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Efficient accelero-summation of holonomic functions. (English) Zbl 1125.34072
Summary: Let \(L\in\mathbb K(z)[\partial]\) be a linear differential operator, where \(\mathbb K\) is the field of algebraic numbers. A holonomic function over \(\mathbb K\) is a solution \(f\) to the equation \(Lf=0\). We also assume that \(f\) admits initial conditions in \(\mathbb K\) at a non-singular point \(z\in\mathbb K\).
Given a broken-line path \(\gamma=z\rightsquigarrow z'\) between \(z\) and \(z'\), which avoids the singularities of \(L\) and with vertices in \(\mathbb K\), we have shown in a previous paper [J. van der Hoeven, Theor. Comput. Sci. 210, No. 1, 199–215 (1999; Zbl 0912.68081)] how to compute \(n\) digits of the analytic continuation of \(f\) along \(\gamma\) in time \(O(n\log^3n\log\log n)\). In a second paper [J. van der Hoeven, J. Symb. Comput. 31, No. 6, 717–743 (2001; Zbl 0982.65024)], this result was generalized to the case when \(z'\) is allowed to be a regular singularity, in which case we compute the limit of \(f\) when we approach the singularity along \(\gamma\).
In the present paper, we treat the remaining case when the end-point of \(\gamma\) is an irregular singularity. In fact, we solve the more general problem to compute “singular transition matrices” between non-standard points above a singularity and regular points in \(\mathbb K\) near the singularity. These non-standard points correspond to the choice of “non-singular directions” in Écalle’s accelero-summation process.
We show that the entries of the singular transition matrices may be approximated up to \(n\) decimal digits in time \(O(n\log^4n\log\log n)\). As a consequence, the entries of the Stokes matrices for \(L\) at each singularity may be approximated with the same time complexity.

MSC:
34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
34-04 Software, source code, etc. for problems pertaining to ordinary differential equations
Software:
Mmxlib; Mathemagix
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[1] Borel, E., 1928. Leçons sur les séries divergentes, 2nd edition. Gauthier-Villars. Reprinted by Jacques Gabay
[2] Borodin, A.; Moenck, R., Fast modular transforms, J. comput. system sci., 8, 366-386, (1974) · Zbl 0302.68064
[3] Braaksma, B., Multisummability and Stokes multipliers of linear meromorphic differential equations, J. differential equations, 92, 45-75, (1991) · Zbl 0729.34005
[4] Braaksma, B., Multisummability of formal power series solutions to nonlinear meromorphic differential equations, Ann. inst. Fourier (Grenoble), 42, 517-540, (1992) · Zbl 0759.34003
[5] Brent, R., 1976a. The complexity of multiprecision arithmetic. In: Complexity of Computational Problem Solving
[6] Brent, R., Fast multiple-precision evaluation of elementary functions, J. ACM, 23, 2, 242-251, (1976) · Zbl 0324.65018
[7] Candelberger, B., Nosmas, J., Pham, F., 1993. Approche de la résurgence. Actualités mathématiques. Hermann
[8] Chudnovsky, D.; Chudnovsky, G., Computer algebra in the service of mathematical physics and number theory, (), 109-232
[9] Écalle, J., 1985. Les fonctions résurgentes I-III. Publ. Math. d’Orsay 1981 and 1985
[10] Écalle, J., 1987. L’accélération des fonctions résurgentes (survol), unpublished manuscript
[11] Écalle, J., 1992. Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. Hermann, collection: Actualités mathématiques · Zbl 1241.34003
[12] Écalle, J., Six lectures on transseries, analysable functions and the constructive proof of dulac’s conjecture, (), 75-184 · Zbl 0814.32008
[13] Gosper, R., Schroeppel, R., February 1972. Artificial intelligence memo. Tech. Rep. 239, MIT AI Laboratory, item 178 on the evaluation of functions, See http://home.pipeline.com/ hbaker1/hakmem/hakmem.html
[14] Haible, B., Papanikolaou, T., 1997. Fast multiple-precision evaluation of elementary functions. Tech. Rep. TI-7/97, Universität Darmstadt · Zbl 1067.11517
[15] Hendriks, P.; Singer, M., Solving difference equations in finite terms, J. symbolic comput., 27, 3, 239-259, (1999) · Zbl 0930.39004
[16] Karatsuba, E., Fast evaluation of transcendental functions, Probl. inf. transm., 27, 339-360, (1991) · Zbl 0782.65017
[17] Karatsuba, E., Fast evaluation of Bessel functions, Integral transforms spec. funct., 1, 4, 269-276, (1993) · Zbl 0827.65022
[18] Karatsuba, E., Fast calculation of the Riemann zeta function \(\zeta(s)\) for integer values of the argument \(s\), Probl. inf. transm., 31, 353-362, (1995) · Zbl 0895.11032
[19] Karatsuba, E., On the computation of the Euler constant \(\gamma\), J. numer. algorithms, 24, 83-97, (2000) · Zbl 0958.33001
[20] Martinet, J.; Ramis, J.-P., Elementary acceleration and multisummability, Ann. inst. H. Poincaré, 54, 4, 331-401, (1991) · Zbl 0748.12005
[21] Mitschi, C., Differential Galois groups of confluent generalized hypergeometric equations: an approach using Stokes multipliers, Pacific J. math., 176, 2, 365-405, (1996) · Zbl 0883.12004
[22] Moenck, R., Borodin, A., 1972. Fast modular transforms via division. In: Thirteenth Annual IEEE Symposium on Switching and Automata Theory. Univ. Maryland, College Park, Md. pp. 90-96
[23] Poincaré, H., Sur LES intégrales irrégulières des équations linéaires, Acta math., 8, 295-344, (1886) · JFM 18.0273.02
[24] Ramis, J.-P., Dévissage Gevrey, Astérisque, 59-60, 173-204, (1978) · Zbl 0409.34018
[25] Schönhage, A.; Strassen, V., Schnelle multiplication grosser zahlen, Computing, 7, 281-292, (1971) · Zbl 0223.68007
[26] Stanley, R., Differentially finite power series, European J. combin., 1, 175-188, (1980), mR # 81m:05012 · Zbl 0445.05012
[27] Thomann, J., Procédés formels et numériques de sommation de séries d’équations différentielles, Expo. math., 13, 223-246, (1995) · Zbl 0831.34003
[28] van der Hoeven, J., 1997. Automatic asymptotics. Ph.D. Thesis, École polytechnique, France · Zbl 0874.65047
[29] van der Hoeven, J., Fast evaluation of holonomic functions, Theoret. comput. sci., 210, 199-215, (1999) · Zbl 0912.68081
[30] van der Hoeven, J., 2001a. Complex transseries solutions to algebraic differential equations. Tech. Rep. 2001-34, Univ. d’Orsay
[31] van der Hoeven, J., Fast evaluation of holonomic functions near and in singularities, J. symbolic comput., 31, 717-743, (2001) · Zbl 0982.65024
[32] van der Hoeven, J., Effective complex analysis, J. symbolic comput., 39, 433-449, (2005) · Zbl 1126.68105
[33] van der Hoeven, J., Around the numeric – symbolic computation of differential Galois groups, J. symbolic comput., 42, 236-264, (2007) · Zbl 1396.34054
[34] van der Hoeven, J., et al., 2002. Mmxlib: The standard library for Mathemagix. http://www.mathemagix.org/mml.html
[35] van der Put, M.; Singer, M., Galois theory of linear differential equations, Grundlehren der mathematischen wissenschaften, vol. 328, (2003), Springer · Zbl 1036.12008
[36] van Hoeij, M., Formal solutions and factorization of differential operators with power series coefficients, J. symbolic comput., 24, 1-30, (1997) · Zbl 0924.12005
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