Computable error bounds for collocation methods

internat. 3. Math. & Nath. Sci. VOL. 18 NO. (1995) 89-96

COMPUTABLE

ERROR

BOUNDS FOR COLLOCATION METHODS

89

A.H.AHMED

Department

of

Mathematics

Ash-Sharq University Khartoum, Sudan

(Received August 12, 1992 and in revised form February 27, 1994)

ABSTRACT.

This paper deals with error bounds for numerical solutions of linear ordinary

differentialequationsby globalorpiecewisepolynomialcollocationmethodswhich arebasedon

consideration of theinvolveddifferential operator, relatedmatricesandthe residual. Itisshown

that significant improvement may be obtained ifdirectbounds for theerror in the solution are considered. The practical implementation of thetheory is illustratedbyaselectionof numerical

examples.

KEY

WORDSAND PHRASES:

1980AMS

SUBJECT

CLASSIFICATION:

1. INTRODUCTION

This paperis concerned with computable error bounds for the solution oflinear ordinary

differential equations by global and piecewise polynomial collocation methods. Thiswork is

motivatedbyanabstractapproachoferroranalysissuchasdescribedforexample byKantorvich andAkilov 1,Chapter

XIV]

andAnselone

[2,

Chapter

I].

Itextendspreviousworkinthisareaby Cruickshankand Wright

[3]

and considersdirect errorboundsonthesolutionandits derivatives

ratherthanviatheerrorboundsonthehighestderivativesasdescribed in

[3].

Thebasic idea here is tomakeuse of thematrix involvedin thenumerical solutionofthe

differentialequationincomputingerrorbounds. Therelationshipbetween various matrices and the inverse differential operator has beenconsideredby Wright

[4],

Gerrard andWright

[5]

andAhmed and Wright

[6]. In

particularthesepapersare concernedwithasymptotic relationshipsbetween inverse operatornormsand those ofmatrices relatedtothenumerical solution. The theory leads

naturallytotheuseofmatrixnormsinboundingthe inverseoperatornorms. Then computation of

the residual norm leadsimmediatelytocomputationoftheerrorbounds.

Theuseof theseideasin error estimation is discussed in

[7].

Theirpracticalimplementation

inselection criteriaforadaptivecollocationalgorithmsis examinedin

[8].

Referencestoalternative

approaches

in collocationalgorithmsand erroranalysissuchasDelves

[9],

DeBoorandSwartz

[10],

De Boor

[11]

andRussell and Christiansen

[12]

maybefoundin

[7].

Detailedcomparisons withthealgorithmsof Ascher, Christiansen and Russell

(used

in the well known codeCOLSYS for solving boundaryvalue

problems)

[13],

ispresentedin

[8].

Throughoutthispapertheanalysisand illustrative resultsuseinfinity norms,

though

someof

theideascould be extendedtoother norms. Theassumptionsabout the methodsandequationsare introducedwhenneeded,toemphasizetheirsignificance. The theoreticalbackgroundispresented only brieflyasit isreasonablywell known. The results are testedon aselection ofproblemsand

comparisonsaremadewiththose in

[3].

The workpresentedinthispaperisbasedonAhmedthesis

90 A, H. AHYlED

2. ASSUMPTIONSAND NOTATION

Inorderto beable to treatboth global and piecewise polynomial collocation in a uniform manner wefollowverycloselytheassumptions andnotationsused in

[7].

We consider the linear ruth orderdifferentialequationof the form:

xt")(t)

+

,-o

P’(t)xO)(t)

"y(t)

(1)

With m associatedhomogeneous boundaryconditions. Withoutloss ofgeneralityweassumethat

theequation holds in

[-1,1].

Theequation maybewritten inoperator form:

(D’-

T)x y

(2)

where

D

denotesthe differential operator.

In

(2)

wesupposethatxE

X

andyC

Y

where

X

and

Y

are suitableBanachspaces. The operators

(D

T)

and

D"

withthe associated conditionsareboth assumedtobe invertible. Theapproximatecollocation solution istakeninasubspace

X,,,

CX. To definethis,wefirstdefineasubspace

Y,,

CY.

Suppose

theinterval[-J,

1]

issubdividedbythebreak points-1

to

<

t

< <t,, 1.

In

each sub-intervalqcollocation pointsareused chosenas:

+k--((tk--tk_l)

+(tk

+

t_))/2

j 1 q

(3)

k 1,...,n

where

{’}+

j 1,...,q,aregiven reference pointsin

(-1,1).

Thespace

Y,,,

consists of functions whicharepolynomials ofdegree q 1 in each of the intervals

Jk-

[t_t

+,t-

1],k

1 n. No

assumptions regardingcontinuityatthe break pointsismade. The solutionspace

X,,

isthen taken

as

(D’)-Y.

Theprojectionoperator

9,,,

is defined asthe operator whichgivestheinterpolantin

X,,,

basedonthecollocationpoints

{g}.

Withtheseassumptionstheapproximatesolution

x,w

satisfies:

(D"

,r)x,

y

(4)

In

[6]

certain matrices

Q

were introducedand theirpropertiesexamined.

Here

weuse aspecialcase of this and denoteitby

Q,,,.

Thisis mostconvenientlydefinedby consideringtherighthandside

y and the solutionatthe collocationpoints. Then thereis amatrix

Q,,,

such that

xt)= Q,

wY

(5)

and this can beregardedas adefinitionof

Q.

In

asimilarwaywedefinethe matrix

Q.<)

by

x(r)

(r).

Q.q,,

r 1,2,...,m 1

(6)

and the matrix

Q)by

x<,,,)

Q,,,y

Thenunder suitable conditions it wasshownin

[6]

forr 0,1 m 1that

t’)ll

IID’(D’-

r)-ll

as n o_{,q fixed,and}

OOll

.,,

lID’(D"

r)-ll

as n ,n fixed, and in

[4]

and

[5]

that

O0’011

II:.,,

II(

T(D’)-’)II

as n oo, q fixed, and

as n--, n fixed.

COHPUTABLE ERROR BOUNDS FOR COLLOCATION NETHODS 91

Theseconditionsconcerned thelocationof thecollocationpointsandrequiredthe continuity of the

coefficients

Pj(t)

in

(1).

Inparticular, theglobal case(q %n 1)assumed that thepoints

{’}

were zeroofcertainorthogonalpolynomials.

Some further definitions and notations are needed before constructing the bounds. It is convenientheretodefinethe compact operatorsKand

K

by

K

T(D")

-1

K,

D’-"

for r 1,2,...,m 1.

Wealso introduce theevaluationoperator

.q:Y

R

"’

togivea vectorconsisting ofthe values of

afunction atthecollocationpoints. Further an extension operatorqJ’.R"q

Y

_{canbe definedto}

give

afunctionwhosevaluesatthecollocationpointsagreewiththe components of the vector,withy

theproperties

.q[I

tIJ.qll

1,

.qC,,q-.q

and

(nqLIlnq-

I.,,

_9.q

.W.q.,

asdescribed in

[4,5]. Operator

expressionsfor thematrices

Q.

can now beobtained. From

(4)

-1

andapplying

.

gives

-,.T)

,.y,

(8)

and hence

Q.q

.qCD"

In

asimilarwayweexpress

Q,,’,

and

W,,,

in thefollowingoperatorforms

-1

and

(9)

(10)

(11)

TI-IE

ERROR

BOUNDS

Suppose

anapproximatesolution

x.q

of the differentialequation

(2)

has been found and

x.,

satisfies

(4).

Lettheresidual

r,q

be definedby

r,,

(D"

T)x,,

y and the errorby

Using

(2),

wehave

(12)

(D"

T)e,,q

r,,,

or

e,q

(D"

r)-lr,

q.

(14)

e.,ll

"=

(O"

T)-II

r.

(15)

Notethat

I1,’.11

canbeevaluatedatany pointwithoutdifficultysince

x,,,

isapiecewise polynomial.

Thus

r.ll

maybe estimatedbyevaluationatasuitablyfinegridofpoints. Nowonce aboundon

W’-

T)-ql

has beenobtained,

(15)

canbe usedtobound

e.ll

To bound

W’-

r)-ll

directly weneedtoextend thetheoryin

[3]

in thefollowing way. Ifwedenote the inverses

(D"

-$,,,T)

-I and(I-

q,,rK)

-1

when restrictedtothesubspace

Y.,

by

(D"

-#hqT)-lY,

and

(I-

,,rg)-’Y,,,

then the

followingrelation canbe seen betweenthem. It immediatelyfollows that

92 A.H. AHMED

whichimplies

(I

,,qK)

-

I+(I

,,,K)-’Yq9,,,K

(D"

#,,qT)

-1

D"

+

(D"

q,,,T)-tY,,,,,qK

(16)

(17)

(D"

-9.,T)-’

I111

(D’)

-t /

(O"-

.,T)-’Y.q

.g

(19)

Now (D" T)

-

canberelatedto(I

-.qK)

-

and(D"

-.,T)

-1.

By

applyingAnselone

[2,

page

59]

tothe identity

(D"

t)(D")-)(l

+K+...+Ka-+(I

%K)

-a)K

,

I+

(0.,

I)K(I

+

,,.,K)

-

K

d

weconclude that

(D"

T)-t

I111

(D")

-

+

(D’)-’

K

+...+

(D’)-’

K

-’11

(D"

.T)

-x

K

6-II(%-l)g(1-*.g)-’gq

< d-1,2

(20)

Nowtoapplytheextendedprojection theory suggestedin

[3],

define

W,,d

by

W,,

(D’)-y

+

(D

Tx,,

By

definitionof

,q(D")-x,,D

isaboundedlinearprojectionfrom

X

to

X,.

Define

T,:X

Y

by

T. T(D’)-,

".

Thenit caneasily be verifiedthat,

(o"

r.)w.

y,

(D"

L)-’

(D’)

-x+

(D’)-’T(I .T)-’,

(21)

and

(K

K.q)-’

I

+

K(I

.,rK)y.q

Now byapplying Anselone

[2,

page

59]

totheidentity

(D"

T)-’

(D")-’

(I+

K

+ +

K

a-’

+

(I

-KO,,,)-’Ka)(I

+

K(O.,

1)(I

-Kc#.,)-’K

a-’)

weconclude that

(D’)-’

+

(D’)

-1

K

+...+

(D’)-’

K-’

+

(D"

T.)-’

K"

(D"

T)-’ I1":

1

A.

a if

A.-

II(%-I)K(I-

.)-Kq

<1 d-1,2,...

Nowifwerecallequation

(9)

Q,,,

q),,(D"

-x

anduse

q.tIJ.,qs.,

q.,

q.,rx.,

x.,,

,,,V.

I.

and

.. .

weget

-1 -1

sothat

D’-.T)-’Y.

IIll

.

Q.

(22)

C0IPUTABLE ERROR BOUNDS FOR COI,LOCATION NETHOI)S 93

Ifwerecall

(11)

and usethesame idea weget(D T)

-

<

-,,qT)

Ynq

I111

(D)

%

Q.q

II.

(24)

Again,if we recall

(21)

and usethesame ideaof

(23),

we get

(D"

T,,,)

-t

I111

(D’)-’T,

Q,

+

(D’)-’

II.

25) Inasimilarway it canbeshown thatK(I

Q,K)

-

K,q,Q,q

andhence

,.K)

K

(,.

I

d

’]l

+

(,.

z,.ll

o.ll

K"II

(26)

and

IIK(%-)(-K%)-KII

IIKII

II(*.-ZII

+IIKII

II(.-Z.II

I1.%11

IIKq.

(27)

Now if we denote the bounds in

(20)

by

QP

and use

(18), (19), (23), (24)

and

(26)

we get

QPd

(D’)-’[[

11K’II

+

*.11

I1Q.II

*.Kd.’l[

(28)

,-o

-and

-I +I

wp,,-ii

(D’)-’

:

K’

+

(D)

,.,

Q."

*.de"

i-o

and ifwedenotetheboundsin

(22)

by

Aa

anduse

(21), (25),

and

(27)

weget

i-o

-A

(29)

(3O)

zx.

-II

K

(*

-r)

K

/

r

(*.

-OK,.

?.

K

I1

1 d 1,2,....

Followingthearguments of

(28),

(29)

and

(30)

andusing

(10)

insteadof

(9)

similarbounds forthe

derivativesof thesolutionn be derived.

Nowin ordertocompute these boundsweneedtocompute bounds for

(,.

II,

*.

II,

and

D -’T.

Boundsfortheprojectedtermscanbe

computed

using

Jacon

theorem

[15]

whereas

boundsfor theconstant tescan beevaluated directlyas described in

[3].

3.

NUMERICAL RESULTS

Wepresent herea selectionofnumericalexamples illustratinghowtheideas canbeapplied

in practice and whatsortof results can beobtained. WeusezerosofTchebychev polynomialsas

collocationpointsandforthepiecewisecasewework withtwopointsonly. For comparisonwe

/-4 A. H. AHN[’I)

Problem x"+c( +

t2)x

y x(__.l) 0

Problem2 x" ctr y x(+_l) 0

2a Problem3

x"-(t

+5)

2x

y x(__l) 0 2coc" 2tx

Problem4 x"+ y x(_l) 0

+3

(t

+3)

The parametertis included tovarythe stiffness of theproblem. Intable wepresent thevalues

ofPd, WPd, QPd,

Ad

and

QAa

for theglobal collocationmethod. In comparing these boundswe

observe thefollowinginequalities:

Problem ct--.5 A <

QAI

Po

<

WPo

<

QPo

ct .5,1

A:,

<

QA

<

WPt

<

QP

<

Pt

Fromtheseinequalitiesand the results in the tableswe note:

(1)

The extended projection bounds

(QAt,2,A1,2)

are much more accurate than the projection method bounds

(QPo,

IPo,).

This confirmsexpectation based ontheexpressions for these boundsas

aPt

and

ed

includetheprojectionnorm

,q

which is notuniformlybounded.

(2)

If we consider each projection method separately, we observe that

(with

the extended

projection

method)

in all problemsexceptproblem1,

QA1,2 <A,2.

Forthe usualprojection

method inmostcases

QPo

and

WPx

havenoimprovementsover

P0,

while

QP

and

WPI

are better than

P.

In

general,if

K

I1<

1/2,it canbe seen thatWPa <

P,t

andsuperiorityof bounds

usingthe matrix

Q

becomes moreclear when

a.,

I1:11

W"-x)

w.,

asjustifiedbythe

theory.

(3)

Ifwecomparebounds with small values of dwiththoseofhighervaluesof dweobserve that thelatterare

always

better. This isobviouslyduetothe bounds derivedfor

(z-.,)

r

as

mentioned in

[3].

Resultsforthepiecewisecollocationmethod arepresentedintable 2. Alltheproblems

therehave consistentlysatisfiedtheinequality

QAI,,

<

QPo,1

<

WPo,

<ml,2

<

P0.1"

Thisconfirmsthesuperiorityof bounds using the matrix

Q

and the improvement of

Pa

by

WP,

as expected bythetheory.

NOTE: The missing values in the tables are duetothe deltasnotbeing <1.

Problem2 ct .5 A <

QA

Po

<

WPo

<

QPo

a-

QA

<At

aPo

<

WPo

<

Po

ct .5,1

QA

<

QP

<A <

WP

<

PI

Problem3 .-.5,1

QA

<A

<Po

<

WPo

<QPo

ct-.5,1

QA

<A <

P

<

WPa

<

QP

Problem4 t-.5

QA

<A

96 A.H. AHMED

4. CONCLUSION

Thenumerical results show thatimprovedcomputablebounds for the inverseof thedifferential

operatorcanindeedbefound ifweconsiderthe differentialequationin theoriginal form,

(D"

T)x y, insteadofthetransformedone

(l -K)x y

Theintroductionofthe matrix

Q,,

whosenorm as shownin

[6],

tendsto the normofthe inverse

differential operatoris a mainfactor of the closeness of these bounds. The improvementis more obvious withthepiecewisecollocation method thanwiththeglobalcase. Within theglobal case

there is much improvement with the extended projection method. This is clearly due to the

involvementof the projectionnorm(whichis notuniformly

bounded)

in thelatter case. However in that case we havegot

WPd

boundswhich use (D

)9,,,

w,,

[[,

insteadof

@,,

Q,,

[[,

as alternatives. Despite the closeness of thesebounds,unfortunatelythe conditions ofapplicability (5,d

<1,A,d

<1),which weremajor problemswiththepreviousanalysis

[3],

have turnedout tobe the

same.

An

alternative approachtodeal with suchproblems is tosplitthedifferential operatorin

somedifferentway. Forexample

(D")

in

(2)

maybereplaced by

a,,D"

+a,,

D

+...+

aid

where

al,a2 a,,are someconstantstobe chosentogivethehighestpossible applicability. This idea is justified bythe resultdescribedin

16]

andmaybeinvestigatedinaseparate paper.

REFERENCES

[1]

KANTORIVICH,

L. V. and AKILOV, G.

P.,

Functional Analysis in Normed

Spaces,

Pergamon

Press,

NewYork

(1964).

[2]

ANSELONE,

P.

M.,

Collectively

Compact Operator

ApproximationTheory,Prentice-Hall,

EnglewoodCliffs,NJ

(1971).

[3]

CRUICKSHANK,

D. M. and

WRIGHT, K.,

Computable errorbounds for polynomial

collocationmethods, SIAM J. Num.Anal. 15,134-151

(1978).

[4]

WRIGHT,

K., Asymptotic properties of collocationmatrix norms 1- global polynomial

approximation,

I.A.A.J.

Num.

Axial, 4, 185-200

(1984).

[5]

GERRARD,

C.and

WRIGHT,

K.,

Asymptotic properties ofcollocation matrix norms 2:

piecewise polynomial approximation,

I.M.A.J. Num.

Anal, 4,363-373

(1984).

[6]

AHMED,

A.and

WRIGHT, K.,

Further asymptoticpropertiesofcollocation matrixnorms,

I.M.A.J. Num.Anal.5,235-241

(1985).

[7]

AHMED,

A.and

WRIGHT, K.,

Errorestimationfor collocationsolutionof linearordinary

differentialequations,

Comp. &

Maths. withAppl;,,Vol.128, No.5/6,1053-1059

(1986).

[8]

WRIGHT, K., AHMED, A.

H. and

SELEMAN,

A.

H.,

Criteria for mesh selection in collocation algorithms forordinarydifferential boundaryvalueproblems, I.M.A.J. Nurrl, Anal. 11, 7-20

(1991).

[9]

DELVES,

L.

M.,

inModern NumericalMethods forOrdinary_{Differential Equal;i0ns}

(Edited

by G.Hall andJ.M.

Watt),

Chap. 19, OxfordUniversity

Piess

(1976).

[10]

DE BOOR,

C. andB.

SWARTZ,

Collocation atGauss points, SIAM J. Num. Anal. 10, 582-606

(1973).

[11]

DE

BOOR

C.,

Goodapproximation by splineswith variableknotsII.

In

proceedingofa conferenceonthe NumericalSolutionof Differential Equations

(Edited

by G. A.

Watson),

lecturenotesin Matheamtics 363,Springer,Berlin

(1974).

[12]

RUSSELL, R.

D.and

CHRISTIANSEN,

.l., Adaptivemesh selectionstrategies

for

solving

boundary

valueproblems, SIAM J. Num. Anal, 15,59-80

(1978).

[13]

U.

ASCHER, CHRISTIANSEN,

J.and

RUSSELL,

R.

D.,

COLSYS

A

Collocation

forBoundary-ValueProblems. Codes forBoundaryValue Problems inOrdinary_ Differential

Equations. gd.B.Childs etal., New York,

Spriner

Verlag,164-185

(1979).

14]

AHMED, A.,

Collocationalogrithmsand erroranalysisforapproximatesolutionsof ordinary

differentialequations,Ph.D.Thesis, Universityof Newcastle

Upon

Tyne

(1981).

[15]

CHENEY,

E.

W.,

IntroductiontoApproximationTh.eory,McGraw-Hill,NewYork

(1960).

16]

AHMED,

A.,

Asymptotic propertiesof collocationprojection norms,Vol. 19,No.4,45-50,