e-01 (0.59) 4.2361e-01 (3.05) 2.0320e-01 (0.56) 60 4.7472e-01 4.6612e-01 (0.94) 5.7724e-02 (3.05) 3.9628e-03 (0.63) 120 8.5163e-02 7.9048e-02 (0.94) 9.1171e-03 (2.75) 5.2002e-06 (0.68) 240 9.2787e-03 8.0863e-03 (0.94) 1.4085e-03 (2.05) 1.2257e-13 (0.70) 480 6.2903e-05 5.4821e-05 (0.20) 3.4604e-04 (2.05) 4.1744e-14 (0.51)

with

δ

=√0.5



for a parameter



which characterizes the steepness off1 . Here we have computed withthe two cases



=104 and



=106 .Thefrontofthisfunctionisin

θ

= π₂ ,asmaybeseeninFig.2.

Toestimatetheerrorweconsidered2000uniformlyspacedpointsin[0,2

π

]andwecomputedthemaximumdifference betweentheinterpolantandthefunctioninthesepoints.

InTables1and2wemadeacomparisonusingvariouskindsofpoints.Thefirstsetofpointswerenequidistantpoints in[0,2

π

).Weusedthemap(6)derivedfromthatofKosloff andTal-Ezer(KTinthetable)andthatofBaylissandTurkel (8)(BTinthetable):asparameter

α

wechosethebestvalues(writteninparenthesesinthetables)throughatrialanderror methodwithinthe sets[0.05:0.01:0.99]and[2.05:0.05:15],respectively;asparameter

β

inBayliss andTurkelwe took thevalueforwhichthemapclustersthepointsatthefront

π

/2,i.e.,

β

=1/√2.Sincein[7,11]theresultsdidnotnoticably dependonaperturbationoftheparameter

β

aboutthelocationofthefront,werefrainedfromtryingothervalues.Forthe lastcolumnweusedthemap(11)presentedabove(BRT—for“barycentricrationaltrigonometric”—inthetables),wherewe chose thebest

ρ

within theset [0: 0.01:0.99].(Onecould gofurther byestimatingan optimal

ρ

,inthespirit of[7],or numericallyoptimizeit,asin[11],butouraimhereisonlytodemonstratetheeffectivenessoftheproposedmapinsome situations.)Forthesamereasonasfor

β

above,wefixed

ρ

asthelocationofthefront.

The approximate squaring of the error when N is doubled reflects the exponential decay of the error predicted by Theorem1.

The reasonwhythe map (8)does not leadto exponential convergenceis that its construction doesnot provide con-formality.Indeed,g

(

0

)

=0inview ofthefactorsin(

θ

/2)intheinnerderivative;andan analyticfunctionwithvanishing derivativeisnotconformal[1,p.133].

Fig. 3. The two test functions.

Table 3

Error with the function f 2 .

n Equi BT BRT

60 4.7774e-01 3.6941e-02 (13.3) 2.1843e-02 (0.77) 120 8.6081e-02 5.3467e-03 (12.7) 9.8636e-05 (0.79) 240 9.3970e-03 5.7084e-04 (6.00) 1.5925e-09 (0.79) 480 6.3819e-05 2.7993e-05 (2.05) 6.6613e-14 (0.74)

Table 4

Error with the function f 3 .

n Equi BT BRT

35 1.1156e + 00 3.0810e-01 (12.35) 1.9638e-02 (0.90) 70 7.3624e-01 1.0701e-01 (13.70) 1.0129e-03 (0.87) 140 1.9318e-01 8.6162e-03 (14.65) 2.0570e-06 (0.86) 280 2.2589e-02 2.1048e-04 (10.10) 1.9059e-12 (0.85) 560 2.8214e-04 3.0953e-05 (2.80) 9.7700e-15 (0.89)

4.2.Twofrontsfunction

Asanexampleofafunctionwithtwofrontswehaveconsideredamodificationoff₁ : f2

(

θ

)

=e1 /(sin (θ+ π2)+1 .6)+cos



π



sin



θ

+

π

2



+0.5



+erf

(

δ

(

sin

(

θ

+π3

)

+1

))

erf

(

δ

)

+ erf

(

δ

(

sin

(

θ

+6 π 5

)

+1

))

erf

(

δ

)

,

with

δ

=√0.5



,and



=106 ;thisfunctionhasonefrontin

θ

1 =7 6

π

andanotherin

θ

2 = 10 3

π

. Thefunction f3

(

θ

)

=tanh



50cos



θ

+

π

3



,

hastwofrontsaswell,respectivelyin

θ

₁ =π₆ and

θ

₂ =7 ₆

π

. Thetestfunctionsf2 andf3 areplottedinFig.3.

As forf1 ,weperformedatrialan errorsearch fortheparameterofthemap(16),inthesamesetasforf1 andsetting thesame

ρ

forbothfronts,asthesameparameter

δ

isusedinthetwoerf–functionsinf₂ andthesamevalue50multiplies thetwozerovaluesofthecosineinf3 .

In Tables 3and4,we display acomparisonwiththe equidistantnodesandwiththe periodicBayliss andTurkelmap withthesamevalue oftheparameter

α

forthetwofronts,chosen witha trialanderrorinthesamesetasforf₁ .Again, the

β

–valuesarethecosinesofhalftheabscissaeofthefronts.Sincewedonothaveanexplicitformulafortheinverseof thismapwithmultiplefronts,weapproximatedthe

φk

bymeansofthefunctionfsolveinMATLAB,i.e.,asthevalueswhich solvetheequation ¯g−1

(

φ

)

=

θ

, where ¯g−1 is thefunctionin(12)withJ=2andg−1 _q theinverseoftheBaylissandTurkel map.

5. Conclusions

Inthisworkwehaveintroduced aperiodicconformal mapforshiftingnodesontheintervalofinterpolationwiththe purposeofapproximatingperiodicfunctionswithfronts.Ourexperimentsdemonstratethatthesuggestedmapindeedleads to exponentially convergent barycentric rationaltrigonometric interpolants, as to be expectedfrom the theorem by Bal-tensperger. Suchan exponential convergenceis fasterthananypower of1/n,in other wordsspectral. Inthe caseoftwo frontswehaveananalyticalformulaanddonotneedtonumericallysolveanequationforeverypoint,aswithothermaps suggestedintheliterature.Inmanycasesthisleadstoanenormoussavingofcomputation.Whenmorethantwofrontsare present,wemerelyhavetosolveapolynomialequationofcorrespondingdegreeforeverynode.

Acknowledgement

TheauthorsthankRichardBaltenspergerandarefereefortheircomments,whichhaveimprovedthepresentwork.

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