Proceedings of the First International Colloquium on Numerical Analysis: Plovdiv, Bulgaria, 13–17 August 1992 [Reprint 2020 ed.] 9783112314111, 9783112302842

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Proceedings of the First International Colloquium on Numerical Analysis

Proceedings of the

First International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 13 -17 August 1992

Editors: D. Bainov and V. Covachev

///VSP///

Utrecht, The Netherlands, 1993

VSPBV P.O. Box 346 3700 AH Zeist The Netherlands

© VSP BV 1993 First published in 1993 ISBN 90-6764-152-9

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed, in Great Britain by Bookcraft

(Bath) Ltd., Midsomer

Norton.

CONTENTS

Preface

Orthogonal polynomials associated with a non-standard measure A. Arteaga and J. Muhoz Masqué

1

The numerical treatment of ordinary and partial Volterra integro-differential equations H. Brunner

13

Numerical methods for y" = f(x,y) J.P. Coleman

27

Numerical-analytical method for finding periodic solutions of nonlinear impulsive singularly perturbed differential systems V. Covachev

39

Special algorithms for the numerical integration of problems in orbital dynamics J.M. Ferrândiz and P. Martin

51

Polynomial approximation with side conditions: Recent results and open problems H.H. Gonska and X.-i. Zhou

61

Continuous numerical solutions and error bounds for matrix differential equations L. Jôdar and E. Ponsoda

73

Convergence rate estimates for the finite difference schemes compatible with the smoothness of data B.S. Jovanovic

89

Numerical solutions of differential-function problems Z. Kamont An iterated method of computing the inverse mapping J. Muhoz Masqué and G. Rodriguez Sanchez On using high orders finite elements for solving structural mechanics problems T. Scapolla

97 113 123

Numerical solution of differential equations for the analytic singular value decomposition K. Wright A new trust region algorithm for nonlinear optimization Y.-X. Yuan

Preface The First International Colloquium on Numerical Analysis was organized by UNESCO and Plovdiv Technical University with the help of the Austrian Mathematical Union, the Canadian Mathematical Society, Hamburg University, Institute of Mathematics of the Bulgarian Academy of Science, Kyushu University, the London Mathematical Society, Plovdiv University 'Paissii Hilendarski', Technical University-Berlin, Union of the Bulgarian Mathematics - Plovdiv Section, and sponsored by Bulgarian Mails and Telecommunications Ltd., and the firms 'Eureka' and 'Microtechnika Ltd.' It was held on August 13-17, 1992 in Plovdiv, Bulgaria. The subsequent colloquia will take place each year from 13-17 August in Plovdiv, Bulgaria. The Address of the committee is: Stoyan Zlatev, Mathematical Faculty of the Plovdiv University, Tsar Assen Str. 24, Plovdiv 4000, Bulgaria. The Editors

First International Coloquium on Numerical Analysis pp. 1-12 (1993) D. Bainov and V. Covachev (Eds) © 1993

ORTHOGONAL POLYNOMIALS ASSOCIATED WITH A NON-STANDARD

MEASURE

Angel Arteaga* and Jaime Munoz Masque** National Research Council of Spain (CSIC). * IMAFF, Serrano 123, E-28006 Madrid, Spain, e-mail: [email protected] **IETEL, Serrano 144, E-28006 Madrid, Spain, e-mail: [email protected] Abstract Let Z f be the group of infinite binary sequences, and let us denote by p. the Haar probability measure on Zj". By means of the standard homeomorphism from TL^ onto the Cantor triadic subset K C [0,1], p induces a measure PK on K, which in turn defines a measure v on the unit interval by simply setting i/(/) = PK(/\K), for every continuous function / 6 C ([0,1]). According to the representation theorem, v is a Lebesgue-Stieltjes measure. Hence, there exists a non-decreasing function a on [0,1] continuous on the right, such that "(/) = fo /(*) 0. T h a t is, h' p(x) = p(x) + Y , ~ xk) • k=I

From this point of view, the measure we consider is, clearly, a non-classical one, in the sense that it does not correspond to any perturbation of a classical measure. In a forthcoming paper we will try to extend the present results to more general pro-finite abelian groups.

2

2

A. Arteaga

and J.M.

Masque

H o m e o m o r p h i s m b e t w e e n t h e group of sequences zeros and ones and t h e Cantor triadic set

Let TL2 be the set {0, 1} and X ^ be the set of sequences of 0's and l's with the topology product, considering in 'EI2 the discrete topology. Clearly TL2 is a compact topological group. Let K be the Cantor triadic set. As is well known, K is identified with the points of the segment [0,1] whose irrational expansion in base 3 does not contain 1; i.e. x € K if, and only if, . 0-2 , a3

«1

an

and all the a;'s are zero or two. In this way, a homeomorphism between the sequences of 0's and l's and t h e Cantor triadic set is obtained: 9 : TL^ -> K , defined by ([6]): =

3

2oi

2«2

+

+

+

2a,,

,

+

(a,

_ „

£ TL2) .

,„.

(1)

T h e Haar measure on TZ^

As is well known, any compact abelian group G has a distinguished measure fi: the unique invariant measure for which the mass of G is equal to one; or, in other words, ¡i is t h e standard Haar measure of the group G. T h e Haar measure can be built in the following way: Let / : ^ ^ —> R be a continuous function only depending on a finite number of components of the sequences of i. e., there exists n £ IN and a continuous function g : ^ -> R such t h a t : , x 2 , . . . , xn, f o r a l l x = ( x j , x 2 , . . . , xn,

xnJr\,...)

xn+\,...)

—

,...,

xn),

(E

Let T be the set of such functions. By the Weierstrass approximation theorem, T is dense in the space C of continuous functions of the group so every continuous function / : ¡Z^ —> IR can be expressed as the limit of a sequence of functions /,, £ T with respect to t h e uniform convergence: / = lim fn. Again, let / £ T . We define ,«(/) to be: = ^

i.e.,

£/(*);

(2)

M / ) = ^ ( 5 ( 0 , . . . , 0 ) + --- + « / ( l , . . . , l ) ) . T h e measure defined above has the following properties:

i) \n{f)\

< m a x | / ( x ) | = 11/11-

Proof:

=

m a x

,-

£

r ; lf(-T)l

=

m a x

x-

e

^\f(*)\-

Orthogonal

ii) Let

fl6Sf,

l i f e Proof:

and let ta :

-> ^

be the translation ta(x)

then / o ta G T and /i(/ o i „ ) =

J7,

3

polynomials

~a + x.

p{f).

With the same hypotheses we get: (f ot

a

)(x

u

...,x

n

=

,...)

/(fl!+a;,,...,a

= flf(ai + x i , . . . , a „ + i n ) . . . ) =

(g ° t

a

)(xi,...

,xn,...).

Besides, we get /¿(/ 0 < a) = ^ - Y,

(Jo1.)W = ^

E

«/(a + a;)-

Let us do the change of variable y = a + x. As far as if, and only if, y runs over W^. Thus

/*(/

=

^

9(a

fTOTl

+ x)

=

x^llu^

-L

^

T7TI

is a group, x runs over