Polynomials arising in factoring generalized Vandermonde determinants: an algorithm for computing their coefficients

MATHEMATICAL AND COMPUTER MODELLING PERGAMON

Mathematical and Computer Modelling 34 (2001) 271-281 www.elsevier, n l / l o c a t e / m c m

Polynomials Arising in Factoring Generalized Vandermonde Determinants: An Algorithm for Computing Their Coefficients S. DE MARCHI D i p a r t i m e n t o di M a t e m a t i c a e I n f o r m a t i c a U n i v e r s i t y of Udine, V i a delle Scienze, 206 33100 Udine, I t a l y demarchi©dimi, uniud, it

(Received November 2000; accepted December 2000) Abstract--We

consider generalized Vandermonde determinants of the form

vs;.(xl . . . . ,xs) = Ix~'~l,

1_< i,

k < s,

where the x~ are distinct points belonging to an interval [a, b] of the real line, the index s stands for the order, the sequence # consists of ordered integers 0 <_ #1 < ~2 < • " < Ps. These determinants can be factored as a product of the classical Vandermonde determinant and a homogeneous symmetric function of the points involved, that is, a Schur function. On the other hand, we show t h a t when x -- xs in the resulting polynomial, depending on the variable x, the Schur function can be factored as a ~ - 1 it x - x i ) , two-factors polynomial: the first is the constant 1-[~=11x i~1 times the (monic) polynomial "W[ 11i=1 while the second is a polynomial PM(X) of degree M ----m s - 1 - s + 1. Our main result is then the computation of the coefficients of the monic polynomial PM(X). We present an algorithm for the computation of the coefficients of Phi based on the Jacobi-Trudi identity for Schur functions. © 2001 Elsevier Science Ltd. All rights reserved. Keywords--Generalized ces.

Vandermonde matrices, Schur functions, Interpolation, Toeplitz matri-

1. I N T R O D U C T I O N The study of generalized Vandermonde determinants goes back to the end of the 1920s when Heineman proved some interesting formulas for different types of generalized Vandermode determinants (cf. [1]). L a t e r o n , a few p a p e r s w e r e d e d i c a t e d t o t h e s u b j e c t b u t m o s t o f t h e m concentrate minants

o n f i n d i n g e x p l i c i t f o r m u l a s for particular t y p e s o f g e n e r a l i z e d V a n d e r m o n d e

deter-

(see e.g., [2-4] a n d r e f e r e n c e s t h e r e i n ) . I n 1975, K i n g [5], p r o v i d e d s o m e u s e f u l f o r m u l a s

b a s e d o n S c h u r f u n c t i o n s , for f a c t o r i n g m o s t g e n e r a l V a n d e r m o n d e

determinants

of the same gen-

e r a l t y p e w e a r e c o n s i d e r i n g h e r e . T h e r e h e p r o v e d t h a t t h e s e S c h u r f u n c t i o n s c a n b e e x p r e s s e d as This work was done during my stay at the University of Dortmund (Germany) with the support of the Italian Research Council (CNR) within the program "short-term mobility", pos. 140.2, year 1999. Furthermore, I would like to express my special thanks to Dr. U. Maier and Prof. M. Reimer who invited me to spend a pleasant and productive time in the Fachbereich Mathematik der Universit~it Dortmund. 0895-7177/01/$ - see front matter (~) 2001 Elsevier Science Ltd. All rights reserved. PII: S0895-7177(01)00060-7

Typeset by A.A.~S-TEX

272

S. D E bIARCHI

linear combination with positive coefficients of symmetric monomial functions. T h a n k s to these results, we explored the possibility of explicitly c o m p u t i n g the coefficients of these factors and in this paper, we present our investigations. We start from these observations. Let X~ = { x l , . . . , x ~ } be a set of s > 2 distinct points belonging to an interval [a, b] of the real axis. Let us denote by V D M ( x l , . . . , X s ) the classical V a n d e r m o n d e determinant of order s, t h a t is 1

xl

•••

x s1- - 1

1

x2

...

x2

1

X s

. ..

X s

8--1

V D M ( x l , . . . , x~) = s--1

It is well known t h a t (cf. [6, p. 25, (2.1.7)])

V D M ( x l . . . . , x8) = V D M ( x l , . . . ,

x,)).

Xs--1)

(2)

Now, let Vs-l(X) = V D M ( x l , . . . , x , - 1 , x) be the polynomial of degree s - 1 obtained when x is used instead of x~. Then, formula (1) says t h a t V s - l ( x ) is the p r o d u c t of the classical Vanders--1 m o n d e of order s - 1 times a monic polynomial of degree s - 1, t h a t is, 1-i,=1 (x - xi). This idea can be used to c o m p u t e generalized Vandermonde determinants of the form: = IxU l,

1

i,

k

(2)

where the xi are s distinct points belonging to an interval [a, b] of the real line; s stands for the order and the sequence # consists of ordered integers 0 _< #1 < #2 < "'" < #~. Notice t h a t the sequence # recalls the monomial basis on which (2) is based. In fact, while the classical V a n d e r m o n d e determinant of order s is based on the monomials { 1 , x , . . . , x*}, the generalized ones are based on the basis { x m , . . . , x l~.'} for some positive integers 0 _< #1 < #2 < "'" < #~We also point out t h a t this different basis in some cases is obtainable from the basis of bivariate polynomials of degree s restricted to some curve y = x p, p < s (cf. [7]). Generalized Vandermonde determinants are strictly connected to Schur functions. In fact, t h e y can be factored as a p r o d u c t of the classical Vandermonde determinant and a Schur f u n c t i o n (see T h e o r e m 3.2 or for a different proof [5]): Vs;,(x, . . . . ,xs) = V D M ( x l , . . . , x 8 - 1 ) " S , ~ . _ , ( x l , . . . , x , ) , where m8-1 = #s - #1. Moreover, when x = xs, we shall prove t h a t

1,x,: /

) ,=1

(3)

with 114 = m s - 1 - s + 1, for some polynomial PM (x) of degree/1.I. O u r main purpose is the c o m p u t a t i o n of the coefficients of the polynomial PM (X). Indeed, the polynomial V s , , ( x l . . . . , xs, x) is completely determined once we know- the coefficients of PM (x), since V D M ( x l , . . . , x8-1) is a constant term and the coefficients of I ] i8-1 = t (x -- xi) are simply the elementary s y m m e t r i c functions of the points X l , . . . , xs-1. In Section 2, we present some results concerning the c o m p u t a t i o n of the coefficients of the factors for a simple class of generalized Vandermonde determinants, cf. (4). We start from the polynomial division algorithm which is intrinsically connected with the inversion of a lower triangular Toeplitz matrix, we prove t h a t the coefficients of the polynomial P n - k ( x ) are complete

Generalized Vandermonde Determinants

273

elementary symmetric functions (see T h e o r e m 2.2). Then, we also propose an algorithm to c o m p u t e these coefficients. In Section 3, we introduce Schur functions and by using some of their properties, we c o m p u t e the coefficients of the polynomial PM (x) (cf. (3), for the m o s t general V a n d e r m o n d e d e t e r m i n a n t s of the t y p e (2). A n algorithm to accomplish this task is also presented. In every section, we provide some examples in order to render the presentation as clear as possible.

2. T H E P O L Y N O M I A L DIVISION ALGORITHM AND THE INVERSE OF A LOWER TRIANGULAR TOEPLITZ MATRIX Let us start by considering the polynomial associated to a simple (k + 1) x (k + 1) generalized V a n d e r m o n d e determinant: 1

Xl

...

1

x2

...

z~ -1 xk2 - 1

x~

z~ -1

n Xk

xk-1

Xn

(4)

Vn;k(X ) = 1

Xk

...

1

x

...

where 2 < k < n - 1, so t h a t the n - k powers k . . . . , n - 1 are missing. This polynomial vanishes O11 X l , X 2 , . . . , X k , then y ~ ; k ( x ) = (~ - x l ) ( z - x : ) . . .

(5)

(~ - ~k) p ; _ ~ ( x ) ,

for some nonmonic polynomial Pn_k(x) of degree _< n - k. Therefore, in order to completely determine Vn;k(x), we should know the coefficients of the polynomial P~_k(x). It is not difficult to see (cf. [9])

Vn;k(x) =

,Xk)(X -

VDM(xl,...

Xl)(X

x2)'"'

(x -

Xk)

(6)

Pn-k(X),

for some monic P n - k ( x ) . n--k Let P n - k ( x ) = z n - k + ~-~i=1 bi+lxn-k-~ a n d Q k ( z ) = I l k 1 (x - x.i). Since V~;k(x) is a polynomial of degree n and bl -- 1, then dividing b o t h sides by V D M ( x l . . . . . xk) (which is not zero by hypothesis on the nodes), we have

P n - k ( x ) " Qk(x) =

x n-k + ~ i=1

bi+lx '*-k-i

(x

-- Xi)

-~" X n - -

/~k_l(X),

(7)

i=1

where Rk-l(X) is a polynomial of degree < k - 1 and bl = 1. Moving Rk-l(X) to the left side, the equation also says t h a t the polynomial P n - k ( x ) is the quotient of the division of x n by Qk(x). Thus, by means of the division algorithm for polynomials in which the dividend is the monomial x n and the divisor is Qk(x), we m a y determine the coefficients of P n - k ( x ) . But this is not a g o o d way to proceed: we can do better. In fact, as proved in [9], the coefficients of P n - k ( x ) can be found as the solution of a linear s y s t e m whose matrix is a lower triangular Toeplitz matrix generated by the elementary symmetric functions of the points involved. Since lower triangular Toeplitz matrices are structured matrices (generated by a vector, see below), then our problem reduces to finding the first c o l u m n of the inverse of the lower triangular Toeplitz matrix of order n - k + 1 generated by the vector whose first k + 1 elements are the coefficients of the polynomial Qk(x) and the remaining n - 2k are set to zero (see also, cf. [10, C h a p t e r 2, Section 5]), and it is this point t h a t we want to investigate a little bit more deeply.

274

S. DE IVIARCHI

DEFINITION 2.1. (See [10].) T w r i t t e n as

=

is a lower triangular Toeplitz m a t r i x i f it can be

(ti,j)l<_i,j<_n

T = ~

aiS i-l,

i=1

where a = ( a l , . . . , a s )

is a real or c o m p l e x vector a n d t h e m a t r i x S = $1 is the down-shift

matrix, that is the m a t r i x

S =

(i00i/ 0 1

0 0

... ...

• .o

"..

"°

...

0

1

.

Thus, a lower t r i a n g u l a r Toeplitz m a t r i x is completely determined by its first column or its last row. Moreover, this set of m a t r i c e s is an algebra g e n e r a t e d b y t h e down-shift m a t r i x 8 . If a lower t r i a n g u l a r T o e p l i t z is invertible, t h e n t h e inverse is still a lower t r i a n g u l a r T o e p l i t z m a t r i x . Hence, t h e inverse of an invertible lower t r i a n g u l a r T o e p l i t z m a t r i x is c o m p l e t e l y defined b y t h e knowledge, for e x a m p l e , of its first column. G i v e n t h e lower t r i a n g u l a r n x n Toeplitz m a t r i x g e n e r a t e d by t h e v e c t o r a=

(1,al,a2,...,an_l)

T

t h a t is, t h e m a t r i x

A=

1

0

al

1

a2

al

an-2

an-3

an-1

an-2

......... 0

0~

......

0

1

0

...

0

"..

"..

:

:

••

al

1

0

an-3

"'"

al

1

•

(8)

t h e first c o l u m n of t h e m a t r i x C = A - 1 , t h e v e c t o r c = ( c i , 1 , . . . , c,~,1), can b e c o m p u t e d s i m p l y by solving t h e s y s t e m Ac el, =

w h e r e e l = ( 1 , 0 , . . . , 0 ) T. T h e s o l u t i o n provides t h e following useful recurrence: 01,1 =

1; i-1 i = 2,...~n.

el,1 = - - ~ _ ~ C j , l a i - j ,

(9)

j=l

Hence, t h e recurrence (9) a p p l i e d to t h e p o l y n o m i a l division x n : (x - x l ) . . . (x - x k ) gives t h e coefficients bi in (7) of t h e p o l y n o m i a l P n - k ( x ) , in t e r m s of t h e coefficients a v , p = O, 1 , . . . , k of the polynomial Qk(x): bl = 1; (10)

i--1

bi=-~bjai_j,

i= 2,...,n-k+l.

j=l

F u r t h e r m o r e , as = O, k + 1 < s < n - k. To c o n c l u d e t h i s p a r t , we give a simple e x a m p l e . EXAMPLE 2.1. Let us t a k e n = 5 a n d k = 2, so t h a t we consider t h e division of x 5 by Q 2 ( x ) = x 2 + a l x + a2. T h e q u o t i e n t is P3(x) = x 3 - axx 2 + (a 2 - a2)x + ( - a 3 + 2 a l a 2 ) .

Generalized Vandermonde Determinants

275

Now let us consider the lower triangular Toeplitz matrix A of dimension 4: lO

al as O

A =

1 al a2

0 1 al

(11)

'

T h e first column of the inverse C is determined by using the recurrence (9) applied to the vector a = (1, al, a2,0) T, obtaining (21,1 : 1; c2,1 = - a l ; c3,1 = a 2 - a2; c4,1 = - a 3 + 2 a l a 2 . T h e above steps are then equivalent to find the polynomial of degree 5, 1

X1

v~;2(z)= 1

x2

x5

1

x

x5

Since Q2(x) = x 2 - (Xl + x 2 ) x + x l x 2 , P 3 ( z ) is P3(x)

-~ 323 -I- ( X l -~- x2)2 ;2

X5

(12)

then a = (1, - ( x l + x 2 ) , X l Z 2 , 0) T and by means of (10),

H- (322 -4- XlX2 H- x22) x --~ (x32 H- Xl x2 -~ x21x2 H- x31) .

Finally, Vs;2(x) = V D M ( x l , x 2 ) . ( x - X l ) ( X - x,2) . P3(x). 2.1. A n A l g o r i t h m

for Computing

the Coefficients and Some Consequences

We recall t h a t the coefficients of the polynomial k Qk(~)

= II(.~

- x,,)

i=1

in (7) are up to the sign, the e l e m e n t a r y so defined:

symmetric

functions,

er(k) , of the points x~,, i = 1 , . . . , k,

e(0k) = 1,

e(~~) =

~ z~<,.x,~, 'i: <...< i~:

l < r < k,

e(f ) = 0 ,

r<0orr>k.

T h e latter is usually assumed by d e f i n i t i o n . T h a n k s to this definition, we are ready to present a simple algorithm t h a t c o m p u t e s the coefficients of the quotient polynomial P n - k ( X ) in (6). ALOORITHIvl 1. I n p u t s : n, k and the points xi, i = 1 , . . . , k . 1. Form the n - k + 1 dimensional array a, t h a t is

(k)

a:

e(0k),-(2? ), (22(k'

, ' " ,

4

(k)

' ("2 k + l ' ' ' ' ' (o' 2n-k[

2. B y means of the recurrence (10) c o m p u t e the array b.

;

276

S. D E hIARCHI

Output : the array b of the coefficients of Pn-k(X). Letting A be any partition of length < r, a multivariate polynomial is usually defined as mA(xl,..

•

Xr) = E

,

x~

.

.

. a,. X

r

where a varies on all distinct permutations of A. This definition helps us to introduce the complete elementary symmetric functions of the set of q points, that is, the homogeneous symmetric polynomials

h~q)

O,

=

S < 0 or s > q.

Thus, h! q) is the sum of all monomials of total degree s in the variables X l , . . .

,

Xq. In particular,

h~:)=4 q), h?)= ~iq) LEMMA 2.1.

h~q) = d e t ~ ,[e ~(q)- i + j ] ,

1 < i,

j ~_ s,

and A = ( 1 , . . . , 1). Y 8

PROOF. The proof is based on [11, p. 21, formula (2.6~)] by solving for h~ (cf. also [11, p. 28]). | Finally, we state our first theorem. THEOREM 2.2. Given k distinct points, then the coet~cients of the quotient polynomial P~-k (x) in the factorization (6), i.e., the elements of the array b, are the complete elementary symmetric t'unctions bi h(k) i = 1, . . ,n . -- . k + 1 'Oi--l~ PROOF• Let us consider the array a as above, made up of the elementary symmetric functions of the k points xl . . . . ,xk. From the recurrence (10), we recognize that

( 4~) ~) ...... el~'I e~k)

e~k)

•

.

_(k) ~i-2

......

bi = det

• " " ..

......

,~_~

~i--3

".

•

4 ~) 4~) |

which by the L e m m a 2.1 , is exactly ~(a) '~i--l" COROLLARY

2.3.

The quotient polynomial Pn-k(x) can be written in detemninantal form, that

is,

(k)

eO

P~-k(x) =

(k)

--el

.

.

--el(k)

.

.

..

0

e (k)

•

.

o

o

...

4 ~)

1

X

. . .

X n - k - 1

.

''' .

(

)~--l'n-ke(k) n-k

(k - l- ~ nJ _ k _ l e ( kn -) k - 1 .

-e?) X n-k

(13)

Generalized Vandermonde Determinants

277

EXAMPLE 2.2. Given the polynomial

=

1

X1

1

x2

X~

X~

X~

(14)

1 1

x4

x~

x~

x~

1

x

x2

x3

x7

T h a n k s to our results, we can say (i) VT;4(x) = V D M ( x l , x 2 , x 3 , x 4 ) (x - Xl)(X - x 2 ) ( x - x3)(x - x4) P3(x), and (ii) Pa(x) = x 3 -1- h~a)x 2 + h (4)x + h~4).

. SCHUR FUNCTIONS VANDERMONDE

AND GENERALIZED DETERMINANTS

A Schur function is defined as follows (cf. [11]). DEFINITION 3.1. Given a partition A = ( / ~ 1 , . . . , / ~ n ) , [)~1 <- ?)~, the Schur function s~, defined on R n, is

8 A ( X l , . . . ,Xn)

=

aet(x:J+' dot( :

,

1 <_ i , j <_ n.

(15)

Notice t h a t the d e n o m i n a t o r is nothing else t h a n the classical V a n d e r m o n d e determinant. Schur functions provide a very useful tool to s t u d y and c o m p u t e generalized V a n d e r m o n d e d e t e r m i n a n t s (cf. [5,8,9]). To fit out' setting in this fi'amework, we simply set x,~ = x. EXAMPLE 3.1. Let us consider the determinant

V4:2 =

1

xl

x41

1

X2

X4

1

X3

X4

(16)

F r o m our results and after a few calculations, V4;2 - - ( X l - x 3 ) ( x 2 :

- x3)[(x

3 - x31) Jr- (22 2 - x 2) x3 -~- ( x 2 - Xl)X32]

( x 1 - x 3 ) ( z 2 - x 3 ) ( x 2 - X l ) [x 2 -1- X l X 2 -~ x 2 -~ x 2 x 3 -t- X l X 3 Jr z 2 t ,

which shows once more t h a t we can factor the classical Vandermonde V D M ( x l , x2, x3) out of the determinant. Hence, let ct = (4, 1,0) and 5 = (2,1,0) be the partitions consisting of the degrees of the monomials involved considered in reverse order, then letting A = c~ - 5 = (2, 0, 0), the above d e t e r m i n a n t divided by V D M ( x l , x2, x3) is the following Schur function: 1

xl

x4

1

x2

x4

1

x3

x4

1

Xl

X2

1

x2

x2

1

X3

X2

-- X22 Jc X l X 2 -t- X 2 ~- X 2 X 3 -t- X l X 3 ~- X 2 ---- 8 A ( X l , X2, X3).

(17)

REMARK. We recognize t h a t the above Schur function is h~3). This again is a general setting for these determinants, as we shall see in the next example. EXAMPLE 3.2. T h e case of a group of k missing powers can easily be m a n a g e d in the framework of Schur functions.

278

S. DE MARCHI

PROPOSITION 3.1. L e t us consider the generalized Vandermonde determinant, ... X~ - k - 1 X? 1 X1 X21

Yn;n_ k ~

1

x2

:

:

1

x~

.

.

.

: 2 Xn_ k

Xn-k

•

n-k-1 Xn_ k n-k-1 Xn_k+ 1

. .

2

1 X n _ k + 1 X n _ k + 1 • .. when the k powers n - k , . . . , n - 1 a r e missed• T h e n V~;~-k = sx • V D M ( x l , . . . , x ~ _ k )

with A = ( k , 0 ('~-k)) = ( k , 0 ,

•

.

.

= h~n-k+1)

n

Xn - k ,n

Xn-k+l

VDM(xl,...,xn_k)

,0) a n d it,('~-k+l) which is t h e k th complete s y m m e t r i c fimction vk

n--k

o f the n - k + 1 variables X l , . . . , x ~ - k + t . PROOF. T h e Jacobi-Trudi identity (cf. [11, f o r m u l a (3.4), p. 41]) says [h(n-k+l)~ 1
W e c l a i m t h a t sx = '°k

. In fact,

h(n-k+l) h(n-k+l) k k+l h(;~l- k + l )

h(n-k+l) 0

s,X =

(18)

t,(n-k+l) '~k+2 h(n-k+l)

t (n-k+l) tn h(n-k+l) n-k-1 h(n-k+l) n-k-2

'Vl

h(-n2- k + l )

h(_l- k+ 1)

t,(n-k+l)

h(n-k+l) -n+k

h(n-k+l) -n+k+l

""•

Iv 0

h(n-k+l) 0

a n d since h}n - k + l ) -- O, p < - 1 , we get t h e c l a i m e d result. REMARK. F r o m L e m m a 2.1, we also observe t h a t h2 n - k + l ) is a function of t h e e l e m e n t a r y s y m m e t r i c functions, t h a t is, k

=det\

A~-i+j ] ,

1_
j_
(19)

w h e r e here A* = ( 1 , . . . , 1)/ Therefore, we we m a y express ha as a d e t e r m i n a n t of a (Hessenberg) n--~+l

m a t r i x whose e l e m e n t s are tile e l e m e n t a r y s y m m e t r i c functions e0, • • •, ek. N o w we are ready to extend our results to very general Vandermonde d e t e r m i n a n t s and its associated p o l y n o m i a l s as we did in the previous section. To be precise, given a sequence of integers, 0 _< >1 < It2 < " ' ' < /Zn, and a set of n distinct p o i n t s , X l , . . . , xn, we can consider d e t e r m i n a n t s of t h e form Vn;tt = Ix~t9 II<_i,j<_n"

(20)

It is obvious t h a t these d e t e r m i n a n t s can be r e w r i t t e n Vn# =

Xi

\

i

]l
i=1

--

--

and so for o u r purposes, it is enough to consider t h e a s s o c i a t e d p o l y n o m i a l s of t h e t y p e .?'Yl 1

1

x1

1

X n,,~- 11

• "

Xn--

1

1

x m~

• •

X m

. -- 1

w i t h obvious n m a n i n g for t h e s y m b o l s used• We need another concept related to partitions:

p• 2]).

•

.

X l

"-1

.m II- 1

(22)

that is, the diagram of a partition (cf. [11,

279

GeneralizedVandermondeDeterminants DEFINITION 3.2. 1
Given a partition

A, its diagram is the set of points (i,j)

E Z2 such that

I xi.

DEFINITION 3.3.

The conjugate of a partition A, is the partition A’ whose diagram is the trans-

pose of the diagram of X obtained by reffection in the main diagonal. For example,

if X = (3,2,0),

then A’ = (2,2,1).

The definition of conjugate

allows to introduce an equivalent relation to the Jacobi-Trudi

of a partition

identity for Schur functions.

That

is, for any partition X of n, (23)

SA = det (ex:-%+j) l
THEOREM 3.2.

of length n associated Then,

Vm,,_l;n-I(X) where M = m,_l

mn-2,.

. . , ml, 0) and S = (n - 1, n - 2,. . . , 1,0) be the partitions

to the determinant

(22). Let X = LY- S and A’ be its conjugate partition.

= VJW~l,.

(24)

. . ,

- n + 1. Moreover, if X1 2 n, then Ad > n and (-1) A:+“‘-leX;+M_l

.

(-l)X;ex;

ew(x) =

l-1) “:n-“exI,_nr

(-l)~a-“~+‘ex~,_nr+l

..

1

X

...

otherwise A’ has n - X1 zeros and M < n. PROOF. The equality

(24) is simply proved by resorting to the definition of Schur functions

when x = xn. Since X1 > n, then A’ has no zeros. To get the determinant determinant of the matrix obtained by bordering the matrix

(25), we observe that PAI

is the

(26) with the column consisting of the elementary symmetric starting from the first column of the matrix (26), i.e.,

((-l)X:-le+l,

functions translated

. . . , (-l)X:~-““ex;,-*r,

by 1 to the left

l)T,

and with the last row consisting of the vector of the powers (1, IC, . . . , x”). If X1 < 11, then A’ has n - X1 zeros. Therefore, sponding rows.

in the matrix (26), there are not the corre-

I

EXAMPLE 3.3. Given the polynomial

vj;3(X) =

1

x;

xi

x2’

1

x;

x;

x3’

I1

x3

x4

x7

S. DE MARCHI

280

By Theorem 3.2, V+3(2) = VDM(xi, x2, x3) (x-x~)(x-22)(x -x3) P4(2). Since a: = (7,4,3,0), 6 = (3,2,1,0), X = (4,2,2,0), and X’ = (3,3,1, l), then the determinantal form for P4(x) is -3

e2

P4(2) =

-3

e2

-1

-e-l

e0

-e-3

e-2

-e-l

1

X

e-2

-5

e4

et5

e4

-5

e2

-1 eo

x2

-1

x3

x4

Finally, since ej = 0, j > 3, and j < 0, then e2

P4(x) =

0

-es

-ei

e2

0 0 1

0 0 5

-es

1 0 x2

0 0

0 0

-cl 1 x3

-ei x4

e2

EXAMPLE 3.4. Let us consider the polynomial

1/4;2(.r) =

1

Xl

x;’

1

x2

x;

1

x

x4

By Theorem 3.2, v4;2(x) = VDM( x1,x2) (x -x1)(x - x2) Pz(x). Since cr = (4,1,0), X = (2,0,0), and X’ = (l,l,O), then the determinantal form for Pz(x) is

e0

P2(2) =

-1

-e-l

eo

1

.x

e2 -el x2

S = (2,1,0),

.

REMARKS. In Example 3.3, Xi = 4 > 7%= 4, then X’ has no zeros and M = 71 = 4. Example 3.4, Xi = 2 < 11,= 3, then X’ has some zeros and consequently, AJ = 2 < 11.

In

Finally, we can write down an algorithm for computin g the coefficients of PA{(X) based on Theorem 3.2. ALGORITHM 2. Inputs:

the partitions a, S of n/r and the points x,, i = 1,. . , M.

1. Let X = a: - 6 and X’ its conjugate. Build the matrix

E = (e~:r+&,,~
1

where eP is the p th elementary symmetric fuuction of M points. 2. Build the matrix El by bordering the matrix E: add as firstTcolumn the array

((-l)X;-‘ex;-i,

. . , (-l)X~~-h’ex:,_fi~,

I)T

and as last row (1, x, . . , xnr). Output : the coefficients of PAI(

which are the minors of order iW - 1 of the matrix El.

Generalized Vandermonde Determinants

281

REFERENCES 1. E.R. Heineman, Generalized Vandermonde determinants, Proc. Royal Soc. Edinburgh, 464-476, (1929). 2. R.P. Flowe and G.A. Harris, A note on generalized Vandermonde determinants, S I A M J. Matrix Anal. Appl. 14 (4), 1146-1151, (1993). 3. L. Kuipers and B. Meulenbeld, Symmetric polynomials with non negative coefficients, Proc. A M S 6, 88-93, (1956). 4. P.C. Rosenbloom, Some properties of absolutely monotonic functions, Bull. A M S 52, 458-462, (1946). 5. R.C. King, Generalized Vandermonde determinants and Schur functions, Proc. A M S 48 (I), 53-56, (1975). 6. P.J. Davis, Interpolation ~4 Approximation, Dover Publications, (1963). 7. L. Bos and S. DeMarchi, Fekete points for bivariate polynomials restricted to y = x m, East J. Approx. 6 (2), 189-200, (2000). 8. S. De Marchi, On computing the factors of generalized Vandermonde determinants, In Recent Advances in Applied and Theoretical Mathematics, (Edited by N. Mastorakis), pp. 140-144, World Scientific Engineering Society, (2000). 9. S. De Marchi, Generalized Vandermonde determinants, Toeplitz matrices and the Polynomial Division Algorithm, Universit~it Dortmund, Ergebnisberichte Angewandte Mathematik, nr. 176, (June 1999). 10. D. Bini and V. Pan, Polynomial and Matrix Computations, Volume 1, Fundamentals Algorithms, Birkh/iuser, Boston, (1994). 11. I.G. Macdonalds, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, Second Edition, (1995).