Peano, his School and … Numerical Analysis

  • Giampietro Allasia


Giuseppe Peano, an outstanding mathematician of unusual versatility, made fundamental contributions to many branches of mathematics; in numerical analysis, noteworthy results concern representation of linear functionals, quadrature formulas, ordinary differential equations, Taylor’s formula, interpolation, and numerical approximations1. Many results are still of great interest, whereas a few others appear obsolete.


Quadrature Formula Divided Difference Numerical Calculus Midpoint Rule Linear Homogeneous Differential Equation 
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  1. 1.
    Some of these works (e. g. those on differential equations and Taylor’s formula) are considered by some people as pertaining mainly to mathematical analysis; this is not astonishing because numerical analysis is based on mathematical analysis. The book by J.B. Scarborough, Numerical Mathematical Analysis, Baltimore, Johns Hopkins Press, 6th ed. 1966 (1st ed. 1930), amongst the earliest courses in numerical analysis, must be remembered. The title is clearly motivated by the following statement in the Preface to the first edition: “Applied mathematics comes down ultimately to numerical results, and the student [… ] will do well to supplement his usual mathematical equipment with a definite knowledge of the numerical side of mathematical analysis.” These words sound quite similar to the following considerations by Peano (1918c), 693, referring to the ordered list of Peano’s publications in the CD-ROM L’Opera omnia e i Marginalia di Giuseppe Peano (with English version), C.S. Roero (ed.), Torino, Dipartimento di Matematica, 2008, see the URL “Lo scopo della Matematica è di dare il valore numerico delle incognite che si presentano nei problemi pratici [… ] tutti i grandi matematici sviluppano le loro mirabili teorie fino al calcolo numerico delle cifre decimali necessarie. È di somma importanza che le teorie matematiche che si insegnano nelle scuole di vario grado siano coronate dal calcolo numerico” (The aim of mathematics is to give the numerical value to the unknowns arising from practical problems; all the great mathematicians develop their wonderful theories up to the numerical evaluation of the required decimal digits. It is extremely important that the mathematical theories, which are thought in schools of various levels, are crowned with numerical calculus).Google Scholar
  2. 4.
    “Indice, per quanto possibile completo, de auctores italiano, que […] pertine ad schola de Peano” (A list, as complete as possible, of Italian authors, who pertain to Peano’s school), in general, is drawn up by U. Cassina (1932), 124 and a shorter one, restricted to numerical analysis, is given by C. Migliorero (1928), 36, who writes: “Peano reincipe studio de […] questiones [de calculo numerico], et suo labores fundamentale da origine ad serie de alio studios interessante facto per suo discipulos de […] ultimo periodo” (Peano restarts studying problems of numerical calculus, and his basic work gives rise to a series of other interesting contributions by his followers of the last period). Further information on the most significant contributions by members of Peano’s school is supplied by U. Cassina in the notes to the list of Peano’s works (1932), 133–148. Peano’s scientific work as a whole is illustrated by Cassina (1933), 323–389.Google Scholar
  3. 6.
    Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Leipzig, Teubner, 1898-1935; Encyclopédie des sciences mathématiques pures et appliquées, Paris, Gauthier-Villars, 1904–1916. Interesting viewpoints are expressed in G. Fano (1911), 106–126, and in F.G. Tricomi (1927), 102–108.Google Scholar
  4. 7.
    D. Brunt (1923); E.T. Whittaker, G. Robinson, The calculus of observations. A treatise on numerical mathematics, London, Blackie & Son, 4th ed. 1944 (1st ed. 1924); J.F. Steffensen, Interpolation, New York, Chelsea, 1950 (reprint of the 1st ed., Baltimore, Williams & Wilkins, 1927); J.B. Scarborough (1930); L.M. Milne-Thomson (1933), further ed. 1951; H. Levy, E.A. Baggott (1934).Google Scholar
  5. 9.
    The former is reviewed by Peano in (1928h), and by G. Vivanti in U. Cassina (1929), 199–201; the latter by Tricomi in G. Cassinis (1928), 74–75: “Nel complesso mi sembra che questa del Cassinis sia un’opera veramente notevole, completamente nuova nella nostra letteratura scientifica” (As a whole, this book by Cassinis seems to be a really remarkable work, completely new in our scientific literature).Google Scholar
  6. 11.
    In the Preface of the book by Whittaker and Robinson (1944), vi, we read: “The material equipment essential for a student’s mathematical laboratory is very simple. Each student should have a copy of Barlow’s tables of squares, etc., a copy of Crelle’s Calculating Tables, and a seven-place table of logarithms. Further, it is necessary to provide a stock of computing paper […], and lastly a stock of computing forms for practical Fourier analysis. […] With this modest apparatus nearly all computations hereafter described may be performed, although time and labour may often be saved by the use of multiplying and adding machines when these are available.”Google Scholar
  7. 14.
    Hence, the following comment by S. Di Sieno (S. Di Sieno, A. Guerraggio, P. Nastasi (eds.) 1998), 5, must be corrected: “Altri sono i suoi [di Peano] interessi prevalenti, sin dall’inizio del secolo. Né la situazione viene modificata dalla redazione di due brevi Note [Peano 1918d and 1919b], che rimangono le uniche del periodo in questione ascrivibili in qualche modo alla disciplina [Analisi matematica]” (Ever since the beginning of the century Peano has other prevalent interests. The situation is not even changed by the writing of two short notes, which are the only ones in the considered period somehow pertaining to mathematical analysis). Indeed, U. Cassina (1932), 122–123, states more precisely: “In anno 1903 Peano initia studios philologico et opera interlinguistico, ad que illo post vol dedica, usque ad morte, parte extra grande de suo activitate […]. Tamen, hoc non porta (ut, in modo vero erroneo, aliquo crede) ad relinque studios mathematico, que, ad contrario, recipe novo impulso. In vero, circa anno 1913, illo initia et cultiva cum grande successu novo campo de investigationes mathematico: illo dedicato ad Calculo numerico, neglecto et considerato quasi cum dedignatione ab aliquo mathematico moderno (per quanto jam mathematicos illustre […] ne habe omisso de dedica scriptos ad quaestiones de calculo numerico)” (In 1903 Peano begins some philological studies and interlinguistic works, whom he devotes the greatest part of his activity till his death. Nevertheless, this fact does not imply, as some people think quite wrongly, that he leaves mathematical studies, which on the contrary receive a new impulse. As a matter of fact, nearly in 1913, he starts to investigate with great success a new mathematical field: the one that is devoted to Numerical Calculus, which is neglected and thought almost unworthy of consideration by some modern mathematicians (even though previously famous mathematicians did not omit writing papers on topics of numerical calculus).Google Scholar
  8. 18.
    E.T. Whittaker, G. Robinson (1944), 32–33.Google Scholar
  9. 21.
    H. Poincaré (1886), 295–344.Google Scholar
  10. 23.
    Tricomi, discussing in detail the theorem of existence and uniqueness for the solution of Cauchy’s problem see F. Tricomi (1948), 18–29, writes (note at p. 26): “Al metodo delle approssimazioni successive si associa generalmente il nome del grande matematico francese E. Picard (1856–1941) che ne ha fatto vedere tutta l’importanza. Tuttavia già qualche anno prima che dal Picard, esso era stato usato dal nostro Peano” (The method of successive approximations is generally associated with the name of the great French mathematician E. Picard (1856–1941) who showed the full importance of the method. Nevertheless a few years before Picard, it had been used by our Peano). A chronologically ordered list of references to the papers by Peano, Picard and Lindelöf, which are pertinent to the topic, is given by U. Cassina (1943), 190, note 354. Interesting considerations are also made by B. Levi in (1955, 14–18), or (1932), 253–262, and by T. Viola in (1985), 33–35. Note that Peano seems to prefer the term “method of successive integrations”, see (Peano 1897c), 12.Google Scholar
  11. 25.
    G. Peano, Opere scelte. Volume I: Analisi matematica-Calcolo numerico, Volume II: Logica matematica — Interlingua e algebra della grammatica, Volume III: Geometria e fondamenti — Meccanica razionale — Varie, a cura di U. Cassina, edito dall’Unione Matematica Italiana col contributo del Consiglio Nazionale delle Ricerche, Edizioni Cremonese, Roma, 1957–59.Google Scholar
  12. 27.
    E.J. Rémès (1939), 21–62; (1940a), 47–82; (1940b), 129–133.Google Scholar
  13. 28.
    A. Sard (1948), 333–345.Google Scholar
  14. 29.
    H. Brass, K.-J Föster (1998), 175–202. See also H. Brass, K. Petras (2003), 195–207, and references therein.Google Scholar
  15. 30.
    G.D. Birkhoff (1906), 107–136; J. Radon (1935), 389–396; R. von Mises (1936), 56–67.Google Scholar
  16. 31.
    E.J. Rémès (1940b), 130: “On obtient ainsi, comme cas particuliers, diverses représentations intégrales des termes complémentaires, qui ont été indiquées, comme conséquences de différentes considérations théoriques, par Mises [‘Über allgemeine Quadraturformeln’, 1936], Radon [‘Restausdrücke bei Interpolations-und Quadraturformeln’, 1935], et par d’autres auteurs (Peano 1913g, […], Kowalewski [G. Kowalewski, Interpolation und genäherte Quadratur, Leipzig, Teubner, 1932] […], Birkhoff [‘General mean value and remainder theorems’, 1906]).”Google Scholar
  17. 32.
    J. Radon (1935), 396: “Zusatz bei der Korrektur […]: Für den Fall Dn[f(x)] = f(n)(x) (polynomiale Annäherung) findet sich die Formel (3.2) bereits bei G. Peano [Peano 1913g].”Google Scholar
  18. 33.
    It is odd, for example, that in 1944 Whittaker and Robinson (1924) do not quote Peano’s theorem, in spite of their knowledge about the researches of Peano’s school. In fact, they cite by Peano the papers Applicazioni geometriche (1887b), and ‘Resto nelle formule di interpolazione’ (1918d), and by Cassina, ‘Formole sommatorie e di quadratura ad ordinate estreme’ (1939a), 225–274, and ‘Formole sommatorie e di quadratura con l’ordinata media’ (1939b), 300–325. W.E. Milne, in (1949a) and (1949b) gives no cross-reference to Peano.Google Scholar
  19. 34.
    U. Cassina (1943), 183–186, and also 168–177.Google Scholar
  20. 37.
    General hypotheses are considered by Sard in the quoted works. Here we follow the presentation by P.J. Davis, Interpolation and approximation, New York, Dover, 1975, and P.J. Davis, P. Rabinowitz, Methods of numerical integration, New York, Academic Press, 2nd ed., 1984. A more recent and interesting chapter on Peano’s theorem was also written by G.M. Phillips (G.M. Phillips 2003, in particular 147–162).Google Scholar
  21. 38.
    Rémès starts with the Taylor formula as well. A few possible inaccuracies in Peano’s proof are pointed out by Sard (‘Integral representations of remainders’, 1948, 339).Google Scholar
  22. 40.
    From a computational viewpoint a little more can be said; see, e. g., G. Allasia, M. Allasia (1976), 353–358; G. Allasia, P. Patrucco (1976), 263–274; G. Allasia, C. Giordano (1979), 1103–1110; G. Allasia, C. Giordano (1980), 257–269. See also P.J. Davis, P. Rabinowitz (1975).Google Scholar
  23. 41.
    See the proof in P.J. Davis, P. Rabinowitz (1975), 290–291.Google Scholar
  24. 42.
    See Peano (1913g). Peano was interested in this example because in 1887 he gave first the remainder of Simpson’s rule in the form (3.9) (see) Peano 1887b, 204). In fact, Peano (and Rémès too) also consider the functionals relating to divided differences and numerical differentiation.Google Scholar
  25. 43.
    Indeed, Peano observes (Peano 1914b, 7): “Non es necesse que formula de approximatione, de que nos determina residuo, contine in modo explicito uno integrale. Suffice que es lineare in functione f” (It is not necessary that the approximation formula, whose remainder has to be determined, contains explicitly an integral term. It is sufficient for the formula to be linear with respect to the function f). A precise characterization of the considered remainder functionals is given in Peano 1913g, 563.Google Scholar
  26. 44.
    U. Cassina (1943), 186.Google Scholar
  27. 46.
    P.J. Davis, Interpolation and approximation, 1975, 72–73; L. Schumaker, Spline functions: basic theory, Krieger Publ. Co., Malabar, Florida, 1993.Google Scholar
  28. 47.
    U. Cassina (1943), 184–185.Google Scholar
  29. 48.
    G. Kowalewski (1932), 21–24, or P.J. Davis (1975), 71–72.Google Scholar
  30. 49.
    Davis (1975), 72.Google Scholar
  31. 50.
    A. Sard (1948), 341–343.Google Scholar
  32. 51.
    The reference is to the following paper on the error terms for the Gauss—Lagrange formulas: P. Mansion (1886), 293–307. P. Mansion, editor of the journal Mathesis, had a close relationship with Peano. In particular, he noticed the interest of Peano’s remainder theorem and wrote a note on the topic: P. Mansion (1914), 169–174. About this point see G. Allasia (2005), 43–61. E. Picard, the editor of Hermite’s Collected Works, wrote (see Euvres de Charles Hermite, É. Picard (ed.), vol. 1–4, Paris, Gauthier-Villars, 1905–1917, in particular, vol. 4, Avertissement): “J’ai encore le devoir de rappeler l’aide que m’a apportée l’esquisse biographique et bibliographique écrite quelques semaines après la mort d’Hermite par M. Mansion, professeur à l’Université de Gand. […] Puisse mon souvenir atteindre le vénéré doyen de la science mathématique en Belgique dans la ville où il est retenu depuis près de trois ans.”Google Scholar
  33. 52.
    Peano (1913g), 562: “Alcune formule di quadratura hanno il resto espresso mediante un’integrale definito. Tale è la formula di Taylor. Anche la formula sommatoria di Eulero ha un resto calcolato da Jacobi sotto forma di integrale definito, e da cui si deducono le espressioni con valori medii. Caso particolare è la formula del trapezio […] Di altre formule di quadratura si conosce solo il resto espresso mediante il valore medio di una derivata. Tale è la formula detta di Simpson, e le formule di quadratura di Gauss, il cui resto fu calcolato dal prof. Mansion nell’anno 1887. Di tutte le altre formule di quadratura, non si conosce alcuna espressione del resto. Il resto in ogni formula di quadratura si può sempre ridurre ad integrale.”Google Scholar
  34. 53.
    Peano (1887b), 202–205.Google Scholar
  35. 54.
    Peano (1893h), vol. I, 238.Google Scholar
  36. 55.
    Peano (1887b), 219.Google Scholar
  37. 56.
    See, e.g., E.T. Whittaker, G. Robinson, The calculus of observations, 1944, 160, and M. Abramowitz, I.A. Stegun (1964), 916.Google Scholar
  38. 57.
    U. Cassina (1943), 174.Google Scholar
  39. 58.
    Peano (1887b), 209, note.Google Scholar
  40. 59.
    J. Boussinesq, Cours d’analyse infinitésimale, vol. 2, Paris, Gauthier-Villars, 1890, 70–73.MATHGoogle Scholar
  41. 60.
    This is asserted in E.T. Whittaker, G. Robinson, The calculus of observations, 1944, 156, and in U. Cassina (1943), 169, but see also A.A. Markoff (1896), 59.Google Scholar
  42. 61.
    C. de la Vallée Poussin, Cours d’analyse infinitésimale, vol. 1, 7e éd., Louvain, 1930, 332; U. Cassina (1943), 169.Google Scholar
  43. 62.
    H. Laurent (1885–1888), vol. 3, 498–499.Google Scholar

Copyright information

© Springer-Verlag Italia 2011

Authors and Affiliations

  • Giampietro Allasia
    • 1
  1. 1.Department of Mathematics “G. Peano”University of TorinoItaly

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