Alin Bostan
ABSTRACT
We provide algorithms computing power series solutions of a large class of differential or q-differential equations or systems. Their number of arithmetic operations grows linearly with the precision, up to logarithmic terms.
- W. Balser. Formal power series and linear systems of meromorphic ordinary differential equations. Universitext. Springer-Verlag, New York, 2000.Google Scholar
- M. Barkatou, G. Broughton, and E. Pflügel. A monomial-by-monomial method for computing regular solutions of systems of pseudo-linear equations. Math. Comput. Sci., 4(2-3):267--288, 2010.Google ScholarCross Ref
- M. Barkatou and E. Pflügel. An algorithm computing the regular formal solutions of a system of linear differential equations. J. Symb. Comput., 28(4-5):569--587, 1999.Google ScholarDigital Library
- D. J. Bernstein. Composing power series over a finite ring in essentially linear time. J. Symb. Comput., 26(3):339--341, 1998. Google ScholarDigital Library
- A. Bostan, F. Chyzak, F. Ollivier, B. Salvy, S. Sedoglavic, and É. Schost. Fast computation of power series solutions of systems of differential equations. In Symposium on Discrete Algorithms, SODA'07, pages 1012--1021. ACM-SIAM, 2007. Google ScholarDigital Library
- A. Bostan, L. González-Vega, H. Perdry, and É. Schost. From Newton sums to coefficients: complexity issues in characteristic p. In MEGA '05, 2005.Google Scholar
- A. Bostan, F. Morain, B. Salvy, and É. Schost. Fast algorithms for computing isogenies between elliptic curves. Math. of Comp., 77(263):1755--1778, 2008.Google ScholarCross Ref
- A. Bostan, B. Salvy, and É. Schost. Power series composition and change of basis. In ISSAC'08, pages 269--276. ACM, 2008. Google ScholarDigital Library
- A. Bostan and É. Schost. Polynomial evaluation and interpolation on special sets of points. J. Complexity, 21(4):420--446, 2005. Google ScholarCross Ref
- A. Bostan and É. Schost. Fast algorithms for differential equations in positive characteristic. In ISSAC'09, pages 47--54. ACM, 2009. Google ScholarDigital Library
- R. P. Brent and H. T. Kung. Fast algorithms for manipulating formal power series. J. ACM, 25(4):581--595, 1978. Google ScholarDigital Library
- R. P. Brent and J. F. Traub. On the complexity of composition and generalized composition of power series. SIAM J. Comput., 9:54--66, 1980.Google ScholarCross Ref
- D. G. Cantor and E. Kaltofen. On fast multiplication of polynomials over arbitrary algebras. Acta Inform., 28(7):693--701, 1991. Google ScholarDigital Library
- D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. J. Symb. Comput., 9(3):251--280, 1990. Google ScholarDigital Library
- C.-É. Drevet, M. Islam, and É. Schost. Optimization techniques for small matrix multiplication. Theoretical Computer Science, 412:2219--2236, 2011. Google ScholarDigital Library
- Fischer and Stockmeyer. Fast on-line integer multiplication. J. of Computer and System Sciences, 9, 1974. Google ScholarDigital Library
- J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, 1999. Google ScholarDigital Library
- V. Guruswami and A. Rudra. Explicit codes achieving list decoding capacity: Error-correction with optimal redundancy. IEEE Trans. on Information Theory, 54(1):135--150, 2008. Google ScholarDigital Library
- J. van der Hoeven. Relax, but don't be too lazy. J. Symb. Comput., 34(6):479--542, 2002. Google ScholarDigital Library
- J. van der Hoeven. Relaxed multiplication using the middle product. In ISSAC'03, pages 143--147. ACM, 2003. Google ScholarDigital Library
- J. van der Hoeven. New algorithms for relaxed multiplication. J. Symb. Comput., 42(8):792--802, 2007. Google ScholarDigital Library
- J. van der Hoeven. Relaxed resolution of implicit equations. Technical report, HAL, 2009. http://hal.archives-ouvertes.fr/hal-00441977.Google Scholar
- J. van der Hoeven. From implicit to recursive equations. Technical report, HAL, 2011. http://hal.archives-ouvertes.fr/hal-00583125.Google Scholar
- J. van der Hoeven. Faster relaxed multiplication. Technical report, HAL, 2012. http://hal.archives-ouvertes.fr/hal-00687479.Google Scholar
- K. S. Kedlaya and C. Umans. Fast polynomial factorization and modular composition. SIAM J. Comput., 40(6):1767--1802, 2011. Google ScholarDigital Library
- P. Kirrinnis. Fast algorithms for the Sylvester equation AX − XB<sup>t</sup> = C. Theoretical Computer Science, 259(1-2):623--638, 2001. Google ScholarDigital Library
- R. Lercier and T. Sirvent. On Elkies subgroups of ℓ-torsion points in elliptic curves defined over a finite field. Journal de Théorie des Nombres de Bordeaux, 20(3):783--797, 2008.Google ScholarCross Ref
- A. Schönhage and V. Strassen. Schnelle Multiplikation großer Zahlen. Computing, 7:281--292, 1971.Google ScholarCross Ref
- V. Shoup. NTL 5.5.2: A library for doing number theory, 2009. www.shoup.net/ntl.Google Scholar
- A. Stothers. On the Complexity of Matrix Multiplication. PhD thesis, University of Edinburgh, 2010.Google Scholar
- V. Vassilevska Williams. Breaking the Coppersmith-Winograd barrier. Technical report, 2011.Google Scholar
- A. Waksman. On Winograd's algorithm for inner products. IEEE Trans. On Computers, C-19:360--361, 1970. Google ScholarDigital Library
- W. Wasow. Asymptotic expansions for ordinary differential equations. John Wiley&Sons, 1965.Google Scholar
Index Terms
- Power series solutions of singular (q)-differential equations
Recommendations
Linearization techniques for singular initial-value problems of ordinary differential equations
Linearization methods for singular initial-value problems in second-order ordinary differential equations are presented. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus ...
Series solutions of coupled differential equations with one regular singular point
Special issue: Proceedings of the 9th International Congress on computational and applied mathematicsWe consider two linear second-order ordinary differential equations. r = 0 is a regular singular point of these equations, Applying the classical Method of Frobenius, we do not obtain any indicial equation and therefore no solution, because the ...
Analytical approximate solutions for nonlinear fractional differential equations
We consider a class of nonlinear fractional differential equations (FDEs) based on the Caputo fractional derivative and by extending the application of the Adomian decomposition method we derive an analytical solution in the form of a series with easily ...
Comments