APPROXIMATION
OF THf, BESSEL EIGENVALUf, PROBLEM
BY FINITE DIFFERf,NCES*
HERBERT L. DERSHEM1
Abst rci. Diferencecqudrionprobtensalesrudiedwhoscsolutionsaie estinarcsofthe sotutions
of the ctenvrlue probl€n
.,= , -l,*a,:,;r.
,(0) = 0,
rero.r,
t(11= 0,
for0-,
t .Rerenroperaro^.t,.arelonstrr!,eJsorhd,L-p11r.,-I@nrrl
1 . I h r e ep o r n d
..N.wfcre,d., ndgrlenserotfundion,andtr
I
l . 2 . J . r =|
,,\ rti
An uppe' boundon th€ disrancebelweenan cis€nvalu€,i, ofL od rheser
or
{ J}l oreisenralues
an! three-pointdifle.encaoperatorar whosecoemcientssalistypre$ribed cooditiotu is obrained.
The bound is a producl ola conslantand lhe nax norm oi rhe tru.calion error oI t and L. wirh
r c s p c , 'r u r h e e . B e n l u n f ' u no\ l L c o r r e i p o n d r nrgo ^ 1 t r o a p p r o p n e t j ( o n n r u c r e dd , f l e r e r c c
opcratorsare shown lo bave cocfiicienrswhich satislt tbe plescribedconditions,and for lhese.
nlnrl,l Ar isshown ro bc o(/,,)
l. Introduction.The eigenvalues
of the Sturm Liouville problem
(1)
(p(x)y')'- slx)!' +,t/(r)jr = 0,
i€(0,l),
a0J,(0)ar)'(0):0,
bor,(1)
+ rr)'(l) = 0,
l d o+
l d,l *0,
ar10,
brf 0,
l r o i+ l r l l + 0 ,
are&equentlyapproximatedby theeigenvalues
ofa corresponding
6niaedifl€rence
problem,wherethe finite difrerence
problemis constructedin sucha way that the
difrerentialoperators in 1l) are r€placedby suitably chosenfinite diflerence
operatorsover an equallyspacedmeshof lengthl. If the coemcientsof (l) are
suliciently well-behaved,
that is, if p, r > 0, q > 0, and p, 4 and r are suliciently
smooth on 10.ll, Rnite differenceapproximationscan be found such that the
eigenvaiues
of the diflerenc€problem are within O(rr) of the eigenvalues
of (t).
For example.Keller has shown this to be true if p',4 and r are continuouson
10,1l and all derivativesare replaced by their standard central diflerence
approximations[4. pp. 135 t37]. Hubbard [3] has shown rhar a similar resDtt
holds when p,4, r are piecewis€continuouslydifferentiable.the coemcienlsare
replacedby averagesof the coemcientsover the mesh,and each derivativeis
replacedby its forwarddiilerenceapproximation.
Variationalmethodsyield higherorder approximationsto the eigenvalues
of
(l). See,for example,lll. The assumptionthat p and r are positiv€in the closed
int€rval 10,1l is also madefor det€rminationof error boundsof aDDroximations
obtainedin tht way.
StDrm Liouvill€ problemswith singularitiesarisingfrom p vanishingat one
ofthe endpointsof the intervalarenot includedin the error analysessuchas those
mentionedabove.It might be expectedthat for sucb operatorsth€ order of the
t Reccivcd
by theedironr_ebruary
3, 1970.
iDepartnentofMalhemalics,
HopeCollege,
Holland,Michigrn49423.
106
AP}ROX]MATION BY FINITE DIFFERENCIS
107
of
operatorsAn example
errorboundmightbelowerthanit is for nonsingular
problem
problem
is
the
Bessel
eigenvalue
sucha
(2)
(rI')' -
.)'+,ix}, = 0.
x € (0,1),
. i ( 1 :) 0 ,
which has a regularsingularpoiDt al i. : 0. The resultsgiven in Table I in | 4
probl€m approxjmating(2) which is definedbv
show that the finite dilTerence
replacingeach derivativeby the standardceniral differenceapproximatioDhas
smallesteigenvaluewhich difrersfrom the smallest€igenvalueof(2) bv O(r'?")for
0 < 1,< 1 and O(h'z)for r'= 1. We seethat Keller's result doesnot extendto
problem(2)ior 0 < v < L
To studythe effectof sucha singularpoint, we limit our study(basedon [2])
of the tridiagonal
of 12)by the eigenvalues
to approximationof the eigenvalues
(2)
by the finite difference
N x N matricesobtainedfrom the approximationsof
problem
j:1,..,N,
L r u i= d t u i t + P l u j + t j u j * t : / \ u i ,
r
\
' '
ar*r :0,
,n :0,
.y(0):0,
with conslantmesh length h = l/(N + 1), N a positive integer,where L' is a
6nite diflerenceapproximationto the differentialoperatorI givenby
(41
1 v
L y = - ) i "- ? , ' +
2
1 r' r
Note that (2) is the eigenvalueproblem for the operator I wilh approprrale
boundaryconditions.
1,, of (2) from the set of
A bound on the distanceof any given eigenvalue,
ot(3) is givenby TheoremI as a constanttimesth€ max norm ofthe
eigenvalues
with ,l.t
lruncationerror of I and L, with respeclto the eigenfunctionassociated
whenthe coefrcientsofrl satisfycertainconditions.
we study threeditrerenceoperatorsdenotedby Lll), Itz), Ll,t).The first of
thes€is the standardcentraldifer€nceapproximationto L on the mesh We show
that the max norm ollhe
that the operatorsLlt)and I are inconsistentin the seDse
truncationerror is Dnboundedas}i J 0 when0 < v < 1. Theselwo operatorsare
consistentwith respeclto a difer€nt norm. but this facl is irrelevantto lhe present
sludy. The truncationerror of rft) and I is zero ior any quadraticpolvnomial
,(x) = dx'1+ t x + .. but quadratic polynomialsyield poor approximationsto
ofl. whichbehavelike x"as xJ0lor0 < r < I This is the
ihe eigeniunctions
we
rcasonthat th€ max norm olthe truncationerlor is unbounded.Nevertheless,
of the diferenceproblem conv€rgeto thoseof the
obs€rvethat the eigenvalues
dif€r€ntialproblem,but at a dower ratethan for the nonsingularSturm Liouville
problems.
The operatortl') is derivedin sucha way that accountis takenofth€ b€havior
of (2) near the singularpoint. Th€ coemcientsd, p, l are
of the eigenfunctions
determin€dsuch that for any function u(x): axv+'] + bxl+r + cx", the valueof
ltz)r(r) is the sameas the valueof lr(x) at eachmeshpoint, i e.,the truncatron
HERBERTL. DERSHEM
708
of (2)ar€constant
error ior Ltz)andt is zerolor suchfunctions.Theeigenfunctions
rnultiplesofJ"(,11/'?jr),
wher€,Lis the corespondingeigenvalue,
and J"(x) is
I
k!I(v+l+l)2'+2k
Thus,the operatorLt'z)isconstructedso that a betterapproximationto the eigenfunctions near .j( : 0 is achiev€dthan wh€n quadratic polynomialsare used.
Furthermor€,awayfrom r : 0, functionsofthe form u(x): dx"+': + b-{"+I + cr'
It follows,as it is
also yield good local approximationsto the eigenfunctions.
shown in S2, that the max norm of the truncationerror is O(r?),and sincel,t')
of
satisfi€sthe hypothesesof Theor€m 1, it is a corollary that the eigenvalues
ll'z) difer irom lboseof I by at most o(h'?).
TheoperatorxL is self-adjoint,
but theoperatorirl;'1)is not.To bercrapproximate this property of l, the differenceoperator It' is constructedby finding
: ar"+'? + bjl', the
coemcienlsd, /J,? suchthat for any functionofthe form 1)(.,r)
valu€ofrtr)u(x)is th€ sameastbevalueofLu(r)at the meshpointsand,in addition,
the operatorr-Lr3)isself-adjoint.
The truncationerror of L[3)and, is shownto be
o(h'z),anditisacorollary ofTheoremI that theeigenvaluesof Ll,3)diflerfrom thos€
of L by at most O(r'z).
ln $ 4 numericalresultsare displayedwhich illustratetheseconclusjons.
2. Truncrtion €rror analysis. The operator tt1) constructed so that
f;')u(x) : Lu(r.)at all mesh points for all quadratic polyDomials, is givcn by
/
I
r\
/ .
' , i
|
{ 5 rL 1 ' r rlr-,7a- + . n 1 , , , r l i , - , , 1 " , 1 /, ; \
ln-
I \
tnf,,
j = l
A
for any setof numbers{tl.,}.We definethe setof functions9" by
t6t
. q ,=
(
2
r
= a I c r r x '' : r . ) , r ' (xr e C a l 0l l. l .
tl)ixr
we notethat all eigenfunctions
of (2)belongto ,tr".Application
of Taylor\
gives
lheorem
withremainder
to anyfunction
-.,4"
Jr
(7)
t2
h2
24
h2
r-x,it ,; , lJr'rtr- J''ur].
t41n
otn
L;"y(.x,- lr(xi) = ---tr't() -;),
j - r.
. N.
w h e r \er . | < , t < ( < e < ) r - r . A s r l 0 . t h e n x j - i , ? . s i l h l 6 x e d . i ' s ul hc a
hl
x, I 0. Hence.for frxedi, Jj (xrr= Orlr'-rr and yi'txtt = oth'r a. r, 0. Therefore,by (7), w€ havein generalthat
(8)
llt')y(jr) .- rJ,(xj) - = +!(! - t)'(t'
2)to.i-1h-1+
i:1,.
o(h'),
,N. ))e9",
where
- denotesthe max norm. Since0 < r < 1, we have that Ltt) is not
consistentwith I with respectto the classof functions.4" and in the max norm.
= lu(.x) for r =ir,
The operator l[2) is constructedso that Lj,'?)r(x)
j = 2, .., N, and for all I of the form {9:
r""-, + b.x"*r+ cx". Sucha
rx?RoxD&d&ona&rrEriFF$rpFms\tcFs
(bllttustiob is
(e)
lot q,81,11.lt
ducesto three
equdtiotrs.h
t = 1, reducibg
oxootlyt{ith ,
Tho opor&tor
to foake
syliom to flv6
-' *:]$"'-'
=[-J;+1
+$*'
-! -!)*^
.,(,-t-*
Ji11.'
-
Fotanyyeg,, )(r) ir in c410,1l. ff ;ii{rr * J #d xl",{xi- r) arqcoaputea
i.in withlcEoitrder,,atd
by T6ylor'g
+rbstitute!h (10),oncobtaino
(11)
rffbcoJ - 1,Yra
(12)
Therofore,(11)
ve4.
03)
in g1,theditrele@l'obdid@id&0iod by
04)
,^,6=S-*',1"i!u" '
P< x < l ,
t-p_i+qoastruct€dso that it agreas
is eeu-a4joiot. fqitc dif€sanceqnDls1oq
fuwa
for
tunotioDs
dt
s*"' + bx', and lhe op€mtor
thd
with L o!, the fh
,IP d€6dod
0t
(x43)dr=t,r1%,
'll0
mRDERTL. DERSHEM
is self-adjoint. The operator rLl3r is self-adjoint if and only if its coefrcients are
s u c ht l a t d . / + r : \ j / \ j + r ) ) r j , j : 1 , . . . , N
1 . T h e c o e m c i e not sf L t r ) a r e
therefore defined by the recunion relations
4
P , - . ; - ( v+ l ) .
rn
?,=
( 1 6 ) d r - "t - l t i , .
j
e,-
ltt'4
l , l , , , ( y r -l J .
\tl ,t'
/i-l\" 41
4l!+ll
\ ' l l t n f , * , ar ' i * , .
4r'I t)/ r i'
l j t \ ' ( 2 -j D 0 - t )
t,-U-'/
v ' - , - r z +i t n , \ i t r f
t2j.tj
ru
r::
The first order difrercnce equation for y, in (16) can be used to generate the
sequence
of valuesyj,j : 1, . , N, and this resultusedto obtain
._
_iri,)
a,=a,j"tn"[,,_1;i,r,
(t7)
:
-4'1,;rr[j-,"it'*,'-,, --,''].
i;]
z)
j
fh' Ll"
.?,
j-r... .N.
wherelq is lessthansomepositiveconstantK for all valuesofj.
Wenowshowthat
)
pt:
n8t
;,
1
- -ott.
Thequantityin braces
in (17)is equalto the€stimate
of theintegralof x,"+rrr"
fromx = 0 to r = j by thetrapezoid
ruleappliedat eachof thej + I equalsubdivisionsof 0 = jr S j. Sinc€thegraphof thispositiveintegrand
is concave
upward,the estimateis an upperboundon the valu€of the integral,whichh
j1/(2't+ 2).The yaltteoI thesumin thebraces
in (I 7)is equalto theestimate
of ahe
integmlofx"+rr" fromr : 'to x = j - * by themidpointruleappliedat each
of thej equalsubdivisions
of li,j - ;1. Sincethispositiveinregrand
is concave
upward,this €stimate
is a lowerboundon the valueol the integral.Hence,an
upperboundon thequantityin rhebraces
in (1?)isjr/(2y+ 2) + c/, wherelcJ is
j,
lessthansomepositiveconstant
c for all Therefore,
thetwo boundion thevilue
yield(18).
in thebraces
Suppose
J € 4. Then,(r) = jr- "t(x)is in C'[0,1].Ifwe useTaylor'stheorem
with remainder
to expand,(J(,.J and u(.r,-,, abour.r - x,, andusethefacrrhar
and
are
derermined
so rhatLl3ta.x'-r- bx't : Ltax'''z+ ,x') on rhe
ldrl
t/,]
LLf'|-Lllrrjr:
0e)
t
Lz\r
tr
-;pil
2!_t
.'"-yr',r
Lr,lx/
-l
v ( v "''vr.,r
2)
+::;
I
- #{l-Su'.'t:t^'lur',
Y,rro
-"'"t04,
i: 1,'N,
2l-#o'."i.lu't,ir
lll
APPROXIMATION BY FINITE DIFIERENCIS
where x.r 1 < ( < t < 4 < rj+ 1- But from (18), th€ fact that
2 1 1 . . - ' ( u r' ' j
- vlx,l - O{l lxi': '.
rrl.{ri
v t.{il ^i
^j
and r''(x) : xo(l), we havefrom (19),aftersomemanipulation,that
t20)
llLfht - Llyl-:
o(h'z),
J eg,.
W€definethevectornormsI ll, andll ll- in theusualway'
3. Errorhounds.
that is.
/
!
-\,/,
At, I Lzil
\ r=1 l
zl- - mdxrzr:
and a matrix norm subordinat€to the vectornorm I ' 2, for any r€al symmetric
matrix,4,is
I .4l, = 1.i.,,.
of ,4 with largestmagnitude
where,tr.,"is An eigenvalue
since the diference €igenvalueproblem (3) has homogeneousboundary
dl and ?rvin (3)canbe setequalto zero.Thentheeigen_
conditjons,the coefficients
value problem (2) is equivalentto the eigenvalueproblem for the N x N tridiagonalmatrix ,4 whoseelementsare givenby
qt,J+t -
tl,
q/+r,j -
qj+r,
j = 1 , . . , N - 1 ,
j:r,
,N.
of Kelleis 14,Theorem5 3 21resultwhichholds
Theorem1 is an extension
,4
onlyfor symmetric.
LEMr"qL Fof an! N x N real tnatrixA, { thercexistsa positiuedefnite
thenthe eigenaolues
matl.xD suchthatDAD-| is qtmhetric,
ltr)i=1 oJ,l are
-lettot
N
any
honth)ial
real
number
1and
real,andlor any
t,
(21)
.*i^,-,rr
sll144fi#!L
_
of '4
theeigenvalues
B : D.4D r is symmetric,
P/oot Si ce,by hypothesis,
Then
areall real.For anynontrivialJ,,wesett - (,4- .tr/)),.
(DAD \ - ll)Dy = lB - II)DY: Dr.
a,tr.
,{,mini^r -,ll = 0,andhence(21)ho]dsforsuch
lf ,[isaneigenvalueof
If ,l is not an eigenvalueof .4, then (B - ,l,l) is invertible and
(B - ,U)-' is symmetdc,
wehav€
by hypothesis,
Dy:\B1I) 1Dr.Since,
|
Dy, = ll(B- ).1) I )lDr l,
|
D r-, .
s ' er -\ {^ r^- , , 1,/ l D r r-sl -m -n +
l ^ x - / t,
- .l arenonzeroby hypothesis,
weobtain
Furthemore,since Dtl , andminrl.Ak
minl^k.- ,fl a lotl'll Dyllz.
712
. . rnRelatrtr!.
DiEBa@M- i
Bythe dofinition t, ihis istftGsEqealqFdr}
llh]iG,oaqtrl€t€s
theproofof Ledrma1.
LEitM 2. F
ary N x N reql fticlEBbtralhdtix A",for ythichtha proilucts
. .. ,:Nt @eWitiaefdre$e\?,niff"s
p pasitfueddnite rtatrix D
suchthat DAD- |
Proof.Il D
eleEent denoted
synmetlic.
an,IV,,x .i\r.posi6h{€ltt6,trit€.di€garal,{qatrix, wirh ,(th diagonal
(A
i =? . . , N .
y positivonumber,the4ogoputstionshowstlat E,4D-t is
Dplct€$
h€.proof"of[,r44q2.'
I raj'joar-|,y'a -t usedin tlE proolof Lernma
2 by Rjtr,
rEicecwha$!qig .v3lwrpfoble$prc nsocialedv{ithrhose
) rcsp.ctivcly.
From(5),(10),andl(I5),
wehavethat
with dr &r ar
syEmctric.This
Wo donotc
nlr),R13)
Jorthe
oir,lttitPta"a
.l-
Weobtain
byttc toqqwios.
LEMMA.
3. F
positiaelnteget s
?=f-=-)';q,**'l
co,fittqtrts
c, ald czsqch
ry$
c,l 3d3 Sczf,
,
wult4, an4
whereD = di€g ' , . .... d'i i' {s 1r.,@4;
sssoptoted
d$,!ti ttt4Nrhj
m-t+li-,@r
- b
,
Ptool,Frcn 22),wc have
t I'* k+.a. .,.--,! * {t+a^
a i - d | L --l
dif'z-'-l-l,AE-t;'
r- 2"' N'
r-2"'N'
tl JJ,F;A=qrz-'Il"qE-#'
JJ,ri'=
The value of dl
and honc€,
arhseEry,pnd
wFcbooqe
so th4.dZ= 2t,
it to be,lqetflLt22u'1
(23)
Itr telms,oftrh
{urrction[6,p,O3tL,,
n0+4)-,['(k+4)A@.i.
An applicationof
Q4)
to (23)diflos
Il
APPRO'Ctr!@IIQ}TNSAFq$E&qqFFERENCTS
'For
rj'!.,rho p!odiupqfXiid''1,r(?.r'6J,1F@
+ dJis a g lEncon'
$alrt
,ft
rU+ a)1.flffil
n1= 3-11!*5,1,
to zetgesj + 6. Appltingthisto
wh€re{r, is a
(24),wehave
andwhor6rj -
'
,i r ft/qfd + ,),'
sJ + c;. qeahoDr4i -- trldf|,(r+ ,, ando' = K 3{pll + ,i),
3io.9l!tn!FtF+
aodttreproof Le4nsa
= 1,. ,Nr0 < I < l, ard ar! rc41positiee
,tyr= J,(11t2xl,J
.
(dr
real
dr)
suohthat thbre exl$s o norneg,atiDe
)
diag
ulmbet 1, Mil
,
,
mm$erm and
$lpt ,
awpegg's1tQlallaoh
LBMMA4,
cr/,,4d,tcaj^,
Qs\
thereexlstsq
sufrcietly
Ptoof.Fot
Q6)lDylz=
Wenotc that
irn, D.r;(ff .h,l);1 =.1 .,U,tdld'r.= r<o,
"--
{-rf/2
,t vz
for sode Ko > . ,HelasqJifi,h,ir,oheano
@aDcrq$b Xd-di, ll > +{<o(N+ 1).
,
thed'wo1h€r/o,.iioDi84v.
ffi, i$,thst
-"'mE
>."o.9il1-|lP,[ll,
f'".
Iteg,RBit
Sordd)).Lh
ils, ,w,
2 tiiid'l ti;iia*w
\Wfurt aq eigqtuBke
I of (21dtd
't,!W:i@i
a,s,6rld;ni;il. ,
"*'ran,*
- b,!E01tu4n
Iot h
714
IIERBERT L. DIRSHEM
,nhereC is sone positil)econstunt,{i.L\X=| are the eigenwluesof the tridiagonal
matrix A oj coefrcients of Lh, andy is the uector\J
J, .., J"(,lt/'?xn)).
"(,t1/':x
ProoJ:By Lemma2,weknow that the hypotheses
ofthe theoremare sumcient
for the existenceof a positive definite diagonal matrix which symmetrizes,4.
Lemma 1 impliesthat
(28)
minl^
tl = D(A - 1I):A1,/Drl2.
By L€mma3, therc existpositiveconstants.r and c, suchthat
1.,r z.
Butbyhypothesis,
n + 2 > 0,sothatLemma
4 impli€s
([',3dj
(2e)
D J i2 > K N ^ + t t ' z ,
)D\A
j:2,
^t)yl,=c,N^+u,\@-ltD
.,N.
-
for someconstantK independentof i and & sufficientlysmall.
We combine(28)and (29)to obtain
min A, - ,l < :l tA
K
ltty
' _ - CttA - ).r)tt-.
"
This compl€testhe proofofTheorem L
Theorem1 can be appliedto the diflerenceoperatorslt2) and Lf)to obtain
the followingcorollary.
CoRoLlARyl. The eige aaluesl{k\[= | oI the tridiagonalna*ix A oJcoellidents o! Lfj (respectil)ely
Lf;t) are suchthat,for ahy giueneigenralue). oI l2l and
Jor h sufrcientlysnall,
min,Ar-,1 = O{l':).
4. Numerical experiments.In the numerjcal experimentsconducted in
relationto this study,the smallesteigenvalueAr of the threedifrerence
operators
Lrtt, Ll"', and Lltt werccomputedwith error no greaterthan l0-3. ln Tables I
througb 3, th€ amounttheseapproximationsdifferedfrom the small€steigenvalue
,11of (l) is givenfor eachoperatorfor ! : *, j, i, and N + | = l/h = 4, 8, 16,32,
64, 128,For any two successive
valuesof N + 1, a valu€ofthe experim€ntalorder
(EOC)
oi convergence
of the error as given by log(e{N*rrr/cN+r)/log2,where
:
?x* ! l,l, Arl, is also tabulated.The experimentalord€r of convergence
is
the power of I by which the error is decreased,
computedfrom th€ valuesofthe
erroral a givenvalueofl and halItharvalue.
Thehypothes€sofTheorem
I aresatisfiedwhenLi isanyofthethreeoperators
discussedin S2. Hence,we expectan error bound for the €igenvalues
ol these
operatorsand any eigenvalues
of(l) to be givenby TheoremL
Sinceby (8),Ll!)and L arenol consistentwith resp€ctto the classoffunctions
,4, whichcontainsthe eigenfunctions
of(1),Theor€m1do€snotimply convergence
olthe eigenvalues
ofl,t')to thoseofl. The resultsdisplayedin Table I indicatethat
such convergence
does occur in the caseof the smallesteigenvalueof (1), with
apparentorder of convergence
2!. Suchbehaviorofthe eigenvalues
ofLll)can be
verifiedby a techniqueof weinberger[7].
Corollary 1 isillustratedby the valuesin Tables2 and 3.
715
APP&@xtr{![MOllrlNjIlA{EE1@lBrqENcEs
IL
- 3,E28E
- r I
-6.9SE'* 2 ]
- 2
- 1,0058
l6
2.80
9!04 - 5 l.
-3.12
€.096r- 4
0,?l
32
64
49468 - 4
128
Tr!I.32
otdq' d co tt'lstr,4. kr thr shl4tP't.t|.wv//,'.s d L\'t o$nL
-49704 - I
193
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