Page 1

ElectronicEdition

F astinversion of Cheb yshev?V andermonde

matrices

I?Gohb ergandV?Olshevsky

SchoolofMathematical Sciences?Raymondand Beverly SacklerFacultyofExactSciences?Tel

AvivUniversity?Ramat Aviv ?????? Israel

Received January ????????Revisedversion receivedMay ???????

Summary?This papercontainstwo fastalgorithms forinv ersionof Cheb yshev?

V andermonde matricesof the?rst and secondkind?They arebasedonspecial

represen tations of theBezoutiansofChebyshevp olynomialsofb oth kinds?The

paper also contains theresultsofnumerical exp eriments which show that the

algorithms proposed here arenotonlym uch faster? butalsomore stablethan

other algorithmsavailable?Itisalsoe?cientto usetheabovetwo algorithms for

solvingCheb yshev?Vandermode systemsof equationswith preprocessing?

MathematicsSubje ct Classi?cation ?????????A?????A?????B??? ??F?????T???

??Y?????Q??

??In troduction

In thispaperweconsiderp olynomialV andermondematrixof theform

V

P

?t??

?

?

?

?

P

?

?t

?

?P

?

?t

?

????P

n??

?t

?

?

P

?

?t

?

?P

?

?t

?

????P

n??

?t

?

?

?

?

?

?

?

?

P

?

?t

n??

?P

?

?t

n??

????P

n??

?t

n??

?

?

?

?

?

??t??t

i

?

n??

i??

??

wheret

?

?t

?

?????t

n??

are di?erent complexn umb ersandsystemfPg?fP

?

????P

?

????

????P

n??

???gforms the basisin thelinear spaceC

n??

??? ofall complexpoly?

nomialsin? whose degreedo esnotexceedn? ??The conceptofpolynomial

V andermondematrix isa generalizationofthe ordinaryV andermondematrix?

whereP

i

?????

i

?ForordinaryV andermondematrices thenumb er of fastalgo?

rithmsisav ailableforsolving linearsystems? Bj?orck andP ereyra ???????T ang

and Golub???????Gohberg?Kailath? Koltracht andLancaster??????? andfor

inv ersion?T raub ???????Heinig andRost ???????Gohb ergand Koltracht???????

Moregeneralclasses ofpolynomialV andermonde matricesappearindi?erent

areasandwereconsideredandanalyzed in literature ?see? forexample? Karlin

and Szeg?o ???????V erde?Star????????The analysisof then umericalprop er?

ties andalgorithms connectedwithp olynomialV andermondematricescanbe

found inv arious sources?InGautschi ?????? theconditionn umb er ofp olyno?

mialV andermondematrixw asestimated forvariousc hoicesof thesystemfPg?

Correspondence to? I? Gohberg

Numerisc heMathematikElectronic Edition? pagen umb ersmay di?erfrom theprintedversion

page ??of Numer?Math? ???????? ??????

Page 2

for solvinglinear systemswithp olynomialV andermondematricesw ere deduced

and analyzed?

Inthepresent pap erthe useof the Bezoutianallo ws to deducethefollo wing

equality whichconnectstwopolynomialV andermondematrices corresp onding

totwo systemsofp olynomialsfPg andfQg?

V

P

?t?

??

?B?V

Q

?t?

?

? diag??

?

?

Q

n??

i??

i??k

?t

k

?t

i

??

?

n??

i??

???????

whereb

ij

aredeterminedfrom theequality

B?

F????F???

???

?

n??

X

i?j ??

b

ij

?P

i

????Q

j

???

withF????

Q

n??

k??

???t

k

?? and

B??b

ij

?

n??

i?j??

?

Numericalfeatures of thealgorithm forcomputingV

P

?t?

??

basedon equal?

ity????? dep endessentially on thecomplexityof thematrixB?Weconsiderin

this paper thecasewherefPgandfQg are systemsof Cheb yshevp olynomi?

als?In the latter caseit turnsout thatthe matrixB isa Hankel ?oralmost

aHank el?matrix? Thisleads tothe factthat thealgorithm forinv ersion of

Cheb yshev?Vandermonde matricesbased onthe equality ?????hascomplexity

?n

?

op erations?The proposed algorithmscompare fav orablyinb oth time ander?

roraccum ulationwithalgorithmsfromHigham ???????Reic hel andOpfer ??????

andalsowithGaussian elimination?

The paperconsistsoftwo sections? The?rstcon tains thedeductionof the

formulae used laterfor thedesignof the algorithms?Thesecondsectioncontains

thedetailed descriptionof thealgorithms andpresen ts theresultsofnumerical

experimen ts?

Afterthesubmission for publication ofthepresentpap erthe authorsre?

ceiveda preprint ofCalv ettiandReic hel??????? which contains anotheralgo?

rithm for inv ersionofpolynomialV andermondematrices?Thisalgorithmisa

generalizationof the scheme forordinaryV andermonde matrices? suggestedear?

lier inT raub???????The Calv etti and Reichelalgorithmdo es not restrictitself

to Cheb yshev?Vandermondecase and isv alidformore generalclasses ofp olyno?

mialVandermonde matrices?At the sametime itisa littleslow erandrequires

??n

?

?O?n?arithmeticoperations?

??F orm ulae forinv ersesofChebyshev?V andermondematrices

????Notations? de?nitions andr elevantfacts

Matrices? The algebraofalln?nmatriceswith complex en tries willbedenoted

byC

n?n

?

Tr ansp ose?T ransposeof any matrixA?C

n?n

willbe denotedbyA

?

?

Numerische MathematikElectronic Edition? pagen umb ersma ydi?er from theprintedv ersion

page ?? ofNumer?Math? ??????????????

Page 3

n?N willbe denotedbyC

n

????

Chebyshevp olynomials?Let

fTg?fT

k

??g

n??

k ??

?fUg?fU

k

??g

n??

k??

stand for thefamilies ofChebyshevpolynomials

T

k

???? cos?k? arccos?????U

k

????

sin??k? ??? arccos????

sin?arccos????

?

of the?rstand ofthe secondkind? respectiv ely?

Re curr encerelations for Chebyshevpolynomials? Asit isw ell known?Chebyshev

p olynomials canalsobe de?nedby thefollowingrecurrencerelations?

T

?

??????T

?

???? ?????T

n??

????T

n

????T

n??

?????????

U

?

??????U

?

??????????U

n??

????U

n

????U

n??

?????????

Connectionsb etween Chebyshevp olynomials?It isw ellkno wnand immediately

follo wsfrom????? and ?????that

T

n

????

?

?

?U

n

????U

n??

????? ?????

U

n

??????

?

n

?

?

X

j??

T

n??j

????

????

n

??

?

??????

Associated ve ctors?F oreachvectort??t

k

?

n??

k ??

?C

n

willbe asso ciatedtwo

v ectors

a?t???a

k ??

?t??

n??

k ??

andb?t???b

k ??

?t??

n??

k ??

?????

whosecoordinates aredetermined from thefollo wingequalities?

n??

Y

k ??

???t

k

??

n

X

k ??

a

k

?t??T

k

????

n

X

k ??

b

k

?t??U

k

????

Bezoutian? LetF????G???betwopolynomials fromC

n

????Thebilinearform

B

?F?G?

??????

F????G????F????G???

???

?

n??

X

i?j??

q

ij

??

i

??

j

?????

is referredtoas theBezoution ofF????G????

PolynomialVandermondematrix? Lett??t

k

?

n??

k ??

?C

n

andsomesystem of

p olynomialsfPg?fP

?

????P

?

???? ????P

n??

???g is given?The matrix

V

P

?t??

?

?

?

?

P

?

?t

?

?P

?

?t

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????P

n??

?t

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P

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????P

n??

?t

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?

?

P

?

?t

n??

?P

?

?t

n??

????P

n??

?t

n??

?

?

?

?

?

Numerisc he MathematikElectronicEdition?pagen umb ers may di?erfrom theprin tedversion

page ??ofNumer?Math???????????????

Page 4

systemfPg andtov ectort?

Chebyshev?V andermondematric es?MatricesV

T

?t? andV

U

?t?willbe referred

to as Chebyshev?Vandermondematrices of the ?rst andof the secondkind?

resp ectiv ely?

Hankelmatrix?F or somevectorc??c

k

?

n??

k??

?C

n

byH?c? willbe denoted the

follo wing triangular Hankel matrix

H?c??

?

?

?

?

?

formulae

?

?

?

c

?

c

?

???c

n??

c

n??

c

?

?

?

?

?

?

?

?

?

?

?c

n??

?

?

?

?

?

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?

?

?

c

n??

c

n??

?

?

?

?

?

?

c

n??

????????

?

?

?

?

?

?

?

?

?

Greatest inte gerinn is denotedby?n??

???? Mainr esults

Theorem????Lett??t

k

?

n??

k??

andt

?

?t

?

? ????t

n??

ben di?er entc omplexnum?

b ers? Thenthe fol lowinghold?

V

T

?t?

??

???D? H?d?t???D?V

T

?t?

?

?diag?c?t??? ?????

V

U

?t?

??

? H?e?t???V

U

?t?

?

?diag?c?t????????

V

T

?t?

??

???D? H?a?t???V

U

?t?

?

? diag?c?t????????

V

U

?t?

??

??? H?a?t???D?V

T

?t?

?

? diag?c?t??? ?????

where

D? diag?

?

?

????? ???????v ectora?t? is asin ????? andcoordinates ofv ectors

c?t???c

i

?t??

n??

i??

?d?t???d

i

?t??

n??

i??

?e?t???e

i

?t??

n??

i??

arede?nedas follo ws?

c

k

?t??

?

Q

n??

i??

i??k

?t

k

?t

i

?

?d

k

?t??

?

n?k??

?

?

X

i??

a

k ????i

?t??k????? ????n? ???

e

n??

?t??a

n

?t??e

n??

?t??a

n??

?t??e

k

?t??a

k ??

?t??a

k ??

?t??k????? ????n????

Theorem ????L ett??t

k

?

n??

k ??

andt

?

?t

?

? ????t

n??

ben di?er entc omplexnum?

b ers? Thenthe following formulaehold?

V

T

?t?

??

Electronic

???D?H?f?t???D?V

T

?t?

?

? diag?c?t??? ?????

V

U

?t?

??

??? H?b?t???V

U

?t?

?

?diag?c?t??? ?????

V

T

?t?

??

???D?H?g?t???V

U

?t?

?

?diag?c?t??? ?????

V

U

?t?

??

??? H?g?t???D?V

T

?t?

?

? diag?c?t????????

Numerisc heMathematikEdition?pagenumbersmaydi?erfromtheprintedversion

page??ofNumer?Math? ??? ???????????

Page 5

as in theThe or em????Theco or dinates ofthe vectorsf?t???f

i

?t??

n??

i??

and

g?t???g

i

?t??

n??

i??

are de?ned by

f

k

?t??

?

n?k??

?

?

X

i??

?i????b

k ????i

?t??g

k

?t??

?

n?k??

?

?

X

i??

b

k ????i

?t??k?????????n????

The pro ofofb othabove theoremsisbased onin terrelationsb etw eenCheb y?

shevp olynomialswhich aregiv en in thenext theorem?

Theorem????F orn?? the fol lowingformulae hold?

T

n

????T

n

???

???

???

?

n??

?

?

X

k ??

n????k

X

j ??

???

T

n????k?j

????T

j

????

??

?

n??

?

?

X

k ??

T

n????k

??????

?

n??

?

?

X

k ??

T

n????k

????

??????

n??

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T

n

????T

n

???

???

?

n??

X

j ??

U

n???j

????U

j

????

n??

X

j??

U

n???j

????U

j

???? ??????

T

n

????T

n

???

k

???

???

n??

X

j ??

T

n???j

?U

j

????T

?

????U

n??

???? ??????

U

n

????U

n

???

???

???

?

n??

?

?

X

k ??

?k????

n????k

X

j ??

T

n????k?j

????T

j

????

???

?

n??

?

?

X

??

?k? ???T

n????k

????

??????

n??

?

??n? ?????????

U

n

????U

n

???

???

???

n??

X

j ??

U

n???j

????U

j

??????????

U

n

????U

n

???

???

???

?

n??

?

?

X

k??

n????k

X

j ??

T

n????k?j

????U

j

????

???

?

n??

?

?

X

k ??

T

?

????U

n????k

???? ??????

Theorems ???????and Theorem??? willbeprov ed in the nexttwo subsec?

tions?

NumerischeMathematikElectronicEdition? pagen umb ers may di?erfrom theprin tedv ersion

page?? of Numer?Math? ??????????????