finite-differences and interpolating polynomials

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Leslaw Bieniasz

Posted: Thu Mar 15, 2007 6:38 am

Guest

Cracow, 15.03.2007

Hi,

It is known that classical finite-difference quotients
representing approximations to derivatives can be interpreted
as derivatives of a Lagrangian interpolation polynomial
passing through a set of discrete function values.
I am looking for historical and other good references
discussing this particular issue, including finite-differencing
on non-uniform grids, and possible generalisations onto
Hermitian (compact) finite-difference schemes, where the
interpolant should be Hermitian, instead of Lagrangian.
Any pointers will be appreciated.

L.B.

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Alex. Lupas

Posted: Sun Mar 18, 2007 2:05 am

Guest

On Mar 15, 1:38 pm, Leslaw Bieniasz <nbbie...@cyf-kr.edu.pl> wrote:

Quote:

Possibke references;

[A]=BOOKS

[1-a]. R.P. Agarwal and P.J.Y. Wong,
Advanced Topics in Difference Equations,
Kluwer, Dordrecht, 1997.

[1-b] Agarwal R.P.,
Difference Equations and Inequalities Theory,
Methods, and Applications,
Second Edition, Revised and Expanded ,
Marcel Dekker, Inc. New York , Basel,2000.

[2] Boole G.,
Calculus of Finite Differences, 4th ed.,Chelsea, New York, 1958.

[3] Cogan E.J. and Norman R.Z.,
Handbook of Calculus,Differences and Differential Equations,
Prentice-Hall, Englewood Cliffs, N.J., 1958.

[4] Fort T.,
Finite Differences and Difference Equations
in the Real Domain,
The Clarendon Press, Oxford, 1948.

[5] Gelfond A.O.,
Differenzenrechnung,VEB,Berlin,1958,
see also
Calculus of Finite Differences,
Hindustan, Delhi, India,1971.

[6] Jordan Ch. (=Karoly), Calculus of Finite Differences,
Chelsea, New York, 1947, 3-rd.,edition 1965.

[7] Markoff A.A., Differenzenrechnung, Sankt Petersburg, 1889/1891;
see also the translation in German
by Th.Friesendorf und E.Pruemm, Leipzig, 1896.

[8] Miller K.S. ,
An Introduction to the Calculus of Finite Differences
and Difference Equations,
Holt, New York, 1960.

[9] Milne Thomson L.M.,
The Calculus of Finite Differences,
Maximilian, London,1960.

[10] Norlund N.E., Vorlesungen ueber Differenzenrechnung,
Berlin , Springer Verlag, 1924.

[11]Norlund N. E., Lecons sur les series d'interpolation.
Gauthier-Villars et C-ie, Paris, 1926.

[12] Richardson C.H.,
An Introduction to the Calculus of Finite Differences,
Van Nostrand, New York, 1954.

[13] Spiegal M.R.,
Calculus of Finite Differences and Difference Equations,
McGraw-Hill, New York, 1971.

[14] Steffensen J.F.,
Interpolation ,
New York, Chelsea,1950.

[15] Whitatker J.M. ,
The interpolatory function theory,
Cambridge Univ. Press, 1

Alex. Lupas

Posted: Sun Mar 18, 2007 10:15 am

Guest

[B]PAPERS/surveys/(also Books):
==========[1]P.R. Allaire and R.E. Bradley,
Symbolical Algebra as a Foundation for Calculus:
D.F.Gregory's Contribution,
Historia Mathematica 29 (2002), 395--426.

[2]A.-M. Ampere, Essai sur un noveau mode d'exposition des principes
du calcul diff´erentiel, du calcul aux diff´erences et de
l'interpolation
dessuites, considerees comme derivant d'un source commune,
Annals de mathematicques pures et appliquees
de Gergonne (Paris, Bachelier, in (4o)16(1826)329--349.

[3] S.Avdonin and W. Moran,
Ingham -type Inequalities and Riesz bases
for divided differences,
Int. J. Appl. Math. Comput. Sci., 2001, Vol.11, No.4, 803--820.

[4] C. de Boor,
On calculating with B-splines,
J. Approx. Theory 6(1972), 50-62.

[5]C.de Boor,
A divided difference expansion of a divided difference,
J.Approx. Theory 122(2003), 10-12.

[6]C.de Boor,
A Leibniz formula for multivariate divided differences,
SIAM J. Numer. Anal. 41(3),(2003)856--868.

[7]Carl de Boor,
Divided Differences,(preprint),
ArXiv Math (1 Febrruary 2005)
see also Surveys in Approximation
Theory 46 Vol.1, (2005) 46--69.

[8] C.de Boor and A. Pinkus,
The B-spline recurrence relations of Chakalov
and of Popoviciu,
J. Approx. Theory 124(2003) 115--123.

[9]C. Brezinski,
The Muhlbach-Neville-Aitken algorithm and some extensions,
BIT 20 (1980) 444-451.

[10] R.Bulirsch, R. and H. Rutishauser,
Interpolation und genaeherte Quadratur,
in Mathematische Hilfsmittel des Ingenieurs,
Teil III, S. Sauer & I. Szabo, eds,
Grundlehren vol. 141, Springer-Verlag, Heidelberg,1968, 232-319.

[11]A.Cauchy,
Sur les fonctions interpolaires,
C. R. Acad. Sci. Paris 11,(1840)775--789.

[12]L.Tchakaloff,
Sur la structure des ensembles lin´eaires d´efinis par une
certaine propriete minimale,
Acta Math 63(1934)77--97.

[13]L.Chakalov,
On a certain presentation of the Newton divided differences
in interpolation theory and its applications (in Bulgarian),
Annuaire Univ. Sofia,Fiz. Mat. Fakultet 34(1938), 353--394.

[14]H.B.Curry and I. J. Schoenberg,
( On P´olya frequency functions IV): the
fundamental spline functions and their limits,
J. Analyse Math. 17(1966) 71--107.

[15]P.Erdos, P. and P. Turan,
On interpolation, III. Interpolatory theory of
polynomials,
Appl. Math.(2) 41,(1940) 510--553.

[16] M.Floater,
Error formulas for divided difference expansions and numerical
differentiation, J. Approx. Theory 122(2003) 1--9.

[17]C.D.Fraser,
Newton's interpolation formulas,
J. Instit. Actuaries LI(1918) 77--106.

[18]C.D.Fraser,
Newton's interpolation formulas. Further Notes,
J.Instit. Actuaries LI(1919)211--232.

[19]C.D.Fraser,
Newton's interpolation formulas. An unpublished
manuscript of Sir Isaac Newton,
J. Instit. Actuaries LVIII(1927) 53--95.

[20]G. Frobenius,
Ueber die Entwicklung analytischer Functionen in Reihen,
die nach gegebenen Functionen fortschreiten,
J. reine angew. Math. 73(1871) 1--30.

[21]A. Genocchi,
Relation entre la difference et la d'erivee d'un meme ordre
quelconque,
Archiv Math. Phys. (I) 49(1869), 342-345.

[22]A. Genocchi,
Intorno alle funzioni interpolari,
Atti della Reale Accademia delle Scienze di Torino 13(1878) 716--729.

[23]A. Genocchi,
Sur la formule sommatoire de Maclaurin et les fonctions
interpolaires,
C. R. Acad. Sci. Paris 86(1878) 466--469.

[24]Ch.Hermite,
Sur l'interpolation,
C. R. Acad. Sci. Paris 48(1859) 62--67.

[25]Ch.Hermite,
Formule d'interpolation de Lagrange,
J. reine angew. Math.84(1878) 70-79.

[26]E.Hopf,
¨ Ueber die Zusammenhaenge zwischen gewissen hoeheren
Differenzenquotienten reeller Funktionen einer reellen Variablen
und deren Differenzierbarkeitseigenschaften,
Dissertation, Universit¨at Berlin, 1926 ,(30 pp).

[27] J.L.M.V. Jensen,
Sur une expression simple du reste dans la formule
d'interpolation de Newton, Bull. Acad. Roy. Danemark xx(1894)246--xxx.

[28]W. Kahan,
Divided differences of algebraic functions,
Class notes for Math 228A, UC Berkeley,1974, Fall.

[29] S. Karlin and W. Studden,
Tchebycheff Systems (Interscience, New York, 1966).

[30]A.Kowalewski,
Newton, Cotes, Gauss, Jacobi: Vier grundlegende Abhandlungen ueber
Interpolation und genaeherte Quadratur",
Teubner, Leipzig,1917.

[31]G. Kowalewski,
Interpolation und genaeherte Quadratur,
B. G. Teubner,Berlin,1932.

[32]D.Loeb and G-C Rota,
Recent Contributions to the Calculus of
Finite Differences: A Survey.
(preprint ArXiv Math. 9 Feb -1995 )

[33] A. de Morgan,
The Differential and Integral Calculus,
Baldwin and Cradock, London,1842, UK.

[34] G. Muhlbach,
A recurrence formula for generalized divided differences
and some applications,
J. Approx.Theory 9 (1973) 165-172.

[35] G. Muhlbach,
Newton- und Hermite Interpolation mit Chebyshev-Systemen,
Z. Angew. Math. Mech.54 (1974) 541--550.

[36] G. Muhlbach,
On interpolation by rational functions with prescribed poles
with applications to multivariate interpolation,
J. Comput. Appl. Math. 32 (1990) 203-126.

[37] G. Muhlbach,
Interpolation by Cauchy-Vandermonde systems and applications,
in J. Comput.Appl. Math. (2000).

[38] G. Muhlbach,
A recurrence relation for generalized divided
differences with respect to ECT-systems,
Numerical Algorithms 22 (1999) 317--326.

[39] G. Muhlbach and L. Reimers,
Linear extrapolation by rational functions, exponentials
and logarithmic functions,
J. Comput. Appl. Math. 17 (1987) 329-344.

[40] I. Newton,
Epistola posterior (Second letter to Leibniz via Oldenburg),
24 oct.1676.

[50] I. Newton,
Philosophiæ naturalis principia mathematica,
Joseph Streater,London,1687, (see
ftp://ftp.math.technion.ac.il/hat/fpapers/newton1.pdf and
ftp://ftp.math.technion.ac.il/hat/fpapers/newton2.pdf)

[51] N.E. Norlund,
Vorlesungen ¨uber Differenzenrechnung,
Grundlehren Vol. XIII, Springer,1924, Berlin.

[52] G. Opitz,
Steigungsmatrizen, Z. Angew. Math. Mech. 44(1964) T52--T54.

[53] T.Popoviciu,
Sur quelques proprietes des fonctions d'une ou de
deux variables reelles,
Dissertation, presented to the Faculte des Sciences de
Paris,1933,
published by Institutul de Arte Grafice "Ardealul" (Cluj, Romania).

[54] T. Popoviciu,
Introduction `a la theorie des differences divisees,
Bull.Mathem., Societea Romana de Stiinte, Bukharest 42(1940)65--78.

[55]H.E. Salzer, Divided differences for functions of two variables
for irregularly spaced arguments,
NUmerische Mathematik 6(1964)68--77.

[56]H.A. Schwarz,
Demonstration ´elementaire d'une propriete fondamentale
des fonctions interpolaires,
Atti Accad. Sci. Torino 17(1881), 740-742.

[57]J.F. Steffensen,
Note on divided differences,
Danske Vid. Selsk. Math.Fys. Medd. 17(3),(1939)1--12.

[58] A. R. Stanford and G. Lopez Lagomasino,
Approximation of transfer functions of infinitedimensional
dynamical systems by rational interpolants with prescribed
poles,
Report 433, Institut fuer Dynamische Systeme,
Universit¨at Bremen (1998).

[59]B. Taylor,
Methodus incrementorum directa et inversa,
London,1715,England.

[60] E.T. Whittaker, and G. Robinson,
The Calculus of Observations, A Treatise on Numerical Mathematics,
2nd ed., Blackie, Glasgow, 1937,UK.

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