A TheEis
SUbni ttF.l
f.r
+hF r'^.',1+r'
I
Purdue University
'bv
L1
ACXNOWI,EDGTIENTS
r wiah to thank Professor
research and for providing
intEgral
come an
r!
my
nany of the ideas which have be-
of thie
theEls.
I am grateful
fo!
has devoted to digcuEsion and reading of the
the time he
trrarruogr
part
R. E- Lynch for directing
u .
I would }ilie
to thank ny alway8 understanding,
Eonetlnee neglected,
wife,
though
Kay. f,Qr hef suPport, both noral
and financ ial.
I
acknowledge the fi,nangial
aEsl-stance of the United
StateE Government through a National
F9l]owship,
and Honel4rell. corPoration
corporaLion
FeltowshiP.
Defense Education Act
through
a llone!4''e 11
rLtr
Page
v i :
CIIAPTERI :
CH3PTERII!
2.1
2.2
2.3
PREL
fhe Differential
The Difference
Rel-ations B
Diff,erence opera
CHAETER III
1
XMRODUCT
I APPROXIMAT
PROBI,EM
BOUNDARY-VAI,UE
Dif,ferential
r9......
and
OF SO',I}TIONSOF A
.,
BESSET'SOPERATOR.
3.1
3.2
3.3
uatl.on ProDtem.
rhe Pifferential
a t i a n P r o b l e n . . . , ." . . ;
the PJ.ff,erence
E
T
t
r
9
r
naryEr9.
Cruncation
3,4
Proof, that
3.5
An Error Bouiid..
CHASIER IV:
4.1
4,2
4. 3
4.4
4.5
L;"
BESSEIJEI
r... '.
EXAC{ E
A BES6EII DITFERENCE OP
Properties of !h
RePrggentation o
Eiqenvaluee....,
17
l9
26
2A
30
':
Matrix Theory.
A General Found
f the coefficients of q. . . . '..
Typical Behavior
Error Bounds €or i g e n v a l u eE E t l m a t i o n , . . ; . . . . . , .
A n a I Y € i s o f t h e runcatlon Error for a class
ratols...,....
of Difterence
Dif ference Sche$e.
Setf-Adjoint
Errof, Bpund for
CIIAPDEBv!
5.1
5,2
a l 4 o n o t o n eO P e l a t g r . . ; , .
16
39
40
IONS AND EIGENVAI.qESOP
roR..,..........,,.,...,....
Eigenvqlues. .
Exact Eigenfunctions
and
60
J-V
C H A P T E RV I :
NUIIERICAL RESILTS..^..,
6.1
6,?
Rate of Convergence....,
ExFerlnental Rqsu.lts'...
6.3
Numerical Technique....,
64
65
' 76
.t.9
h
Table
1.
o
Errorg and Experinental
f,or Exanple6.2.L,.....
Rates of Convergenca
'
" " i : " : " i i " '
2.
Errors and Experimental Rates of Convergence
f , o r E x a m , p 6l e. 2 . 2 .. . . ; . . , j . . . , . . ' , .
3,
Errors and Experlmental Rates of convergence
6.2.3.,......,.......
fo! Exadple
73
vi
Ph. D., Purdue University,
tlersbern' Hgrbert l€wis.
opof
APproxination
Besset'E Differential
1969.
August,
Operators erator of trraqtional order by Flnite'Diffelence
Rob€ft E. Lynch.
t{ajor Professort
Problena are etudiEd uhose solutions
Diff,erence €quation
of ths solutions
arq estinatea
'
yalue. probrem fy = f(x)
value probl$r
Jy - fxlv
thg Bessel op€rator
L_-
O < v < l.
structad so that
, y(o) = o,
v(f)
= A and the eigen-
- O,
y(1)
. O, wh€re I is
, y(o)
of fractional
diffeience
thr6€-Point
\p(k)ttr)
boundary-
of the ttto-Point
oraler,
i
= -xt d" /dxt -xd/,t*+v" ,
op€rator3,
- r+(klt*r),
Lh, are con-
k = 1,2,3, j = r,
is a given eet of functions and
...,N, nlrere to(kl l;.,
xr=i/(N+l)'
.
For alifferential
eguation boundary-value problens whose
solutions behave llke
.
ax- + px""
BeEsel functlon ity), it
as x - o (that is'
like the
i8 sho\"n that the operator constlucted
by choosing g(k) - xv+I-l
and t have trqncation error which
i9 bounded, and the solution of the resutting
difference eq-
uation probleB and the solution of the differentr.al
problem have discretization
error
which is o(h')
eqqation
as h - O.
respect
to
tha
ef,al differenqe
sati6fy
€i-genfUnctiorig
oPerat9rs
of t
obf,lesp9nding
to
).
are Ehown to have coefficients
the prescribed conditions,
Sev-
which'
and for these, mln,lI-A,
I
ie sllown to be O(ha)
Exact ieFreEentatioiri:
of the eigenfunctions
valueg of one of the diff,elence
r]
f\
operatols
and eigen-
are obtained
in terms
In this
lnvolving
single
treat
of Problems
work rte approximate aolutions
a Beesel dif,ferdntial
regular
€ingular
the tlto-point
(1 - Ia)
point
operator
at x=o'
boundary-value
t,
which has a
rn Palticura!,
ne
Ploblem
Cy(x) . f(x)
,xc(0,1)
and the el,genvaluQ probtren
we conelder.only
pr9blerne fgr which there
one Folutlan y euch that
(r - 3 !
i9 one and anly
2
vrhele m is
some potlllve
solutiqna
qf certain
technigues
'
O <m<l
lees smqothness were assluted,
If
of ana1y9i9 wguld yield
of convetrgence. We treat,
amount of smooth-
equatian problems as the
dif,ference
negh length h tends to zero.
this
0(h2) convergence to y of
we obtaii
neEE of, the salution,
.
nudber. with
a cortesPonding
in palticula!,
lower rate
cases in 'rhich
s o t h a t a l e r i v a t i v e s . o f y ( x ) a r e u n b o u n d e da s x '
fne e:tample of, Euch an gperator
ls the BeEeel operator
whj.ch !^'e tleat
0.
in detail
defined by
of orde
I
(l--4)
f
ry(x) =
-n-y"(x, - xy' (x) + v-y(x)
l
6^e n < r' < l.!
w e a P P r o x l m a t et h e s o l u t i o n
In Chapter III
to problen (1-I) with x defirred by (1-4), a=o in (I-Ib),
Xn qhapier IV, we aFProxinate
.
(1-2) for J given b,
(f-4) '
the eigenvalue
and
Prablem
In both cases, the solutioo,
Y,
is of tbe forn (f-3) with nt=Y'
we congider approxit$afione to the Eolution, y' af, a.problem Euch as (l-l)
r\
obtained by finite-di
ffelenqe aPproxiBatian
nesh
on an equalfy
1,...
ii
o p e r a t o r s , L , a l e s t u d i e d w h i c h ( in.
mesh points.
t at the interior
, j = 1,"',t't,
IJu. = f.
l
l
(1-6a)
u N + r= B '
(1-6b)
w h e r e t j = I ( x r , , iE taken as an egtimate
.4
).
'ooi-nts x.7 ,
A qammonnethoC f,or abtaining
vative
tral
ai each intellor
Then if
r B e s hp o i n t .
v
I/ !6 to !€place
oPelator
in thE Fecond-order differential
divided difference.
of,
y€cr[0,1],
Keller
each de!i-
t by a cen-
Sv - Ly = O(h'?)
(r91, p.?3i gave an
exanple 9f guch a sshene aPpl{ed to the problem
(1.-?a)
(r-7p)
-y"(x) + e(x) y'{x) + r(x) y(x} = f(x)
y(a)=e
,a < x < b
,v(l)=F,
,dhere q . r , f a r e c g n t i n u o u E o n [ a , b ] a n d q i 5 p o s i t l v e t h e r e '
I t I F shgwn that
irl
for eucb a dJ.fference. operator L, the solu-
t10n u sf tbe corrQFpanding diff,erence equatj-on problem is
n
l' u). - -v (.x . ) | r c n ' at each nesh point and fof
i
smaLl , \^there c is sone positive constant
h sufficienlIv
such that
a1l
j-nale-
pendent of li,
An oxaqpte o! ei.genvarue approxination
raallar rfc1. nh
(1-8a)
l?1-l?qI
for
fha
^i^h
was. considered by
l an
(p(x) y'(x))r - q(x) y(x) + trr(x) y(x) = O, a < x < b,
, Y ( b )= o ,
Y(a) = o
(1-8b)
h ' h e r ep > O , p ' a n d q a r e c o n t i n u o u s , ! > O a n d q I O o n i a , b ] ,
lie proveC tlat,
c'Ia,b],
f^\
for
with
thq eigenvalueg of the dlfference
flom leplacj.lng all
differences,
derivative8
in
ej.genfunctionF
(1-8a) by centlal
eingqlu"
order at eiqher
(L4J) extended Kelle!'s
in
operator resulting
oonverge to the eigenvalues of (r-8)
Bulirgqh
to th.
the problem€ (f-8)
result
divided
as h',
h - O,
for Froblen (l-g)
case where q and r eaih have a pole of filst
of the enalpoints.
values of, t|tre dlfference
ploblernr
ioblen, conEtructed
constructed
verge to thdse of the dlfferential
Under thg ahove conditions
A tbree-Foint
He proved ihat
finite-di
problen
on q and r.
ffarenqe
the eigen-
above, conas abov(
aE
(I-8)
as h2, h - O.
the eigenfunctions
opgrator
are
L, which approxi-
n a t e e a s e c q n d - o r d e rd + f f e r e n t i a r e p e r a i o r x o n ( o , 1 ) c e n a l s q '
be constructed
p'-'
tp'-',
O"',
in the folloning
\iray.
be a Eet of function8 i,rhich i.s linearly
indepenalent on any set of three
(0,1,).
Let the functions
points
dLEtinct
Then L is codstructad so that
r , r ( k )( * . ) = s o ( k )( * . )
l!-ql
l
,k = !,2,3
l
for every mesh point in
(O,I),
x i = j h = j , Z ( t r i+ 1 ) .
A c o n m o nc h o i c e o f t h e s e f u i c t i o n s
!'or J of the form (Li7a),
identical
Allen
exponential
examplee.
i"
divideq
([!])
constructed a diffetence
lkl
the O""
He gave no proofs
eqch derivative
oFerator by use of
and gave gome numerical
of convelgence.
One expects rapid eonvergence of, the solutian
difference
equation
to the Bolution
equal spaced nesh, provided
the (,''-'.
derivatives
that
the latte!
over the interval
of the
of, the differential
tion with operators conatructetl u"ing ,(k)1*;
approxinateal well
way is
approxLmation.
difference
f,unqtion€ faf
= *k-1,
r(k)(*)
the L obtained in this
to the one obtained by replacing
by a central
= *k-r
eguaon an
sol-ution can be
6y Linear conbinations
of
For example, any function '^,ith thlee continuous
can be epproxinated
of lengtlt ?h by a qua4ratic
(1-l)
in
and (1-2) witb
!o order hs over an intelval
polynoni-al.
Fingular
operqtors
But for ploblems
whose y are Eucn tna!
6
x
- u- y € C ' [ 0 , 1 ]
with 0 < v < 1, local polynomial apptoximaticn
cannot be guaranteeal to be as good as orde! h3. as h goes to
zero,
quently,
it
solutions
ical
of the derivative
near the singularity
is doubtful
that
at x=O-
probLens have
such difference
which cobverge as fast
Conse-
of a nurner-
as h?, and results
e x p e r i i n e n t f o r s u c h a c a s e a r e p l e s e n t e d i n E x a m p L e6 . 2 . 1
of Chapter VI.
time, the use of a set of O'-'
We consider; for the first
which does apploximate eolutions y for which x-vycc"iO,t1,
O < y < l,
to orcler hs locaLly,
the problen which results
X. as 1n (1-4),
from leplacj-ng the Bessel operato!
in problen (1-l)
(f!5) .
*itn
The function f in (l-la)
The value A in
(l-l-b)
oper-
by a finite-difference
ator constructed. aE specified by (f-9)
for k = 1,2,3.
we consider
In ctrapter III,
o(k) (*) = *k*v-l
is taken to saci.sfy
is chosen to be zero in order
to guarantee the boundedngss of the solutign.
It
in the ploof of Theorem 3-1, that the sofution
of this
is shown,
differ-
ence problem converges to the sol-ution qf the differential
pf,obl"en with
order of convelgence two fo!
f sueh that
- - - v - 2 -r r -L^ t r n , .
i
Lv,r.l.
The Be€sel eigenvalue
(1-4) and p(x) = x',
functions
L
,1..(I').)
are
problen
(l-2)
i{ith
t defined by
is considered in chapter IV.
the
eigenf,unction6
of
this
The Bessel
proble'n
so
that
y is of the form
the solution
y(x) =.E_ cr* x""^
k=0
(I-rO)
In Section 4.3 we ohtain
f,ixed eigenvalue
a bound on the distance
set of eigenvaluee
of, gl0l three-point
whoEe coeff,icients
satiEfy
.:+1^n
arr.r
given conditions.
J and Lh ale conEislent,
vergence of tbeir
.eigenvalues ,
to
in chapter xIIr
to obtain
values of the
difference
of
eigenvalues
one
(1-2).
L-
fhe bound is
.8.
result
This
then one obtains
of the difference
con-
/ lhe rate of con-
fufthernore
vergence is the same as the orde! of consistency.
j"s then applied
l
and the max-norm of th€ t!un-
n
if
operatpr
differeirce
of I"- and the Begael operato!
€hol^'ethat
betvreen any
probleB and the
of the 8e€sel eigenvalue
given ag a protluct of a canstant
.
,O<x<1.
This result
operatorE
discussed
a rate of convelgence of the eigenequation oigenvalue
rn pelticular,
ploblem to the
the pioblem lrhich results
florn leplacing t.by Lh cbnstructed as in (1-9) with
(k),
p i ' - ' ( x ) = xk' -+' r - "l , k . L t 2 , 3 , i s s h o w nt o h a v e e i g e n v a l " u evsh i c h
are convergent to any eigenvalue
(l-2)
of
with
order of con-
yerqe4ce two.
In Seetion 4.4r we conEider lhe approximation of the PeEsel
operator.
trl
gt"(*)
{,
,
= *",
by a flnite-all
Itl
g"'
fference
tt')
(x) = x'''
oFerator
l1l
und 9'"'(x)
as in
arbitrary.
(}-9),
vith
we Ehc{
8
l 2 l
that if
= x'
@\"'(x)
v + 2, then the difference
consistent
operator
vrith reEpect to a class
of the eigenfunctions of
all
flon
for any real ,l different
and Bessel's
v and
operator
are
which incluiles
of functions
(1-2), r,rith order of consistency
at least two.
r?l
For 9'-' chosen so that the resulting
convelgence of the eigenvalues of the differ-
is self-adjoint,
ence opelator
least
dif€erence oPerator
to thoge of
(1-2) rtith
two i3 shor,rnin section
S o m er e s q l t s
order of convergence at
4.5.
ale availabLe ih the literature
on exact
representations of eigenfunctiong and eigenvalues of Bessel's
difference
operator of order zero.
Parri8h ([6],[7])
value problen
e\Jt
- xr-L.
dlfference
Gelgen, Dressel, and
constructed the exact solutions
for the diffelence
of the ei'gen-
operatof, f,ormed by (l-9)
v].th
rhey sho\ted (16l) that the eigenvalues of the
problen convergie to those of the differentiaL
and
that the eigenf,uncttons also converge (17:l), both with ratP
^F
a^a\,afdon^^
,
Another study of approximation of BesseLs oPerator of
n o n - z e ! o o r d e ! b y f i n i t e - d i f , f el e n c e s i s g i v e n b y P e a r s o o ( : 1 l l ) "
Ee has shor^rn,by consideration
of the solutians
tion
i -1
xJ ",
of the j-ntegral
of the alj-fference equation,
of the dif,ference
equation
converges to the solution
representatLons
that
the sol-ui i )
f,omed by choosiog o"'(x)
=
of Bessel's eguation which is
9
bounded
at
the
origin.
he does
Although
the rate
he tloes not detelnine
of this
We extend some of the above results
tion.
Prove
converger l c e .
on exact represenia-
In Chapter V we PreEent rePresentations
values and eigenfunctions
considered in Chapters III
/il
c h o o s i n go " ' ( x )
The results
, r ' i - ' l- ,
= x''
of the finite
for
diffelence
the eiqenoPelator
anat Iv r,thich is constfucte'l
by
) = L,2,3.
of Eo4€ nu![6rica1 experinents
nunber o! the acheneg. qonstructad
in ChaPter vI.
convergencei
ln ChaPter IV
aPPlying a
ale given
l
CHAPTER II
PREI,IMINARIES
In thiE chapter,
problens
equation
we descrLbe the class of, diffelential
to rrhlch our analysis
alescribe the typical
diff,erence
the terns
error,
truncation
can be applied,
equation ploblem and deline
consigtency,
error
diEcretization
anal convergence,
t
We treat
i.,
oetlnect
l
rnha
niFFaranfir1
aibAr.f^-
the Becond-order ordinary
operato!
diff,elential
Dy
ly(x) = n(x) y" (x) + q(r) y'(x) + r(x) y(x)
o
rrhere the coeffi.cient
q(x) =
@
kPlqkx--,
eff,iciente
regular
k
functionE
o
r(x)
=*E.rnx"
are euch. that
Einqqlar
point
L
on 0 5x
3
at the origin,
and
1
lda
:c<!1na
operato!
+hA
xy(x) = f (x)
^r,idin
boundary-value problen
, x € ( 0 , l - ),
flt6
J has a
only singular point of J in [0,r'].
12-2al
,
K=2' K
the differential
We consider the two-point
!
are of the r o r m p
' (x, = -:-p" x ,
ie
+ha
-^-
1l
y(o) = A,
(2-2bl
we n@, tlescribe
for
Y(1) = B-
the behavior
of the solution
the caae $trich we Etualy in the follorting
For the case that
the coefficient
(2-2al
of
chaPters.
pr is non-zero,
direct
of
substitution
v Z
k
v ( x ) = x Z c . x
K=U
lnto
( 2 - 2 a ) w i t h t t 0 yields
ck for which this
equatlons for the coef fi cients
expanaion iE convelgent
fhe value of v iB a root
( [14 ], p. 197) .
of, the i n d i c i a I e q u a t i o n .
Bhus, ?
is given by
I
. r-
v = i \ L -
Pe
fi r - 9r1e
Pa'
only the case that
We treat
with O <,
sirlgular
4to
P2
rtJ
one of these values is between 0
and the seaond i9 negativg.
anC l,
tion
9rr
I'or this
case, lhe series
< 1 iE zero at x = 0 analthe series tith
at x = O.
v < 0 is
Hence, tbe uEe of a zero boundary cond!-
at x = O excluqes singular
Eolutions
pf the llonageneous
eguation.
If
the right
has the forrn f(x)
(2-2a)
Eide of, the nonhg$ogeneous equation
h c
k
= )t'kPoakr<" f,or sgme nonneltative real
nunber n, anC +f ao I 0 and n I v. one easlly
obtains
the
.
general
v s
= ar<
Y(X,
lr cL*
k=O
of p,q.r
and f
ary condition
zero if
is
+ *
k
m g .
zJ Dvra ,
k=O "
(provided the coefficients
ak
With 0 < v < l,
the
enough aa kl'?).
tend to zera rapidly
val,ue of y(O)
l i
-
bk can be exPre€seil in tenns of the
where the coef,f,icientg
coefficients
n > O and bo if
= B qan be satisfied
y(1)
2
(2-2a) which i-5 bounded at x=O to be
of
solutibh
L
m=0. Tne bound-
by ploper
choice of
the value of a.
For Fa I 0 and the qPecial
L
mcv, the solution
of the nontronogeneouEproblem (2-2a) has a logarithnic
Eingulerity
obtalng
'
at x=0 if
aolo.
a bounded solutton
the aperator
+ .fx"o,
t of
(2-1)
one
behave like
Thus, vre lnclude
ranges of values F?'
rootE af the indiciiil
and the ather,
of the forn *t1po"**k
ary condition
as x - o.
We incluale functionE
m > y+2 when 0 <- v f l'
the case m-/,
the caEe ao=o.
f,or aPPropriate
real
ane of, which is negative
l?^2al
only for
O 3 v < L
q1 and ro which yietd
O < v < L.
Hence, for
(2-2) whose Eolutions
We stualy Problens
ax". * Bx"-'
.)
case that
equatj.on
v, satisfies
f on the right
6ide of
for vthich m = v +l er for which
Eor euch Problens we uee the bgund-
y(O) = A = O for
the caEes that
O < v < 1.
13
2.2 The Difference
we limit
our stuily
to unifolrn
Operator
meshes on the l-nterval
L O , I J w a t h n e' s n p o l n r a x 4J ,
x r= j h
)
, j =0, . . . , t { + t ;
h= I / ( N + I ) .
. | | for any function
either
DI - , w h e r e
t . or,l ,
the interval
f defined on tbe donain
lo,fl
or the Eet of nesh
is defLned by
ll I nax lf(x, l.
x€D-
The dlifference
I
operatora L-,
nhich we treat/
are of the
form
, j '
IJLur = a u -r + b.rul + c{u. .
r
.
J
J
r
.
)
+
L
(2-4)
wnere u;,
J = v t J ' t . ! . r N"
have Epgelfic
i
l,
ls any sEt of reaL nunberg, and
choices of the coefficlents
of our choicep of coefficients
alepends on the ne*t
length
1,.'.,N,
a,b,c
a, b and c'
Eacb
in subsequent chapterg
h.
For a given rA as in (2'41 , we can take the Folution
(if
ii
(?-54)
exisEs) of
L"or
f(xJ) .
J = 1,...,N,
14
u' ^ = A , u - - - . = B
(z-)D,
AI+I
at mesh Points
as an apProxinalion
y(x)
of (2-2).
T h e solution
if, the tridiagonal
of
B x N natrix,
x = xr of the aolutign
J
(2-5) exists
and is unique
A, with elements
= a - , a - . .
K
K.I,X
= C " ,
K
k = 2 , . . . , N , iE non-aingular.
.
Definition
Diffelence
feren
2.3 RelationE
2-1:
O
rator6
For a given €et g of f,unct ions in the
alonaln of S, the truncat{on
.9-:9,,
1, [r,rith respecd t o t I o f
the opelators t anal L, i8 alef,inealby
12-61 7j (h r u) t 1u(x3l - Iu(xr), i = r , . . . , N ,u € t .
Def,inition
Fot a given set s
of t? the oPerator8 t anal \
dq|ain
flrlth
2'2.
respect
to 3l if
o!
f,unctions in the
are s a i d t0 be consistent
and
I t n l l T ( h r u ,l l = u
h'o
ru(&.
Consisteot Fchenes are said to have oraler of,
I--eaet!
[Eag! resPect to U], tf and only if ,
l l r ( n r u )l l - - o ( h P )
ash-0, u's'
Definition
2-3:
The aliacretization
problems (2-2) anal (2-5)
e.(h) Eu.J
\^rhe!e u ie
the
Definition
-..-:+
ls definetl by
,j = o,I'...,N+ 1,
y(x.)
J
J
solution
2-4;
error,
of
(2-5) anil y is the solution of
The solutions of probLems (2-2 ) aad
(2-5) are Eaid to be ggryeIgglE
lf
and o n l y l f
l i n l l e ( h )l l = o
h'0
Convergent solutlons
gL LeaEt E if
are gaid to have order 9f, conve!qence
and only if
l l e ( h ) 1=1 o t n P ) ,
as h - 0.
r-\
,
CIIAPTFR TtrI
APPRO)GMATTONOF SOIIIIIIONS OF A BOIINDARY-VAL9E
In this
chaPter we Etudy a trDrticular
boundary-
two-point
:
2
-x"
value problem ryith Bessel'e olt€rator l =
:;t
-x
:
+ v''
ft
f,inite-
rn section 3.2, vre ghon that a cordnon,three-pqint
differEnse apProxinqtion bas oraler of consletency Lesg than
I
two over a clasE of functione contalning the solution of the
l^
r \
dif,ferenttal
Equation problent in qontraEt, this sane differ-
ence apProxinatlon
diff,elential'
used for problens in\ olving ngn-singurar
operators yields order of conslstency two'
ive a new. threE-point differ:l:e
6how in Section 3.3 that
at' l€agt
problen'
is shoivn, in Section 3.4,
wj,th the faqt
'
arl
have o!'ler
{''
that
the galution
tlhe ll-near diff,erence
to be nonotonic ' and this'
t2l
are coneiFtent
i and {''
an't
of consistency
two over a cl"asg of f,unctions contalning
of, ihe diiferential
.
I antl
I2\
aPeratorr Ia-"
with
opelator
togethdr
order of
consiqteney two (ges!!on 3'3), i's uged to Ehowthat the
ali€cretiaatign errar lF arder h?'
tle
11
In
tanl.te
,o < x < I
iv=t
Y(r) = A
y(o) = 0,
(3-lb)
where J ig the BesEel operator
iy(x)
(3-2)
= -x'y"(x)
defined bY
. xy'(x) .+ YaY(x)
for sone fixed , ln (0.1).
Particular
problenE which are of, intereEt,
and for
14hlch one night ltish to attenPt aPproxination by finltedj.ff,erenceE,ale Be6ael'E Eguation
( 3 - 3 a)
iy = xay
( 3-3b)
y(o) = 0, y(l-) = A
,O<x<L
qnal the BeEsel eigenvalue
(3-4a)
Jy = Ix-Y
(3-4b)
Y(0) - 0
anil lra want te include
prqblen
,0 < x < L.
, v ( 1 ): o
these gFeclal eaEes in our analyeis,
The solution
of the
nultiple
(3-3).
of
function
tk
.rv (x) = xl
L
.tu(I{x)
sideB of
right
ZR
- ^
K=U
of
and the eigenfunqtiong
the functionE
, ' 'k
E c^, x--' .
.rr
(3-4)
are constant muLtiPles of
(3-3a) anil (3-4a) are'
in our analyEis
to incluae
we lrould lihe
theref,ole,
(3-4}'
The
the F!o-
- v+2
anal x
function
ducts of an analytic
}', of
eigenvalues,
all
for
I
k I f( v+k+l ) 2 v + 2 k
functions
f
of, the form
f 1x) = *Y
i a,.xk.
k=O "
ao I O, thEn.the Eolution of (3-ta) has a logaritlmic
But if
at x=0.
Eingularlty
Bence, we conei''ler only right
sides f
Euchthat*-Pf-Oasx-0.
The generaL
Ealution
of
bhe homogeneous equation
n
ly(x) - 0
v
<
+ bx-v for any conatants a and b.
1.9 ax
\^'ith
a
f aE ln
l
Fo! Prablen (3-t)
(3-5) with ao = O and the boundary canditj'on
Y ( o )= 0 , + f t h e s o l u t i o n Y ( x ) i s
(3-7)
glen direqi
11= !,13tt,..
v 9 . , k
y(xl =x
L cLrt
k-o ^
gubstltutlon
,0 < x < 1r
yieldB co arbitrary
and ck = lk' +zv]r)
\,
The value of co is deteg$ined by the gqgond
L9
boundary condition
= e.
y(1)
The aerres
(3-7) converges if,
antl only. if,
+ 2l,rk) convergea.
,_E,a,_(k"
K=I K
Our analy6is appliee to problons (3-I)
right
Eide has the farrn
3
v+N
s
+ S(x),
*!1u**-'''
f(x) =
,oh.r. ge*v+4c4to,rl,
solution;
that
x-v-Agec4fo,rl.
of the fanily
we denote such a function
&, that
the
aa an element
3
.,.L
l v l v ( x ) = -K=U
E - cK- x " ' ^ + c ( x ) , c ( x " " * c * : o , r - , o < x < r , '
3.2
The Difference
A conmon technique
for
to a differential
a difference
J is to replace
alivided-difference
then on a unifonn nesh (c.f.
the difference
and for J.., c1 = 63 = g.
v
'
Equation Problem
obtaininE
operator
by standard O(h')
(3-2),
f,
iB,
W e n o t e t h a t \ T . , ( f lf o r a n y p o s i t + v e ,
v
'
tives
Fo! this
*i^"**'+k + c(x)
where cex"*c*lo,l].
(3-s) r=
ie
o s x < O.
y, of tbe problen ha6 the f,orn
y1*) =
tion
in which the
chapte!
operator,
S-',
I).
approximaat}
deriva-
approximations
to
Fo! I defined by
obtained in this
way is
- u i _ . rr u . . ,
u. , - 2u. + u.rr
rrl
r-r
(3-9) {"u,
= -x2l --J-=!----------J---Js I : - x. ri
i+] }J
n
*
I
.., "jL
h"
2h
= (-j'
+U) uj_l + \2)z + v"l u. * (-i" -':j)
u5*t,
t,zu
_j.
j=r..,.,ti.
20
y € r,\d h e re 3 ts . gi v enby ( 3- 8' l . F or s uc hay , : | .l .
,
(3-10)
\-'
,
r{xrt = ry(xj, +h- t-i;y'
-lxj,
(t)-z.y
+ o(h-),
v
^. v+I.
Sinca y(x) behaveslike x' + 9(x''-)
where xr , < t < x..,.
I--r
l+r
then as h goes to zero and j remainE
a9 x goea to zero,
f i x a a l , w e n o t e t h a t y- ( o ) ( * , ) - 7 o ( n v - 4 ) - y ( 3 ) 1 x l. 1 = o ( r r v - 3 ) .
"nd
Therefore.
tr'l
L ' - ' v ( x - l = t -v ( x l. ) +
oth- )
,h-0, j fixed.
Away fron zero, the derivativ. gs of y(x) are bounded, Eince
1r-r
H e n c e , f o r a n y p o s i c i v e n w n b e rc i n
y(x) is in c"(o,f).
t 1 l
(o,r),
{"y(xr)
By (3-ff),
I and {"
l 1 l
order of consistency
Eo Ehow that
u€e th9 function
j
h-0, x. > c > 0.
= ty(xj) + o(he) ,
are conEietent
with
reEpect to &, '.tith
at leaEt v.
the above order of consistency
y(x)
=
"v.
At th"
firgt
is exactly
v,
meEh Point,
= r, 4'', 1g
r . ( r ) *lv = l - 3 ^ z v - L + a + u ' l h Y .
n
Tbe value in the brackets
non-zero.
in the above expres6iotr j.s alwal's
Since x- i8 a golution
of the honageneous equation,
'j:j
2!
ix'
'r
r1l
= 0.
at j=l
Hence, Ial-' x- is equal to the truncation
n
and so there exists
at Least one function,
for which the oraler of consistency
for any function vttose selies
For example, this
te!m.
positive
In
(4-l),
y
is also exactly
includes a nuftiple
of x-as a
i€ true of y(x) = J,,1).:'x; for any
v
real tr.
Chapter
IVr
we sho\ir that
the eigenvalues
replacing
it
oamely xv,
y. Since the
is exactly
= x',
is exactly v for y(x)
order
error
the
fo!
the
of a diffelence
eigenvalue
ploblem
problem obtained
pro-
differentiaL
operator
blem by an approximating
difference
operator,
with
bounded by the order
the eigenvalues
consi€tency
respect
of
(4-l)
of the dif,ference
to the eigenfunctions
form .1, (tr-'?x) for
some ),.
the
b!,
in
erro!
eigenvalue
converge to
and diffelentialof
(4-1),
v{hen the
operator
of
with
which are all" o,f the
difference
operator
used
( 1 )
is
r,.)'',
we hence have a bound O(h-).
truncation
vergence
nurerical
of
error
apparently
the
ej-genvalues
experinents
Ihe reason tbat
order
of
consistency
explains
at
\irith
di.fference
that
the usual difference
..ra J fail
least
apploximation
the
relatl-veIy
anal eigenfunctio!1s
(see Chapter
r,fl)
This behavior
VI
of the
sfolf
con-
obser!'eC in
Example 6.2 .l) .
to be consistent
tr^ro, rtrhich is
the
to a non-si-ngular
approximation
is
wirh
order
obtained
operator,
constructed
in
is
such a way that
where p(x)
that
at mesh pointa,
is any quatlratic
the values of l p ( x . ) ,
equals
polynonial,
{1)rt"rt
i8,
l,\*,o(x-)
= Jp(x. )
to yietd
One expects such apProxinationB
soLutions which alo behave tike
quadratic
polynonial
Poltrnomiatg loca1ly,
on equally
interpolation
has local accuracy o(hs).
cannot bs approxineted
good estimates
of
since
sPaced mesh
But edlutions of the form (3-?)
locally
to o(h3) by quadlatic
poly-
nomiala near x=0,
.
In geneFat, a three-Foint
difference
constlqcteal to approximate a diff,erential
oPerato!
can be
qperator
I for
r t l
any given set of three functionE
l T \
' o"')'
linear ly
i F d e p e n d e n to n t h e € e t o f n e s h P o i n t s x , = j h , j = 1 , . . . , N
+1,
Eo lhat
e Q \ , e p l I c R " rl = x e ( k ) t x r ) , j
"n{e(t),
= 1 , . . . . N ,k .
Given three such functions.
can obtain the coeff,icientE
of the difference operatar,
given by
u. = cluj-r * SJoj * TJuJ*r
by sgtving the system of equationE
t J ' L t r . . t L \ l
1,2,3.
23
( 3 - 1 2)
ILl
1Ll
lLl
d . o ' ' ' ' ( x . - ) + 'EI - r p " " ( x - ) + y . r D ' " ' ( x . , , ) = l o ' . ' ' ( x . ) , k = 1 , 2 , 3 .
I
when the solution
difference
k = 1,2,3.
is in class
the system (3-L2) leduces
in two unknowns.
only two of the function"
*',
*"r,
usealte conatruct. the aPproximatingoperator.
and x"".
This operator,
.
I
y+k= x - " I- ,
lk I
Note that when j = l,
l)\
of
a, we apploximate I by a
operator constructed as above with o""(x)
to a Fygten of three equations
thiE point
l
l+r
in order to approximate the solution
In particular,
(3-l)
I
r,thich r,re tilvestigate
Hence, at
*'*2
b"
""n
.
v
We choosexin this
chaPte!.
We uEe the notation
we ilenote lV f,,-'.
(3-l3a )
L ( 2 ) u . = t 'h. {- * ' , * v + r , * " 2 J ' ,
( 3 - 1 3 b)
h
I
f,of any €et of nqnbers
,j = 2,...,N,
l
I u j] .
ComPutation of the c o e f , f i c i e n t s b y s o l v i n g ( 3 - 1 2 ) g i v e s
IJ\
(3-l4a)
t?\
r t r - ' u , + ( 4 u + + ) [ q t - l { v u eJ / 3
( 3-14b)
2,,
-j .
-
1
1--J-1Y
'j
+ 1'
j = 2,...,N.
-Ln
teqns of x. = jh,
this
xi
..'
.
can be written
)
xi
,r .
|. ,. - - - ) , (-,n
I
) u..
-j = --JL \ *, r ,'/
n
l-I
^l " e
^
J
-
/ xi
l
,v
-2u.+i::a;u.,
\ x.r1./
I
J
'
r
-
J+I r
= 2,...,N.
we define
If
thE olterator
x bY
y(x) = -14y"(x) - x(2v + I) y'(x)
(3-15)
then lxzy(x)
= x'Xy(x) .
each derivative
If
,0 < x < L,
lte aPProximate X by replacing
by an O(h-) divided-di
fference
approxina-
'rre obtain R- given by
!lon,
( 3-l6a)
-u. _ + u. -
\oi
x.(2v+ tt( -Jffi)
-
= ( - J +j v + L J ) u , - t + 2 J r u , t ( - 1 " - j v - l j ) o j * r , j =
, we note that
for any eet ot real nunbere [u.i'
Irt
u
I ,I jt- ' j - u , = j - Yrvr uJ . , f , o r j = 2 , . ' . , N .
Hence we night
exPect
J
Itl
I{-'
to be a good approxination
a p p E e x i n a t i a nt o . x a n c t [ o , r ] .
to
on 3 since (h is a good
25
A6 We noted
ot 4"",
above,
the
= i-K u- : f=2,..
operators
,N.
\h and
h
"(2
) . r " re Iated
f o o b t a i . n a n ope!ator
'
for
r t l
rrhich tlriE correaPondenceholds even at j=1, we define Kl"
{rl
( 3-15r) xi- 'u,
= *i'oji)[r*'"lr] = tn, +
I
For three-point
alif,ference equation ProblemE $ith
values 9f u at the boundary Points
an eqqivalent
trialiagonal
ing leNna nhich
Ll
4) (u1 - u2)/3,
j = 2 ,' . . , N '
r ( 2 ) u . = x n- ol .
h
ly
natrix
j=0 and j=N + f'
Problem.
given
one has
we plove the follo$'-
we need in ctraPter V.
The matricee A and C, equivaLent to the dif!L
,"
rrt
and
K;-', r e s p e c t i v e l y , a r e s j , n i l a r .
li"
f,erenceoperators *J
x2
t
l
The matrix c is non-singular' and FII eigenvaLuaB of c have pos!ru9_1,!:
itive
real part.
natrix
D lE alefined by
o = diag(1,2Y,...,NY),
glen c - D-!AD,
fhe natrix
c ls irreducibly
wblch can be Feen frpn (3-16),
all
a)
of, its
therefore,
eigenvalues have lto8itive
1 " . 8 ,p . 2 3 ) .
diagonaLly dominant,
c i5 nonsj.ngqlar and
real Part ([l:],
fleorerr
3.3
Truncati.on Error Anal-vsis
por X anal
T,erhna3-2:
?
Irl
d,-'
Si'r"n by (3-2) and (3-14) and
u ( s s o t h a t u ( x ) = , - D ^ c , - x " + r+ c ( x ) ,
G(xv+4c4[O,I],
K=U K
r1-l?l
(
(J
t - o,1')l u(xr) = c1o(5Y+t)
* (".*ltl
{2)t ot*rl = (.s*!*r* *l*21o ( h ? ) , J
Pfoof:
^f
+
F o r u ( x ) a s glven,
* Y + 2 )o ( n 2 )
= 2,...,N.
we have, by the cons !ructi.on
Itl
?.r-'
n
1r - oj')l u(xr)= (, - {2)r r*!^ .**f*k+
since f,xP+k
2
l v 2 - 1 v +k) ] r.,*k
c ( x r) l
anar,l2)x!+k=
- k+v
3
t + r + 4 )G - 2 K )
w e h a v e & . n a l ( 2 ) o f G ( x t ) a r e b o t h o l h v + 4 1s i n c e c . * ' + 4 c 4 [ o , l ] ;
End hence
t.r - oj')l 't*, l. ril .*lv2 - l, + k)2lhv+k
-
*io "* $t'*rl tr-e*t nY+k+ o16Y+4;
1
: { - 2 v + I ) r r - - ' * .o j1ro, + t) + o(rrv+4).
= c1h
P+l
v+I
+ (, c a l l
+
xlrz) o(n?)
21
Hence, (3-17) holds at j = l.
Since K is conatructed
n
, defined by (3-r5),
the derivatj-ves
by replacing
by the usual divideq-di
fference
app!ox-
then
inations,
=
( , ,-( \ )
"t*r) $
j * l, ...
* r r , 2 . , (t 3* r)l * ! n 2 ' , ( 4( 6) ) + o ( h 3 ) ,
"N,
f o r x . - < t < x . . . a n d f,or any f,unction v(x) ( c 4 [ o , l : ,
I-r
Jrr
v(x) iq given by
I
in
v(x) = x-Yu(x) -
.nrtk + *-rct*)
Jl
,
rf
0 <.x < l,
w e n o t e t h a t u ( t ) ( * ) = ca + xg(x) ' f,or sonE function g in
(a)
c +[ 0 , 1 ] a n d. v ec [ 0 , r ] . Bhis followg 6ince c r x v + 4 c 4 1 ot ,l .
we have
l'!ar4 this
(
-
\)
v ( x i ) = c , x . o ( h ' ) + x l . o( h ' ) ,
gince i;'s111 =
l
"'N.,(") "na
l
)
-
L t , . . 1 L 1 .
r.l2) xYv(x.) = x;\v(xj)
fo!
ac noted i,n Seoti0n 3 . 2 , w e hqve
($
_ r.
(2)
I u(xj) -
x ! t r - r q tu t r r t =( " -. * .l ' * l * x . Yl * 2 1
2,...,N,
ThlE ee$FleteE
of l,enna 3-2,
o(h1),
Proof, that Ll2) is a I'{onotonic oeerator
3.4
linear
A non-singulatr
nonotonic since, if
arbitraly
u>O for
Tg>O imPli.es that
if
operator
T, iE said to be a nonotonic
operator,
A linear
u in the donain of T'
inverse
t{hich has a Posltive
operator
Tu = v>O then u = !-tv
and if
T-'>0'
is
then
T-1v>o,
It
!9g43.-!l:
l)\
>o for
{-'v
j = 1 ', . . . , N a n d v o = Y - r.r, = C ,
N
then v.>0 fb! j = 1,...,N'
.)
the natrix
!g99!r.6inee
that
itE
'
inverge
lE poeitive
Ehe natrix
tonic.
to a matrix which is ilreducibly
dominant ( Lenrna 3-I)'
diagonally
I
is sinirar
r{2)
operator
A equivalent to the difference
4"'
t"
we show
non-singula!'
and therefole
that
t2\
L;-'
is ncno-
Al1 has colqmns wittr elements ukj = ttjk),
11.l
w r r e r eu - ! K l, J = 1 , . . . , N ,
is tbe Eol\rtion Eo the Problen
J
(3-I8a)
l ' ) o j *)' u j *)
( 3-l8b )
J=r,....w,
N+.l
'r,.ru 6jk) = r ir j = r..na ojk) = 0 is j I k'
since
=
4''
'
{'
ixy
it
= o
,rl
ri'aB can8tlucted
r,i"
so that
- v
$x: o
"na
totto"" t h a t u . = * Y i " o n . s o l u t i o n o f
)
f
"Y,
u = O. Futhesnorer x ll e a t i e f l e s t h e b o u n d a r y c o n d j ' t i c t ' l
29
rdc have that
Therefore,
u o = 0 .
(k)
= a xv.
u'
for
j = 0,...,k,
some conatant a.
a function
If
f(x)
mesh, then the pair
i€ not ialentieally
of function€ *!..ra
by dlrect
dlfference
:
i
-U
aubstitution
*f
-. r
independent on the meBh. rf
)
eguatlon which iE satisfied
)
on the
are l1nea!1.1,
f(x.)
xl f(x.) satiEfies
l
l
n
and slmplification,
L,y = O, then
-
we obtain
the following
by f!
v - \ 1 f . . ,+ 2 ) f . - ( j + v JJL
congtant
J
l) f-., "
= ( J - y - L ) ( f r - f r _ , ) + ( J + v + ! ) ( f .- f . . , ) = 0 ,
J
J
L
J
)TL
j = 2;.'';N
( j + v + L ) f o r j : 2, end b<v<I, then
s i n c e 0 < (j
.
.
l+r
samesign.
teke f]
AlEo, f.
l
- - f. have the
l+t
l
f i€ a etrictly
= l a n d f e = 2.
inc a9q 9i nt
- f . _ l and f,
JUnCEl'On t
monotone functj.on
T h e n f i s a poBitl.ve,
EhuE, nith
thi€
f,
I J - r U , C a n !e written a s
lr.l
u l-"
l
r(ilf'
= |.
I t j r r t [vb + e r J ]
jsk
J:k.
strictly
of
j.
Wa
nongtone
the general soLution ef
To satisfy
b = -cf--..
the boundary
- - - _ _ - _ - - , . - rlll
N+]
r condition
For both definitions
N+I
at j = k. we take
requj-res
c =
that
a.:.u
of utT'
- f"*r).
" ] l(fr.
= O,
- , r.
to be consiscenc
(2 (k)
r h e f a c t t h a t r i ' / d)
1
+ vx + 4xl (fx - f:.*r) ]-'l wnicir
[t'tv(t'
fN*t > f,k. The coefficieot
a is positive
"ir".
is equal to the product, c(f* - f**r),
of rwo neg-
is negative
since it
atj.ve nunlcers. Therefore.
j > k, we have
t 1 z)
u).', = (jh)-
c(f . - f"*r)
J
this
wtlen
> O.
J
conpleteE the proof
o! Lerula 3-3,
An Elror
3.5
We define
the two-point
I t t
operator
difference
= tt*.1
{2) o.
( 3-I9b)
uo - 0,
I)l
where Li-'
show that
"utr
solution,
y. of
(3-19) cgnverges to the solution
-
The solutLon
(3-1) with
We f,irst
af
sj=2(jh)Y- (jn1v+r
0.
(3-19) converges to the
order of convergence at least
define
of
Eolution is in g with cr '
y€5 and cl = 0.
Proof:
the
In the proof of Theorem 3-l? we
as O(h-) when the latter
Theolen 3-l-:
problem for
=
the sol-ution of
^,- e,
boundary-value
(3-14) .
is aE ln
Bound
wir:.cir LE analogous t o ( 3 - 1 ) , b y
4-',
(3-rea)
(3-l)
O for j = 1,...,k.
uj*'t
the nesh functj"on
9 by
j = o,L,,,.,N-+1.
2 when
that
We note
-
r. t
|
,r'l
+(4v+ 4l (ih)
L"'q.=
"
o , l " , : - r t z " ! ' - * 1 * r )= ( 2 v+ l ) ( i h ) v + r , j > 1 '
So g is
a positive
function
on J = 1,.. ',N
qrror,
the qigcfetizati.on
J
t2\
r4'hich Tr;-'
t
o'
"j
e, rE
) = 0,1,..',N+ r'
6 . = y ( x . . )- u *
)
for
J
of the problem
fhen e iE the solution
I
n
j = 1 ,' . . , t r
Eo.0:
. N * 1o o
For canvenience, we drop the palanete!5 of, ?
'
throughout th€ re-
y'
nainder of lh9 proof, Einc€ they are a!'ways h and
FY Lirnrna3-2' we hav6r elnce c1 = 0'
r. = 11" o1n'; '
We define s bY oJ t
-ilt
_L
xr
J
iE
1,"',N'
r. and theref,ore o. = o(he)'
)
7
vte note
I
that
of
e is also the solution
- frf i
*
,^,
1 ,n( z ' gl .= ol .
j
j=
,
r,...,N,
e ^ = O , e - = 0 .
N+I
--J-
w by w. r lloll sr -.r'
wedef,ine
)
fo!
J
/tr
s i n c ex r + I 4 ' ' n j
^
J
J
j = .1,...,N,
o,l'' ",
--_I_
v+1
xi
> l l o l l - o . > o
J
,
j = 1 , . . . , N .
Theaeforer by LenunE3-3, lte have
lle define v by v, = lloll
J
9, + er, we once again obtain
'
J
J
1
-- j9 + 1 . I i - '
v.i > 0
J = 1,..',N
,
and, by r.emna3-2,
(3-zr)
u . = l l ql l g . = e r > o
J
I
t t,
J
J
,
j=1,.'.,N.
33
combining (3-20) and (3-21). we have:
l ;'
lurl'llole
lloll llsll
j= r,"',N'
But since lloll i" orderh2 ana llvll is 2, tnen
llell = o(h'?)
is
and the Froof
cornPleted.
we observe that
y,
of
(3'1)
is
if
-)x-z-v
f(c4[O,I],
of, the necessaly
form for
then the solution.
Theorem 3-1 to apply'
In tbis
eiqenvalueB Qf the Beasel eigenvalue
,\ *,- r 4 1
^\
,,
to the
chapter r"e cbnE'iiter approxinatlons
e ' ,( ^t i* 1 = - x € y " ( x )
*J
PlobIem
- x y ' ( x ) + r " y ( x ) = A x - y ( x )'
0 < x < I,
(+-tE,
.,ta'
Y\tJ, = u,
Ytr,
= v'
. We tahe as eqtirhateB gf eigenvalue8 I ot (4-1)'
the eigen-
. valueg, A, of
i = 1,'.,,N,
,
(4-2b,
uo-u
, tlv,' =-,
Where 1,* i'i a diff,erence
I it-J,
oPerator glven py
L' hu . = . 4 . u . . + p . u . + 7 Jr uJ .' r, r ,
J J
I
I l"r
j=1,""N
get 9f valu9g q'P'7'
for anv Eet cf nwlb€tF u., ana!gQmegiven
si.nce the ei.genvalue
we note that
probtem
(4-2)
homogeneousbountlary condlitl-ong. the coefficients
y,. can be set equal to zero.
(4-2)
tridiagonal
-*-I
x.
r,,.
rr
natrix
Problem
Problem fpr
the NxN
A whose elenants
rnat ie, if
ot anil
Tlren the eigenvalue
to the eigenvalue
ig equivalent
has
are the coeff,icients
of
A and u are an elgenvallre an'l cor-
J
ie€ponding eigenfunction
of
(4-2),
then
Au = A.l
_ . t l
The vector norrn I l. I l. defined bY
n
N
l , l l "= ( .KD= r- zr:
rIt
used throughout thiE ohaFter.
A natrix
norm €ubordinate
t h i g v e c t o r n o r n , i e , f,or any real g!4nnetric natrlx
A,
l l e l l " = l r ; a *| ,
where tr$ax a9 art eigenvalue of A wlth
largest
magnitude.
We nQw Prove the follort'ing.
Lenna 4*11 !'or any N x N Feal natrix
'.t.i.'-|-
A, if
there exists
|l
matrix
oefinite
a Dosrllve
.
then the eigenvalues
. .real nwber
nL!rk t.l<
D such that
.N
tlrll=f
DAD
_
I and any non-trivial
N-vector
symmeEllc I
and tor
A are rear'
"f
i3
any
y.
- 1r 3 IL.IP49i13rJ-&.J.-La
llDYlle
Proof:
gince,
the elqenvalueg
vector
r -
bY hyPotheers,
[ll.J
)
is
B = DAD
of A are a1t real.
lor any non-trivia I
y, l|te €el
(A - II)
(DAD .
y,
l I ) . D y = ( B . AI) PY -
eigenvalue o f A , t h e n (B - II)
tf I
s}4Nnetrlc,
is lnv€rtible
antl
Dy
ginae, hy hypotbeeis ,
(
t
(B - II)
- t-
j.9 evnmetri'c, we have
l l o v l l "t l l ( s - r r ) - ' l l " l l o rl ,l
I
< mar<-/i l-i-----i r
\
rr--rr s
lllr' ll"
./
mrnk l \ - ^ r I lor 1l .
I A - I I are non-zero by
and mi5
F u r t h e r m o r e ,s i n c e l l n v l l a
K
we obtain
hypothesis,
pvll"
l l n r f l _ a=--TE'1-1"
ll(pA?;',,rr)
(4-4\ mi.nu
I\- Il 't1frfl; =
If
proof
y.
any tr and any non-trivial
holds for
- Il
of A, the rninU l\
I is an eigenvalue
= o, so ( 4 - 4 )
Thie completes the
of Lelruna4:-1'
For any N x N leal
Lemma4-2:
matrixr
tridiagonal
A,
for \rhich the product a]<,]<-tat-tiX is positj.ve, k = 2,-",N,
natrix D such that olp-' is
there exists a pogitive definite
tri c.
syrlune
ror any n o n - s i n g u l a r N x N d i a g o n a l n a t r i x D ,
!I99!:
\^'e have, since A
the kth tliagonal elenent denoted by {,
\.tith non-zero
t h a t B = DAD-E: is tridiagonat
1s tridiagonal,
with
v-"
given bY
elenents
-l<
" k + t ,' k
-
-k+r,k '
dK"
k = 1,...,N -
If
bt
,k=
K+t
K,K+L,
Positive
numbe! and take
elenentE af D to be
d,
,
(4-s)
F*
b1,1,g = a11,1t
I,
we choose dr to be an arbitraly
the renaining
' bk,k*1
t:.,k
v-t
.k
k= I,.'',N-r,
4K-f , . = ( : * ) ' d ' - ,
K+L'r
then
B is
4,2
.
flnite
ical
We now examine
differeRce
in Chapter I1I,
completes
This
s\4[netric.
behavi.or
aPProxinations
(r)
Li-'
:r
t2)
and L:
tr
proof
of Lenlnla 4-2'
Coef,fiei-cnts of
avLo
lhe
the
of the coefficients
to.lj,
of
namely, the two s t u d i e d
operator
difference
For the
note that the ratio
j = 1,....N-1.
n
r.re
,r".q to obtain
of, coef,ficientE, 5,
4r,r'
aliagonal
lhe aynnetrizing
rtl
lr-r-' , g i v e n b y ( 3 - 8 ) ,
D is
natlix
tl4 =
. ' +, ' 3ri / 2
i'
"j*i
A s J { 6 ' t h i 6 r a t i o of polynomials
+
tends
one:
t2l
The correEponding ratio for the operator Id-'; which is
Yt
j-l--Ll--|
(
j = 2,...,
d e f i n e t rb y ( 3 - I 4 ) , i s A l : = \ F1 = )I 2 Y + r j - ; - : r ' ,
Ae j a o,
N"- l.
i+l
this ratio
ators whieh we conaider.as
ter
l
ia\
the ratio
ale Euch that
Y;
--..)-
4..,
wfrele I
/ I i
\ l+r
= l.
lF €Oqe
v
r(J) ,
algo tends to one.
All
thc oper-
aPProximatlgnE 9f I in thj's chaphas the f,orn
1
= J....,N :
rr
reaI nunber Bnd r ( x ) is a rational
function
qf the forn
( 4 - 5)
(x + e').,. (x + as)
r (x) * ( x + b
r) . . , ( x + b " )
foF sone poEttive integer s and aone sets o f v a r u e s f aK" l ,
;*
39
suctr that ak, bkt -2 fot k = 1...'.,S.
[l*]
night
examine all
we derive
Euch operators,
for operators
the eigenvalue e€tination
In order that we
elror
bounds on
lthich satisfy
such a
condition.
4.3
ErroE Bgunda for
Eiqenvalue Estimation
discussed in the preceding
l.-t
/.
i
rl
;- .j = = ( f i ,
r ( i ) = \/ l : if
vrhere alr,bk > -2 a\d L
itive
of the folln
section.
For any IJ- aE in
!g@.31!
opelatol6
for
We now prove the following
(4-2) which is such that
s
(j +aL)
J r.nrri14i'
some leal
o'
j=3,...,N,
nunberr there exlst
Pos-
constants cl anal C a such that
c1k2+!I 3 4
" ""t2*t
3l99lg:
From
k = 2,...,N,
, d " ) i s as in ( 4 - 5 ) f o r A , t h e n a t r i x
where D = diag
aEEocialed with
,
l
x.
I
-
, and m
( 4 - s ) we
have
b.),
K
n
K
We can choose dr to be any positive
it
nunber and we choose
to be
q-
' 1 6^ L / 2 - l
= /| a z: z
)
: rL , -
With thi6
choice,
.
r
.
then d! = 2' and hence
k
l4-7')
4* . = x " + -
., o 4
. r(j),
.,
k=2,...,N.
B y a w e l l - k n o w n F r o p e r t y o f t h e g a n m af u n c t j , o n ( : I 4 - - , p . 2 3 1 ) ,
.(x+Ir=xr,(Xr,hence
k
-! -!r ,
tp + a,) = r(p +k + arl / t'la.\,
and thug
I
(4-8) j!3
r(j) =
!
rrr;r+ai)
1!riffiffif{t'ff
l ' ( b i )=
- ,!,fsi#r ,i,itti*ifi '
o
ak, bk > -2,
Since, by hypothesis,
index I is sorne positive
An inportant
whj-ch follows
fact
constant
the first
K.
from the theoly
of the gammafunction,
formula ([lO],
fron Stirling's
product L'ith
p. 254-255r, Ls
= i"-b(r*6.).
+!++
r(l + D,
l
' vfhere {6.1 is a Eequence of valueg which tends to zelo as
)
j - o.
k
i!3
Applying
(4-g),
thiE to
we have
a-+.,.+ a - b.-...-b
s
r ( j ) = r q (r
r
"tt+o*),
I
n
where 6. r O as k i
t<
t.
If
we
combine this
result
with
(4-7),
we have
a i = r o . z * t (*r o * ) ,
I{l!h a choj.ce of cr - K i[f(l
k=2....,N.
+ 0k) and ce = K stP(I
+ 6k),
the proof is conplete.
LEnma
4 - 4 ! F o r y- l. = y- ( x . ) = J t f l * . ) ,
v
I
J
any real poeitive
that
there exigta
I = 1,...,N and
nunber trf and D = diag (d1,.,',dR)
a nonnegative leal
such
number m and Positive
;
'
a
)
numbers caand c{ such that
csj^"5drrc.j"'
,
l=r,...,N,
there existB aone positive
con5tant K such that
KN"''' fqr h . I / N+l sufficiently
small,
llOyll"
.
Furthermole,
nrl",,
for any vectqr rr, I lDwlla < crtl"" 'l lwl I,
proof! we first
examine llDylle.
we frave
N
N
N
'
llDyllS= r.Pr9i vi . ,.&zzai ,i .
vi
{
r.3rz:
*,,lll"*
'
^
> ^?rNzcrZ[
qu! \,re nole that
if
we let
N
t
..4
kAt,/p ,k .
N - .r, tha!
is,
the meshLbecone
ama J.Ie I,
u*n r ; " = l [ y ( x )] ' a x = r r ,
and It > 0 eince. by hypotheEiq, y is *r,(Ilx) and )\ > O so
tnat ly(x)13 ie positive
except at a finite
nurher of Foints
in | < x < L. Eenca, if we chooee.h small enough, then
N
yi b : klt.
If h is ehoEen that snall, theo
r.sczz
t
l
43
llovllB = "3(n/z)2tc*rl ,Az, vfl r' ' og(!)2t*rKoN2**l
'mererore,
' for
l l D y l l ? : K!f-
AI5o, we bave
N
N
E, {"K wfK < ma:<.
I lo1|'ll" = ,K=I
4'j( .k=IE- t( - N" wl lrl t"=
K
","-..1
= a ? N 2 t * 1l l r , l l ?
fhis
cdnpletes the proof of Lerutla4-4
l.rhich hae
n
coeffLci.ent€
suqh that
, i*u* .,
rlE;/
, J - 3 ,' ' . , N ,
vrhere a,-,b,- > *2 and 4 ia sone real nunber r and for any
K
K
eigenvalqe,
(4-l)
of
I,
and cqrreapondj.ng eigenfunction,
a
y(x),
if
,t +
ke(at-
bk) :
-2 and the veclor
r is deflned
by
" i-
=
'|
i ?1 t \ -
t'l y(x.)
then for h aufficientLy
*i\l\-
small,
r l < c l l rIl ,
t J
-
L t , . . ' L ! t
where c is soBe lroEitive
.N
are the eigenand [A_J;__,
',<
K3I
constant
va1ue6 of the trialiagonal nagrLx A of coefficients
"t h\.
'1
Proof. By Lemm€4-2, we know that there exists a posA. we define the
diagonal matrix D which sFuretrize6
itive
y Dy
veccor
,r
y = (y(rr),..,.,yh)),
y(x.) =,tu(I'x ).
a-t,
tnen, by f,ernrna
I
(4-e)
nirI.r -lAK- trl 3
since f = 1a-tr1)v' ay Lennt 4 - ?
'F1r6ra
avi
e+
h^a.i
+i rta
g
(a" - b. ) such that
consEancgc1 and ca anal n - t + -K D=I-K
K
hrt
- n+2
c ,. k " ' . - s d K
: s cq k....-
Bul by hyPolhesia,
, k - 2,...,N
m + 2 > 0 so that
ne can apply
Lenna 4 --1
to obtalD
(4-10)
l l o y l l " t r o t * h,
llo"ll, . ."
f,or aone conEtant I( and h sufficiently
"t*L
smal1.
ll"ll,
(4-9) and (4-I0)
e ean conbine
to obtain
ni\l\- rl . ?l lrll =cllrll.
lhis
conpleteE
of Ttreorern 4 - L .
the proof
4-1 to the o_p e r a t o r L (n2 ) -
We now appLy ftreoren
Corollarv
diegonEl natrix
-i1,
f,or any given
f*!e eLgenualueE t n a l 1 = 1 o r c n e E r r -
4-1!
A af coefficiente
eigenvalue
tr of
of -* ii''
(4-l)
are auch that,
)
and for h suff,iciently
snalI,
min*lr1+l = 01rr')'
Proofr
We have ghown in Sectlorl
4 | 2 t h a t for
the
operator {2) .
7k-t
ok
/k-l\
\ k , /
2 v +' L 5tgl.I.
k -y+ lt
For thie caee, in gheorem 4-l we have
AIeo, from leruna 3-2
= leex!+l+*f2) o(r,") , j = r , . . . N
u11-y
G-il2)l
o
t
l
l
6c,r u{f with cr=o,
thi9
c6=q.
BUt all
f,orm, and further,
Therefore.
all
eigenfunctione o f ( 4 - l )
eigenfunctionE
are of
are such that
'
l7rl=
J
| !
^
r"
-1,,r
l;'l-tr-\
i
t
4
--r-- r| - ^rlz\
, ulx{, | = vtn,
t
6
, l-r,-..,N'
J
and Theoren 4-1 imPlieg
rnin
ll -rl = o(ha).
K' K
ttlis
cortrpletes the proof of, corollary
4-1.
See ExanpLe 5i2.3 in ChaPte! VI for nurnerical resultg
of using
the operator
eigenvalues of
ta\
to obtain
t{-'
of the
approxinations
(4-l).
4.4 Analyeis of S_!re!,!g_3Ig9E_
for a CIa€8 of Difference Operator6
rn t.}}lg seclion,
rre 3!udy t}|e behavior of the local
,v
u +-,o(x)
2
r,
truncatlon error of, I and an operator r,'^tx-,xfor I eome functian in A[0,1] with @(0)=0 and the edditional
rt
resttriction
that the eet x-,
ent on any set of three
^hAr'+^r
(4-rt)
,. f*u
*V42
n
distinct
1
ie
t L
of
independ-
@(x) is linearly
Points
the
in
(0,1) .
forrn
1 1 f x v , x v + 2 , o { x3) . , , = o r o i - 1 * F i u j * r j o j * l
f,or any get of, real
7j
raf*l
tY
\^t
rtL2
x-'-,
are deternined
nunbers [u,]
as in Section
and coefficiehts
4.,
3.2 by ehe equations
/3j,
The
5 {*Y,*Y+2.c{*) I [a*l
. v
ILtx
I
'x
v+2
I
,,.
,9(x, ilaxr
J
v
uf+21 = 5;"*f * -Pxrv + 2.l"
V + 2+
+ Dx{
J
co(xr)1 - staxl +ux!+2teo
t1) i,
j=2""N'
for all a,b, c.
I € t u ( x ) = J ( I ' x ) w i t h u ( 1 ) = 0 denote
Eheoren 4 -2:
en eigenfunction
v
of (4-1).
t. of r, fx",t'-',o!x\
l|,-2i"1 .Iq
J
-
If I ts auch that the coef,f,icient
I gatisf,tee
, j=2,.'.,N,
for sotne IG indepenalent of h, tben for h sufficiently
(4-L2l
*5
sEall,
(x)].rlu(x.) - o ( h F) , j = l 1 . . . , u :
*'*'.e
[o, [*',
t
r
!
J
Rernark3 The etpregsl-on
-i
$
I
l
is lbe truncation etrror ?J aE in Theoren 4 - L .
Proofr
i n
il
I
since 5{xY.xY+2,,01u, ls tt e
eame as
trl
u1 , by
{-'
3 -2
I€mra
I
i
tt"f*',*'*2,9 (x)).slu (x,)
t5t*',*'*2,p (x)] -c)ur = xlo(rr')=
91i.,2nP1
,
7
elnce u{l witJl cr=o. trense (4-12) is true at j=L'
48
The coefficients
0.
I
and y.
I
of It [x- , x
n'
v+2
, 9 ( x ) , c a n be
expressed in terrna of B- ag
J
L+2j
; ,- , r --':-:=
(:J:)
(r+l) -i JI
a.
i- 4 7 '8.+
7
|
I
l-r
j = 2,.'.,tr
(4-r3)
(v+r)il
v.
' 1 = tJ-,tulL;?i
F--)
J+! L +f
I
If
we uee Taylor'B
and v (x- .)
[treoren with
about x=x.,
coef,f,lclentg,
j
= 2,...,N.
remainder to expand v(x-,,)
lrhere v(x)
= x-v u(x),
and use those
$€ ablain
T .
l
_v
-2
us
i, u, "' "i*'
lru,-*
- . ' ( 4 )( o l ] , j = 2 , . . . , r . r ,
"l["(a)(,i)
| . € < n ( * J-T.+ , .
where xj _1
x.l v(3)t*rl - p.o(a)1e1
We note that
there exists
a conslant K1 Euch that
u'l
l
.2u+L
*
u:
x.
)
1
-
*1*vk724"2k|k'-
k \-x " o *1
< v"-'K,
I
-
f,'he hypothesiE
of
the
theoren
e
- ^ _ - t !' 1
, | = :E121
2x.-. '
2i'-
l' I
implies
^ |
,K^
21'
that
K.
"
2x'
ano nence,
h
a
l
1 - - . i + 'jI = +
o- ( h . ) .
2x'
x:
give
tvro reEulte
fhese
that
Eide of
(4-14) is O(h').
v'-'(x)
= xo(l),
side of
(4-14) t6 o(ha).
yield
the firgl
term on the light
rhe hypothesiF and the fact
that
that the second tern on the right
Thl6 conpleteB the proof, of,
Theorem 4-2.
Thls analysis
b]' (3-I
of,. t-fxv ,xv+z,9(x!
n-
)) as a slrecial
t?\
For L'-',
n
incrudes r*
h
',f
theoren,
of the former,
u
c6x--rc2x
',
rL[x',x'
t,!
we expect that
of the
J in
the operators
(4-l)
since neaf the singularity
lhe eigenfunctions
V+,-+O(x-u+/L
-).
-l , o ( x ) ,
1[*',*v+z,o(x)
vJ€ could have used those operatore
clae€ do not approxinate
equation.
1
-.
9. = 2a't
form 1,,.[x",x"-*,@(x) ].
latler
19i.,.r,
case r,fith the choice rp(x) = x'
Instead of, using the clasE ofoperatot"
in bhis
(2)
of
(4-l)
The trq.nqalion
as well
of the
as those
of the diffeleotial
behave like
error
for
opelators
and J vrith regPect to any eigenfunction
]
50
fr
u(x),
(4-l-).
of
proving
qan be found by the same proqedure used in
rle obtain
Theoren 4-2.
h
-
,41
an expression
|/1\
( 2 y + 1 )[ v ' - ' ( ? ) - v ' = ' ( f ) 1 J
+'i
analogous
, j=2,...,N
I
quantities
where all
are as defl.ned f,ar (4-L41.
If
the
aaEunpc:.on
'l 'B , - 2 J ' l < $
1
, )=2,.,.,x,
iE rqade as in Theoretn 4-2, vre find
o, * t#
o1, :- 4
X.
X:
1
-
Henoe, for
)
j fixed,
-
7
tha!
oI. = * l o r r ) .
-
we have for
l
the flrst
tern of
F.n-
a.
i,,
(u:l - :r' ul) + ?-} = *1-' o{n") ,
(1- ;+E)
.
^
j
,
^
:
)
J
-
. , . . . , 1 1 .
(4-15),
to
As h i
0, then t..
r faila
Thefefore.
j = 2,...,N,
If
j fixed,
can be n o b e t t e r t h a n O ( h ) .
o(ha), unless B. = 2i2,
to be unifornly
)
in which case p(x) = x
ue constEuct the operator
is any real
-
v12
l,-ix-,
n
x
v+2
p1
, X J, wnere I
fron y and. v + 2, we find
nurnber diff,erent
that
yr,
l e , - Z'i ' 1 .
I
a s s u m et h a t P - v i s a n i n t e g e r
'B ), i s s u c h t h a t t h e r e e x i s t E I G 6 u c h t h a t
j = 2,...,t1.
tf
(trrositive or neg?tive,
but different
'i-l
i
-dj
/
i"
the forn (
iiI
"t
'
we have not shol'n lhat
3
4-I.
f,unction r(x)
i,-,
we further
r(j),
in qeneral
has the conditions
is as in (a-6) r and | =
r(x)
:.1
then the ratj.o
as i.s required in Theorem
the above rational
nece€aary for Theorem 4-1
or for what values of-P,
to apply,
from 2),
if
any, it
(.X - bk):
does not.
If
-2. then Theolem
4-1 appliee to rtr{x', *'*2, *P1 and, in particular, lrll = o(h').
Error
4.5
Bound f,or a Self-Adioint
we now treat
the special
Difference
Schene
case of the operator
.v
\[x',
v +- ,2
x'
9 ( x ) J , c o n a t r u c t e d e o a s t o m a k et h e t r i d i a g o n a l
matrix
of coefficj-ents
Theoren 4-I
ot I
f. sy|Iut|etric,
* j n to ttrig operator
is applicable
We Ehow ehat
and that
l lr l l = o ( h " ) .
FirEt,
opelator.
.-
we note tnat
Fince it
-I I
x
is a self-adjoint
can be r^'ritten
in the forn
differential
r
t
l
t*t" t*' = ( - x u ' ( x ) ) ' + v" u ( x ) . .
x
Hence, it
ia reasonable to aqauttrethat if g(x) is chosen so
Y+2
that l - t v
x'.-', elx) ] ie Betf-adjoinr (or equivarentry,
\[x-.
i
so that the trLdiagonal natrix representing the finite-dif:
v+2
f e r e n c e o-p e r a t o r i
L - [ x - ,' x - ' - . o ( x ) J i s s ] m m e t r i c ) . t h e n
x.
11
.
l
the reEulting op""ito. Irhich we denate by
li
"n{*',*'*2,r(*}
is a gaod atrrproxfimtion (in eome aense) to X,
'do
not ieternine
the function g(x) whtch gives Ll,
vre conslruc! lthe .finite-ilfference
Echerne. fhe onerator f,fl
given by
I
(4-L6l
I i u , = d , u - , + 't l. u , + t - u , . .
n I
I l-r
I
I l+f
j=1,...,N,
,
for any
wtrEre the coeffieients
' eet ot nunbers [u,J;
)
lernined
by the relatlo4"
P1" f(v + r)
r
.
rt * -(llY {tr, . rl
t4-L7to, =fr r3-r , u, --G'/t#j
,l, =(j$z
'j
are ile-
(2i -I)
r
\J+r./ (2j + r) (j - r)
-.
-4ja
)-r
. 5fif*+l
(Yr!)/i
\Y
(2J + 1) \j+1./ .
J = 2 . .' . , N '
Eheseare obiained bi firEt
coqPqting aj in quch a way ag tO
j,
solved (see [S], p. 50) to grve
(4-r8)
')
=
4J(r, + l)
t
{rj]i'l-;-ij-Pil + r)T
-
, - . ]- *z'tr
-
L
K-Z
J
j = 2..,.,N.
From (4:J.7) and (4-18),
i
(4-Ie)
P) =
An expansion of
( 4 - 2o )
rdhete
one can calculate
j-r
, .-2v
+e+lrl*
r=7n
!.
and
into por\'er series gives
I-lZ
i' -r -l
*d n"*t * *r, *
k=-l
less than sone posltlve
Tbe value of, the quantily
equal to the estinate
j-2....,n.
*g,*"" ti*rE-,
8
' l. . 4 l v + r l fL t' ..'- 2 v
lnrI is
" . ,. ,
to x = j by nsanE of tlte trapezoid
(4-20)
!..2v _
- 2 v +-,/j-of, xfron
rule
j + t equsl subalivieione of 0 r x 5 J.
J -
real nunber K, for all
in the braces in
of the integral
K.
applied
at each o t
since the graph
the integrand
is concave utrxdatd. this
bound on the value Jt/l2v
estinate
+ 2) of the integral.
[he value of thE sun in the braceE in
equar to tne egtrnate
is an upper
ot tne rntegrar
(4-20)
i6 also
- 2v+I ..Zv ot x
tron
/)
-
x = t to x = J - | by neana of the midpoint
rule
eacb of the J equal subdiviaions of \3xs
j - \,
the graph of, thg" integrand
is concave ugrard,
at
Because
thiE
is a lower bound on the value of the integral.
upper bound on the vbl,ue of, the quantity
applied
estimate.
Hence, an
in the braces
in
(4-20) i8 given by
l-.
L
t . .
2v+2t"
. . 2 V + Z
"
,,,2v+21|
\,,,
-lv=
= 2i v_+ 2
A conbination of thege last
and (4-20) showg that
two re€ult6
tbere are two c o n € l a n t s K r , Ke independent of j,
2 j a + K l s p. t 2jB + rh
J
-
- t J t . . .
such that
.
Thereforef . fheorern 4-2 appliee t o L - .
Wa furlher
lrhich inpliBs
(
l
r
note that far ,* t r ' --:-ia.--l-l
that lheorgn 4-2 aFplies
l
to L*.
n
= 3 ,, . . , N ,
T h u s , ve have
rrre eioenvafuesf\if; k" = l
coett!.clenEs
of
- I
ot.-
14 ale
--
^f
-"-
-----
+h6
i,i^i-
gucn tnaE,
3
(4=1) and for h sufficiently
lor
€nalI.
n l r r -l A - I l = o ( h ? I .
K I
.
K
gEe chapter VL Exanple 6,2.4 {or aonq nune.lical
!e-
Eults of approxinatihg the eigenvaluee of (4-1) by repl.acing
r
Wl-!n
lL
.
CHAP1IER
V
EXGENFUNCI
XONS AND DIGEITVALUES
A BESSEIJDIFFERENCEOPFRSSOR
In thie
chapter l'e again consider
the Bessel dif-
Irl
ference operator
n ti",
and the Fessel
' which waa constructed
operator
g = .-x'aaldi?
Eo that
it
.
xd./dx + ve
bave zelq truncalion
egor
for the functionE ^*' + btv+L+
"ry+2.
we obtaln rdpresentatione , for the exEct
a.hrcf arpitrary.
a{genfunstionE 4nd eiEenvalues of the atifference problem
j = r.....N.
In Chapter Mt
(5-l)
was,Bltownthai a9 N - F, the eigenvalues of
oonverge to the qigenvelues of the BeiEFI differentiEl.
eigenvalue problem
s y ( x ); \ x e y ( x ) , o < t r < 1
y ( a ) " 0 , y ( r )= 0 ,
l
these lepresentationE,
n
are obtained
by the sane technique
onJ-y the casE y = 0.
rreated
recurrenqe
of
(5-1)
relations
for
shgw that
usJd by Boyer
We show that
it
0 < y < l.
(l3l)
follows
th€ Leggndre functions
that
who
fron
soluliona
conbinations
of
Propertiea
5.I
The trialiagonal
natrix
genvectorg ldontical
(5-1),hae
the elgenvalues
of
(5-1) are reat and
(0,4(N + 1)').
in the intervat
,
for
of lhese functions.
We first
t
are valid
can be expanded in gerns of linear
celtain
l
rlhich
A, which hae eigenvalueB and ei-
to the eigenvalueg and gigenfunctions
non-zera soef,f,iclents
.
- - c -t r 9
'*,:<
*k '
of the Eiqenvalueg
of
gj.ven by
i
v^ d
,
"
u k , k * r = yn 7 katta
h = 1,....N,
txti,t< = ak+l '/ (k+1)'hE
k = 1 , . . . , N- 1 ,
lt\
where Li-'
r v r
J
-
is of t h e f o r m o , ! 2 ) r r ,= Q . u , . + f . u . + y . u , . n
I
I l - 1
I I
) f L
f r r . . , a ,
a,
F, y, given by (3-14).
W€ now 6how the
folLowing.
Theeren 5-l;
:Ehe natrix & has eigenvatueswhiCb are all
reelf positlve and boupdedqboveby 4(N + ll?.
By Lenna 4-2, A is sinilar
Proo{3
-
r\
to a matrix
slnilar
real Dart-
eigenvalues ttith
rthich has all
Therefore,
to a real
s}4runetric Br
PositivE
eigenvalues of A are real
aII
and Pos-
itive.
we let
c ilenote the natrix
n = dIaglL,2v ,. . . ,N") .
given by D-IAD, whele
C is the same lnatlix
This natrix
is the matllx
lras analyzed ln t enma 3-1, slnce it
r
lr'l
i e n t s o f : Ex -. Knt - ' ,
zero
I
of
elenentB
cr t
(5-2)
c.
3r
irl
r , r h e r eK " '
n
that
of coef,f,ib-
is given by (3:f6). The non-
C qre
(r+l)
L
-
,
l;7(v+I)
=
"i,l=k '"J,j*r 5#
. =
J 3 2 , . . . , N- 1 .
tN,
t{-l
=
"N,N
N
lt
v < \,
the4 each !o!r sum,
of c ie les6 than or egual toft=
Hencef all
eigenvalues
*D, lc3,rl
nt* + L)2, when o < v < \.
of c qre no larger
Nowwe consialer t < l, < 1.
, j = tr...,N.
than 4(N + l)2 '
The sturm sequence, ier]!=0,,
fox the trlaliagonal rnatrix c is defined (t2)r p. 2o2l b;
a)
fo (+) = I
f).(x) = (x - cr.r) fo(x)
a
I = r, '.
The value f.-(x) is equal to detl-c
N
sinqe the elenelta
of C are real
the
chanqeB
numlcer of
siqn
the number of, eigenvalues
now Ehow that
so that
there
+ II]
and itq
x = 4(N + t)'.
p. 203)
eigenvalues are
lf
I Di<
r!,r
F.n)-'
q
{uar
are 1alger than x.
are no sign changeE in
-
.N
tfi J;' vhen x = 4 ( N + 1 ) ? ,
is an upper bounal on the spectrai
thiE
N-r.
"
and thus (-2r.
'ih l-hF gqFd
E r 4 qrFn.F
ErEs
of C that
rv\
raQius.
"ttr'
t h e s t u r n E e q u e n c ei s
12'vl fo
=1rt.-*otr*
n
l
we uge an induction
4
argument on j to prove that
Stura geqqence does not change Eign.
NoFe lhat
L
tfr"(,-
vl >
I
n
the above
60
For a given value qf j,
putatlon
1
E|rrDDOEe
I .
-- qives
of
I
-
_
L
- v
/_
: J
'furr-h
6
L-A
,
J
)-t
I
1
l
-
t . , J o e, ,
l
n
l
Y > Ii,
hypo-
Eherefole,
that
we have shown by inaluction
>
-?--_-:f
I
the f,act that
and the E.econd fron
-
/
fol,Iows from the induction
inequality
where the f,irst
\ l !-.
L/4 - ve \
/-
t L.+
\
the€is
a coln-
fhen,
-
v t "
I
all
Therefole,
the eigenvalues
Since A iE sinilar
' l ?
of c a r e s m a l ' . | € ! E n a n + \ N + r , .
to c, the Proof of, Eheoren 5*1 is coBplete.
s and Eiqenfu
5.2
r.et nf ana Of, aenote the associated Legendre functions of
degree r and order € of the f,lrst
and second kinds,
resPecij.vely.
It is well-known ([12], p. 165) that any linea! combination,
(s-3)
e
E
S
Yl = oP. + FQr '
of theee two functions EatlBfies
I
diff,erence eguation
( x ) + (' 2 r + l )
( s - r - I ) Y' - r+-L
-l
We no!^, showr for
difference
fixed
equation
x antl E, that
(5-4)
for
x v -r ( x ) - ( s + r )
(x) = 0.
< x < L.
the general
of
solution
r = !0,!o + 1,...,
is given
(s - 3 ) .
Another vell-kndwn . ( [ 12 ], p. 163) recurlence
whtch is satisfied
by any linear
conbination
relatj.on
of Pa and Qs ,
Y- , as in (5.3), ie
( x e_ r ) * ! +
i
r-l
_ , ( r + r )xYt
-1
It
follqrB
frorn (5-5),
that
for
any fixed
6. and x = x.'
the
pafr el (xo) ana.Of (xo) are linearry indepen4enton the set
r = Fo,&
+ L,.';t
EinCe if
they wEre not'
then
(xo)ror k = 0,1,.;;i .tl;*.
p"o*k('o) = aoro+k
a.
xf this were true,
then by (5-5), -=*
=
!ii].if ,
"--*tfBut Eince r-o anC Qlo are both selutions of ihe sane secondarqer linear lronogenequsdiffelential
equation on (-1,1),
we
: (x) = aAl (x) fo! all x in (-1,1).
But
r h e n h a v e r h a t pro
ro
this canlradLcts the f,ast tft.t elo and Qso are an independent
seE o!
Iunctl.ons
x, the genelal
adrr+
i ^h
i o
ltlEn
solution
^i uah
}\i'
of the second-Order
t <-?
- rt rn
f
= 3-'t, {,")
'
v]lrtc"',t
f,
linear
s and
diff,erence
t 5 - . 1 , and make the
i + rrl- i 6n-
.plY, (cos@)
J-''
ih
fixed
l
I f w e s e t s - - y a n d r =
qrrh.+
H e n c e , for
to'x,
regpect
+ha
l|i
tfaF6h^a
a^!rr+i^E
^ - v'
9
j-n
(cost{J) = l
(*) ,
= -i - s . ( s . )
l '
( 5-4)
obtain
ri*, {.} *
\1+ l/
-Yn,
2
cos(€- (s) -
i-T
n
.t
t r 2 ,. . . , N .
(5-6) yield€
t4
(s-r) ol2)'.t*r =\t7
h
tfe cao deternine
that s.('!)
sina f
)
s, t ,")
i€ the general sol-
2,,.,.N.
one of tbe qr.bitrary conEtants l-n
63
(, ",, , v 4
:i'(
(5-e) 3#(v : l-l s1 (&r)= ra.
LTE
Then, Is. ('!) ]'l-'
(@)
Sz {d)
=
.ini + I s,(.)
satisfles
I
l=o
s- (@)=0
),
I, r_(2
x. n
Y + r)
=
l
For
6 -( -' :t i S . ( d )
zJ
3
[atr,r*r)"g i n -
(o,4(N+ l)'),
t h e r e exist€ an (! in (o, ?)
such that
d = 4(N + l)'
gina
u
Hence, sinqe by Theoren 5-1, a l l
b f t h e eiqenvalues.
AK of the
-
problen (5-l) lie in (o.4(N + l ) ' ) ,
(5-l)
can be represented by
fs,r,*rr!11
lhere
a n y eigenfunction of
s,(@) 1s dellneil
J
k = l,,.;,N,
r
by (5-B),
one of the constantE is d.e-
ternined hy (5-9), the other remaining albitrary,
ek-2pin-r#11
and
CIIAPTERVI
NUI.{ERICAI. RESIJI,TS
The nurnerical experinentg conducled in retation
to this
study cgnsiEted of, the estimation of the srnallest eigenvalue
.
and a corresponding eigenfunction to the finite-difference
eigenvalue problen
.l -6 - t_e- ,l
l
|
T h, r- jr
_
,'i = 1,,..,N
= .' _
/ \j x _aju . ,
.
( 6 - I b u) o = o ,
u * , , = 0 ,
N'+
where h = (N + I)-t,
for finite-dlfference
wgre dissussed in ChEpters III
and tv,
the BeFsel operatqr x = -xa d'/dxt
6.1
operators
which
\,
as approxlnations to
- xd/dx + v2 .
Rate of cqnverqence
For each f inite-d j.f f erence opetrator, Lh , consialered,
and for eaeh vaLue of, h-. - N + L t 4,a'L6,32,64,128,
obtained agqurate estinates
of the snallest
and corre6ponding eigenf,unctloh,
lh\
n""
we
elgenvatue,
, nornall-zed by
\
,
65
thI
ui*-i.r,r" = l,
1i6ted
of
(5-I).
The eigenvalue error,
which is
in Tables I through 4, is conputed by l\
a rs rne square of the Emallest zero of .Iy(x),
smal-lest eigenvalue
of problen
are found in
The error
[15].
(4-1).
_ Il,
rrrtrere
\rhich is the
ffre values of I used
i-n the eigenfunction
is taken
EO DE
- ujh)t"l,]l
I iL ,oor,^**r)
where c is a congtant chosen so tnat c,r, { ),},zz - t.
)
cilr(I'x)
I
,
1
then
i6 the nornalized eigenfunction of problen (4-l).
ily(x) was calculated by surruningthe filst
twenty terms of, its
power sef,ies expanslon.
In qrder to exarnine lhe rateE of convergence fo!
eLgenvalqes anal edgenfunetLons; the erq)erinental
vetrgence (ERC) waE conputed and tabulated.
lhe
rate of con_
thi.s nunber is
defined by
ERch = 1o9(e2h,/ebr/rq
2,
wheFEeh i8 the error for 6 t1u"tt neeh s+ze h,
This nunber
is the power of h by whiqh the arrerE are approaching zero.
conFuted frolo the errora
I
l.
wlth
rneehFize h and 2h.
66
Experimental
6.2
(..}
with
is the three-point
the Beseel operator
polynomials.
ratic
vector
estination
5v/3,
operato!
t at the nesh points
ll l
using LJ-'
which agrees
for
in the eigenvalue
are listed
quad-
all
anal eigen-
in Table,1,
fo!
in Section 6.1 f,o! I values of v.
Eee in Table I that
with
L.'^' , as defined bv
clifference
The errors
values of h listed
l r l
The operator,
Exanple 6.2.13
(3-9),
Regults
tlre ei.genvaluea are appalently
order 2v and the eigenfunctionE
fo\ 0 < P < 1,,
with
we
converging
appareflt order
Fron the valueg in Table l.
values and eigenfi4nctions
the
appear to converge with
the eigenorder 2
rvhen ,, > 1.
lhe dif,ference
eigenvalue problen
.
- | v , v +-L, x r lJl .1
operator
\l.x-;x-
for
v - v +
ax' + bx"-,+
the f,inite-difference
the three-point
that iq,
f,erence operator which egrees with
vrhen applied
of solving
In Eabre 2 are the results
SgPlg-!.3r-?.t
the Bessel operator,
to f,unctiong of the forn axv + bxv'I
! !
cx- at x,,
j > 1, f,or v = L/4, I/2,
lt = v - L,O,V + L/?, v + 3 for each val'ue of v.
f,ron Eection 4.4,
that
the truncation
error
over a cla9s of, functj.ons which includes
of, the differential
there
prabte$
tq be no better
Tabtre 2, we gee that
finite-dj.ft,
at xr and
3/4 a\d
we lecalL
of, such a schene
the eigenfunctigns
is guaranteed by the analysis
than o(hy)
at ehe firsi
lhe ohgerved rate
Polnt'
rn
of converqeqce of bath
' t
Table 1. Errors
Example 6.2 .I.
and Experihental
eigenvalue
N+L
8
Ib
v=o.25. 32
64
r2a
l6
z=0.33.. 32
ERC
6 , 4 E- I
4
4E -l
3 . i.B.bl
I
32
64
12a
2.08-1
-0.41
o. 6 4
0.85
0. 94
o. 9 7
3.78-2
1. 9E-?
L rE-2
L2A
5.08-2
3 . 7 8- 2
D-1.0
32
128
0.39
0.43
o,46
o. 5 2
I - 88-2
4'74
-0,82
0'54
1.00
1.r6
3,9E-3
U.5I
J . -tti -J
l,0s
7,OE-4
1 .r 0
A
u-0.75
-o.1l
a. 2 I
3 . 9 E- 2
o. 6 1
A4
l6
32
64
12B
eigenfunction
o,02
r.5E-l
1.0E-1
9.08-3
16
v=o,66., 32
0.30
0.50
0.53
0.53
eigenfunction
error
9.5I'-2
a . 3 8- 2
4.0E-1
1"6
v=o.s
eigenvalue
Rates of Converqence for
I
l. 0E-2
2.79
L lE-4
4.99-4
0 ,7 1
l,6E-1
4.OE-2
1. 0E-2
2.58-3
6 , 3 8- 4
1 1
1,23
r,24
2 .'18-4
1 )4
2.Q0
I. 0E-3
r. 97
2.00
2.00
2.00
2.684
2.O2
.
l
I
:
'
Table I.
64.
(co[t'd)
s
16,
v=L.s
32
64
L28
I
Iti
v=2.O
@
128
2.4D-1
6.-!!,-2
l.5E-2 ,
3.88-3
e.4I"4
3 : 4 F- 1
8,4n-2
2.\E-2
5 . 3 8- 3
1.3E:3
2.O0
2.OO
2,OA
2.OO
2.00
?.00
2.00
2.O0
2.OO
4.rE+
2.10
2. O0
1-98
l.0E-4
r.99
1.99
9,7E-3
2,04
5. 9!-4
1.58-4
2.00
2.00
.l
69
the eigenvalues
than order h',
and the eigenfunctions
brrt th-t
better
the rate of convergence does appear
to increase as y is increased.
the begt experfunental
is considelably
rate
We note from Table 2, that
of convelgence
occurs when
P = v + O.5.
Exanple 6.2.3:
inental
In Table 3 we have the errors
rateE of convergence for
the difference
problen (6+1) rfith I{^ = t -t*u ,*v*2,*ll,
n
n
difference
operator
lrhen applied
first
!o functionE
ne€h point
point.s,
v + 6,
which agreeE with
D
- v+2
anal ax- + bx
*
the three-point
the renaining
"*F "t
The operalor L.ixv,x
in Section 4.4 as an app5elination
tl,ro operators
finite
the Bessel operator,
ar,d l t = v - 2 , 0 , L , v +
fot each value of y,
invegtigated
eigenvalue
l,
of the form ax/ + bxP+2 at the
v = I/4,L/2,3/4
for
and exper-
were noted to have truncation
mesh
v + 4,
,x J was
to J, and the
elror
order h2.
3n Table 3, \re have apparent h2 convergence for both the eigenvalues and eigenfunctions
& = I.5.
problen
In this
in all
cases except v = 0.5,
case. the eigenfunctions
are identical
of the dif,ferential
to thoae af the diff,erence
the rBesh. Hence, the error
given in Table 3
for
problem on
this
case
which grows aE h becomes s.nalleer,
is only the round-off
erroxr
as one night
The cases ln Table 3 where p = y + I rep-
resent
the
expect.
operator
which
we denote
l)\
by Ii-'
and which
is
given
70
n
fable 2. Errols
Example 6.2 .2.
N+l
8
16
32
6
P=-O,75 4
I28
.
v=o.25
a
16
v=t.2s
32
p= 0.0
64
.
l2s
and Exp€rinental
elgenvalue eigenvalue eigenfunction eigenfunction
error
ERC
errgr
ERC
1.48-l
L.46
t.6E-3
0.S5
4.3F..2
1.56
r.2E-3
0.3?
1.38'2
L.1g
s. rE-4
r .2.6
3.48-3
I.8?
l.7E-4
I.58
9.10-4
1,91
5.18-5
L.74
9:68-2
2.58-2
6.8E-3
1.,78-3
4.48-4
v=0,25 t6
1tr1.7s
64
1.sO
1.88
1.93
1,95
L,9',t
2.5E-4
2.4E-4
l.0E-4
3,38-s
9.7E-6
2.LO
2. I 7
3.0E-4
1.4E-I
'
3
2
6
4
113,25
,
r.l!;-5
+.+f
0 .3 ?
^ ; ^
! ,'78-4
g.La-4
4 .r F - 1
Y = 0 ,5
15
32
P=-0. 5
64
2.55
o.Os
r.26
l.60
r.76
s.?E-4
I .9E-4
I
n
Rate of Convergencelfor
l.?E-t
t. au-a
0.14
L;at5
1.91
o.oo
v=o.5
tFo. 0
r F o ,5
{|1" O
l6
32
128
1.48-3
3,6D-4
32
64
9 ,39-2
1.98-2
4, 38- 3
1.0E-3.
l?9
2,4F-4
2.Q',l
2,O4
2.02
3.48-4
L .28-4
0.83
1.49
9.88-6
0.00
2,27
2.L7
2.10
2.O9
6.48-4
2 ,28^4
5,68-5
]. FE- 5
1.51
1.89
11
rable 2 (cont 'd)
.eigenvalue
N+I
error
.8
r.8E-L
16
32
64
1.2I
5.38-2
l.4E-2
3.78-3
9 .4E-4
8
v=O.75 15
32
u--0.25 64
r2a
1,4S.-1
3.28-2
7.78-3
1.98-3
4.78-.4
v=0.5
p-3.5
v = O . 75
f-0. o
II
1.60
0.50
t.aD-J
2.O5
2.O2
32
ItaL,25 6 4
128
I
v = 0 , 75
2.O2
5Z
lo
.l()
32
64
t2q
I.
J!I-I
z. tE-z
aRc
0.02
l. 8E-4
2.2A
l.8E-3
.4F-4
.4
eigenfunction
2.r4
r.4E:-t
3 . O E - 2.
54
128
eigenfunction
erfor
2,28-3
4.OE-4
L.2E-4
3.28-5
I
15
E
u-0.75
eigenvalue
ERC
1
2.rE-4
2.oL
2. 3E-1
r . 70
0.48-a
I
l.6E-2
4.rE-3
r,0E-3
1 , 95
l .. 6 8
0.45
2 .r e
6. 9E-3
l.7E-3
4.28-4
1',l
1 ,t 2
r.,69
r. atc
0.46
1,32
A't
3. 3E-4
1 q n
'72
by (3-I
).
w e h a v e , by Corollary
sho$rn thaE its
4-I,
vatues converge with
those of the aliffelential
order
at least 2.
of convergence
perinental
rates
genfunctions
of
problem with
In Table 4, $re have the errors
Exanple 6.2.4 r
of convelgence
the
operator
the
for
L;
eigen-
and ex-
eigenvalues
and ei-
, which lras discussed
in
s e c t i o n 4 . 5 ,- w n e r e r f
i s t h e o pne r a t o r o f t h e f o r m L n- t . a v , v v + 2 , e 1 x \ ' 1,
fot v = \,
where g(x) is determined so that :t{ is se}f-adjoint,
By coroll.ary 4-2. ne have that the eigen-
L/3,f/2,2/3,3/4.
values
ential
fable
k-.,
1 e
of -iI{
converge with
J
problem with order of
4 this
convergence
eigenvaJ-ue I of
c
L- is
n
identical
the
is
lrl
I,.'-'
n
y = 0,5 and t! = I.5,
that is,
For all
least
to occur
equation
differtvo.
.fo!
probfem.
the
ln
smallest
when v=0,5,
6ame behaviol
aa noted in Exanple 6.2,3 with
errors
the errors
convergence
at
the
, and hence we have the
of, the eigenfunction
ference and differential
of
observed
diffelential
to
any eigenvafue
the eigenfunctions of the dif,-
operators
in Table 4 are round-off
are identical,
errors
for
anil hence
thi.s case,
values of 1,,in Table 4 except u - 0.5, \^renote that
the eigenfunqtion€
appear to be convelging lrith
vergence 2 + 2v. l nuch faster
served for
otder of con-
rate of convergence than ob-
any of the ottler operators
\i/hich were used.
anal Exp€rl$ental
Table 3. Errors
Exanple 6.2.3,
N+l
8
L6
v=O.25
'
32
64
lt'=-L.75
128
v=O.25
,tso.o
I
16
32
64
L2a
v = o. 2 5
P.r.o
32
54
128
eigenvalues
error
3.08-2
5.6E-3
r.6E-3
3.98-4
9.88-5
6lgenvaluea elgenfunction
xRc
2.44
2.18
2.06
2.OL
2.00
8.lE-2
2.OE-2
5.IE-3
1.38-3
3,28-4
1.93
1.98
2.OO
2.OO
2.OO
8.6E-2
2.28-2
5. 5A-3
1 .4E- 3
r.92
v40.25
s.4E-2
2.LE-2
5.3E-3
ll.L ,25
r.
Jli-
5
3.34-4
2.OO
2.OO
2.OO
r.93
r.98
2,00
2,00
2.00
1.4r
v - O. 2 5
Ita4.25
v=O.25
l*6.2s
t2a
l6
32
64
128
Rates of, Cgnvergence for
error
7 .58-4
r.5E-4
3,6E-5
8 , 7E - 6
2.04
2.OO
2.94
2.8E-5
6. 3E-6
1. 5E-5
2.38-5
2,O3
3.0E-7
1. 5E-3
3,0E-4
1. 8E-5
4,68-6
z.zE-L
2.20
2.OO
2.OO
3.6E-3
8.08-4
t.9E-4
2.00
2,O3
2.OO
3.00
r.93
r.99
2.00
2.00
2.OO
2.03
2.Ot
L.4E-5
s.8E-3
2 .28-3
5. 5E-4
L.4E-4
5,4P-2
l.4E-2
3.48-3
8. 5E-4
eigenfunction
ERC
2,4?
3.03
2,24
2.O4
2.r7
2.O3
2.Ol
2.00
74
Table 3.
e = O. 5
,l=-I. 5
've0.5
,r=0.0
v=O.5
(cont 'd)
N+l
I
15
32
64
128
eigenvalu6
error
9.lt-2
2.28-2
5.5E-3
l.4E-3
3.4E-4
eigenvalue6
ERC
a
t6
32
64
l2s
l.4E-1
3.48-2
.8.5A-3
2. tE-3
5.3E-4
,1.95
1.99
2.OO
2.O0
2.OO
8
r. 8-r
1.96
1.99
2.00
2.OO
2.00
l.0B-4
r.98
3.0E-12
3. lE-12
3.3E-12
3.58-12
4.6E-L2
15
32
64
r2a
3.48-2
8. 5E-3
?.lE-3
5.38-4
s
16
32
64
l2S
1.3E-t
3.28-2
7.98-3
2.08-3
s. oE-4
YrO.5
Q
15
32
6.78:..2
r,78-2
4.lE-3
lF4i 5
o+
J.. uu-J
L2e
2.68-4
l=1. o
v-O.5
P-I. 5
a
v=4.5
16
t*6,5
32
64
r28
?
ill-l
8.18-2
2.Oa-2
5.0t-3
r.3E-3
eigenfunction
eigenftrnction
ERC
^
4.3E.4
2.O4
2.OL
2.OO
2.00
1.99
2.00
2.OO
5.IE-6
-i
z .40
'
2.Os
2.O2
r. Jl;-o
l. oE-4
2.2E=5
2.69
2.27
2.O7
5 .2E- I
2.00
2.26
l o ?
-0.05
-0. 0.6
-0.09
L.97
2.AI
2, 00
2.00
2.00
4.rE-4
l,0E-4
2,58-5
6.48-5
2;O0
2.OO
2,5L
2,03
2.00
4. OE-3
9.7E-4
2.48-4
2.44
2.O2
2.00
2 .O 0
L
2.00
3.29
tl;-'
z. !o
Table 3. (cont'd)
N+l
I
v=4.75 16
32
p=-L.25 64
L2S
v=O.75
I
15
f=0.0
64
v=O.75
,r=1. o0
v=O.'15
p=1.75
v = o . 75
I
15
eigenvalue eigenvalue
error
ERC
1.78-l
2.O2
4.38-2
2,OL
t. rE-2
2.OO
2.78-3
2.OO
6.88-4
2,OO
2. rE-t
5.28-2
L,3E-2
3.3E-3
a.lE-4
2.OO
2.OO
2.OO
2.oE-r
5,oE-2
r.99
7.88-4
2.OO
2,OO
2.OO
64
r28
8
15
64
L2A
I
l5
't
'tE!- l
4.38-2
l.lE+2
2.78-3
l.2E-l
7. OE-3
1i=4.75
64
128
4.48-4
1*5.15
2.25
2.O3
2. 0 1
2.OO
2.33
2.O4
2.Ol
2. OO
2.00
2.48-3
32
2 .98-2
2.Or'
128
l. aE-3
2.OO
2,OO
6l
2,OO
L.784
2.66
2.O4
2.49
2,L6
2.O4
a
2.00
2.OO
2,OO
2.00
2.OO
4.8E-1
eigenfunctj.on
I. 9E-4
2.98-5
I
v=O.'15
eigenfunction
2. 3E-6
).lE-t
L.4E-4
2,OO
2,00
2.OL
2,OO
2.00
2.OO
3.08-4
2.OO
g,!ee
.9.J---Nu!grfscf-L99
The technique useal to eEtinate
the srnallest eigenvalue
of, the N x N Matrix A asEoeiatetl with
(6-l),.
vaLue problen
an initial
associated
corresponding guess for
conputing
y.!0) = .i.tzjn
approxination,
the eigenvector
in this
is described
with
rnt
we chase
Eection.
of
as an estinate
eigenvalue-
the snallest
A
)t(0) , ,uu" ^ad" py
the eigenvalue,
the Ray1ei.gh quotient,
eigen-
the diffelence
is,
that
ldl
r(0) = ll:j-j.-:11)
(.,(ol .,(o) r
.)
\r
An inproved estimate
tained by Eolving the linear
purpose, and hence stas used.
Rayleigh quotient
t l t
of
valueg
succegsive
= I(0)y(0)'
l r !
y'-',
until
1(N).nU
St each etep of the above iterationr
90 that
its
vergence criterlon
nitltlle
on tht
when N + I r^raslarger
for
this
was then
an improved estimate
lA-----:--4--i
the eigenvector
component rtas unity.
vrlte
was nor!!a-
fhe above con-
,a(t) tt4s alvJdys satisf:ed
than 8, fFe fLtal
<IO-' '
l r t n )I
"tnl
lized
sin"u
works well
This estinate,
to obtain
, was then ob-
The above Process was rePeated to
,. (n)
, (n_I)r
of the eLgenvalue, tr'-'.
f ind
systen Ay(I)
Gauss elinination
A iE alwayE tri.tliagonal.
used in this
yr^'
of the elgenvector,
lar t: - 2t
,rlr') an<l veclor
77
Table 4. Errors
Exanpte 6.2.4.
N+l
g
16
v=O.25 32
64
.
l2a
.
,
eigenfunction
errox
l,2E-s
2.3E-6
4.28-7
7.sE-8
l-,3E-8
eigenfuction
ERC
2.21
2.3'7
2.44
2.48
2.49
9.48-6
r.7B-5
2.aE-7
4. 5E-8
7. LE-g
r.2E-r
3.2E-2
1.98
1.99
3 .O E - r 2
3.2E-12
32
?. 9 8 - 3
2. O O
3.3E-12
-0. 05
I28
5.0E-4
2.OO
4.28-12
-0.32
8
t.7a-t
t6
4.?E-2
v=0.666,:32 1.18-2
54
2.68-3
I28
6.68-4
t.9S
2.OO
2.oo
2.00
2,00
7.tE-6
7.88-7
8.3E-8
8,5E-9
8.7E-10
2.99
3.r8
3.24
3.27
3'30
1, 9E-l
4.8E-2
L.2s'2
3,0E-3
7,5A-4
1. 98
2.00
2.00
2.OO
2.OO
7,68-6
1.68-7
7.58-7
6.88-9
6.3E-10
3. 16
3.32
3.37
3,4L
3.43
I
16
v=0.75
32
64
128
.
l''*
eigenvalue
ERC
1.95
L.g1
1.98
1.99
1,99
1,96
1.9S
I.99
r.99
2 .OO
v=0.5
'
eigenvalue
error
7.48-2
L.SE-Z
4.78-3
r.2F-3
3.oE-4
g.o$-2
2,38-2
5.7E.-3
l.4E-3
3.58-4
. s
16
y+0.33...32
64
L2A
' 1 6I
/-r
and ExperimentaL RateE of Convelgence for
2,05
2.41
2.60
2.64
2.66
1 .0 3
-O.ll
qrM
gtuvdrqs
svrrs.lluqurv
etYerrvsrLv!
rvraF.rasv
rrix A
natri
'
It
ehould be.noted that
no knotrn result
guaranlees the
convergencaoftheaboveiterationschemef,orconputingthe
eigenvalues and eigenvectors
natriceg
to 'rhlch
slmunetric.
it
r.re have applieil
casea considered),
the value to which it
the veclor
I
lr-,
it'
since
ghe above process
Ilowevsr ' if
did in all
of a given matrix A, for
the
they are all
doeE qonverge
one can easily
verif,y
non(and
that
convergieE is indeed an eigenval.ue End
generated a cor'reEPonding eigenvector.
gpection of the eigenvector
revealE that
aII
when in-
of itE comPo:lents
are of the Earneeignr then the elgenvalue muat be hhe snallest
one of A.
Ihq above nunsrical
at Purilue University.
hr
experinent was tun an the CDC 6500
BIBT,IOGRAP}IY
BlBLIOGRAPITY
'1.
,
,.2,
!,
Allen, D. N. de G., "A suggested approach to finiteqf, differential
alj-fference representation
eluations.
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butlone near a Bliding contact, " Qualt. ,t. Mech, and
Appl. Math., fE (1962), II-33.
Berezin,
'Vol, II,
I. S. and 4ridkov, N. P., "Computtng t4ethods, "
Perganon Press, Oxford, I955.
3.
Boyer,.R. H.r "Discrete BeEEel f,unction. "
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4.
Fulirech, R., rrExistenznach\deis und Approximation
von Eigerwerten und Eigenfunktlonen nit Hil.fe Eines
Diff,erenaenwerfahrens bei Singutlren sturn-LiouviLleE-i'
chen Randwertaufgaben. " Arch, Rational !Gch, Anal.,
9 ( _ r . 9 6) 2, I 1 1 - 1 2 0 .
5,
Courant. R. anal Hilbert,
D., "Methods of l.lathematical
{
Physlc€,
Vol. I, Interaqience.
New York, 1953i
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Dressel,
Gergen, 't. J.,
F. G., and Parrish, c. 8.,
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AppI., t3 (1966), Io2-1I7
7.
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J. Itlath. Anal. Appl,, 20 (L9671, 269-29a.
8.
Henrici. P., "Elenents of Nurnerical Analysis,"
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9.
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lI
lii
,t, l.lsth.
,tohn
lO.
llilne-Thonson,
I,. lrl., "The Calculus of Finite
ferences, " Macmillan and Co., London, I933.
11.
Pearson, C. E. "Integral
representatj-on for solutions
of Beagelrs dif,ference equation, " ,t. Math. and phys. r
Dif,-
e ^6l 0 ) , 2 e 7 - 2 9 2 .
:a!69 (r l1d 9
24.
i-.,
12.
Robin. L., "l'oncti.ona
Fontions
Spheroidales,
Pari6, 1958.
13.
VaFga, R. S., "Uatrix Iterative Analysis,"
Iiall, Englelraod CLif,fs, Ne\d ,tersey, 1962.
14,
Whittaker, E. T. and WatEon, c. N., "A Course.of,
Modern AnalyEis, " Canbridge University Press, 1927.
15.
National Bureau of, Standalalg, 'Tables of, BeEaeL Funqtions
of Fractional Order,',
CoLwbia University press. Nev
York, 1948.
Spheriques de Legendre et
cauthier-Vj.Ilars,
" Tome If,
I
Pren!.!ce-
l
I
V I'IA
1943.
gchoot in 1961'
He graduated from Troy High
ceived
a Bachelor
UniverEily
fron
of ScLence degree j'n l4atbgnatics
of Daygon in 1955 and a l'taster
Purdlue Univerelty
in
vror'l< in Conputer science
united
StateF.
196?r after
at Purdue '
lle refron
of gcienqe
'
tlle
degFee
!"hich he began doetoral
He is a cilizen
of the
: .