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
?
????P
n??
?t
?
?
P
?
?t
?
?P
?
?t
?
????P
n??
?t
?
?
?
?
?
?
?
?
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??
?
?
?
?
?
?
?
?
?
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 20
T able??Equidistant no des on???????Right handside?f? ??????
i??
??????comperr??????????????err????????
n G?O NHR?OGauss G?O NHR?O Gauss
? ???e??? ???e??????e??????e??????e??? ???e??? ???e??????e???
?????e??? ???e??????e??????e??????e??? ???e??????e??? ???e???
?????e??????e??? ???e??????e??? ???e??? ???e??????e??? ???e???
?? ???e??????e??????e??? ???e??? ???e??????e??? ???e??????e???
?? ???e??????e??? ???e??????e??? ???e??? ???e??????e??? ???e???
?? ???e??? ???e??? ???e??????e??? ???e??? ???e??????e??? ???e???
?????e??? ???e??? ???e??????e??????e??? ???e??????e??????e???
T able ??Equidistant nodes on ??????? Co ordinatesof the right?handsidef? random integers
on ????????
???????comp err??????? ????????err???????
n G?ONH R?OGauss G?O NH R?O Gauss
? ???e??????e??????e??? ???e??? ???e??? ???e??? ???e??? ???e???
?????e??? ???e??????e??? ???e??? ???e??????e??? ???e??? ???e???
?????e??? ???e??????e??? ???e??????e??? ???e??????e??? ???e???
?? ???e??????e??????e??? ???e??????e??? ???e??????e??????e???
?? ???e??????e??? ???e??????e??? ???e??? ???e??? ???e??????e???
?????e??? ???e??????e??????e??? ???e??????e??? ???e??????e???
?????e??????e??????e??? ???e??????e??????e??? ???e??? ???e???
T able ??Nodes? extrema ofT
n
????Coordinates oftherigh t?handsidef? randomintegers on
????????
??????comperr??????? ????????err???????
n G?ONH R?OGauss G?O NHR?O Gauss
? ???e??????e??? ???e??????e??????e??????e??? ???e??? ???e???
?? ???e??????e??? ???e??????e??? ???e??????e??????e??? ???e???
?????e??????e??? ???e??????e??? ???e??? ???e??????e??? ???e???
?????e??? ???e??? ???e??????e??? ???e??????e??????e??? ???e???
?????e??????e??? ???e??????e??? ???e??? ???e??? ???e??????e???
?? ???e??????e??? ???e??????e??? ???e??????e??? ???e??? ???e???
?? ???e??????e??????e??? ???e??? ???e??????e??????e??????e???
T able ???No des?zeros ofT
n
????Coordinates ofrigh t?hand sidef? randomintegers on?????
???
??????comp res????????????????res?????? ??
n G?O NHR?O GaussG?O NHR?O Gauss
? ???e??????e??????e??????e??????e??? ???e??????e??? ???e???
?????e??? ???e??????e??????e??? ???e??????e??? ???e??? ???e???
?????e??????e??? ???e??????e??? ???e??????e??? ???e??????e???
?????e??? ???e??????e??? ???e??????e??????e??? ???e??? ???e???
?????e??????e??? ???e??????e??????e??????e??? ???e??????e???
?? ???e??????e??? ???e??????e??? ???e??? ???e??????e??????e???
?????e??????e??????e??????e??? ???e??? ???e??????e??????e???
Numerisc heMathematikElectronic Edition?pagen umbers may di?erfrom theprin tedv ersion
page?? of Numer? Math????????? ??????
Page 21
??????compres???????? ???????res????????
n G?ONH R?O Gauss G?ONH R?O Gauss
? ???e??? ???e??????e??? ???e??????e??? ???e??????e??? ???e???
?????e??????e??? ???e??? ???e??????e??? ???e??? ???e??????e???
?????e??????e??????e??? ???e??? ???e??????e??????e??? ???e???
?????e??????e??? ???e??? ???e??????e??? ???e??????e??? ???e???
?? ???e??? ???e??????e??????e??? ???e??? ???e??????e??????e???
?????e??? ???e??????e??? ???e??????e??????e??? ???e??? ???e???
?????e??????e??? ???e??????e??? ???e??????e??? ???e??? ???e???
to solve morethantwo linearsystemswiththe same Chebyshev?Vandermonde
matrix?
We also compared theaccuracy in thecomputedsolutions of ??????de?
terminedby G?O?NH? R?O and Gauss?Again? letx
?sp?
??x
?sp?
i
?
n??
i??
and
x
?dp?
??x
?dp?
i
?
n??
i??
stand for the solutioncomputed insingle anddouble pre?
cision arithmeticand
err?
kx
?dp?
?x
?sp?
k
?
kx
?dp?
k
?
? comp
err? max
??i?n??
jx
?dp?
i
?x
?sp?
i
j
jx
?dp?
i
j
?????
T able??? Nodes?extrema ofT
n
???? Co ordinates ofrigh t?hand sidef? random in tegerson
????????
??????comp
F
res????????
algorithms
????????res????????
ha
n G?O NH R?O Gauss G?O NH R?OGauss
? ???e??????e??? ???e??? ???e??????e??? ???e??????e??????e???
?????e??? ???e??????e??????e??????e??????e??? ???e??????e???
?? ???e??????e??????e??????e??????e??????e??????e??????e???
?????e??? ???e??? ???e??????e??????e??????e??? ???e??? ???e???
?????e??????e??? ???e??????e??????e??? ???e??????e??? ???e???
?? ???e??????e??? ???e??????e??? ???e??? ???e??????e??????e???
?? ???e??????e??? ???e??????e??????e??? ???e??????e??????e???
denotethe in?nitenorm and comp onent wise relativeerrors in thecomputed
solution? or all fourdiscussedwevecomputed solutions ofa large
n umb erofsystems ????? withv ariousc hoicesofno dest
?
?t
?
? ????t
n??
andrigh t?
handsidesf? Ineach casewedetermine therelativeerrors?????accompan
i??
ying
eachalgorithmfrom four mentioned? The conclusionwhich follo wsfrom the
resultsofn umericalexp eriments isthat forreasonablevaluesofnalgorithm
G?Ocomputesthesolutionof ?????accurately?F or thesolution ofChebyshev?
Vandermonde systemswith prepro cessingG?Oshouldbepreferredalgorithm of
thoseconsidered?The data oncomputedexamples aregiven inTables????
Wealsodeterminedtheresidualin?nite?normandcomponentwiseerrors
res?
kjV
T
?t??x
?sp?
?fk
?
kfk
?
?comp
res? max
??i?n??
jg
i
?f
i
j
jf
i
j
?????
by accum ulationsums indoubleprecisionarithmetic? Hereg??g
i
?
n??
?V
T
?t??
x
?sp?
? Thedata oncomputedexamples aregiveninTables??????
Numerisc heMathematikElectronicEdition?pagen umb ersmaydi?erfrom the prin tedversion
page ??ofNumer?Math????????? ??????
Page 22
?? Bj? orck?A??P ereyra?V? ??????? Solution ofVandermonde systemsof linearequations?
Mat?Comput? ??? ???????
?? Calvetti?D?? Reic hel? L? ???????F ast inv ersion ofVandermonde?like matricesinv olving
orthogonalp olynomials ?preprint?
??Gautsc hi? W? ???????The conditionofVandermonde?like matrices? inv olving orthogonal
polynomials? LinearAlgebraAppl? ??? ???????
?? Gohb erg?I??Kailath? T??Koltrach t? I?? Lancaster?P? ??????? Linearcomplexity parallel
algorithms forlinear systemsof equationswith recursive structure? LinearAlgebra Appl?
?????? ???????
??Gohb erg? I?? Koltrach t? I???????? Mixed?comp onent wise andstructuredconditionn um?
b ers ?preprint?
?? Gohb erg? I?? Olshevsky? V? ???????F astalgorithms with preprocessingformultiplication
oftransp oseV andermonde matrixandCauchy matrixwithv ectors?submitted?
??Heinig? G??Rost? K????????Algebraic metho ds forTo eplitz?like matricesand operators?
OT?? ?I?Gohberg? ed???Birkh? auserV erlag?Basel
??Higham?N????????F ast solutionofV andermonde?like systemsinv olvingorthogonal
p olynomials?IMA J?Numer?Anal??? ???????
??Higham?N????????Stability analysis of algorithmsforsolving con?uentV andermonde?
like systems? SIAMJ? Matrix Anal??? ???? ?????
???Karlin? S??Szeg?o G? ??????? Oncertain determinantswhoseelementsareorthogonal
p olynomials?In? G?Szeg?o? Collectedpap ers????????? ???????Birkhauser?Boston
???Lander? F???????? TheBezoutian and theinv ersion ofHank el andTo eplitzmatrices ?in
Russian?? Mat?Issled?Kishinev?? ?????????
??? MA TLAB User?s Guide???????TheMathW orks?Inc?? Natick MA
???Press? W?? Flannery? B??T eykolsky? S??V etterling?W? ???????Numerical recipes inC?
Cam bridge Univ ersity Press? Cambridge
???Reic hel?L?? Opfer? G????????Chebyshev?V andermondesystems? Mat? Comput??? ??????
???????
???T ang? W??Golub?G? ??????? The block decomp ositionofaV andermonde matrix andits
applications?BIT ??????????
???Traub? J????????Asso ciatedpolynomials and uniformmetho dsforthe solutionoflinear
problems? SIAM Review?? ???????
???V erde?Star?L???????? Inv erses ofgeneralizedVandermondematrices? J?Math?Anal?
Appl????????????
This articlew aspro cessedby theauthor usingtheL
a
T
E
X style?lecljour? fromSpringer?Verlag?
Numerisc heMathematikElectronic Edition? pagen umb ersmaydi?er from theprintedversion
page ?? of Numer? Math? ??????????????