Alphonse P. Magnus

From a photograph full photo (jpg file 61K) showing me between Walter Van Assche (left) and Ken McLaughlin (right). Photography taken in front of the library of exact sciences in Louvain-la-Neuve on 13th May 1996 by Therese McLaughlin. Obrigado!

In front of the Muslim castle of Cassis (France). Picture by Diego Dominici, July 2007.

Institut Math. Univ. cath. Louvain
Chemin du Cyclotron 2
B-1348 Louvain-la-Neuve
Belgium
mail to alphonse.magnus@uclouvain.be

(From Altdorfer's "The battle of Alexander at Issus")
web museum 1 .. web mus. 2 .. web mus. 3 ..

(Thanks to Bruno Vroman)
The full, much more explicit picture ..

" Vêtu de probité candide et cousu de fil blanc "
" Bold, math, and italic! "

Best wishes deux mil 24, pdf file
(2023, (2022, 2021,2020,2019,2018,2017,2016,2015,2014,2013,2012, 2011, 2010, 2009, 2008, 2007, 2006, 2005, 2004, 2003)

CV (html)

Pascal Maroni 1933-2024 Maroni2024.pdf Much more in opsfnl312.pdf

André Ronveaux 1932-2023.
It is my sad duty to report the passing of André Ronveaux, professor of mathematical physics in Montréal (Canada) and Namur (Belgium). He wrote many papers, most on orthogonal polynomials, books and proceedings, such as ' Heun’s Differential Equations', . Oxford U. P., 1995, Bar-le-Duc meeting proceedings, Springer lect. notes math. vol. 1171, 1985, and lesser known books on lectures, such as 'Équations aux différences finies et équations différentielles pour tous. Expérience pilote d'enseignement au Québec (1965-1969)', Lidec, Montréal, 1966, 1968, and Univ. Montréal 1969, 2nd printing: presses universitaires de Namur 2017, a healthy reaction against the 'new math' rage of the period.
André was a competent scientist, a devoted teacher, and a benevolent human being.
Alphonse Magnus.
Much more in OPSF 31.1, in opsfnl311.pdf

2023 Chasing special biorthogonal rational functions, poster presented at "Integrable Systems, Random Matrices and Special Functions at UBI 20 November, 2023 University of Beira Interior, Covilhã, Portugal" https://asimoe1.wixsite.com/workshop
The poster (pdf) The slides (pdf)
First comments, April 2024.
Satoshi Tsujimoto, Luc Vinet and Alexei Zhedanov already gave a full description of the Hahn case in "An algebraic description of the bispectrality of the biorthogonal rational functions of Hahn type", Proceedings of the American Mathematical Society, 149 (November 2020), pp.715-728..
The same authors also expanded the Masson's constructions in "The rational Heun operator and Wilson biorthogonal functions", The Ramanujan Journal, (May 2021), https://doi.org/10.1007/s11139-020-00383-7 (23pp).
"An extended scheme of classical special functions" by Vyacheslav P. Spiridonov tells the last word on elliptic hypergeometric expansions of biorthogonal rational functions, up to ₁₂V₁₁ expansions (amazing!), to appear in a future book in honor of R. Askey (G. Andrews et al., eds; Springer, 2024 or 2025), arXiv:2403.16264v1 [math.CA] 24 Mar 2024.

Jean Meinguet, 1930-2023 We are very sorry to report the passing away of Jean Meinguet, professor of numerical analysis and numerical computer science at the université catholique de Louvain (Leuven, Louvain-la-Neuve, Belgium). He graduated in engineering (1954), went to Zurich to learn numerical analysis with H. Rutishauser and E. Stiefel, worked in the "Comité d'étude et d'exploitation des calculateurs électroniques (Belgian Science Foundation, Brussels, Belgium)" on programming a Bell Telephone co. computer, got a phd on filtering problems, advised by Vitold Belevitch (1960). He organized, and taught, numerical analysis, numerical functional analsis, numerical algorithmics, numerical linear algebra (with emphasis on the theory of determinants), algol 60 and other languages. He published results on an amazing scope of subjects: rounding error analysis, such as stability of Cholesky factorization, multivariate spline interpolation, optimal recovery, rational interpolation (the Cauchy problem), Adamjan-Arov-Krein-Takagi theory, extremal measures determination. The most often heard, and most exact, statement on him is: "He knows everything".
Alphonse Magnus, Paul Van Dooren, Yves Genin
NA Digest Wednesday, November 01, 2023 Volume 23 : Issue 23

202?:
A. P. Magnus, J. Meinguet, Strong asymptotics of best rational approximation to the exponential function on a bounded interval, to appear in Mat.Sb. https://www.mathnet.ru/eng/sm8815
Key words: rational approximation, exponential function, complex potential
2010 Mathematics Subject Classification:
30E10 Approximation in the complex domain 30E15 Asymptotic representations in the complex domain 31A15 Potentials and capacity, harmonic measure, extremal length 33E05 Elliptic functions and integrals 33F05 Numerical approximation and evaluation 41A20 Approximation by rational functions %41A21 Padé approximation 41A25 Rate of convergence, degree of approximation 41A80 Remainders in approximation formulas
Abstract. We apply recent findings of complex approximation theory to best rational approximation of degree n to the exponential of -(n+n)x on [0,c]. The error norm behaves like the nth power of the main approximation rate times the n th power of a secondary approximation rate. The computation of the first rate is a consequence of works of A.A.Gonchar, E.A. Rakhmanov, and Herbert Stahl done in the 1980s; the complete asymptotic description has been achieved by A. Aptekarev in the first years of the 21st century. The solution is given in terms of elliptic integrals of the three kinds.

The last preprint .

2020:
Alphonse P. Magnus, François Ndayiragije, André Ronveaux, About families of orthogonal polynomials satisfying Heun's differential equation, Journal of Approximation Theory, 263 (2021) 105522,
https://doi.org/10.1016/j.jat.2020.105522
The last preprint.
Abstract. We consider special families of orthogonal polynomials satisfying differential equations. Besides known hypergeometric cases, we look especially for Heun's differential equations.We show that such equations are satisfied by orthogonal polynomials related to some classical weight functions modified by Dirac weights or by division of powers of binomials. An appropriate set of biorthogonal rational functions, or 2-point Padé approximations, is also described.
Keywords: orthogonal polynomials, Padé approximation, Heun's differential equation.
2010 Mathematics Subject Classification: 34M35, %Differential equations in the complex domain, singularities, monodromy, local behavior of solutions, normal forms, 34M55, %Painlevé and other special equations; classification, hierarchies; 41A21, %Padé approximation 42C05, %Orthogonal functions and polynomials, general theory, 81Q05. % Closed and approximate solutions to the Schr\"odinger eq. etc.

15 Jan. 2019:
Why did we live almost 200 years with the wrong face of Legendre? BMS-NCM Newsletter #121, January 15, 2019, 11-12. http://bms.ulb.ac.be/BMSNCM/BMSNE121.pdf

2017:
Alphonse P. Magnus, François Ndayiragije, André Ronveaux: Heun differential equation satisfied by some classical biorthogonal rational functions. Preliminary notes. Poster shown at the 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA-14), University of Kent, Canterbury, UK, July 3-7, 2017.
The pdf file

April 2015, Aug. 2016, Dec. 2016, Apr. 2017, Nov. 2017, Feb 2018 Gaussian integration formulas for logarithmic weights and application to 2-dimensional solid-state lattices
Journal of Approximation Theory 228 (2018) 21-57
https://doi.org/10.1016/j.jat.2018.02.001
Key words: Numerical analysis, Integration formulas, Logarithmic singularities, Solid-state physics, 2-dimensional lattices, Graphene.
2010 Mathematics Subject Classification: 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series, 42C05 Orthogonal functions and polynomials, general theory, 42C10 Fourier series in special orthogonal functions (Legendre polynomials, etc.), 47B36 Jacobi (tridiagonal) operators (matrices) and generalizations, 65D32 Quadrature and cubature formulas, 81Q05 Closed and approximate solutions to the Schrödinger eq. etc.
Abstract. The making of Gaussian numerical integration formulas is considered for weight functions with logarithmic singularities. Chebyshev modified moments are found most convenient here. The asymptotic behavior of the relevant recurrence coefficients is stated in two conjectures. The relation with the recursion method in solid-state physics is summarized, and more details are given for some two-dimensional lattices (square lattice and hexagonal (graphene) lattice).

Last preprint grapheneR6.pdf

The 69999 first recurrence coefficients of graphene
graphene69999.txt (2M)

Research lectures: Postscript & pdf files of lectures
Analyse numerique 3
Special topics in approximation theory.

1997: CF-Hankel-BMO(A) Rational approximation. 216K
1997: more of above, 120K
1997: still more ("1/9"), 68K
MAPA3011A 1997-98 New difference calculus. 500K
MAPA3xxxA 1998-99 Padé approx. branch points. 526K
1999-2000: Semi-classical orthogonal polynomials on the unit circle.
Abstract: Semi-classical orthogonal polynomials on the unit circle are examined. Special care is given to generalized Jacobi polynomials.
MAPA3072A 1999-2000 Semi-classical orth. pol. on unit circle. pdf (1999-2000, 2002-2003, 2012-2013)
2000-2002: Abstract: several ways to build rational approximations to various species of analytic functions are examined. Special emphasis is put on strong asymptotic estimates of the form $ f(z) -R_n(z) \sim \sigma(z) \rho^n(z)$, when such estimates are available, and on approximation to exponentials of polynomials.
MAPA3071 2000-2002 Complex approximation and interpolation. ps-gz 183K (as of 11 Oct. 2001)
MAPA3071 2000-2002 Complex approximation and interpolation. pdf 521K (as of 11 Oct. 2001)

2000-2001: Abstract: several ways to build rational approximations to various species of analytic functions are examined. Special emphasis is put on strong asymptotic estimates of the form $ f(z) -R_n(z) \sim \sigma(z) \rho^n(z)$, when such estimates are available.
1st lecture: Friday 13 October 2000, room b101-102 as above, 14h30-16h.
2001-2002: Abstract: several ways to build rational approximations to various species of analytic functions are examined. Special emphasis is put on approximation to exponentials of polynomials.
1st lecture: Friday 12 October 2001, room b101-102 as above, 14h30-16h.
ps and pdf files (Oct. 2001):
pdf file (521K)
gzipped ps file (183K)
March 2002): a summary (19 pages) pdf file (240K) ps file (349K)
The slides (5 pages) pdf file (121K) ps file (216K)

2002-2003: Abstract: Generalized Jacobi orthogonal polynomials on the unit circle are examined: differential equations, recurrence relations, asymptotic properties.
1st lecture: Friday 11 October 2002, room b101-102 as above, 14h30-16h.
pdf file Mar. 2003 (408K)
(March 2003): bits and pieces of a solution of Grünbaum-Delsarte-Janssen-Vries problem in m3xxx99.pdf

2006-2007. MAPA3071. Elliptic lattices. (pdf 640K)

Colloque OPSFA "Orthogonal polynomials, Special functions and Applications" 2 au 6 juillet 2007 http://www.cirm.univ-mrs.fr/liste_rencontre/Rencontres2007/Valent07/Valent07.html
Rational interpolation to solutions of Riccati difference equations on elliptic lattices. The slides (pdf) The pre-preprint (pdf) The preprint of Nov. 2008 (pdf)

Rational interpolation to exponential-like functions on elliptic lattices, 3èmes Journées Approximation May 15-16, 2008 Salle de Réunion, Bâtiment M2 Laboratoire Paul Painlevé UMR 8524 Université de Lille 1, FRANCE
The slides (pdf) The pre-preprint (pdf)

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 http://www.hausdorff-center.uni-bonn.de/elliptic-integrable-systems

Elliptic difference equations. The elliptic logarithm. The slides (pdf) http://www.emis.de/journals/SIGMA/Elliptic-Integrable-Systems.html
The preprint of Nov 2008. (pdf)
Alphonse P. Magnus, Elliptic Hypergeometric Solutions to Elliptic Difference Equations, SIGMA 5 (March 2009), 038, 12 pages , http://www.emis.de/journals/SIGMA/2009/038/sigma09-038.pdf

OPSFA-10 (Orthogonal Polynomials, Special Functions and Applications), Leuven 2009 http://wis.kuleuven.be/OPSFA/OPSFA10.html

Rational Freud equations, my slides Freud2009s.pdf

Approximation Days 2012 2-3 July, 2012, Leuven, Belgium http://wis.kuleuven.be/events/ad12
Threepieces2012s.pdf
The Polignac problem has been solved long ago, it is in Whittaker & Watsons's book! see PolWhit.pdf
More on gen. Jacobi pol. asymptotics on unit circle in Semi-classical orth. pol. on unit circle. pdf (1999-2000, 2002-2003, 2012-2013)

Special topics in numerical linear algebra 2000

Seminar series on rational approximation and systems theory 2002
SEMINAR SERIES ON ITERATIVE METHODS FOR LARGE SCALE PROBLEMS (2005)

A piece of Belgian literature
A piece of computer science (1970)

The Journal

Journal of Approximation Theory at Ohio State.
Journal of Approximation Theory at Technion.
www.academicpress.com/jat

Papers

Alphonse P. Magnus, Rational interpolation to solutions of Riccati difference equations on elliptic lattices, Journal of Computational and Applied Mathematics, available online: 26-FEB-2009, http://dx.doi.org/10.1016/j.cam.2009.02.047

Alphonse P. Magnus, Elliptic Hypergeometric Solutions to Elliptic Difference Equations, SIGMA 5 (March 2009), 038, 12 pages , http://www.emis.de/journals/SIGMA/2009/038/sigma09-038.pdf

Alphonse P. Magnus, Viviane Pierrard, Formulas for recurrence coefficients of orthogonal polynomials related to Lorentzian-like weights, Journal of Computational and Applied Mathematics, Volume 219, Issue 2, Pages 431-440 (1 October 2008) , Special Issue dedicated to Prof. Claude Brezinski, on the occasion of his retirement from the University of Sciences and Technologies of Lille in June 2006, Edited by M. Prévost, M. Redivo-Zaglia and F. Wielonsky .
The pdf preprint, with the % in the acknowledgements!

Germain, M.; Magnus, A.; Steenberghe, V., How to design and use the clean development mechanism under the Kyoto Protocol? A developing country perspective, Environmental and Resource Economics , Volume 38, Number 1, September 2007, pp. 13-30.

M.Germain, A.Magnus: Prices versus quantities: stock pollution control with repeated choice of the instrument, J. Comp. Appl. Math. . 197 (2006) 437-445 the pdf (145K) preprint, Oct. 2005.

A.P. Magnus, " Freud equations for Legendre polynomials on a circular arc and solution of the Grünbaum-Delsarte-Janssen-Vries problem" , Journal of Approximation Theory, Volume 139, Issues 1-2, This Volume is Dedicated to Barry Simon on the Occasion of his Sixtieth Birthday, March-April 2006, Pages 75-90. Keywords: Unit circle orthogonal polynomials
The pdf (200K) preprint, May 2005.

A.P. Magnus, J. Meinguet, The elliptic functions and integrals of the `1/9' problem, presented at Antwerpen international conference on rational approximation June 6--11 1999 ICRA99, Numerical Algorithms 24: (1-2) (2000) 117-139. See more here.

M. Germain, A.P. Magnus, Progrès technique et durée de vie du capital, IRES Discussion Paper 9911 (March 1999) http://www.ires.ucl.ac.be/DP/Biblio_html/9911.html

In preparation: V.Pierrard and A.P.Magnus: Lorentzian orthogonal polynomials
The FORTRAN77 program

Papers by A.P.Magnus
see also in my CV

All about the "1/9" constant!

see here

Software: Freud machine

Freud programs and doc

NEW (2012) Bar-le-Duc stuff BarlD.htm

NEW (June 2003): pre-preprint Freud equations for Legendre polynomials on a circular arc and solution of the Gr\"unbaum-Delsarte-Janssen-Vries problem.
The pdf file
The java program