Comparison between Borel-Padé summation and factorial series, as time integration methods

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April 2016, 9(2): 393-408. doi: 10.3934/dcdss.2016003

Comparison between Borel-Padé summation and factorial series, as time integration methods

Ahmad Deeb ^1, , A. Hamdouni ^1, and Dina Razafindralandy ^1,

1.	Laboratoire des Sciences de l'Ingénieur pour l'Environnement - UMR 7356, Université de La Rochelle, 17042 La Rochelle Cedex 1, France, France, France

Received April 2015 Revised November 2015 Published March 2016

Abstract
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We compare the performance of two algorithms of computing the Borel sum of a time power series. The first one uses Padé approximants in Borel space, followed by a Laplace transform. The second is based on factorial series. These algorithms are incorporated in a numerical scheme for time integration of differential equations.

Keywords: Padé approximants, Divergent series, numerical time integrator., factorial series, Borel summation.

Mathematics Subject Classification: Primary: 40G10; Secondary: 37M0.

Citation: Ahmad Deeb, A. Hamdouni, Dina Razafindralandy. Comparison between Borel-Padé summation and factorial series, as time integration methods. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 393-408. doi: 10.3934/dcdss.2016003

References:

[1]	V. Adukov and O. Ibryaeva, A new algorithm for computing padé approximants,, \arXiv{1112.5694}., ().
[2]	G. Baker, J. Gammel and J. Wills, An investigation of the applicability of the Padé approximant method, Journal of Mathematical Analysis and Applications, 2 (1961), 405-418. doi: 10.1016/0022-247X(61)90019-1.
[3]	B. Beckermann and A. Ana Matos, Algebraic properties of robust Padé approximants, Journal of Approximation Theory, 190 (2015), 91-115, arXiv:1310.2438. doi: 10.1016/j.jat.2014.05.018.
[4]	J. Boyd, Exponentially convergent Fourier-Chebshev quadrature schemes on bounded and infinite intervals, Journal of Scientific Computing, 2 (1987), 99-109. doi: 10.1007/BF01061480.
[5]	C. Brezinski, Rationnal approximation to formal power serie, Journal of Approximation Theory, 25 (1979), 295-317. doi: 10.1016/0021-9045(79)90019-4.
[6]	C. Brezinski and J. Van Iseghem, Padé approximations, in Handbook of Numerical Analysis (eds. P. G. Ciarlet and J. L. Lions), Elsevier, 3 (1994), 47-222, doi: 10.1016/S1570-8659(05)80016-X.
[7]	A. Bultheel, Recursive algorithms for nonnormal Pade tables, SIAM Journal on Applied Mathematics, 39 (1980), 106-118. doi: 10.1137/0139009.
[8]	O. Costin, G. Luo and S. Tanveer, Divergent expansion, Borel summability and three-dimensional Navier-Stokes equation, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 366 (2008), 2775-2788. doi: 10.1098/rsta.2008.0052.
[9]	P. J. Davis and P. Rabinowitz, Ignoring the singularity in approximate integration, Journal of the Society for Industrial and Applied Mathematics: Series B, Numerical Analysis, 2 (1965), 367-383. doi: 10.1137/0702029.
[10]	A. Deeb, A. Hamdouni, E. Liberge and D. Razafindralandy, Borel-Laplace summation method used as time integration scheme, ESAIM: Procedings and Surveys, 45 (2014), 318-327. doi: 10.1051/proc/201445033.
[11]	E. Delabaere and J.-M. Rasoamanana, Sommation effective d'une somme de Borel par séries de factorielles, Annales de l'institut Fourier, 57 (2007), 421-456. doi: 10.5802/aif.2263.
[12]	W. Gautschi, Gauss-type quadrature rules for rational functions, Numerical Integration IV, the series ISNM International Series of Numerical Mathematics, 112 (1993), 111-130, arXiv:math/9307223. doi: 10.1007/978-3-0348-6338-4_9.
[13]	W. Gautschi, The use of rational functions in numerical quadrature, Journal of Computational and Applied Mathematics, 133 (2001), 111-126. doi: 10.1016/S0377-0427(00)00637-3.
[14]	W. Gautschi, Quadrature formulae on half-infinite intervals, BIT Numerical Mathematics, 31 (1991), 438-446. doi: 10.1007/BF01933261.
[15]	J. Gilewicz, Approximants de Padé, vol. 667 of Lecture Notes in Mathematics, Springer-Verlag, 1978.
[16]	J. Gilewicz and Y. Kryakin, Froissart doublets in Padé approximation in the case of polynomial noise, Journal of Computational and Applied Mathematics, 153 (2003), 235-242. doi: 10.1016/S0377-0427(02)00674-X.
[17]	J. Gilewicz and M. Pindor, Padé approximants and noise: A case of geometric series, Journal of Computational and Applied Mathematics, 87 (1997), 199-214. doi: 10.1016/S0377-0427(97)00185-4.
[18]	P. Gonnet, S. Güttel and L. Trefethen, Robust Padé approximation via SVD, SIAM Review, 55 (2013), 101-117. doi: 10.1137/110853236.
[19]	N. Hall, Interview of sir michael berry by nina hall: Caustics, catastrophes and quantum chaos,, Nexus News, (): 4.
[20]	M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems, and An Introduction to Chaos, Elsevier, 2013. doi: 10.1016/B978-0-12-382010-5.00001-4.
[21]	H. Kleinert and V. Schulte-Frohlinde, Critical Properties of $\Phi^4$-Theories, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812799944.
[22]	V. Kowalenko, The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation, Bentham, 2009. doi: 10.2174/97816080501091090101.
[23]	R. Kumar and M. K. Jain, Quadrature formulas for semi-infinite integrals, Mathematics of Computation, 28 (1974), 499-503. doi: 10.1090/S0025-5718-1974-0343549-5.
[24]	D. Lubinsky, Reflections on the Baker-Gammel-Wills (Padé), in Analytic Number Theory, Approximation Theory, and Special Functions (eds. G. V. Milovanović and M. T. Rassias), Springer New York, 2014, 561-571.
[25]	D. S. Lubinsky and P. Rabinowitz, Rates of convergence of Gaussian quadrature for singular integrands, Mathematics of Computation, 43 (1984), 219-242. doi: 10.1090/S0025-5718-1984-0744932-2.
[26]	D. Lutz, M. Miyake and R. Schäfke, On the Borel summability of divergent solutions of the heat equation, Nagoya Mathematical Journal, 154 (1999), 1-29.
[27]	G. Lysik, Borel summable solutions of the Burgers equation, Annales Polonici Mathematici, 95 (2009), 187-197. doi: 10.4064/ap95-2-9.
[28]	G. Lysik and S. Michalik, Formal solutions of semilinear heat equations, Journal of Mathematical Analysis and Applications, 341 (2008), 372-385. doi: 10.1016/j.jmaa.2007.10.005.
[29]	W. Mascarenhas, Robust Padé approximants can diverge,, , ().
[30]	N. Nielsen, Recherches sur les séries de factorielles, Annales Scientifiques de l'E.N.S. 3è série, 19 (1902), 409-453.
[31]	N. Nielsen, Les séries de factorielles et les opérations fondamentales, Mathematische Annalen, 59 (1904), 355-376. doi: 10.1007/BF01445147.
[32]	N. Nielsen, Sur les séries de factorielles et la fonction gamma (extrait d'une lettre adressée à M.-N. de Sonin à Saint-Pétersbourg), Annales Scientifiques de l'E.N.S. 3è série, 23 (1906), 145-168.
[33]	N. Nörlund, Vorlesungen Über Differenzenrechnung, Srpinger Verlag, 1924.
[34]	N. Nörlund, Leçons Sur Les Séries D'interpolation, Gauthier-Villard et Cie, 1926.
[35]	S. Pincherle, Sulle serie di fattoriali. nota II, Atti della Reale Accademia dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali. Series 5, 11 (1902), 417-426.
[36]	J.-P. Ramis, Séries divergentes et théories asymptotiques, in Journées X-UPS 1991, 1991, 7-67.
[37]	J.-P. Ramis, Les développements asymptotiques après poincaré: Continuité et... divergences,, Gazettes des Mathématiciens., ().
[38]	D. Razafindralandy and A. Hamdouni, Time integration algorithm based on divergent series resummation, for ordinary and partial differential equations, Journal of Computational Physics, 236 (2013), 56-73. doi: 10.1016/j.jcp.2012.10.022.
[39]	H. Stahl, Conjectures around the Baker-Gammel-Wills conjecture, Constructive Approximation, 13 (1997), 287-292, doi: 10.1007/s003659900044.
[40]	H. Stahl, Spurious poles in Padé approximation, Journal of Computational and Applied Mathematics, 99 (1998), 511-527. doi: 10.1016/S0377-0427(98)00180-0.
[41]	J. Thomann, Resommation des séries formelles. Solutions d'équations différentielles linéaires ordinaires du second ordre dans le champ complexe au voisinage de singularités irrégulières, Numerische Mathematik, 58 (1990), 503-535. doi: 10.1007/BF01385638.
[42]	J. Thomann, Procédés formels et numériques de sommation de séries solutions d'équations différentielles, in Journées X-UPS 1991, Séries divergentes et procédés de resommation (ed. C. de mathématiques), 1991, 101-114.
[43]	J. Thomann, Formal and Numerical Summation of Formal Power Series Solutions of ODE's, Technical report, CIRM Luminy, 2000.
[44]	F. Thomlinson, Generalized factorial series,, Transactions of the American Mathematical Society, 31 ().
[45]	M. Thomson, The Calculus Of Finite Differences, Macmillan and Company, 1933.
[46]	J. van Deun, A. Bultheel and P. González Vera, On computing rational Gauss-Chebyshev quadrature formulas, Mathematics of Computation, 75 (2006), 307-326. doi: 10.1090/S0025-5718-05-01774-6.
[47]	G. N. Watson, The transformation of an asymptotic series into a convergent series of inverse factorials, Rendiconti del Circolo Matematico di Palermo, 34 (1912), 41-88.
[48]	E. Weniger, Summation of divergent power series by means of factorial series, Applied Numerical Mathematics, 60 (2010), 1429-1441. doi: 10.1016/j.apnum.2010.04.003.

show all references

References:

[1]	V. Adukov and O. Ibryaeva, A new algorithm for computing padé approximants,, \arXiv{1112.5694}., ().
[2]	G. Baker, J. Gammel and J. Wills, An investigation of the applicability of the Padé approximant method, Journal of Mathematical Analysis and Applications, 2 (1961), 405-418. doi: 10.1016/0022-247X(61)90019-1.
[3]	B. Beckermann and A. Ana Matos, Algebraic properties of robust Padé approximants, Journal of Approximation Theory, 190 (2015), 91-115, arXiv:1310.2438. doi: 10.1016/j.jat.2014.05.018.
[4]	J. Boyd, Exponentially convergent Fourier-Chebshev quadrature schemes on bounded and infinite intervals, Journal of Scientific Computing, 2 (1987), 99-109. doi: 10.1007/BF01061480.
[5]	C. Brezinski, Rationnal approximation to formal power serie, Journal of Approximation Theory, 25 (1979), 295-317. doi: 10.1016/0021-9045(79)90019-4.
[6]	C. Brezinski and J. Van Iseghem, Padé approximations, in Handbook of Numerical Analysis (eds. P. G. Ciarlet and J. L. Lions), Elsevier, 3 (1994), 47-222, doi: 10.1016/S1570-8659(05)80016-X.
[7]	A. Bultheel, Recursive algorithms for nonnormal Pade tables, SIAM Journal on Applied Mathematics, 39 (1980), 106-118. doi: 10.1137/0139009.
[8]	O. Costin, G. Luo and S. Tanveer, Divergent expansion, Borel summability and three-dimensional Navier-Stokes equation, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 366 (2008), 2775-2788. doi: 10.1098/rsta.2008.0052.
[9]	P. J. Davis and P. Rabinowitz, Ignoring the singularity in approximate integration, Journal of the Society for Industrial and Applied Mathematics: Series B, Numerical Analysis, 2 (1965), 367-383. doi: 10.1137/0702029.
[10]	A. Deeb, A. Hamdouni, E. Liberge and D. Razafindralandy, Borel-Laplace summation method used as time integration scheme, ESAIM: Procedings and Surveys, 45 (2014), 318-327. doi: 10.1051/proc/201445033.
[11]	E. Delabaere and J.-M. Rasoamanana, Sommation effective d'une somme de Borel par séries de factorielles, Annales de l'institut Fourier, 57 (2007), 421-456. doi: 10.5802/aif.2263.
[12]	W. Gautschi, Gauss-type quadrature rules for rational functions, Numerical Integration IV, the series ISNM International Series of Numerical Mathematics, 112 (1993), 111-130, arXiv:math/9307223. doi: 10.1007/978-3-0348-6338-4_9.
[13]	W. Gautschi, The use of rational functions in numerical quadrature, Journal of Computational and Applied Mathematics, 133 (2001), 111-126. doi: 10.1016/S0377-0427(00)00637-3.
[14]	W. Gautschi, Quadrature formulae on half-infinite intervals, BIT Numerical Mathematics, 31 (1991), 438-446. doi: 10.1007/BF01933261.
[15]	J. Gilewicz, Approximants de Padé, vol. 667 of Lecture Notes in Mathematics, Springer-Verlag, 1978.
[16]	J. Gilewicz and Y. Kryakin, Froissart doublets in Padé approximation in the case of polynomial noise, Journal of Computational and Applied Mathematics, 153 (2003), 235-242. doi: 10.1016/S0377-0427(02)00674-X.
[17]	J. Gilewicz and M. Pindor, Padé approximants and noise: A case of geometric series, Journal of Computational and Applied Mathematics, 87 (1997), 199-214. doi: 10.1016/S0377-0427(97)00185-4.
[18]	P. Gonnet, S. Güttel and L. Trefethen, Robust Padé approximation via SVD, SIAM Review, 55 (2013), 101-117. doi: 10.1137/110853236.
[19]	N. Hall, Interview of sir michael berry by nina hall: Caustics, catastrophes and quantum chaos,, Nexus News, (): 4.
[20]	M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems, and An Introduction to Chaos, Elsevier, 2013. doi: 10.1016/B978-0-12-382010-5.00001-4.
[21]	H. Kleinert and V. Schulte-Frohlinde, Critical Properties of $\Phi^4$-Theories, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812799944.
[22]	V. Kowalenko, The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation, Bentham, 2009. doi: 10.2174/97816080501091090101.
[23]	R. Kumar and M. K. Jain, Quadrature formulas for semi-infinite integrals, Mathematics of Computation, 28 (1974), 499-503. doi: 10.1090/S0025-5718-1974-0343549-5.
[24]	D. Lubinsky, Reflections on the Baker-Gammel-Wills (Padé), in Analytic Number Theory, Approximation Theory, and Special Functions (eds. G. V. Milovanović and M. T. Rassias), Springer New York, 2014, 561-571.
[25]	D. S. Lubinsky and P. Rabinowitz, Rates of convergence of Gaussian quadrature for singular integrands, Mathematics of Computation, 43 (1984), 219-242. doi: 10.1090/S0025-5718-1984-0744932-2.
[26]	D. Lutz, M. Miyake and R. Schäfke, On the Borel summability of divergent solutions of the heat equation, Nagoya Mathematical Journal, 154 (1999), 1-29.
[27]	G. Lysik, Borel summable solutions of the Burgers equation, Annales Polonici Mathematici, 95 (2009), 187-197. doi: 10.4064/ap95-2-9.
[28]	G. Lysik and S. Michalik, Formal solutions of semilinear heat equations, Journal of Mathematical Analysis and Applications, 341 (2008), 372-385. doi: 10.1016/j.jmaa.2007.10.005.
[29]	W. Mascarenhas, Robust Padé approximants can diverge,, , ().
[30]	N. Nielsen, Recherches sur les séries de factorielles, Annales Scientifiques de l'E.N.S. 3è série, 19 (1902), 409-453.
[31]	N. Nielsen, Les séries de factorielles et les opérations fondamentales, Mathematische Annalen, 59 (1904), 355-376. doi: 10.1007/BF01445147.
[32]	N. Nielsen, Sur les séries de factorielles et la fonction gamma (extrait d'une lettre adressée à M.-N. de Sonin à Saint-Pétersbourg), Annales Scientifiques de l'E.N.S. 3è série, 23 (1906), 145-168.
[33]	N. Nörlund, Vorlesungen Über Differenzenrechnung, Srpinger Verlag, 1924.
[34]	N. Nörlund, Leçons Sur Les Séries D'interpolation, Gauthier-Villard et Cie, 1926.
[35]	S. Pincherle, Sulle serie di fattoriali. nota II, Atti della Reale Accademia dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali. Series 5, 11 (1902), 417-426.
[36]	J.-P. Ramis, Séries divergentes et théories asymptotiques, in Journées X-UPS 1991, 1991, 7-67.
[37]	J.-P. Ramis, Les développements asymptotiques après poincaré: Continuité et... divergences,, Gazettes des Mathématiciens., ().
[38]	D. Razafindralandy and A. Hamdouni, Time integration algorithm based on divergent series resummation, for ordinary and partial differential equations, Journal of Computational Physics, 236 (2013), 56-73. doi: 10.1016/j.jcp.2012.10.022.
[39]	H. Stahl, Conjectures around the Baker-Gammel-Wills conjecture, Constructive Approximation, 13 (1997), 287-292, doi: 10.1007/s003659900044.
[40]	H. Stahl, Spurious poles in Padé approximation, Journal of Computational and Applied Mathematics, 99 (1998), 511-527. doi: 10.1016/S0377-0427(98)00180-0.
[41]	J. Thomann, Resommation des séries formelles. Solutions d'équations différentielles linéaires ordinaires du second ordre dans le champ complexe au voisinage de singularités irrégulières, Numerische Mathematik, 58 (1990), 503-535. doi: 10.1007/BF01385638.
[42]	J. Thomann, Procédés formels et numériques de sommation de séries solutions d'équations différentielles, in Journées X-UPS 1991, Séries divergentes et procédés de resommation (ed. C. de mathématiques), 1991, 101-114.
[43]	J. Thomann, Formal and Numerical Summation of Formal Power Series Solutions of ODE's, Technical report, CIRM Luminy, 2000.
[44]	F. Thomlinson, Generalized factorial series,, Transactions of the American Mathematical Society, 31 ().
[45]	M. Thomson, The Calculus Of Finite Differences, Macmillan and Company, 1933.
[46]	J. van Deun, A. Bultheel and P. González Vera, On computing rational Gauss-Chebyshev quadrature formulas, Mathematics of Computation, 75 (2006), 307-326. doi: 10.1090/S0025-5718-05-01774-6.
[47]	G. N. Watson, The transformation of an asymptotic series into a convergent series of inverse factorials, Rendiconti del Circolo Matematico di Palermo, 34 (1912), 41-88.
[48]	E. Weniger, Summation of divergent power series by means of factorial series, Applied Numerical Mathematics, 60 (2010), 1429-1441. doi: 10.1016/j.apnum.2010.04.003.

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