Confluent Vandermonde Matrices Using Sylvester's Structure

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Confluent Vandermonde matrices using Sylvester’s structure Lamine Melkemi

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Lamine Melkemi. Confluent Vandermonde matrices using Sylvester’s structure. [Research Report] LIP RR-1998-16, Laboratoire de l’informatique du parallélisme. 1998, 2+14p. hal-02102110

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ConuentVandermonde matrices using

Sylvesters structures

March

Lamine Melkemi

ResearchReportN

Ecole´ Normale Superieure´ de Lyon 46 Allée d’Italie, 69364 Lyon Cedex 07, France Téléphone : +33(0)4.72.72.80.00

T´el´ecopieur : +33(0)4.72.72.80.80

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ConuentVandermonde matrices using

Sylvesters structures

Lamine Melkemi

March

Abstract

In this pap er we rst show that a conuentVandermonde matrix maybe

viewed as comp osed of some rows of a certain blo ckVandermonde matrix As

a result we derive a Sylvesters structure for this class of matrices that ap

p ears as a natural generalization of the straightforward one known for usual

Vandermonde matrices Then we present some applications as an illustration of

the established structure For example weshowhow conuentVandermonde

and Hankel matrices are linked with each other and also we describ e an On

algorithm for solving conuentVandermonde least squares minimizations prob

lems

Keywords conuentVandermonde matrix Sylvesters equation structured

matrices

Resume

Dans cet article nous prop osons une structure de Sylvester p our les matrices de

Vandermonde conuentes qui parait comme une generalisation naturelle de celle

connue dans le cas dun systemedeVandermonde simple La demonstration

de ce resultat tire prot dune proprieteinteressante disant dans un sens que

nous preciserons plus loin quune telle matrice est en fait plongee dans une ma

trice de Vandermonde par blo cks En exploitant cette structure nous montrons

ensuite quil est p ossible dexprimer linverse dune matrice de Vandermonde

conuente comme le pro duit de son duale et dune matrice de Hankel Toujours

a laide de cette structure nous decrivons un algorithme On p ermettantde

resoudre les problemes aux moindres carres bases sur ces matrices Enn nous

montrons comment on p eut etendre les resultats etablis a une classe de matrices

b eaucoup plus generale

Motscles matrice de Vandermonde conuente structure de Sylvester matrices structurees

Intro duction Given n dierentnumb ers x x x that is x

n i

x for i j Wemeanby a conuentVandermonde matrix the following

m p n matrix

p p

V f x f x f x fx f x f x

p n n n

where f xisthevectorial function

m t

f xxx x

and f xf x f x are the derivatives of f x In fact the number

of the derivatives p should vary as x but for convenience and without loss in

generality it is assumed throughout the pap er that it is xed It is the matter

ofawell known concept that naturally generalizes usual Vandermonde systems

and can b e viewed as a matrix representation of some interp olation problems

For example Hermites interp olation amounts simply to solve the system

of linear equations V x b Because of its imp ortance in applications see

for instance such matrices havereceived a particular attention in the literature

In particular several On algorithms have b een prop osed for computing the

inverse of Vandermonde and conuentVandermonde matrices and that of their

duals In Higham considered the more general case where the

monomial x is replaced by a p olynomial P x and constructed what is called

a conuentVandermondelike matrix Then he develop ed On solutions for

computing the inverse of such a matrix for particular p olynomials In he

analyzed the numerical stability of these metho ds More recentlyLu

showed the ability to design asymptotically faster solutions to these problems

and conceived in particular an algorithm with Onplognplogn running time

for inverting np np conuentVandermonde matrices His approachisbased

on fast convolution pro ducts and fast p olynomial divisions

On the other side it is common to put matrices with particular forms suchas

To eplitz matrices into a unied algebraic formalism In other words given two

simple matrices L and U the idea is to nd the Sylvesters equation

LM MU GH

where the concerned matrix M is completely characterized by the generator GH

which is in the essence a summation of a xed numb er of outer pro ducts The

imp ortance of this equation called the structure of M stems from the fact that

it constitutes a p owerful to ol for eciently manipulating such matrices as it is

emphasized in dierent situations Of course the structure is more and more

elegant as long as the matrices L and U are simple and GH is of a small rank

See and the survey pap er with the references therein

In this pap er we consider this asp ect of manipulating conuentVandermonde

matrices via their Sylvesters structure which is deduced from a useful observa

tion that a conuentVandermonde matrix is in a certain viewp ointembedded

into a blo ckVandermonde one After some notations in section and a brief

discussion ab out blo ckVandermonde matrices in section where weshowsome

limitations when trying to extend the concepts available in the usual case and

evidence particular instances allowing to palliate the diculties wepresentin

Section our main result where a Sylvesters equation satised by the conuent

Vandermonde matrices is established and proved Examining it one may assert

that it well characterizes the conuentVandermonde matrices and apparently

can not b e sharp ened further To our knowledge such a structure do es not ex

ist elsewhere In section we study the converse situation where it is shown

that up to a slight mo dication it is p ossible to determine precisely the ma

trices which are solutions of the mo died Sylvesters equation In other words

we establish an interesting canonical structure to which the studied Sylvesters

equation may b e easily transformed and conversely As an application an in

verse formula is given The sections and may b e viewed as interesting

applications of the Sylvesters structure In section a connection b etween

conuentVandermonde matrices Hankel and To eplitzlike ones is shown in a

way similar to that found out for the usual Vandermonde matrices see

Nevertheless it is shown that though the structure established there

are severe limitations as one attempts a direct generalization Fortunatelyit

is p ossible to comp ensate the drawbacks so that one may construct Onlog n

algorithms for solving weighted least squares minimizations settling overdeter

mined Hermites interp olations In section wedevelop an On metho d for

computing the Cholesky factorization of the covariance matrix of the conuent

Vandermonde matrix The algorithm as it is well known may b e exploited for

solving conuentVandermonde least squares minimizations problems Finally

we consider in section the notion of generalized conuentVandermonde ma

trices and explain how the results available in the simple case may b e extended

Notations Tomake the description of the text as clear as p ossible some

notations are adopted As we will deal with rectangular matrices and blo ck

Vandermonde matrices it follows that dierent linear spaces will b e considered

Consequentlycaremust b e taken ab out the use of a same notation

For this reason we denote throughout the pap er bye f

i im j j p

Cm h and g the canonical basis in the linear spaces

k k mp k k mp

p mp np

C C C C is the complex eld In general we and resp ectivelywhere

consider the canonical basis in R where R is a ring with unit element and

e denote it byE E For example if R is the ring of the w C

p p complex matrices then for k m wehave

E h h h

k kp kp

k p

whichisa mp p matrix Now let say that whatever canonical basis one uses

it is systematically understo o d along the pap er that if u is a vector then u

denotes its i co ordinate

Since one of our main ob jectives is to construct a Sylvesters structure for con

uentVandermonde matrices it is natural to exp ect the use the displacement

matrix dened as a square matrix with s along its rst subdiagonal and s

elsewhere If wehaveak k displacement matrix let denote it by Z For

simplicity and unless otherwise stated welet Z Z and Z Z The well

m np

known reverse identity matrix may b e also utilized in such a context and is a

square matrix with s along its antidiagonal and s elsewhere We will denote

by J areverse identity matrix of size k k

In this pap er for reasons to b e seen latter we distinguish b etween vector trans

p ose and matrix transp ose and weusetwo notations u means the transp ose

of the vector u whereas R means the transp ose of the matrix R By the way

wemay use the antitransp ose op erator T dened as follows

T T

A J A J

k k

where A is a k k matrix

Blo ckVandermonde Matrices ClearlytheVandermonde matrices may

b e dened over arbitrary rings with unit element as for example the ring of

matrices and it is of interest to note that in this straightforward generalization

the well known structure of Vandermonde matrices remains safe That is if

R denotes such a ring then wehave

T t

Z V VD E U

where V V B B B isthe m n Vandermonde matrix with el

ements B into the ring R Z is the displacement matrix with R

in the rst sub diagonal and R elsewhere D diag B and U

n n

B B Recall that E stands for the last element in the canonical

basis of R

In the case where R is the ring of p p matrices dened over the eld of the

Cwe obtain what is known as blo ckVandermonde matri complex numbers

ces Here the displacement matrix in the ab ove Sylvesters equation must b e

of the displacement matrix Z with elements understo o d as the pp ower Z

in the eld C Based on this structure it is thus p ossible to construct ecient

algorithms for manipulating blo ckVandermonde matrices On the other hand

there are natural reasons to exp ect that most op erations over Vandermonde ma

trices may b e carried out to blo ckVandermonde ones requiring up to a p factor

the same amount of computation Unfortunately this is not true in general

b ecause of the simple fact that several exible prop erties such as commutativ

ity are missed in the ring R while well veried in the eld C Consider for

instance the problem of solving a linear system of equations V x b where

V is a blo ckVandermonde matrix An approach due to see also is to

transform this system to the following equivalentone

VV x Vb

The idea lies in the observation that in the case where V is a Vandermonde

matrix one may assert that VV is a Hankel matrix Hence the complexity for

solving the considered system is the same as for solving a Hankel linear system

Since there are Onlog n algorithms for computing the inverse of a p ositive

denite Hankel matrix see among others one nds out another

way of constructing Onlog n algorithms for the p olynomial interp olation On

the contrary when attempting to extend this technique to blo ckVandermonde

matrices the diculty arises as so on as one writes the structure

T p T T t

V Z D V UE

T T

of V where in general wehave D D recall that D diag B As

an alternative issue this last remark evidences p ossible ways for comp ensating

such a disadvantage Actuallyifwe assume that the blo cks B are symmetric

matrices then one can checkthatVV is a blo ckHankel matrix Indeed if we

T T

apply the matrix V to the ab ove structure of V where now D D we

obtain

T p T t

VV Z VDV VUE

pT t

Using the fact that VD Z V E U the following structure is easily

deduced

T p pT T t t T

VV Z Z VV VUE E U V

which asserts that VV is a blo ckHankel matrix

Emb edding conuentVandermonde matrices into blo ckVander

monde ones In this section we restrict ourselves to particular blo ck matrices

Given n dierent complex numb ers x x x we will consider the p p

matrices B B B dened as follows

B B x

i i

Let now form the mp np blo ckVandermonde matrix V V B B B

As where for i m and j n its i j blo ckelementis B

it is previously stated the matrix V fullls the Sylvesters equation b elow

t pT

V VD E U Z

where Z is the mp mp displacement matrix D is the blo ckdiagonal matrix

such that D diag B B and b oth U and E are blo ckvectors

n m

dened over R and R R b eing the ring of the p p complex matrices

It is easily seen that E is the last element in the canonical basis of R

t m m

and U B B The structure of V just evidenced will b e a corner

stone in constructing that of conuentVandermonde matrices More precisely

we will show that the matrix V contains in a certain sense a conuentVander

monde one and conversely every conuentVandermonde matrix is embedded

into a blo ckVandermonde one of a same kind This result from whichthe

desired Sylvesters structure is obviously derived is revealed byaninteresting

observation in the b ehavior of the p owers B xk of the matrix B x

Indeed an investigation in the elements of the k p ower matrix B x p ermits

to observe that its rst rowisinterestingly formed by x and its derivatives

In general it is p ossible to showby an inductive reasoning ab out the p ower k

k k

that the i j element of B x can b e expressed as follows we assume for

convenience

j j k

k k k j i k j i

x x

ij j i

i ij ik j i

for i j i k and is reduced to zero elsewhere As a consequence

the conuentVandermonde matrix V is simply that formed bythekp rows

of V in the natural order k m Mathematicallyif P denotes the

p ermutation matrix dened by

X X

P h h

imk

kpi

i k

C thenwe can assert that where h is the canonical basis of

s smp

pT th

On the other hand since Z V upshifts p times the rows of V so that the kp

th pT

rowofV k m will b e the k p one of Z V the following

interesting relation is obviously deduced

Z V

PZ V

In this direction it is implicitly said that one is applying the p ermutation matrix

P to the structure of V Hence it is useful to analyze the asp ect of the generator

PE U For direct matrix manipulations lead us to consider PE as a

m m

blo ckvector where its j blo ckcomp onent j p is precisely the

t t

m p matrix e f Consequently the rst m rows of PE U form the

m m

t t t t

m np matrix e y e f U where its last row is the rst one of U and

m m

t t

otherwise the elements are reduced to zero Clearly y is the rst rowofU

m m

which is simply an alignement of the rst rows of the blo cks B B It

follows from the ab ove observation related to B xthat

mp mp

t m m m m

y x mx m x x mx m x

p p

n n

where m Taking into account these considerations and the fact that

we are concerned with the rst m rows onlyaswe lo ok forward V we rst

observe that

pT t t

PZ V P VD E U PV D PE U

n n

from whichwe conclude that the Sylvesters equation of the conuentVander

monde matrix V is given by

T t

Z V V D e y

p p m

Clearly this structure app ears as a natural generalization of that of the usual

Vandermonde matrices However b ecause of the fact that D isablockdiagonal

matrix and not a diagonal one one may exp ect serious problems as it will b e

seen in section when trying to safely carry out via this equation the results

established in the simple case

Normalization and the inverse formula In the previous section we

haveshown that the conuentVandermonde matrix V is a solution of the

Sylvesters equation b elow

T t

Z X XD e y

Conversely it is p ossible to determine the matrix X for which this equation

is veried Nevertheless for arbitrary vectors y the obtained matrix X has a

queer asp ect and is not a priori representativeinaconvinced manner inasmuch

one feels b eing far away from the conuentVandermonde context As a remedy

weshow here that the vectors y of the form

y t t t

giverisetosolutions X with elements of simple expressions Beforehand we

suggest a normalization pro cedure over the Sylvesters structure which allows

to conclude that there is no loss in generality when considering the particular

y just ab ove

We rst apply over the considered equation a suitable diagonal matrix K so

that the non null elements along the sup erdiagonal of D will b e reduced to

To this end let K and K denote resp ectively the following p p and np np

diagonal matrices

K diag p K diag K K K

It is clear that the blo ckdiagonal and upp er bidiagonal matrix KDK

diag S x Sx has the desired form since

S x K B x K

i i

On the other hand the transformation we p erform over the Sylvesters structure

consists simply in p ost multiplying it by the diagonal matrix K In doing

so one realizes that using the variable change Y XK the Sylvesters

equation b ecomes

T t t

Z Y Y e y K e g

m m

An interesting feature in this normalization step is that the blo cks S x inthe

blo ckdiagonal matrix are upp er bidiagonal To eplitz matrices eachofwhich

therebycommutes with any other upp er triangular To eplitz matrix In the next

transformation wewillmake useful of this imp ortant prop erty

Consider n upp er triangular and To eplitz matrices R R R eachof

order p and set

R diag R R R

that is R is a blo ckdiagonal matrix where each blo ckelementisa p p matrix

and its blo ckmain diagonal is formed bytheintro duced To eplitz matrices R

Clearly if the structure is p ostmultiplied by R then remarking that R

R we obtain

T t t

Z YR YR e g R e w

m m

so that the variable change Y YR do es not alter the overall required

structure of the transformed Sylvesters equation Therefore it is p ossible in

general to cho ose the matrices R in suchaway that the co ordinates w k

i ipk

of the rowvector

w w w w

will b e reduced to zero It is readily seen that this op eration fails if and only

if there exists at least one co ordinate of w of the form w reduced to zero Up

to now it is assumed that the transformations wemay apply do es not lead to

this situation It is necessary to note bytheway that this normalization is re

versible since the the To eplitz matrices R can b e reconstructed systematically

Letusnow pro ceed to determine explicitly the solution X we will denote hence

forth W x x of the Sylvesters equation

T t

Z X X e y

in terms of the numbers x Let a denote the i hentry of the matrix X and

i ih

write h jp k suchthat j n and k p By controlling

the elements of X in its structure it is easily veried that

x a k

j mh

a x a kp

mh j mh

a x a k

mih j mih

a a x a kp

mih mih j mih

Clearly these recurrence relations fullled by the elements of the matrix X

provide the way of computing each a Even more they allow to show

mih

using an inductive reasoning ab out i k that

k i

a x

mih

Indeed by considering the rowa and the column a of X one

m j p

realizes that a and a are geometrical progressions from

mj pk k mij p i

which it is deduced that the formula is true for i k suchthati or

k On the other hand if we assume its trueness for ikandi k

i m k p and weset

b x a

ih j mih

then after simple calculations we can p erform the op erations b elow

k k

b x a x a

ih j mih mih

k i k i

k i k k i

k k k k

x x x x

j j j

k k

k i i k i i

x x

j j

k k

k i k i i

k k

k i i

k k i

k k

and as a It follows directly that a x b x

mih ih

j j

consequence the desired formula holds for all elements of the matrix X

As an application to this result it is p ossible to derive an expressive formula

for the inverse of nonsingular m m matrices which are solutions of the just

established normalized Sylvesters structure Indeed if W W x x

is the solution of the normalized equation then it is easy to check that

T t

W Z W W e y W

and by applying the antitransp ose op erator T we obtain

t T T T T T T

ye W W W Z W

y b eing the mirror vector of y As a consequence it is p ossible to assert that

under the conditions where the normalization just presented do es not fail there

exist an m m lower triangular To eplitz matrix L and n upp er triangular

To eplitz matrices R R R each of order p such that

T T

L W R W x x x

n n

where R diag R R

Transformation to structured matrices We consider here the asp ect

discussed in section where it is shown that if the blo ckelements of a blo ck

Vandermonde matrix V are symmetric matrices then V V is a Hankel matrix

b b

see In our situation however this result might logically fail in general

since the blo ckVandermonde matrix from which the conuentVandermonde

one is extracted has not the symmetry prop erty the blo ckelements are rather

upp er triangular matrices To palliate this dicultywe suggest the approachof

lo oking forward matrices A suchthatV AV is a Hankel or even a Hankellike

matrix Let A be such a matrix that is V AV is Hankel

t t T T T

e a Z ae V AV Z V AV

m p p

Using the structures of V and V it is readily veried under reasonable as

sumptions V is a nonsingular square matrix and after simple calculations

that

T t t

DA AD vu uv

where u V a y v V e Therefore it is natural to investigate

p p

the simple case where the right hand side is reduced to zero ie to deter

mine the matrices A for which DA AD If we set B KAK where K

is the diagonal matrix intro duced in the preceding section then the problem

amounts to compute B verifying B B In this direction wesawthatthe

blo ckdiagonal matrix is formed by the upp er bidiagonal To eplitz matrices

S x Hence it is not hard to show that the matrix

B diag R J R J R J

p p n p

where the R are p p upp er triangular To epliz matrices and J denotes the

i p

p p reverse identity matrix commutes with In the particular case where

for all i R is reduced to the p p identity matrix one nds that V FV is a

i p

Hankel matrix where

F K diag J J J K

p p p

Though this result p ermits to express V or V with the help of Hankel

p p

matrices deducing hence algorithms for calculating V using those invert

ing a Hankel matrix there is unfortunately a disadvantage making it some

what meaningless That is in the real case the matrix V FV is not p os

itive denite so that there is no way to compute the pseudo inverse or the

weighted one of V using such a matrix On the other hand if we assume

T T

that DA AD GH corresp onding to the case where VAV is in general

a Hankellike matrix it turns out that this situation yields more complicated

symmetric matrices A

As an alternative approach we pro ceed now to construct matrices A for which

VAV are To eplitzlike where it is assumed here that the m np matrix

m np V is real and satises the Sylvesters equation b elow

T t

Z V V e y

in suchawaythat VK is a conuentVandermonde matrix K b eing the diagonal

matrix intro duced in the previous section On the other hand let construct the

np np matrix C dened as follows

T T t

Z C C g z

where Z stands for the np np displacement matrix It is clear that the vector

z maybechosen suchthatCK is a nonsingular conuentVandermonde matrix

T T T

Therefore if we set W C and A WW then one may claim that VAV

is a To eplitzlike matrix Toshow this let us write

X Y

for meaning that the dierence X Y of the matrices X and Y is a summation

of r outer pro ducts with r xed Thus using simple arguments one nds that

T T T T T

Z VAV Z Z VWW V Z

T T T

V WW V

T T T

VWZ ZW V

T T

VWW V

VAV

which p ermits to conclude that VAV is indeed a To eplitzlike matrix In

fact it is p ossible to show that the pro duct VW isaTo eplitz matrix so that

its covariance matrix will b e evidently To eplitz like Indeed one has just to

p ostmultiply the structure

T t

Z V V e y

of V by the intro duced matrix W and then use the structure of C in order to

nd that of VW It is not hard to checkthat

T t

W W Z zg

which can b e substituted in the structure of V multiplied by W In doing so

one nds that

T T t t

Z VW VWZ e y W V zg

from whichwe conclude that our task is accomplished

Besides the general interest of this result it deserves to saythatitismore

signicative than the just established one in the sense that A and VAV are

symmetric p ositive denite matrices and the solution x of the linear system

VAV x VWb solves the least squares minimization problem b elow

T T

minjjW V x bjj

The imp ortance of this observation lies in the fact that there are asymptotically

fast metho ds in order to solve it Indeed one may exploit the Onplognplogn

fast metho d of for computing the pro ducts VAV and VWb and on the

other hand it is p ossible to solve the symmetric p ositive denite and To eplitzlike

system VAV x VWb with Omlog m op erations only app ealing the rapid

techniques develop ed in Hence Onplognplogn mlog m arithmetic op

erations are sucient for solving the just intro duced least squares minimization

problem

ConuentVandermonde least squares minimizations In this section

wepresent another application of the Sylvesters structure of the m np m

np real conuentVandermonde matrix V and show that it is p ossible to

construct an Onp algorithm for computing the Cholesky factorization of its

covariance matrix V V The idea is to establish a Sylvesters structure for the

T T

covariance matrix which can b e achieved by computing D V V D using the

structures of V and V We can write

T t T t T t T T T

ZZ V yy Z V e y V Z ye V D V D V

p p m p

p m p p

T t t

Observing that ZZ I e e and denoting by v the rst rowofV it is

m p

readily veried that

T T t T t

D V V D vv V V yy

p p

T T

Let now V V R R b e the Cholesky factorization of the covariance matrix

V V and let substitute it into this relation

T T t T t

D R RD vv R R yy

Therefore we obtain what we can call a rank mo dication of the Cholesky

factorization with the dierence that here we are concerned with the deter

mination of the matrix R It is therefore logical to app eal to the well known

techniques used for and adapt them to our situation see for example As

it is customary in such a case wewillmultiply the augmented matrices b elow

t t

y v

R RD

by appropriate Givens rotations The fundamental prop erty of whichwetake

advantage is that if a step in the metho d stands for a multiplicationby a pair

of Givens plane rotations then eachrowofR and RD is mo died only once

In what follows we describ e the rst step where it is shown how the rst row

of R is determined Let H and H b e the following plane rotations

c s

s c

H H

such that the rst co ordinates v and y of v and y resp ectively are reduced

to zero where

and

According to the prop erty stated ab ove one maintains that the rst rowof

R and that of R are identical and moreover they remain unmo died during

the subsequentGivens rotations Taking into account this and the fact that

v y one may write the equations b elow

a cv sr x

b y r

c cr x sv r y

where r denotes the rst rowofR r the rst co ordinate of r and x x It is

readily seen that the last relation relation c can b e also expressed as follows

x r v r y

from which one obtains the value of r

y v

as a consequence the numbers c s can b e calculated as follows

v r y r x

s d r y c

d d d d

On the other hand since the rst rowofR and that of R are identical it

follows that

t t t t

cr D sv r y

and thereby

t t t

r y sv I cD

t t

is the solution of a bidiagonal system Of course the rowvectors v and y can

b e computed easily as a preparation for computing analogously the subsequent

rows of R Wehave

t t t t

v cv sr D y y r

It is obvious that the up coming steps for computing the rows of the Cholesky

factor R may b e realized identically since the shap e of the augmented matrices

t t

v y

RD R

after eachstepmay b e rearranged in suchawaytohave the original one

Wehavethus sketched a p ossible metho d for p erforming the Cholesky factor

ization of V V using Onp op erations However as it may b e observed it

contains divisions and therefore care must b e taken in its implementation If it

happ ens that a division by zero o ccurs for computing an unknown we suggest

as a solution the use of the classical Cholesky factorization algorithm in order

to circumvent the problem In this p ersp ective it is not hard to remark that if

there are cosines c and corresp onding to a same step say the rst one and an

index k such that cx then the k p co ordinate r of r and only

k p

t t t t

it is missed in the relation cr D sv r y on which the computation

of r is based Since the rows of the factor R are determined bythismethodin

an ascending order and eachcontains at most two co ordinates in dicultyit

follows that the classical Cholesky factorization approach can b e utilized suc

cessfully to compute r without altering the total computational time

k p

We end this discussion by noting that the guidelines of the Cholesky factoriza

tion metho d develop ed here can b e used to compute the Cholesky factorization

of the matrix V SV where S is a symmetric p ositive denite To eplitz matrix

Generalization In view of the formalization established in this pap er it

is natural to consider what wemay call the generalized conuentVandermonde

matrices dened as solutions of Sylvesters structures of the typ e

Z X X GH

where in general the generator GH is a summation of outer pro ducts

b eing a relatively small p ositiveinteger and is the blo ckdiagonal matrix

intro duced in section Such a denition is precisely motivated by the fact that

the linear op erator

X X Z X X

is onetoone so that one is assured of the existence and uniqueness of the solu

tion X of the equation X B no matter what is the right hand side matrix

B The problem that obviously arises in this context is to develop algorithms

manipulating the generalized conuentVandermonde matrices in a way similar

to that available for the simple case where It is readily seen that the

transformation to structured matrices shown in section as well as the Cholesky

factorization describ ed in section can b e directly extended to this case On

the other hand it follows directly from the linearity of the op erator and from

the results of section that if A is a generalized conuentVandermonde matrix

and the solution of the Sylvesters equation just established then there are gen

erally lower triangular To eplitz matrices L L and blo ckdiagonal

matrices R R where each blo ckisap p upp er triangular To eplitz

matrix such that

T k

A L W x x R

As a consequence the complexity of realizing matrix vector multiplications with

generalized conuentVandermonde matrices is up to the multiplicativefac

tor the same as that of computing the conuentVandermonde matrix vector

multiplication It is thus p ossible to construct fast algorithms by exploiting the

fast metho ds develop ed in Combining this observation with the results

of section one may conclude that it is p ossible to construct asymptotically

fast algorithms for computing the inverse of a nonsingular generalized conuent

Vandermonde matrix using the metho ds presented in

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