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

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

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

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.
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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