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 & 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}., (). Google Scholar

[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. doi: 10.1016/0022-247X(61)90019-1. Google Scholar

[3]

B. Beckermann and A. Ana Matos, Algebraic properties of robust Padé approximants,, Journal of Approximation Theory, 190 (2015), 91. doi: 10.1016/j.jat.2014.05.018. Google Scholar

[4]

J. Boyd, Exponentially convergent Fourier-Chebshev quadrature schemes on bounded and infinite intervals,, Journal of Scientific Computing, 2 (1987), 99. doi: 10.1007/BF01061480. Google Scholar

[5]

C. Brezinski, Rationnal approximation to formal power serie,, Journal of Approximation Theory, 25 (1979), 295. doi: 10.1016/0021-9045(79)90019-4. Google Scholar

[6]

C. Brezinski and J. Van Iseghem, Padé approximations,, in Handbook of Numerical Analysis (eds. P. G. Ciarlet and J. L. Lions), 3 (1994), 47. doi: 10.1016/S1570-8659(05)80016-X. Google Scholar

[7]

A. Bultheel, Recursive algorithms for nonnormal Pade tables,, SIAM Journal on Applied Mathematics, 39 (1980), 106. doi: 10.1137/0139009. Google Scholar

[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, 366 (2008), 2775. doi: 10.1098/rsta.2008.0052. Google Scholar

[9]

P. J. Davis and P. Rabinowitz, Ignoring the singularity in approximate integration,, Journal of the Society for Industrial and Applied Mathematics: Series B, 2 (1965), 367. doi: 10.1137/0702029. Google Scholar

[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. doi: 10.1051/proc/201445033. Google Scholar

[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. doi: 10.5802/aif.2263. Google Scholar

[12]

W. Gautschi, Gauss-type quadrature rules for rational functions,, Numerical Integration IV, 112 (1993), 111. doi: 10.1007/978-3-0348-6338-4_9. Google Scholar

[13]

W. Gautschi, The use of rational functions in numerical quadrature,, Journal of Computational and Applied Mathematics, 133 (2001), 111. doi: 10.1016/S0377-0427(00)00637-3. Google Scholar

[14]

W. Gautschi, Quadrature formulae on half-infinite intervals,, BIT Numerical Mathematics, 31 (1991), 438. doi: 10.1007/BF01933261. Google Scholar

[15]

J. Gilewicz, Approximants de Padé, vol. 667 of Lecture Notes in Mathematics,, Springer-Verlag, (1978). Google Scholar

[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. doi: 10.1016/S0377-0427(02)00674-X. Google Scholar

[17]

J. Gilewicz and M. Pindor, Padé approximants and noise: A case of geometric series,, Journal of Computational and Applied Mathematics, 87 (1997), 199. doi: 10.1016/S0377-0427(97)00185-4. Google Scholar

[18]

P. Gonnet, S. Güttel and L. Trefethen, Robust Padé approximation via SVD,, SIAM Review, 55 (2013), 101. doi: 10.1137/110853236. Google Scholar

[19]

N. Hall, Interview of sir michael berry by nina hall: Caustics, catastrophes and quantum chaos,, Nexus News, (): 4. Google Scholar

[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. Google Scholar

[21]

H. Kleinert and V. Schulte-Frohlinde, Critical Properties of $\Phi^4$-Theories,, World Scientific Publishing Co., (2001). doi: 10.1142/9789812799944. Google Scholar

[22]

V. Kowalenko, The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation,, Bentham, (2009). doi: 10.2174/97816080501091090101. Google Scholar

[23]

R. Kumar and M. K. Jain, Quadrature formulas for semi-infinite integrals,, Mathematics of Computation, 28 (1974), 499. doi: 10.1090/S0025-5718-1974-0343549-5. Google Scholar

[24]

D. Lubinsky, Reflections on the Baker-Gammel-Wills (Padé),, in Analytic Number Theory, (2014), 561. Google Scholar

[25]

D. S. Lubinsky and P. Rabinowitz, Rates of convergence of Gaussian quadrature for singular integrands,, Mathematics of Computation, 43 (1984), 219. doi: 10.1090/S0025-5718-1984-0744932-2. Google Scholar

[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. Google Scholar

[27]

G. Lysik, Borel summable solutions of the Burgers equation,, Annales Polonici Mathematici, 95 (2009), 187. doi: 10.4064/ap95-2-9. Google Scholar

[28]

G. Lysik and S. Michalik, Formal solutions of semilinear heat equations,, Journal of Mathematical Analysis and Applications, 341 (2008), 372. doi: 10.1016/j.jmaa.2007.10.005. Google Scholar

[29]

W. Mascarenhas, Robust Padé approximants can diverge,, , (). Google Scholar

[30]

N. Nielsen, Recherches sur les séries de factorielles,, Annales Scientifiques de l'E.N.S. 3è série, 19 (1902), 409. Google Scholar

[31]

N. Nielsen, Les séries de factorielles et les opérations fondamentales,, Mathematische Annalen, 59 (1904), 355. doi: 10.1007/BF01445147. Google Scholar

[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. Google Scholar

[33]

N. Nörlund, Vorlesungen Über Differenzenrechnung,, Srpinger Verlag, (1924). Google Scholar

[34]

N. Nörlund, Leçons Sur Les Séries D'interpolation,, Gauthier-Villard et Cie, (1926). Google Scholar

[35]

S. Pincherle, Sulle serie di fattoriali. nota II,, Atti della Reale Accademia dei Lincei, 11 (1902), 417. Google Scholar

[36]

J.-P. Ramis, Séries divergentes et théories asymptotiques,, in Journées X-UPS 1991, (1991), 7. Google Scholar

[37]

J.-P. Ramis, Les développements asymptotiques après poincaré: Continuité et... divergences,, Gazettes des Mathématiciens., (). Google Scholar

[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. doi: 10.1016/j.jcp.2012.10.022. Google Scholar

[39]

H. Stahl, Conjectures around the Baker-Gammel-Wills conjecture,, Constructive Approximation, 13 (1997), 287. doi: 10.1007/s003659900044. Google Scholar

[40]

H. Stahl, Spurious poles in Padé approximation,, Journal of Computational and Applied Mathematics, 99 (1998), 511. doi: 10.1016/S0377-0427(98)00180-0. Google Scholar

[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. doi: 10.1007/BF01385638. Google Scholar

[42]

J. Thomann, Procédés formels et numériques de sommation de séries solutions d'équations différentielles,, in Journées X-UPS 1991, (1991), 101. Google Scholar

[43]

J. Thomann, Formal and Numerical Summation of Formal Power Series Solutions of ODE's,, Technical report, (2000). Google Scholar

[44]

F. Thomlinson, Generalized factorial series,, Transactions of the American Mathematical Society, 31 (). Google Scholar

[45]

M. Thomson, The Calculus Of Finite Differences,, Macmillan and Company, (1933). Google Scholar

[46]

J. van Deun, A. Bultheel and P. González Vera, On computing rational Gauss-Chebyshev quadrature formulas,, Mathematics of Computation, 75 (2006), 307. doi: 10.1090/S0025-5718-05-01774-6. Google Scholar

[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. Google Scholar

[48]

E. Weniger, Summation of divergent power series by means of factorial series,, Applied Numerical Mathematics, 60 (2010), 1429. doi: 10.1016/j.apnum.2010.04.003. Google Scholar

show all references

References:
[1]

V. Adukov and O. Ibryaeva, A new algorithm for computing padé approximants,, \arXiv{1112.5694}., (). Google Scholar

[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. doi: 10.1016/0022-247X(61)90019-1. Google Scholar

[3]

B. Beckermann and A. Ana Matos, Algebraic properties of robust Padé approximants,, Journal of Approximation Theory, 190 (2015), 91. doi: 10.1016/j.jat.2014.05.018. Google Scholar

[4]

J. Boyd, Exponentially convergent Fourier-Chebshev quadrature schemes on bounded and infinite intervals,, Journal of Scientific Computing, 2 (1987), 99. doi: 10.1007/BF01061480. Google Scholar

[5]

C. Brezinski, Rationnal approximation to formal power serie,, Journal of Approximation Theory, 25 (1979), 295. doi: 10.1016/0021-9045(79)90019-4. Google Scholar

[6]

C. Brezinski and J. Van Iseghem, Padé approximations,, in Handbook of Numerical Analysis (eds. P. G. Ciarlet and J. L. Lions), 3 (1994), 47. doi: 10.1016/S1570-8659(05)80016-X. Google Scholar

[7]

A. Bultheel, Recursive algorithms for nonnormal Pade tables,, SIAM Journal on Applied Mathematics, 39 (1980), 106. doi: 10.1137/0139009. Google Scholar

[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, 366 (2008), 2775. doi: 10.1098/rsta.2008.0052. Google Scholar

[9]

P. J. Davis and P. Rabinowitz, Ignoring the singularity in approximate integration,, Journal of the Society for Industrial and Applied Mathematics: Series B, 2 (1965), 367. doi: 10.1137/0702029. Google Scholar

[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. doi: 10.1051/proc/201445033. Google Scholar

[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. doi: 10.5802/aif.2263. Google Scholar

[12]

W. Gautschi, Gauss-type quadrature rules for rational functions,, Numerical Integration IV, 112 (1993), 111. doi: 10.1007/978-3-0348-6338-4_9. Google Scholar

[13]

W. Gautschi, The use of rational functions in numerical quadrature,, Journal of Computational and Applied Mathematics, 133 (2001), 111. doi: 10.1016/S0377-0427(00)00637-3. Google Scholar

[14]

W. Gautschi, Quadrature formulae on half-infinite intervals,, BIT Numerical Mathematics, 31 (1991), 438. doi: 10.1007/BF01933261. Google Scholar

[15]

J. Gilewicz, Approximants de Padé, vol. 667 of Lecture Notes in Mathematics,, Springer-Verlag, (1978). Google Scholar

[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. doi: 10.1016/S0377-0427(02)00674-X. Google Scholar

[17]

J. Gilewicz and M. Pindor, Padé approximants and noise: A case of geometric series,, Journal of Computational and Applied Mathematics, 87 (1997), 199. doi: 10.1016/S0377-0427(97)00185-4. Google Scholar

[18]

P. Gonnet, S. Güttel and L. Trefethen, Robust Padé approximation via SVD,, SIAM Review, 55 (2013), 101. doi: 10.1137/110853236. Google Scholar

[19]

N. Hall, Interview of sir michael berry by nina hall: Caustics, catastrophes and quantum chaos,, Nexus News, (): 4. Google Scholar

[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. Google Scholar

[21]

H. Kleinert and V. Schulte-Frohlinde, Critical Properties of $\Phi^4$-Theories,, World Scientific Publishing Co., (2001). doi: 10.1142/9789812799944. Google Scholar

[22]

V. Kowalenko, The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation,, Bentham, (2009). doi: 10.2174/97816080501091090101. Google Scholar

[23]

R. Kumar and M. K. Jain, Quadrature formulas for semi-infinite integrals,, Mathematics of Computation, 28 (1974), 499. doi: 10.1090/S0025-5718-1974-0343549-5. Google Scholar

[24]

D. Lubinsky, Reflections on the Baker-Gammel-Wills (Padé),, in Analytic Number Theory, (2014), 561. Google Scholar

[25]

D. S. Lubinsky and P. Rabinowitz, Rates of convergence of Gaussian quadrature for singular integrands,, Mathematics of Computation, 43 (1984), 219. doi: 10.1090/S0025-5718-1984-0744932-2. Google Scholar

[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. Google Scholar

[27]

G. Lysik, Borel summable solutions of the Burgers equation,, Annales Polonici Mathematici, 95 (2009), 187. doi: 10.4064/ap95-2-9. Google Scholar

[28]

G. Lysik and S. Michalik, Formal solutions of semilinear heat equations,, Journal of Mathematical Analysis and Applications, 341 (2008), 372. doi: 10.1016/j.jmaa.2007.10.005. Google Scholar

[29]

W. Mascarenhas, Robust Padé approximants can diverge,, , (). Google Scholar

[30]

N. Nielsen, Recherches sur les séries de factorielles,, Annales Scientifiques de l'E.N.S. 3è série, 19 (1902), 409. Google Scholar

[31]

N. Nielsen, Les séries de factorielles et les opérations fondamentales,, Mathematische Annalen, 59 (1904), 355. doi: 10.1007/BF01445147. Google Scholar

[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. Google Scholar

[33]

N. Nörlund, Vorlesungen Über Differenzenrechnung,, Srpinger Verlag, (1924). Google Scholar

[34]

N. Nörlund, Leçons Sur Les Séries D'interpolation,, Gauthier-Villard et Cie, (1926). Google Scholar

[35]

S. Pincherle, Sulle serie di fattoriali. nota II,, Atti della Reale Accademia dei Lincei, 11 (1902), 417. Google Scholar

[36]

J.-P. Ramis, Séries divergentes et théories asymptotiques,, in Journées X-UPS 1991, (1991), 7. Google Scholar

[37]

J.-P. Ramis, Les développements asymptotiques après poincaré: Continuité et... divergences,, Gazettes des Mathématiciens., (). Google Scholar

[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. doi: 10.1016/j.jcp.2012.10.022. Google Scholar

[39]

H. Stahl, Conjectures around the Baker-Gammel-Wills conjecture,, Constructive Approximation, 13 (1997), 287. doi: 10.1007/s003659900044. Google Scholar

[40]

H. Stahl, Spurious poles in Padé approximation,, Journal of Computational and Applied Mathematics, 99 (1998), 511. doi: 10.1016/S0377-0427(98)00180-0. Google Scholar

[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. doi: 10.1007/BF01385638. Google Scholar

[42]

J. Thomann, Procédés formels et numériques de sommation de séries solutions d'équations différentielles,, in Journées X-UPS 1991, (1991), 101. Google Scholar

[43]

J. Thomann, Formal and Numerical Summation of Formal Power Series Solutions of ODE's,, Technical report, (2000). Google Scholar

[44]

F. Thomlinson, Generalized factorial series,, Transactions of the American Mathematical Society, 31 (). Google Scholar

[45]

M. Thomson, The Calculus Of Finite Differences,, Macmillan and Company, (1933). Google Scholar

[46]

J. van Deun, A. Bultheel and P. González Vera, On computing rational Gauss-Chebyshev quadrature formulas,, Mathematics of Computation, 75 (2006), 307. doi: 10.1090/S0025-5718-05-01774-6. Google Scholar

[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. Google Scholar

[48]

E. Weniger, Summation of divergent power series by means of factorial series,, Applied Numerical Mathematics, 60 (2010), 1429. doi: 10.1016/j.apnum.2010.04.003. Google Scholar

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