Alphonse P. Magnus




 Université Catholique de Louvain

 Institut Mathématique

 Chemin du Cyclotron,2

 1348 Louvain-la-Neuve.



Ingénieur civil en mathématiques appliquées (grade scientifique), UCL, 1969.

Docteur en sciences appliquées (grade scientifique), UCL, 1976.


Assistant, UCL, 1969,

Premier assistant, UCL, 1976,

Chef de travaux, UCL, 1984.

Chargé de cours à temps partiel, UCL, 1997.

Professeur à temps partiel, 2004.


Mathématiques, Analyse numérique, Théorie de l'approximation.

Enseignement, encadrement:

Encadrement exercices analyse numérique en MATH21, MAP21, MAP22 et PHYS 21-22 (aussi INFO21 ou 22 1980-1990).

MATH2171, Analyse numérique 1a : depuis 1972

MATH2172, Analyse numérique 1b : 1972-1994

Encadrement cours et exercices analyse FSA11 et 12: quelques intérims,

Encadrement exercices programmation FSA11 et/ou MATH12 : période 1976-1980.

Directions régulières de mémoires MAP23 et MATH22.


2ème cycle:

``Approximation complexe quantitative'', cours de questions spéciales en mathématique,

Facultés N.D. de la Paix, Namur, janvier-mai 1989.

MATH2171, Analyse numérique 1a : depuis 1995 ( 50% en 1994)

MATH2180, Analyse numérique 2: depuis 1995, (50% en 1995-1998)

MATH2830, Séminaire d'analyse numérique : depuis 1995, 33% (50% depuis 2004).

MATH2900, Séminaire de mathématique: 25% depuis 1998.

INMA 2375, Projet intégré en ingénierie mathématique: 14% 2002-2004.

3ème cycle:

``Méthodes CF en approximation et théorie des systèmes'', partie du cours de formation interdisciplinaire pour doctorands, Facultés N.D. de la Paix, Namur, janvier 1990.

Analyse numérique, special topics in approximation theory:

MAPA3116, Séminaire d'analyse numérique : 1995-2001, 20%

MAPA3079 Séminaire d'analyse numérique (DEA Math.), depuis 2002, 11%.

Encadrement de thèse (G. Bangerezako, dept MATH): 50% 1997-1999.

Co-édition de livre:

C.BREZINSKI , A.RONVEAUX , A.DRAUX , A.P.MAGNUS , P.MARONI, editors: Polynômes orthogonaux et applications Bar-le-Duc 1984 , Lecture Notes in Mathematics 1171 , Springer, Berlin 1985.

Conférences publiées et articles de périodiques:

A paraître:

En préparation:

V. Pierrard, and A. Magnus, Lorentzian orthogonal polynomials

Marc Germain, Vincent van Steenberghe, Alphonse Magnus, Optimal policy with tradable and bankable pollution permits : Taking the market microstructure into account.

J.Lemaire, V.Pierrard, A.Magnus, Overview of models for parallel electric fields distributions and double layer solutions in magnetospheric physics.

A.Magnus, Freud equations for Legendre polynomials on a circular arc and solution of the Grünbaum-Delsarte-Janssen-Vries problem.

Sélection d'abstracts.

87h:42039 (18:11) 42C05

Magnus, Alphonse P. (B-UCL)

On Freud's equations for exponential weights. (English)

Papers dedicated to the memory of Géza Freud.

Journal of Approximation Theory 46 (1986), no. 1, 65-99.

Geza Freud raised two conjectures that have generated considerable interest in the last few years. Both of them are connected with the recurrence coefficients for the orthonormal polynomials with respect to the weights exp(-|x|a), a > 0. The first conjecture was settled by A. A. Rachmanov while the second one has resisted every attempt at solution for several years. This conjecture claims that if the recurrence relation is xpn(x)=an+1pn+1(x)+bnpn(x)+a npn-1(x), then (*)


exists (if it exists then the limit is easy to compute). The paper under review solves Freud's conjecture in the important special case when a is an even integer; more precisely, the existence of (*) is proved for weights of the form exp(-P(x)) where P is a polynomial of degree 2m, m=1,2,¼, with positive leading coefficient. The technique of the proof is very fine; it starts from an identity of Freud for the an's.

It should be mentioned that by a different potential-approximation-theoretic approach Freud's conjecture has recently been fully settled by D. S. Lubinsky, H. N. Mhaskar and E. B. Saff.

See also the following review.

{For the entire collection see MR 87a:41003.}

Reviewed by Totik, V. (Szeged)

Cited in: 88d:42039

Copyright American Mathematical Society 1987, 1995


88i:65022 (19:16) 65D15

Magnus, Alphonse P. (B-UCL)

Toeplitz matrix techniques and convergence of complex weight Pade approximants. (English)

Journal of Computational and Applied Mathematics 19 (1987), no. 1, 23-38.

Starting from the Stieltjes (or Markov) class of functions, the author establishes convergence for diagonal Pade approximants given a choice of comparison weight functions. This class of functions may be generated from spectral investigations of selfadjoint operators. The paper links the convergence established above with that of projection methods of operators and thereafter utilizes well-established results in Toeplitz operator theory, thus providing interesting insight into the linkages between Pade approximants and special operators.

Reviewed by Rodrigues, A. J. (Nairobi)

Copyright American Mathematical Society 1988, 1995


A.P.MAGNUS: On the use of Carathéodory-Fejér method for investigating '1/9' and similar constants, pp. 105-132 in A.CUYT, Editor: Nonlinear Numerical Methods and Rational Approximation , D.Reidel, Dordrecht 1988.

Let En be the error norm of the best L¥ rational approximation of degree n to the exponential function exp(-t) on [0,¥). Grounds are given for setting the conjectured limit En/qn® 2q1/2 when n®¥, where q is the known constant `1/9'= 1/9.2890254919208189187554494359517450610316948677¼, based on the singular values and functions of the relevant Hankel operator (Carathéodory-Fejér's method). Moreover, hints are given according to which a valuable asymptotic expansion of En should also contain nth powers of new constants q1=`1/56¢, q2=`1/240¢, etc.

A vrai dire, cet article-partie d'actes est assez peu cité, on cite plutôt

(cf. A.A. GONCHAR, E.A. RAKHMANOV, Equilibrium distribution and the degree of rational approximation of analytic functions, Mat. Sb.  134 (176) (1987) 306-352 = Math. USSR Sbornik 62 (1989) 305-348,

R.S. VARGA, Scientific Computation on Mathematical Problems and Conjectures, CBMS-NSF Reg. Conf. Series in Appl. Math. 60 , SIAM, Philadelphia, 1990)

une lettre que j'ai envoyée à une dizaine de personnes... Les russes, prudents avec les questions ethniques, ont jugé bon de la référencer comme ``Preprint B-1348, Inst. Math., Katholieke Univ. Leuven, Louvain, 1986''.


95f:33011b (94:16) 33C45 26C05 42C05

Erdélyi, Tamas (3-SFR) ; Magnus, Alphonse P. (B-UCL-PA) ; Nevai, Paul (1-OHS)

``Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials''. (English) SIAM J. Math. An. 25 (1994) 602-614.

Erratum: SIAM Journal on Mathematical Analysis 25 (1994), no. 5, 1461.

It is well known that the Legendre polynomial Pn(x) satisfies the Bernstein inequality
(sinq)1/2 |Pn(cosq)| < (2/p)1/2 n-1/2,    0 £ q £ p.
V. A. Antonov and K. V. Kholshevnikov [Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1980, no. 3, 5-7, 128; MR 82b:33012] obtained a sharper result by showing that the factor n-1/2 can be replaced by (n+1/2)-1/2. For ultraspherical polynomials Pn(l)(x), L. Lorch [Appl. Anal. 14 (1982/83), no. 3, 237-240; MR 84k:26017] proved that
(sinq)l | Pn(l)(cosq)| < 21-l G(l)-1 (n+l)l-1
for 0 < l < 1 and 0 £ q £ p.

Recently, Y. H. Chou, L. Gatteschi and R. Wong [Proc. Amer. Math. Soc. 121 (1994), no. 3, 703-709; MR 94i:33008] have extended this result to nonsymmetric Jacobi polynomials Pn(a,b)(x) with -1/2 < a, b < 1/2, and a+b > 0.

Here, in this interesting paper, a more general result is obtained. For orthonormal Jacobi polynomials pn(w,x) with respect to the weight w(x)=(1-x)a(1+x)b with a ³ -1/2 and b ³ -1/2, the inequality

x Î [-1,1] 


w(x) pn(w,x)2 £


holds for every n ³ 0. The authors, although unable to obtain sharp constants, develop here techniques which are themselves very important. They use generalized algebraic polynomials to estimate the Christoffel functions ln(w,x) and obtain a Riccati equation which yields estimates for pn(w,x)2ln(w,x).

Reviewed by Guadalupe Hernández, José Javier (Logrono)

Copyright American Mathematical Society 1995


A. P. MAGNUS: Freud's equations for orthogonal polynomials as discrete Painlevé equations.

We consider orthogonal polynomials pn with respect to an exponential weight function w(x)=exp(-P(x)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre, in order to study special continued fractions, recurrence relations, and various asymptotic expansions (G. Freud's contribution).

Most striking example is n=2twn+wn(wn+1+wn+wn-1) for the recurrence coefficients pn+1=xpn-wn pn-1 of the orthogonal polynomials related to the weight w(x)=exp(-4(tx2+x4)). This example appears in practically all the references. The connection with discrete Painlevé equations is described here.


Activités d'édition.

Je suis membre de l'editorial board du Journal of Approximation Theory.

Collaboration à l'organisation de congrès.

Polynômes orthogonaux et applications, Bar-le-Duc, 15-18 octobre 1984.

Invitations à des conférences et congrès.

Recursion Conference and Applications, Imperial College, London, 13 & 14 septembre 1984.

Third International Symposium on Orthogonal Polynomials and their Applications, Erice (Trapani), 1-8 juin 1990.

Constructive Methods in Complex Analysis, Oberwolfach, 26 mars-1 avril 1995.

Symmetries and Integrability of Difference Equations II, Canterbury, 1-5 juillet 1996.

  Colloque OPSFA "Orthogonal polynomials, Special functions and Applications" 2 au 6 juillet 2007

3èmes Journées Approximation May 15-16, 2008, Laboratoire Paul Painlevé UMR 8524, Université de Lille 1, FRANCE

Workshop: "Elliptic integrable systems, isomonodromy problems, and hypergeometric functions". Yu.I. Manin, M. Noumi, E.M. Rains, H. Rosengren, V.P. Spiridonov (organizers) Hausdorff Center for Mathematics, Bonn 21-25 July, 2008

Présences à des conférences et congrès.

Padé Approximation and its Applications , Antwerpen 1979.

Padé Approximation and its Applications , Amsterdam 1980.

Padé Approximation and its Applications , Bad Honnef 1983.

Padé Approximation and its Applications , Marseille-Luminy, 14-18 October 1985.

Nonlinear Numerical Methods and Rational Approximation , Antwerpen, 1987.

2nd International Symposium on Orthogonal Polynomials and their Applications , Segovia 1988.

4th International Symposium on Orthogonal Polynomials and their Applications , Evian 1992.

Nonlinear Numerical Methods and Rational Approximation II , Antwerpen, 1993.

Computation in Economics, Finance and Engineering: Economic Systems, Cambridge (GB), 1998.

International conference on rational approximation, Anvers, 6-11 juin 1999.