Reduction to Canonical Forms and the Stokes Phenomenon in the Theory of Linear Difference Equations

Immink, G.K.

doi:10.1137/0522014

Cited by 5 publications

(2 citation statements)

References 7 publications

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“…Remark 3.3. Using the Birkhoff-Trjitzinsky method (see [2], and [15] for complete proofs; see also [24] and [20]) one can obtain that the asymptotics of g n = h n is 0.41349667…”

Section: The Three Variable Casementioning

confidence: 99%

The autoregressive filter problem for multivariable degree one symmetric polynomials

Geronimo¹,

Woerdeman²,

Wong³

2021

Preprint

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The multivariable autoregressive filter problem asks for a polynomial p(z) = p(z 1 , . . . , z d ) without roots in the closed d-disk based on prescribed Fourier coefficients of its spectral density function 1/|p(z)| 2 . The conditions derived in this paper for the construction of a degree one symmetric polynomial reveal a major divide between the case of at most two variables vs. the the case of three or more variables. The latter involves multivariable elliptic functions, while the former (due to [J. S. Geronimo and H. J. Woerdeman, Ann. of Math. (2), 160(3): 2004]) only involve polynomials. The three variable case is treated with more detail, and entails hypergeometric functions. Along the way, we identify a seemingly new relation between

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“…Remark 3.3. Using the Birkhoff-Trjitzinsky method (see [2], and [15] for complete proofs; see also [24] and [20]) one can obtain that the asymptotics of g n = h n is 0.41349667…”

Section: The Three Variable Casementioning

confidence: 99%

The autoregressive filter problem for multivariable degree one symmetric polynomials

Geronimo¹,

Woerdeman²,

Wong³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…We note that many expositions of the Birkhoff-Trjitzinsky theorem warn about possible major gaps in its original proof. However, the consensus among specialists now appears to be that these issues have been resolved in modern proofs [27,62]. Besides, under mild additional assumptions on the Chebyshev recurrence, all the information needed in our analysis is already provided by the more elementary Perron-Kreuser theorem 3 (cf.…”

Section: Elements Of Birkhoff-trjitzinsky Theorymentioning

confidence: 99%

Rigorous uniform approximation of D-finite functions using Chebyshev expansions

Benoit¹,

Joldeş²,

Mezzarobba³

2016

Math. Comp.

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Abstract. A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view.It is well-known that the order-truncation of the Chebyshev expansion of a function over a given interval is a near-best uniform polynomial approximation of the function on that interval. In the case of solutions of linear differential equations with polynomial coefficients, the coefficients of the expansions obey linear recurrence relations with polynomial coefficients. Unfortunately, these recurrences do not lend themselves to a direct recursive computation of the coefficients, owing among other things to a lack of initial conditions.We show how they can nevertheless be used, as part of a validated process, to compute good uniform approximations of D-finite functions together with rigorous error bounds, and we study the complexity of the resulting algorithms. Our approach is based on a new view of a classical numerical method going back to Clenshaw, combined with a functional enclosure method.

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Computing error bounds for asymptotic expansions of regular P-recursive sequences

Dong¹,

Melczer²,

Mezzarobba

2023

Math. Comp.

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Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely a matter of routine, under assumptions that hold often in practice. The algorithms involved typically take a sequence, encoded by a recurrence relation and initial terms, and return the leading terms in an asymptotic expansion up to a big-O error term. Less studied, however, are effective techniques giving an explicit bound on asymptotic error terms. Among other things, such explicit bounds typically allow the user to automatically prove sequence positivity (an active area of enumerative and algebraic combinatorics) by exhibiting an index when positive leading asymptotic behaviour dominates any error terms. In this article, we present a practical algorithm for computing such asymptotic approximations with rigorous error bounds, under the assumption that the generating series of the sequence is a solution of a differential equation with regular (Fuchsian) dominant singularities. Our algorithm approximately follows the singularity analysis method of Flajolet and Odlyzko [SIAM J. Discrete Math. 3 (1990), pp. 216–240], except that all big-O terms involved in the derivation of the asymptotic expansion are replaced by explicit error terms. The computation of the error terms combines analytic bounds from the literature with effective techniques from rigorous numerics and computer algebra. We implement our algorithm in the SageMath computer algebra system and exhibit its use on a variety of applications (including our original motivating example, solution uniqueness in the Canham model for the shape of genus one biomembranes).

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Reduction to Canonical Forms and the Stokes Phenomenon in the Theory of Linear Difference Equations

Abstract: Previous results concerning the existence of right inverses of linear difference operators on Banach Spaces of holomorphic functions are extended. The Stokes phenomenon is analyzed for a class of very singular linear difference equations.

Cited by 5 publications

References 7 publications

The autoregressive filter problem for multivariable degree one symmetric polynomials

The autoregressive filter problem for multivariable degree one symmetric polynomials

Rigorous uniform approximation of D-finite functions using Chebyshev expansions

Computing error bounds for asymptotic expansions of regular P-recursive sequences

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