Variations on computing reciprocals of power series

Faà di Bruno's formula and inversion of power series

An explicit numerical-analytical method is demonstrated for accurate solving the Schrödinger equation in those cases when this equation reducible to a system of n coupled ordinary differential equations with singular points. Fundamental system of solutions is constructed as algebraic combinations of power series, power functions and logarithmic function in the neighbourhood of the regular singular point and as asymptotic expansions of solutions in the neighbourhood of the irregular singular point. The method is based on the calculation of recurrent sequences of constant matrices of coefficients in power series and in inverse power series in asymptotic expansions using derived recurrent relations, that makes possible to calculate solutions at any given point using only algebraic computations and elementary functions. In turn it makes possible to solve accurately the eigenvalue problem and scattering problem and to derive analytical expressions for the wavefunctions. The method is applied to calculations of energies and wavefunctions of the discrete spectrum and wave functions of the continuous spectrum of the hydrogen-like atoms and of acceptors in semiconductors.

Next article Factorization of Lagrange’s Expansion by Means of Exponential Generating FunctionsR. A. SackR. A. Sackhttps://doi.org/10.1137/0114001PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] P. Appell and , J. Kampé De Fériet, Fonctions hypergéometriques et hypersphériquesPolynomes d' Hermite, Gauthier-Villars, Paris, 1926, Appendix 4 Google Scholar[2] H. Bateman, An extension of Lagrange's expansion, Trans. Amer. Math. Soc., 28 (1926), 346–356 MR1501349 Google Scholar[3] Richard Birkeland, Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen, Math. Z., 26 (1927), 566–578 10.1007/BF01475474 MR1544875 CrossrefGoogle Scholar[4] M. M. Carter and , R. A. Sack, Inverse perturbation expansions, to be published Google Scholar[5] A. Cayley, Note sur une formule pour la réversion des séries, J. Reine Angew. Math., 52 (1856), 276–284, Collected Papers, vol. 4, Cambridge, 1891, pp. 30–37 CrossrefGoogle Scholar[6] A. Cayley, Deuxième note sur une formule pour la réversion des séries, J. Reine Angew. Math., 54 (1857), 156–161, Collected Papers, vol. 4, Cambridge, 1891, pp. 54–59 CrossrefGoogle Scholar[7] A. Cayley, On Wronski's theorem, Quart. J. Pure Appl. Math., 12 (1873), 221–228, Collected Papers, vol. 9, Cambridge, 1896, pp. 96–102 Google Scholar[8] J. S. Frame, Power series expansions for inverse functions, Amer. Math. Monthly, 64 (1957), 236–240 MR0085330 0077.28402 CrossrefGoogle Scholar[9] I. J. Good, Generalizations to several variables of Lagrange's expansion, with applications to stochastic processes, Proc. Cambridge Philos. Soc., 56 (1960), 367–380 MR0123021 0135.18802 CrossrefGoogle Scholar[10] E. Goursat, Cours d' analyse mathématique, Vol. 1, Gauthier-Villars, Paris, 1917, Chap. 9, vol. 2, Chaps. 14, 17; English transl. A Course in Mathe-matical Analysis, E. R. Hedrick and O. Dunkel, translators, New York, 1959 Google Scholar[11] C. G. J. Jacobi, De resolutione aequationum per series infinitas, J. Reine Angew. Math., 6 (1830), 257–286, Ges. Werke, vol. 6, Berlin, 1891, pp. 26–61 CrossrefGoogle Scholar[12] Franz Kamber, Formules exprimant les valeurs des coefficients des séries de puissances inverses, Acta Math., 78 (1946), 193–204 MR0014493 0060.15402 CrossrefISIGoogle Scholar[13] Hr. Mellin, Ein allgemeiner Satz über algebraische Gleichungen, Ann. Acad. Sci. Fenn. Ser. A. I, 7 (1915), Google Scholar[14] R. A. Sack, Interpretation of Lagrange's expansion and its generalization to several variables as integration formulas, J. Soc. Indust. Appl. Math., 13 (1965), 47–59 10.1137/0113004 MR0181723 0143.27402 LinkISIGoogle Scholar[15] R. A. Sack, Generalization of Lagrange's expansion for functions of several implicitly defined variables, J. Soc. Indust. Appl. Math., 13 (1965), 913–926 10.1137/0113059 MR0196020 0143.27403 LinkISIGoogle Scholar[16] R. A. Sack, Rayleigh-Schrödinger perturbation expansions to arbitrary order, to be published Google Scholar[17] P. A. Sturrock, Generalization of the Lagrange expansion with application to physical problems, J. Mathematical Phys., 1 (1960), 405–408 10.1063/1.1703671 MR0116836 0091.43503 CrossrefISIGoogle Scholar[18] J. J. Sylvester, Note on Burman's law for the inversion of the independent variable, Philos. Mag., 8 (1854), 535–540, Collected Papers, vol. 2, Cambridge, 1908, pp. 44–49 CrossrefGoogle Scholar[19] J. J. Sylvester, On the change of systems of independent variables, Quart. J. Math., 1 (1857), 42–56, 126–134, Collected Papers, vol. 2, Cambridge, 1908, pp. 65–85 Google Scholar[20] E. T. Whittaker and , G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1927, Chaps. 7, 16 Google Scholar Next article FiguresRelatedReferencesCited ByDetails Degenerate RS perturbation theoryThe Journal of Chemical Physics, Vol. 60, No. 3 | 1 Feb 1974 Cross Ref Perturbation Methods and Lagrange’s ExpansionCooperative Phenomena | 1 Jan 1973 Cross Ref Generalization of Lagrange’s Expansion for Functions of Several Implicitly Defined VariablesR. A. SackJournal of the Society for Industrial and Applied Mathematics, Vol. 13, No. 4 | 13 July 2006AbstractPDF (1277 KB)Chapter 3: Approximate MethodsNonlinear Partial Differential Equations in Engineering | 1 Jan 1965 Cross Ref Volume 14, Issue 1| 1966SIAM Journal on Applied Mathematics History Submitted:10 September 1964Published online:13 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0114001Article page range:pp. 1-15ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

It is shown here the compositional inversion of formal power series has a natural formulation in terms of umbral operators. This fact is used to derive a new proof of the Lagrange inversion theorem by operator theoretic methods.

Inversion of Power Series

This note shows, using elementary properties of ribbon trees, that the universal formula for the inversion of power series can be obtained by counting strata in the compactified moduli space \overline{\mathcal M}_{0,n} .

Presents algorithms for planning feasible and minimum energy paths. The algorithms rely on series expansions to characterize planning problems for polynomial control systems. The resulting inversion problem is solved through an iterative contraction or a power series inversion. While the design methodology is general, our focus is on a class of polynomial control systems for which we provide explicit convergence guarantees. We demonstrate performance and numerical characteristics using one dimensional and six dimensional systems.

We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach builds upon a method proposed by Ferragut and Giacomini, whose main ingredients are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating this power series. We provide explicit bounds on the number of terms needed in the power series. This enables us to transform their method into a certified algorithm computing rational first integrals via systems of linear equations. We then significantly improve upon this first algorithm by building a probabilistic algorithm with arithmetic complexity O ~ ( N 2 ω ) \tilde {\mathcal {O}}(N^{2 \omega }) and a deterministic algorithm solving the problem in at most O ~ ( N 2 ω + 1 ) \tilde {\mathcal {O}}(N^{2 \omega +1}) arithmetic operations, where N N denotes the given bound for the degree of the rational first integral, and ω \omega the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in O ~ ( N ω + 2 ) \tilde {\mathcal {O}}(N^{\omega +2}) arithmetic operations. By comparison, the best previously known complexity was N 4 ω + 4 N^{4\omega +4} arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package RationalFirstIntegrals which is available to interested readers with examples showing its efficiency.

By extending the cyclic reduction technique to infinite block matrices we devise a new algorithm for computing the solution $G_0 $ of the matrix equation $G = \Sigma _{i = 0}^{ + \infty } G^i A_i $ arising in a wide class of queueing problems. Here $A_i ,i = 0,1, \cdots ,$ are $k \times k$ nonnegative matrices such that $\Sigma _{i = 0}^{ + \infty } A_i $ is column stochastic. Our algorithm, which under mild conditions generates a sequence of matrices converging quadratically to $G_0 $, can be fully described in terms of simple operations between matrix power series, i.e., power series in z having matrix coefficients. Such operations, like multiplication and reciprocation modulo $z^m $, can be quickly computed by means of FFT-based fast polynomial arithmetic; here m is the degree where the power series are numerically cut off in order to reduce them to polynomials. These facts lead to a dramatic reduction of the complexity of solving the given matrix equation; in fact, $O( k^3 m + k^2 m \log m )$ arithmetic operations are sufficient to carry out each iteration of the algorithm. Numerical experiments and comparisons performed with the customary techniques show the effectiveness of our algorithm. For a problem arising from the modelling of metropolitan networks, our algorithm was about 30 times faster than the algorithms customarily used in the applications. Cyclic reduction applied to quasi-birth-death (QBD) problems, i.e., problems where $A_i = O$ for $i > 2$, leads to an algorithm similar to the one of [Latouche and Ramaswami, J. Appi. Probab., 30 (1993), pp. 650–674], but which has a lower computational cost.

Abstract. Let G be a complex, affine algebraic group and a meromorphic connection on the trivial G-bundle over , with a pole of order 2 at zero and a pole of order 1 at infinity. We show that the map taking the residue of at zero to the corresponding Stokes factors is given by an explicit, universal Lie series whose coefficients are multilogarithms. Using a non-commutative analogue of the compositional inversion of formal power series, we show that the same holds for the inverse of , and that the corresponding Lie series coincides with the generating function for counting invariants in abelian categories constructed by D. Joyce.

The combinatorial power of the companion matrix

For a ring [Formula: see text] with a derivation [Formula: see text], we determine the Jacobson radical of the pseudo-differential operator ring [Formula: see text] for a large class of rings with a [Formula: see text]-condition. Various types of examples of these rings are provided.

This article draws attention to a comparative study of different members within the inverse Weibull Power Series (IWPS) to analyze the COVID-19 data from South Africa for the period from 27 March to 23 August 2020. A new sibling of the IWPS is introduced, namely the inverse Weibull negative binomial. An EM algorithm is developed for computing the maximum likelihood estimates of the model parameters. The IWPS growth curve model and its special cases are used for prediction of the COVID-19 spread in South Africa. It is found that the IWPS model fits the disease growth of the COVID-19 confirmed cases well with worthy long-term predictions. The IWPS growth curve modeling of South African predicts that the number of confirmed new cases will decrease at the end of November 2020.

A formula is read-once if each variable appears at most once in it. An arithmetic read-once formula is one in which the operators are addition, subtraction, multiplication, and division. We present polynomial time algorithms for exact learning of arithmetic read-once formulas over a field. We present a membership and equivalence query algorithm that identifies arithmetic read-once formulas over an arbitrary field. We present a randomized membership query algorithm (i.e., a randomized black box interpolation algorithm) that identifies such formulas over finite fields with at least $2n + 5$ elements (where n is the number of variables) and over infinite fields. We also show the existence of nonuniform deterministic membership query algorithms for arbitrary read-once formulas over fields of characteristic 0, and division-free read-once formulas over fields that have at least $2n^{3} + 1$ elements. For our algorithms, we assume we are able to perform efficiently arithmetic operations on field elements and compute square roots in the field. It is shown that the ability to compute square roots is necessary in the sense that the problem of computing $n - 1$ square roots in a field can be reduced to the problem of identifying an arithmetic formula over n variables in that field. Our equivalence queries are of a slightly nonstandard form, in which counterexamples are required not to be inputs on which the formula evaluates to $0/0$. This assumption is shown to be necessary for fields of size $o(n/ \log n)$ in the sense that we prove there exists no polynomial time identification algorithm that uses only membership and standard equivalence queries.

Variational calculation of dynamic polarizabilities