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Volumn 217, Issue 20, 2011, Pages 7940-7954

Breaking the limits: The Taylor series method

Author keywords

Automatic differentiation; Box propagation; Freeware software; High precision integration of ODEs; Taylor series method; Variational equations

Indexed keywords

AUTOMATIC DIFFERENTIATIONS; BOX PROPAGATION; FREEWARE SOFTWARE; HIGH-PRECISION; VARIATIONAL EQUATIONS;

EID: 79956062290     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2011.02.080     Document Type: Article
Times cited : (67)

References (43)
  • 1
    • 0012652435 scopus 로고
    • Numerical integration of an orbit and its concomitant variations
    • A. Deprit, and R. Zahar Numerical integration of an orbit and its concomitant variations Z. Angew. Math. Phys. 17 1966 425 430
    • (1966) Z. Angew. Math. Phys. , vol.17 , pp. 425-430
    • Deprit, A.1    Zahar, R.2
  • 2
    • 0043269015 scopus 로고
    • Studies in the application of recurrence relations to special perturbation methods i
    • A. Roy, P. Moran, and W. Black Studies in the application of recurrence relations to special perturbation methods i Celestial Mech. 6 1972 468 482
    • (1972) Celestial Mech. , vol.6 , pp. 468-482
    • Roy, A.1    Moran, P.2    Black, W.3
  • 4
    • 0020143282 scopus 로고
    • Solving ordinary differential equations using Taylor series
    • DOI 10.1145/355993.355995
    • G. Corliss, and Y.F. Chang Solving ordinary differential equations using Taylor series ACM Trans. Math. Software 8 2 1982 114 144 (Pubitemid 12550672)
    • (1982) ACM Transactions on Mathematical Software , vol.8 , Issue.2 , pp. 114-144
    • Corliss, G.1    Chang, Y.F.2
  • 5
    • 0040763312 scopus 로고
    • ATOMFT: Solving ODEs and DAEs using Taylor series
    • Y.F. Chang, and G. Corliss ATOMFT: solving ODEs and DAEs using Taylor series Comput. Math. Appl. 28 10-12 1994 209 233
    • (1994) Comput. Math. Appl. , vol.28 , Issue.1012 , pp. 209-233
    • Chang, Y.F.1    Corliss, G.2
  • 6
    • 0032262788 scopus 로고    scopus 로고
    • Solving high-index DAEs by Taylor series
    • J.D. Pryce Solving high-index DAEs by Taylor series Numer. Algorithms 19 1-4 1998 195 211 (Pubitemid 128405333)
    • (1998) Numerical Algorithms , vol.19 , Issue.1-4 , pp. 195-211
    • Pryce, J.D.1
  • 7
    • 0041331524 scopus 로고    scopus 로고
    • A simple structural analysis method for DAES
    • J.D. Pryce A simple structural analysis method for DAEs BIT 41 2 2001 364 394 (Pubitemid 33632646)
    • (2001) BIT Numerical Mathematics , vol.41 , Issue.2 , pp. 364-394
    • Pryce, J.D.1
  • 8
    • 13544262350 scopus 로고    scopus 로고
    • Performance of the Taylor series method for ODEs/DAEs
    • DOI 10.1016/j.amc.2004.02.015, PII S0096300304002401
    • R. Barrio Performance of the Taylor series method for ODEs/DAEs Appl. Math. Comput. 163 2 2005 525 545 (Pubitemid 40219901)
    • (2005) Applied Mathematics and Computation , vol.163 , Issue.2 , pp. 525-545
    • Barrio, R.1
  • 9
    • 28144454918 scopus 로고    scopus 로고
    • Solving differential-algebraic equations by taylor series (I): Computing taylor coefficients
    • DOI 10.1007/s10543-005-0019-y
    • N.S. Nedialkov, and J.D. Pryce Solving differential-algebraic equations by Taylor series. I. Computing Taylor coefficients BIT 45 3 2005 561 591 (Pubitemid 41698062)
    • (2005) BIT Numerical Mathematics , vol.45 , Issue.3 , pp. 561-591
    • Nedialkov, N.S.1    Pryce, J.D.2
  • 10
    • 33947232075 scopus 로고    scopus 로고
    • Solving differential-algebraic equations by Taylor series. II. Computing the system Jacobian
    • N.S. Nedialkov, and J.D. Pryce Solving differential-algebraic equations by Taylor series. II. Computing the system Jacobian BIT 47 1 2007 121 135
    • (2007) BIT , vol.47 , Issue.1 , pp. 121-135
    • Nedialkov, N.S.1    Pryce, J.D.2
  • 11
    • 67349205394 scopus 로고    scopus 로고
    • Solving differential algebraic equations by Taylor series. III. The DAETS code
    • N.S. Nedialkov, and J.D. Pryce Solving differential algebraic equations by Taylor series. III. The DAETS code J. Numer. Anal. Ind. Appl. Math. 3 1-2 2008 61 80
    • (2008) J. Numer. Anal. Ind. Appl. Math. , vol.3 , Issue.12 , pp. 61-80
    • Nedialkov, N.S.1    Pryce, J.D.2
  • 12
    • 67349135986 scopus 로고    scopus 로고
    • Systematic search of symmetric periodic orbits in 2DOF Hamiltonian systems
    • R. Barrio, and F. Blesa Systematic search of symmetric periodic orbits in 2DOF Hamiltonian systems Chaos Solitons Fractals 41 2 2009 560 582
    • (2009) Chaos Solitons Fractals , vol.41 , Issue.2 , pp. 560-582
    • Barrio, R.1    Blesa, F.2
  • 13
    • 56049127702 scopus 로고    scopus 로고
    • Periodic, escape and chaotic orbits in the Copenhagen and the (n + 1)-body ring problems
    • R. Barrio, F. Blesa, and S. Serrano Periodic, escape and chaotic orbits in the Copenhagen and the (n + 1)-body ring problems Commun. Nonlinear Sci. Numer. Simul. 14 5 2009 2229 2238
    • (2009) Commun. Nonlinear Sci. Numer. Simul. , vol.14 , Issue.5 , pp. 2229-2238
    • Barrio, R.1    Blesa, F.2    Serrano, S.3
  • 14
    • 33845542060 scopus 로고    scopus 로고
    • Painting chaos: A gallery of sensitivity plots of classical problems
    • DOI 10.1142/S021812740601646X
    • R. Barrio Painting chaos: a gallery of sensitivity plots of classical problems Int. J. Bifur. Chaos Appl. Sci. Eng. 16 10 2006 2777 2798 (Pubitemid 44924961)
    • (2006) International Journal of Bifurcation and Chaos , vol.16 , Issue.10 , pp. 2777-2798
    • Barrio, R.1
  • 15
    • 34247637165 scopus 로고    scopus 로고
    • A three-parametric study of the Lorenz model
    • DOI 10.1016/j.physd.2007.03.013, PII S0167278907000735
    • R. Barrio, and S. Serrano A three-parametric study of the Lorenz model Phys. D 229 1 2007 43 51 (Pubitemid 46669726)
    • (2007) Physica D: Nonlinear Phenomena , vol.229 , Issue.1 , pp. 43-51
    • Barrio, R.1    Serrano, S.2
  • 16
    • 0002265766 scopus 로고    scopus 로고
    • Validated solutions of initial value problems for ordinary differential equations
    • PII S0096300398100838
    • N.S. Nedialkov, K.R. Jackson, and G.F. Corliss Validated solutions of initial value problems for ordinary differential equations Appl. Math. Comput. 105 1 1999 21 68 (Pubitemid 129525434)
    • (1999) Applied Mathematics and Computation , vol.105 , Issue.1 , pp. 21-68
    • Nedialkov, N.S.1    Jackson, K.R.2    Corliss, G.F.3
  • 17
    • 0002061016 scopus 로고    scopus 로고
    • A rigorous ODE solver and Smale's 14th problem
    • W. Tucker A rigorous ODE solver and Smale's 14th problem Found. Comput. Math. 2 1 2002 53 117
    • (2002) Found. Comput. Math. , vol.2 , Issue.1 , pp. 53-117
    • Tucker, W.1
  • 19
    • 22544468094 scopus 로고    scopus 로고
    • A software package for the numerical integration of ODEs by means of high-order taylor methods
    • Á. Jorba, and M. Zou A software package for the numerical integration of ODEs by means of high-order Taylor methods Exp. Math. 14 1 2005 99 117 (Pubitemid 41017979)
    • (2005) Experimental Mathematics , vol.14 , Issue.1 , pp. 99-117
    • Jorba, A.1    Zou, M.2
  • 22
    • 0345371405 scopus 로고    scopus 로고
    • Taylor models and other validated functional inclusion methods
    • K. Makino, and M. Berz Taylor models and other validated functional inclusion methods Int. J. Pure Appl. Math. 6 3 2003 239 316
    • (2003) Int. J. Pure Appl. Math. , vol.6 , Issue.3 , pp. 239-316
    • Makino, K.1    Berz, M.2
  • 23
    • 0004149831 scopus 로고    scopus 로고
    • fourth ed. Wolfram Media, Inc. Champaign, IL
    • S. Wolfram The Mathematica® Book fourth ed. 1999 Wolfram Media, Inc. Champaign, IL
    • (1999) The Mathematica® Book
    • Wolfram, S.1
  • 25
    • 0000241853 scopus 로고
    • Deterministic nonperiodic flow
    • E. Lorenz Deterministic nonperiodic flow J. Atmospheric Sci. 20 1963 130 141
    • (1963) J. Atmospheric Sci. , vol.20 , pp. 130-141
    • Lorenz, E.1
  • 29
    • 0004293209 scopus 로고
    • Prentice-Hall Inc. Englewood Cliffs, N.J.
    • R.E. Moore Interval Analysis 1966 Prentice-Hall Inc. Englewood Cliffs, N.J.
    • (1966) Interval Analysis
    • Moore, R.E.1
  • 31
    • 1142268020 scopus 로고    scopus 로고
    • The fractal property of the Lorenz attractor
    • D. Viswanath The fractal property of the Lorenz attractor Phys. D 190 1-2 2004 115 128
    • (2004) Phys. D , vol.190 , Issue.12 , pp. 115-128
    • Viswanath, D.1
  • 32
    • 33751249935 scopus 로고    scopus 로고
    • Sensitivity analysis of ODEs/DAEs using the Taylor series method
    • DOI 10.1137/030601892
    • R. Barrio Sensitivity analysis of ODEs/DAEs using the Taylor series method SIAM J. Sci. Comput. 27 6 2006 1929 1947 (Pubitemid 44784223)
    • (2006) SIAM Journal of Scientific Computing , vol.27 , Issue.6 , pp. 1929-1947
    • Barrio, R.1
  • 33
    • 67549094812 scopus 로고    scopus 로고
    • Bifurcations and safe regions in open Hamiltonians
    • R. Barrio, F. Blesa, and S. Serrano Bifurcations and safe regions in open Hamiltonians New J. Phys. 11 2009 053004 15pp
    • (2009) New J. Phys. , vol.11 , pp. 053004
    • Barrio, R.1    Blesa, F.2    Serrano, S.3
  • 34
    • 0002402664 scopus 로고
    • Algorithms for higher derivatives in many variables with applications to beam physics
    • SIAM Philadelphia, PA
    • M. Berz Algorithms for higher derivatives in many variables with applications to beam physics Automatic Differentiation of Algorithms (Breckenridge, CO, 1991) 1991 SIAM Philadelphia, PA 147 156
    • (1991) Automatic Differentiation of Algorithms (Breckenridge, CO, 1991) , pp. 147-156
    • Berz, M.1
  • 35
    • 58149488788 scopus 로고    scopus 로고
    • Application of high order expansions of two-point boundary value problems to astrodynamics
    • P. Di Lizia, R. Armellin, and M. Lavagna Application of high order expansions of two-point boundary value problems to astrodynamics Celestial Mech. Dyn. Astron. 102 4 2008 355 375
    • (2008) Celestial Mech. Dyn. Astron. , vol.102 , Issue.4 , pp. 355-375
    • Di Lizia, P.1    Armellin, R.2    Lavagna, M.3
  • 36
    • 49749096777 scopus 로고    scopus 로고
    • Achieving Brouwer's law with implicit Runge-Kutta methods
    • E. Hairer, R.I. McLachlan, and A. Razakarivony Achieving Brouwer's law with implicit Runge-Kutta methods BIT 48 2 2008 231 243
    • (2008) BIT , vol.48 , Issue.2 , pp. 231-243
    • Hairer, E.1    McLachlan, R.I.2    Razakarivony, A.3
  • 37
    • 0038971069 scopus 로고
    • On the accumulation of errors in numerical integration
    • D. Brouwer On the accumulation of errors in numerical integration Astron. J. 30 1937 149 153
    • (1937) Astron. J. , vol.30 , pp. 149-153
    • Brouwer, D.1
  • 38
    • 0007310585 scopus 로고    scopus 로고
    • Global dynamics and fast indicators Global analysis of dynamical systems
    • Bristol
    • C. Simó Global dynamics and fast indicators Global analysis of dynamical systems Institute of Physics 2001 Bristol 373 389
    • (2001) Institute of Physics , pp. 373-389
    • Simó, C.1
  • 39
    • 33749317983 scopus 로고    scopus 로고
    • N-body simulations: The performance of some integrators
    • DOI 10.1145/1163641.1163642
    • P.W. Sharp N-body simulations: the performance of some integrators ACM Trans. Math. Software 32 3 2006 375 395 (Pubitemid 44495247)
    • (2006) ACM Transactions on Mathematical Software , vol.32 , Issue.3 , pp. 375-395
    • Sharp, P.W.1
  • 40
    • 70549086655 scopus 로고    scopus 로고
    • Long simulations of the Solar System: Brouwer's law and chaos
    • K.R. Grazier, W.I. Newman, J.M. Hyman, and P.W. Sharp Long simulations of the Solar System: Brouwer's law and chaos ANZIAM J. 46 C 2004/05 C1086 C1103
    • (2004) ANZIAM J. , vol.5 , Issue.C
    • Grazier, K.R.1    Newman, W.I.2    Hyman, J.M.3    Sharp, P.W.4
  • 43
    • 0002519122 scopus 로고
    • The applicability of the third integral of motion: Some numerical experiments
    • M. Hénon, and C. Heiles The applicability of the third integral of motion: some numerical experiments Astron. J. 69 1964 73 79
    • (1964) Astron. J. , vol.69 , pp. 73-79
    • Hénon, M.1    Heiles, C.2


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