Error estimate in vector Padé approximation

Applied Numerical North-Holland

Mathematics

457

8 (1991) 457-468

Error estimate in vector Pad6 approximation A. Belantari UniversitC des Sciences et Techniques de Lille Flandres-Artois, Laboratoire d’Analyse NumCrique et d’optimisation, U.F. R. I.E.E.A. B6t. M3, 59655 Villeneuve dilscq CPdex, France

Abstract Belantari,

A., Error estimate

in vector PadC approximation,

Applied

Numerical

Mathematics

8 (1991) 457-468.

The aim of this paper is to apply Kronrod’s procedure, which was extended to PadC approximation in Brezinski [2], to vector PadC approximants in Van Iseghem [lO,ll]. Some results on Stieltjes polynomials, which have been defined in the formal vector case, will also be given. Keywords. Gauss-Kronrod quadrature polynomials, error estimate.

formulae,

polynomial

interpolation,

vector orthogonal

relations,

Stieltjes

1. Introduction Kronrod’s method [5] for estimating the error of a Gaussian quadrature formula consists in constructing a better quadrature formula by adding new nodes to the previous ones (the Gaussian nodes) in an optimal way. Then the difference between both quadrature formulas provides an estimate of the error on the Gaussian one. Recently, Brezinski [2] extended this procedure to Pad6 approximation. In [3], he gave a new interpretation of this extension which has led to a large class of error estimates in Pad6 approximation. In this paper we shall extend this procedure to vector Pad6 approximants which have been defined by Van Iseghem [lo]. We shall motivate this extension in the way Brezinski motivates his extension to scalar Pad6 approximants [2]. Let us first recall some main results on vector Pad6 approximants which can be found in [lo]. Let F be a function from Q=to Q:d: F = ( fi, . . . , fd)t, f, : c + @, F is supposed to be analytic in a neighbourhood of zero:

f’(t) =

c r,t’, c= (c:),=,.....,-

A linear functional r is defined from C[ X] to Cd by r( x’) = c., i E N. r has components C” defined by Ca( x’) = Cp, i E fIJ. Let V(x) = lYI~=, (x - xi) be an arbitrary polynomial of degree Y where the x, are not necessarily distinct. r still acting on x, t being a parameter, we set:

W(t) =

016%9274/91/$03.50

r %dx-ti

w

1 ’

@ 1991 - Elsevier Science Publishers

B.V. All rights reserved

A. Belantari

458 W

/ Error estimate in vector Pad6 approximation

is a vector polynomial of degree r - 1. W is the polynomial associated to V. We set I?(f)

V(t) = t’V(t_‘),

= tr-‘w(t-‘).

The vector rational fraction @(t)/?(t) is a so-called Pad&type approximant (PTA) of F. It will be denoted by (r - l/r)p( t). V is called its generating polynomial. Pad&type approximants can be related to polynomial interpolation. Let P be the Hermite interpolating polynomial of (1 - LX)-’ at the points x,, . . . , x,. We have [lO,ll]: T(P)(t)

= ti(f)/p(t)

= (r - l/r)F(t),

F(r)-T(P)(r)=&+$j=&.

.@‘, I 0

Di = r(x’v(x)).

Thus (r - l/r)F may be viewed as a generalized formal interpolatory quadrature formula for (1 - tx)-’ with the zeros of V as nodes. The above error expression means that (r - l/r)F is interpolatory, i.e., exact for polynomials of degree G r, which is the classical property of quadrature rule with r nodes. Let r = nd + k, n > 0 and 0 6 k < d. To get a higher-order approximation, let us take V such that (Rl)

r(x’V(x))

= 0,

(R2)

C”(x”V(x))

i=O )...)

= 0,

cx= l,...,

n - 1; k.

The polynomial I’ exists and is unique if the determinant of the system (Rl)-(R2) Thus,

is nonzero.

# 0,

r’n-i

***

c+:-2

r(k) r

.,.

rCk) n+r_ 1

a condition which will be assumed to be true in the sequel. In this determinant each row (c,. . . , I-.+r_l), i = 0,. .., n - 1, represents d scalar rows, except the last one which represents the first k components of (r,, . . . , r,+,_,). If V satisfies the preceding conditions, it is the polynomial of degree r belonging to the family of vector orthogonal polynomials with respect to the functional r. It will be denoted by P,, assumed to be manic, and it can be expressed as a ratio of two determinants:

r, PJX)

H (x) = -+ r

=

rn-1 r(k) n

1

‘r ...

r

rr+L-lri

...

r(k)

...

Xr

r+n

r,_, -’

0

n

r(k) r

:

1

rr+n-2 r(k) n+r-1

I

.

A. Belantari

/ Error estimate in vector Pad& approximation

The family (P,), verifies a recurrence recurrence relation [lO,ll]:

The PTA of F with the generating F(t)

- (r - l/&(t)

relation

polynomial

of order

459

d + 1, i.e., these polynomials

satisfy a

P,. satisfies

= O(f+“).

It is the best order of approximation in the following sense: it is impossible to improve simultaneously the order of approximation of all the components. Thus this vector PTA will be called a vector Pad6 approximant and it will be denoted by [r - l/r] F. [r - l/rlF can be viewed as a generalized interpolatory quadrature formula using the zeros of P, (an orthogonal polynomial) as nodes. This is exactly the method used to construct Gaussian quadrature rules which shows the connection with vector Pad6 approximation.

2. Extension of Kronrod’s procedure Thus [r - l/v] F is a generalized formal Gaussian quadrature formula. According to Kronrod’s idea, [Y - l/r], may be extended to a better quadrature formula by addings new nodes to the r preceding ones (the zeros of P,.). In other words we shall consider the PTA (r + s - l/r + s)~ with the generating polynomial u(x) = P,( x)V,( x), where V, is an arbitrary manic polynomial of degree s. We set

r still acting on x and t being a parameter. W, is a vector polynomial t. The vector polynomial w associated to u is given by w(t)

=

r 4-4x-t- 4d i

where Q, is the vector polynomial C(t) = t’+%( t-l)

i = w,(t) + associated

Y(t)Qr(t),

with P,. Moreover,

= Fr( t)Vs( t)

with &(t)

= t’P,( t-l),

q(t)

G(t) = t r+s-lw(t-l)

= ty(t-‘),

= t’tis(t)

+ <(t)&(t),

with Q,(t)

=

t-‘Q,(t-‘),

Hence we have proved

tis( t) = t”-‘w,( the following

theorem.

t-l).

of degree at most s - 1 in

A. Belantari

460

/ Error estimate in vector Pad& approximation

Theorem 1. Let (r + s - l/r + s)~ have the generating polynomial u(x) = P,(x)V,(x). (r+s-l/r+s),(t)-[r-l/r],(t)=&Z+,

Then

r

F(t)-(r+s-l/r+s)F(t)=t’+”

1

_ ew%)

.r i

iWv,bJ I-tx

i .

(V,(x) - V,( t))/( x - t) is a polynomial of degree s - 1 in x. Thus, due to the orthogonality property of P,., W, is identically zero if s f n and consequently (r + s - l/r + s)~ is identical to [r - l/rlF. For the same reason, when s > n, W, is a polynomial of degree s - n - 1. Then the smallest value of s is n + 1. Now we shall examine the problem of the construction of Stieltjes polynomials and a suitable corresponding extension of Kronrod’s procedure. Let us consider again the error estimate ew(t) r

=

?$!a

t’.

E(t) v,(t) ’

whose components are given by

with

W,*(t) = ca P,(x) i

Kc4

- v,(t) x-t

1

and ti;( t) = tvy(

t-l).

In the scalar case [2,8], the Stieltjes polynomial corresponds to the smallest value of s. Here, when s = n + 1, due to orthogonality of P, the k first components of the above error estimate are identically zero and the procedure has no interest. Assume s > n + 1. We shall examine the best choice of the integer s(r) such that the corresponding Stieltjes polynomial Vss(,.)realizes the best possible order of approximation (in the sense that it is impossible to improve simultaneously the order of approximation for all the components) and that when d = 1 these results reduce to those obtained in the scalar case in [2]. Thus we have to consider two cases: Case 1: s < d. Obviously this case does not give an answer to our question. Case 2: s>d. s=md+v, ma1 andO
=o,

i=o,...,

n.

A. Belantari

Expanding ( P, ( z ) VCn+l)d(~))/(l Theorem 1, we get e((“+W)(t)

= F(~)

- tx)

[r_

_

r

461

/ Error estimate in vector Pad& approximation

I/]

and

&)

using

the above

relations

defining

Fn+rjd,

by

_&;;l)d+n+l.r( x’+‘pr(l~t;+l)d(x)).

_

(n+l)dW

r

So the required order of approximation seems to be achieved but this choice is not the best in the sense that the Stieltjes polynomials associated to many successive orthogonal polynomials, may have the same degree. We also notice the same problem when taking s = ( n + 1) d + k ‘, k ’ a fixed integer: 0 < k’ < d. r = nd + k, 0 G k < d. The manic polynomial

Definition. Let r be a fixed integer, r + d satisfying the relations: (R)

r(xipr(x)v,+d(x)) C”(x

=

of degree

i=o,...,n,

0,

“+lPr(x)y.+d(~))

c+d

=

0,

will be called the Stieltjes polynomial

a = l,...,

k, to P,.

associated

This choice depends solely on Y and it realizes the desired order of approximation which is the first k components have the order of 2u + d + n for all the components. Moreover, approximation 2r + d + n + 1. Let us examine the existence and unicity of Vr+d. We set r+d v,+,(x)

=

c

with a,.+d=

aixi

1.

i=O

Writing the orthogonality we get the linear system

relations

a,T( x”P,) + . ’ ’ + a,+& a,_J(

(R) which define x’+,)

X’PJ + a,T( x,+1,)

Vr+d and using the orthogonality

= 0,

+ . . . +a,+J(

xr+d+lpJ

= 0,

a,r( x”PJ + . . . + ar+dr( Xr+d+nPJ = 0, + ***

.JyX”+‘Pr)

Hence

V,,,

is uniquely

determined 0

= 0

if the determinant ...

0

T(x”P,)

...

r(x r+d-lC>

0

H,*,,=

r(x”pr 1

ry is nonzero.

+a,+dr(k)(X’+d+n+lPr)

x”+lP,)

... ...

r(x r+d+n-lpr) P( x r+d+npr)

of P,,

A. Belantari

462

/ Error estimate in vector Pad& approximation

In this determinant each row (r( x’P,>, . . . , r( x r+d+rPr)) represents d scalar rows, except the last one which represents the k first components of (r(x”+‘Pr), . . . , ~(x~+~+~P~)). Since K+d is the polynomial of degree r + d, orthogonal with respect to the functional A defined by A(x’) = T(x’P,), i E N, we get an expression of <+d as a ratio of two determinants: 0

V,+,(x) = +

r+d

...

0

r(x”P,)

0

’

***

...

r(xnpr)

‘+d+nPr) P( x r+d+n+lpr)

. ..

r(k)(Xn+lpr)

...

1

r( Xr+dPr)

X

rfd

Thus we can state the following theorem. Theorem 2. Let (2r + d - 1/2r + d) h ave the generatingpolynomial chosen such that

r(h)

= 0,

n+b(x))

cqx

v(x)

= p,(x)V,+d(x).

Ifv

is

i=O,...,n, (Y = l,...)

= 0,

k,

then f2r+d+n+l

F(t)-(2r+d-1/2r+d),(t)= (2r+d-1,‘2r+d),(t)= F(t)

.

U”(t)

r(x;‘-“i$

[r-l,‘&(t)

- [r - yy]

r( x;‘“I,x)

F(t) - t2’i;ri+’

)

OY

(2r + d - 1/2r + d)f,( t) - [Y - l/y] LX(t) t2r+d+n+c+l

=fa(t) - b - v-1 /i(t) -

U”(t)

.r

n+r+lv(X)

x i 1-

tx

1

with &=

1, i 0,

(~=l,...,k, a=k+l,...,d.

Proof. The first result immediately follows from Theorem 1 and the above relations defining Vr+d, by expanding the error term. But, by definition tir+,(t)

= t’+d-‘Wr+d(t-‘) V,+dcx>

=

p 1-

c+,(t)r * (4 i

and the result follows from Theorem 1.

0

x -

c+d(t-l) t-’

tr+d+n+*

r( x;“ui$

)

A. Belantari

/ Error estimate in vector Pad& approximation

When d = 1, we find again the results of Theorem (a,,.

. .) a,)‘(b,,..

.) bJ

= (qb,,

2 [2]. We can define

463

the product

. . .) a,b,)’

by identifying a vector with the diagonal of the diagonal the general principles given in [2] we have:

matrix.

On the other hand, by extending

Theorem 3. (2r+d-1,‘2r+d)&)-[r-l/r].(t) =

[r -

F(t) -

F;;;;;“+;t) .r( xn+lpx~~~t~+d~x)))

l/r] F(t) -

r

=

1,

r

fr+n+d+l.

+

xn+lPr(x).r+d(x) 1-

Proof. The proof is an adaptation

rtd

. (q+db))-‘.

tx

of that of Brezinski

[2].

•I

We shall now extend to the vector case some algebraic properties of Stieltjes polynomials which have been given in [2,8]. Since { P, }; assumed to be manic, forms a basis of the set of polynomials we can write 2r+d Pr(x)V,+d(x)

=

c i=O

where the y, are constants

yiP,(x),

Y2r+d

depending

=

on the index

1

Y. From

the relations

(R) defining

Vr+d, we

get 2rid

r=O 2rid c”(x”+‘p,(x)v,+d(x))

=

c

yica(Xn+‘Pi(X))

=o,

ff =

I,...,

k.

I=0

So YO,..-,

we

have

a

Y(n+l)d+k-1

triangular as unknowns

j=o,...,

n,

ff=l

system (S) of (n + 1)d + k - 1 = r + d - 1 equations where the indexes are >f . ., d

or

,$“(xJP,) The determinant

+

”

*

+Yid+~-_lC~(X’p,d+n-l) = O.

is

a= fI (ii LX=1

J=o

ca(x’qd+.l)}

’

h

a=1

Ca(Xn+lpcn+l)d+a-l).

with

A. Belantari

464

/ Error estimate in vector Pad& approximation

From the determinantal expression of P,, we have: =

C.(X@jd+a-lJ

Hjd+a/Hjd+u-lm

All these determinants have been assumed to be nonzero in order to insure the existence and the unicity of P,.. So A is nonzero too, and finally we obtain Yo’

= Y(n+l)d+k-1 = 0

...

and the result follows.

0

This is an extension to the vector case of a result obtained in [2] which is a generalization of a known property of quasi-orthogonal polynomials [4, p. 641. We have: Theorem 4. Let A be the linear functional n(xj)

=r(x’p,(x))

=A;=

previously

defined by iEN.

(@,...,Q:‘),

In the sequel the generalized Hankel determinants associated to A will be denoted by Hi*, i E N. Then yr+d f 0 if and onb if Vr+d can be expressed as a ratio of two determinants and H,.Td+l Z 0. Proof. A has components Qa, Qa(x’) = Q,:. So we have: 2rid c”(

X’P,v,+,)

=

Qa(

XjF+d

(X))

=

c

yica(

Xjp;).

r+d

Taking (Y= k + 1 and j = n + 1, by the orthogonality properties of Pi, we get Q’+‘(x

‘+‘T+L&))

= Y,t~Ck+‘(X”+‘p~+d(x,)~

From the determinantal expression of Pr+d and v,+d we have: Q’+‘(x

‘+‘l/r+d) = H,*+,+,/K+L,

ck+l (X n+lp,+d)

=

Hr+d+de+d.

Since these determinants have been supposed to be nonzero, the result follows.

0

Remark. If yr+d = 0 either H,*,,+, = 0 or V,+, cannot be given in determinantal polynomial v, +d can also be expressed in the basis { Pi } : r+d

V,+,(x)

= c

with &+d= 1.

&P,(x)

i=O

The relations (R) defining V,,,

lead to the system

r+d ;~oaqXjvi)

=

07

j=O 3* * * > n,

r+d x0

PC(X

“+‘P,P;)=O,

(~=l,...,

k.

form. The

A. Belantari / Error estimate in vector Pad& approximation

It s determinant

465

is ...

0

0

T(x”P,)

...

WrPr+d-J

~bpn-,pr)

...

0

A=

...

w%) r(k)( Xn+lPo) In this determinant

each row represents

+ ..a +&+dCa(~iPrPr+d)

pi_lc”(x~P,_lPr)

+ *. . +&C”(x’PiPr)

except the last one which represents

If A is nonzero,

r(k)( x n+‘PrPr+&,)

d scalar rows: j = 0,. . . , n and i such that i +j = n + 1,

,f3ic”(xiP,_,P,,)

&C”( x ‘+‘,,)

wTP,+d-1)

...

=O,

a = l,...,

k,

+j3r+dCa(xiP,Pr+d)

= 0,

a = k+ l,...,

d,

the first k rows:

+ . . . +&+&Y(x~+~P~P~+~)

we get a determinantal

expression

= 0,

a = 1,. . . , k.

for Vr+d:

0

1 = a

K+d(x)

0 Qx”p,p,)

...

...

...

...

r(k)( x .+1,,>

...

...

...

...

P,

***

...

po

This result generalizes a result of [8]. If now xi,. . . , x, are zeros of multiplicity then we get the system with &+d = 1:

P,-1

m,, . . . , m, respectively

2r+d

(S)

P/‘)(Xj)q$(Xj)

=

c

Y,Pi(l)(Xj)

i=r+d

j=l Its determinant

,**-, v is

We have the following:

and

l=O,...,mj-1.

= 0,

~w(X”+‘prpr+d)

P r+d

of P,, r = m, + - . - + m,,

A. Belantari

466

/ Error estimate in vector Pad& approximation

Proposition. If Pr+d and V;+d can be expressed as ratios of two determinants, A of the system (S) is nonzero.

then the determinant

Proof. Let us consider again the linear functional A previously defined by A( xi) = r( x’P,) = A ; =
&P/‘)(x;)=O,

c

~andl=O

j=l,...,

,...,

m,-1.

i=r+d

Thus z.$‘)(xi) = 0, j = 1,. . . , v and I= 0,. . . , m, - 1. Then u(x) is the complementary polynomial of exact degree r + d - 1. = Pr(X)R,+d-I(X) where Rr+d-1 Using the orthogonality property of Pi, we get:

“‘“-‘&Pi(x). We set u(x) = Ci=r+d

r( X’P,R,+d-1 ) = A( i C”(x “+lP,Rr+d_l)

Xi&.+d_l)

=

i=O,...,n,

0

(Y = 1,. . . , k.

= Qa( x~+~R~+~_~) = 0,

These relations show that Rr+d_-l is the polynomial of degree r + d - 1 orthogonal with respect to A. Moreover, we have: Q”(x”+‘Rr+d-I) Then V,,,

= H,*,,/H,*,,_,

= 0.

cannot be expressed as a ratio of two determinants.

•I

Let us remark that A = 0 implies Hrzd = 0. Now we can give a generalization Christoffel. The proof is an adaptation of that of Szegii and Patterson [7]. Theorem 5. If Pr+d and F+d can be given in determinantal

Pr(x)V,+d(x)

=

;

pr+d(x>

. *’

P,+d(X1)

’ *’

‘2r+d

form,

then

cx>

p2r+d(xv>

.

PJ,“;,+l’( x,)

. **

Proof. From the system (S) coupled with the relation giving P,(x)Vr+,(x)

’

Pr+d(x>

***

P2r+d-1(X)

pr+d(xl)

‘*’

P2r+d-1(X1)

-’ ’

.-*

P,c2;_l),(x,)

0

;

’ =-

P,(x)v,+,(x)

and the result follows by computing the term P,.( x)VT+,( x).

we get:

\

Yr+d

’

YZr+d\ P,‘,“;-;‘(x,)

of a result of

p2r+d(x> P*,+,(x)

1

]

\ P$?$)(

X”)

q

3. Implementation Methods similar to those used in the scalar case [2,3] allow us to implement the Kronrod’s procedure extended to the vector case.

A. Belantari

/ Error estimate in vector Pad& approximation

461

The polynomial P,. is first computed recursively from the recurrence relationship of order d + 1 since its coefficients satisfy a triangular system. The polynomial Vr+d can then be obtained by solving a system of linear equations. We set: P,(x)

= b, + b,x + * * * +b,x’,

v,+,(x)

b,=l,

= a, + a,x + . . . +a,+,x’+d,

ur+d= 1,

u, = C”( x’P,> = boCpl + . . . + b,C;+,. Writing the relations defining Vr+d and using the orthogonality and j such that i +j = n = 1, rid u,up+i

c

=

a=1 >*.*, k,

0,

I=/

r+d u,u;+,

c

=

a=k+l,...,d,

0,

I=.j- 1 and r+d c

a=1 ,..., k.

;+,+,=O,

a,u

i=o The error estimate e!“d’(t) t’ %>

is given by

q+dtt>

.

t+d

where v,+dtx>

-

V,+d(t-l)

x-t-l

i

r+d-n-l

=t ’

c

br+d-n-j-ltJ

j=O

with r+d-i-l Pi=

C

c~,+~+J(xjP,),

i=O ,...,

r+d-n-l.

J=n

The above error estimate becomes e;+d(t)

=

t’+” . ‘rtt)

r+d-n-l

1 C+dCt>

i

c j=O

br+d-n-j-It’

which is our extension of Kronrod’s procedure to the vector case.

of P,, we get for i = 0,. . . , n

468

A. Belantari / Error estimate in vector PadP approximation

Acknowledgement Thanks are due to Professor Claude Brezinski for many constructive suggestions and constant guidance during this work. I am grateful to Professor Jeannette Van Iseghem for her very useful comments and helpful discussions.

References [l] C. Brezinski, Pad&type Approximation and General Orthogonal Polynomials, International Series in Numerical Mathematics 50 (Birkhauser, Basel, 1980). [2] C. Brezinski, Error estimate in PadC approximation, in: Orthogonal Polynomials and Their Applications (Springer, Berlin, 1988) 1-19. [3] C. Brezinski, Procedures for estimating the error in PadC approximation, Math. Comp. 53 (1989) 639-648. [4] T.S. Chihara, An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978). [5] AS. Kronrod, Nodes and Weights of Quadrature Formulas (Consultants Bureau, New York, 1965). [6] G. Monegato, Stieltjes polynomials and related quadrature rules, SIAM Reu. 24 (1982) 137-158. [7] T.N.L. Patterson, The optimum addition of points to quadrature formulae, Math. Comp. 22 (1968) 847-856. [8] M. Prevost, Stieltjes and Geronimus type polynomials, J. Comput. Appl. Math. 21 (1988) 133-144. [9] G. SzegG, Orthogonal Polynomials, AMS Colloquium Publications 23 (AMS, Providence, RI, 1939). [lo] J. Van Iseghem, Approximants de Pad& vectoriels, These d’Etat, Universite de Lille 1, Lille, France (1987). [ll] J. Van Iseghem, Vector orthogonal relations. Vector QD-algorithm, J. Comput. Appl. Math. 19 (1987) 141-150. [12] J. Van Iseghem, An extended cross rule for vector Pad&approximants, Appl. Numer. Math. 2 (1986) 143-155.