Advanced Numerical Differential Equation Solving in Mathematica: References

[AP91] Ascher U. and L. Petzold. "Projected Implicit Runge-Kutta Methods for Differential Algebraic Equations" SIAM J. Numer. Anal. 28 (1991): 1097-1120

[AP98] Ascher U. and L. Petzold. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM Press (1998)

[ATLAS00] Whaley R. C., A. Petitet, and J. J. Dongarra. "Automated Empirical Optimization of Software and the ATLAS Project" Available electronically from http://math-atlas.sourceforge.net/

[BD83] Bader G. and P. Deuflhard. "A Semi-Implicit Mid-Point Rule for Stiff Systems of Ordinary Differential Equations" Numer. Math 41 (1983): 373-398

[BG94] Benettin G. and A. Giorgilli. "On the Hamiltonian Interpolation of Near to the Identity Symplectic Mappings with Application to Symplectic Integration Algorithms" J. Stat. Phys. 74 (1994): 1117-1143

[BZ65] Berezin I. S. and N. P. Zhidkov. Computing Methods, Volume 2. Pergamon (1965)

[BM02] Blanes S. and P. C. Moan. "Practical Symplectic Partitioned Runge-Kutta and Runge-Kutta-Nyström Methods" J. Comput. Appl. Math. 142 (2002): 313-330

[BCR99a] Blanes S., F. Casas, and J. Ros. "Symplectic Integration with Processing: A General Study" SIAM J. Sci. Comput. 21 (1999): 711-727

[BCR99b] Blanes S., F. Casas, and J. Ros. "Extrapolation of Symplectic Integrators" Report DAMTP NA09, Cambridge University (1999)

[BS89a] Bogacki P. and L. F. Shampine. "A 3(2) Pair of Runge-Kutta Formulas" Appl. Math. Letters 2 (1989): 1-9

[BS89b] Bogacki P. and L. F. Shampine. "An Efficient Runge-Kutta (4, 5) Pair" Report 89-20, Math. Dept. Southern Methodist University, Dallas, Texas (1989)

[BGS93] Brankin R. W., I. Gladwell and L. F. Shampine. "RKSUITE: A Suite of Explicit Runge-Kutta Codes" In Contributions to Numerical Mathematics, R. P. Agarwal, ed., 41-53 (1993)

[BCP89] Brenan K., S. Campbell, and L. Petzold. Numerical Solutions of Initial-Value Problems in Differential-Algebraic Equations. Elsevier Science Publishing (1989)

[BHP94] Brown P. N., A. C. Hindmarsh, and L. R. Petzold. "Using Krylov Methods in the Solution of Large-Scale Differential-Algebraic Systems" SIAM J. Sci. Comput. 15 (1994): 1467-1488

[BHP98] Brown P. N., A. C. Hindmarsh, and L. R. Petzold. "Consistent Initial Condition Calculation for Differential-Algebraic Systems" SIAM J. Sci. Comput. 19 (1998): 1495-1512

[B87] Butcher J. C. The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods. John Wiley (1987)

[BS64] Bulirsch R. and J. Stoer. "Fehlerabschätzungen und Extrapolation mit Rationalen Funktionen bei Verfahren vom Richardson-Typus" Numer. Math. 6 (1964): 413-427

[CIZ97] Calvo M. P., A. Iserles, and A. Zanna. "Numerical Solution of Isospectral Flows" Math. Comp. 66, no. 220 (1997): 1461-1486

[CIZ99] Calvo M. P., A. Iserles, and A. Zanna. "Conservative Methods for the Toda Lattice Equations" IMA J. Numer. Anal. 19 (1999): 509-523

[CR91] Candy J. and R. Rozmus. "A Symplectic Integration Algorithm for Separable Hamiltonian Functions" J. Comput. Phys. 92 (1991): 230-256

[CH94] Cohen S. D. and A. C. Hindmarsh. CVODE User Guide. Lawrence Livermore National Laboratory report UCRL-MA-118618, September 1994

[CH96] Cohen S. D. and A. C. Hindmarsh. "CVODE, a Stiff/Nonstiff ODE Solver in C" Computers in Physics 10, no. 2 (1996): 138-143

[C87] Cooper G. J. "Stability of Runge-Kutta Methods for Trajectory Problems" IMA J. Numer. Anal. 7 (1987): 1-13

[DP80] Dormand J. R. and P. J. Prince. "A Family of Embedded Runge-Kutta Formulae" J. Comp. Appl. Math. 6 (1980): 19-26

[DL01] Del Buono N. and L. Lopez. "Runge-Kutta Type Methods Based on Geodesics for Systems of ODEs on the Stiefel Manifold" BIT 41 (5 (2001): 912-923

[D83] Deuflhard P. "Order and Step Size Control in Extrapolation Methods" Numer. Math. 41 (1983): 399-422

[D85] Deuflhard P. "Recent Progress in Extrapolation Methods for Ordinary Differential Equations" SIAM Rev. 27 (1985): 505-535

[DN87] Deuflhard P. and U. Nowak. "Extrapolation Integrators for Quasilinear Implicit ODEs" In Large-scale scientific computing, P. Deuflhard, B. Engquist, eds., Birkhäuser, (1987)

[DHZ87] Deuflhard P., E. Hairer, and J. Zugck. "One-Step and Extrapolation Methods for Differential-Algebraic Systems" Numer. Math. 51 (1987): 501-516

[DRV94] Dieci L., R. D. Russel, and E. S. Van Vleck. "Unitary Integrators and Applications to Continuous Orthonormalization Techniques" SIAM J. Num. Anal. 31 (1994): 261-281

[DV99] Dieci L. and E. S. Van Vleck. "Computation of Orthonormal Factors for Fundamental Solution Matrices" Numer. Math. 83 (1999): 599-620

[DLP98a] Diele F., L. Lopez, and R. Peluso. "The Cayley Transform in the Numerical Solution of Unitary Differential Systems" Adv. Comput. Math. 8 (1998): 317-334

[DLP98b] Diele F., L. Lopez, and T. Politi. "One Step Semi-Explicit Methods Based on the Cayley Transform for Solving Isospectral Flows" J. Comput. Appl. Math. 89 (1998): 219-223

[ET92] Earn D. J. D. and S. Tremaine. "Exact Numerical Studies of Hamiltonian Maps: Iterating without Roundoff Error" Physica D. 56 (1992): 1-22

[F69] Fehlberg E. "Low-Order Classical Runge-Kutta Formulas with Step Size Control and Their Application to Heat Transfer Problems" NASA Technical Report 315, 1969 (extract published in Computing 6 (1970): 61-71)

[FR90] Forest E. and R. D. Ruth. "Fourth Order Symplectic Integration" Physica D. 43 (1990): 105-117

[F92] Fornberg B. "Fast Generation of Weights in Finite Difference Formulas" In Recent Developments in Numerical Methods and Software for ODEs/DAEs/PDEs (G. D. Byrne and W. E. Schiesser, eds.). World Scientific (1992)

[F96a] Fornberg B. A Practical Guide to Pseudospectral Methods. Cambridge University Press (1996)

[F98] Fornberg B. "Calculation of Weights in Finite Difference Formulas" SIAM Review 40, no. 3 (1998): 685-691 (Available in PDF)

[F96b] Fukushima T. "Reduction of Round-off Errors in the Extrapolation Methods and its Application to the Integration of Orbital Motion" Astron. J. 112, no. 3 (1996): 1298-1301

[G51] Gill S. "A Process for the Step-by-Step Integration of Differential Equations in an Automatic Digital Computing Machine" Proc. Cambridge Philos. Soc. 47 (1951): 96-108

[G65] Gragg W. B. "On Extrapolation Algorithms for Ordinary Initial Value Problems" SIAM J. Num. Anal. 2 (1965): 384-403

[GØ84] Gear C. W. and O. Østerby. "Solving Ordinary Differential Equations with Discontinuities" ACM Trans. Math. Soft. 10 (1984): 23-44

[G91] Gustafsson K. "Control Theoretic Techniques for Stepsize Selection in Explicit Runge-Kutta Methods" ACM Trans. Math. Soft. 17, 533-554

[G94] Gustafsson K. "Control Theoretic Techniques for Stepsize Selection in Implicit Runge-Kutta Methods" ACM Trans. Math. Soft. 20, 496-517

[GMW81] Gill P., W. Murray, and M. Wright. Practical Optimization. Academic Press (1981)

[GDC91] Gladman B., M. Duncan, and J. Candy. "Symplectic Integrators for Long-Term Integrations in Celestial Mechanics" Celest. Mech. 52 (1991): 221-240

[GSB87] Gladwell I., L. F. Shampine and R. W. Brankin. "Automatic Selection of the Initial Step Size for an ODE Solver" J. Comp. Appl. Math. 18 (1987): 175-192

[GVL96] Golub G. H. and C. F. Van Loan. Matrix Computations, 3rd ed. Johns Hopkins University Press (1996)

[H94] Hairer E. "Backward Analysis of Numerical Integrators and Symplectic Methods" Annals of Numerical Mathematics 1 (1984): 107-132

[H97] Hairer E. "Variable Time Step integration with Symplectic Methods" Appl. Numer. Math. 25 (1997): 219-227

[H00] Hairer E. "Symmetric Projection Methods for Differential Equations on Manifolds" BIT 40, no. 4 (2000): 726-734

[HL97] Hairer E. and Ch. Lubich. "The Life-Span of Backward Error Analysis for Numerical Integrators" Numer. Math. 76 (1997): 441-462. Erratum: http://www.unige.ch/math/folks/hairer

[HL88a] Hairer E. and Ch. Lubich. "Extrapolation at Stiff Differential Equations" Numer. Math. 52 (1988): 377-400

[HL88b] Hairer E. and Ch. Lubich. "On Extrapolation Methods for Stiff and Differential-Algebraic Equations" Teubner Texte zur Mathematik 104 (1988): 64-73

[HO90] Hairer E. and A. Ostermann. "Dense Output for Extrapolation Methods" Numer. Math. 58 (1990): 419-439

[HW96] Hairer E. and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd ed. Springer-Verlag (1996)

[HW99] Hairer E. and G. Wanner. "Stiff Differential Equations Solved by Radau Methods" J. Comp. Appl. Math. 111 (1999): 93-111

[HLW02] Hairer E., Ch. Lubich, and G. Wanner. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer-Verlag (2002)

[HNW93] Hairer E., S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd ed. Springer-Verlag (1993)

[H97] Higham D. ""Time-Stepping and Preserving Orthonormality" BIT 37, no. 1 (1997): 24-36

[H89] Higham N. J. "Matrix Nearness Problems and Applications" In Applications of Matrix Theory (M. J. C. Gover and S. Barnett eds.). Oxford University Press (1989): 1-27

[H96] Higham N. J. Accuracy and Stability of Numerical Algorithms. SIAM (1996)

[H83] Hindmarsh A. C. "ODEPACK, A Systematized Collection of ODE Solvers" In Scientific Computing, R. S. Stepleman et al. eds., North-Holland (1983): 55-64

[HT99] Hindmarsh A. and A. Taylor. User Documentation for IDA: A Differential-Algebraic Equation Solver for Sequential and Parallel Computers. Lawrence Livermore National Laboratory report, UCRL-MA-136910, December 1999.

[KL97] Kahan W. H. and R. C. Li. "Composition Constants for Raising the Order of Unconventional Schemes for Ordinary Differential Equations" Math. Comp. 66 (1997): 1089-1099

[K65] Kahan W. H. "Further Remarks on Reducing Truncation Errors" Comm. ACM. 8 (1965): 40

[K93] Koren I. Computer Arithmetic Algorithms. Prentice Hall (1993)

[L87] Lambert J. D. Numerical Methods for Ordinary Differential Equations. John Wiley (1987)

[LAPACK99] Anderson E., Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorenson. LAPACK Users' Guide, 3rd ed. SIAM (1999)

[M68] Marchuk G. "Some Applications of Splitting-Up Methods to the Solution of Mathematical Physics Problems" Aplikace Matematiky 13 (1968): 103-132

[MR99] Marsden J. E. and T. Ratiu. Introduction to Mechanics and Symmetry: Texts in Applied Mathematics, Vol. 17, 2nd ed. Springer-Verlag (1999)

[M93] McLachlan R. I. "Explicit Lie-Poisson Integration and the Euler Equations" Phys. Rev. Lett. 71 (1993): 3043-3046

[M95a] McLachlan R. I. "On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods" SIAM J. Sci. Comp. 16 (1995): 151-168

[M95b] McLachlan R. I. "Composition Methods in the Presence of Small Parameters" BIT 35 (1995): 258-268

[M01] McLachlan R. I. "Families of High-Order Composition Methods" Numerical Algorithms 31 (2002): 233-246

[MA92] McLachlan R. I. and P. Atela. "The Accuracy of Symplectic Integrators" Nonlinearity 5 (1992): 541-562

[MQ02] McLachlan R. I. and G. R. W. Quispel. "Splitting Methods" Acta Numerica 11 (2002): 341-434

[MG80] Mitchell A. and D. Griffiths. The Finite Difference Method in Partial Differential Equations. John Wiley and Sons (1980)

[M65a] Møller O. "Quasi Double-Precision in Floating Point Addition" BIT 5 (1965): 37-50

[M65b] Møller O. "Note on Quasi Double-Precision" BIT 5 (1965): 251-255

[M97] Murua A. "On Order Conditions for Partitioned Symplectic Methods" SIAM J. Numer. Anal. 34, no. 6 (1997): 2204-2211

[MS99] Murua A. and J. M. Sanz-Serna. "Order Conditions for Numerical Integrators Obtained by Composing Simpler Integrators" Phil. Trans. Royal Soc. A 357 (1999): 1079-1100

[M04] Moler C. B. Numerical Computing with MATLAB. SIAM (2004)

[Na79] Na T. Y. Computational Methods in Engineering: Boundary Value Problems. Academic Press (1979)

[OS92] Okunbor D. I. and R. D. Skeel. "Explicit Canonical Methods for Hamiltonian Systems" Math. Comp. 59 (1992): 439-455

[O95] Olsson H. "Practical Implementation of Runge-Kutta Methods for Initial Value Problems" Licentiate thesis, Department of Computer Science, Lund University, 1995

[O98] Olsson H. "Runge-Kutta Solution of Initial Value Problems: Methods, Algorithms and Implementation" PhD Thesis, Department of Computer Science, Lund University, 1998

[OS00] Olsson H. and G. Söderlind. "The Approximate Runge-Kutta Computational Process" BIT 40 (2 (2000): 351-373

[P83] Petzold L. R. "Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations" SIAM J. Sci. Stat. Comput. 4 (1983): 136-148

[QSS00] Quarteroni A., R. Sacco, and F. Saleri. Numerical Mathematics. Springer-Verlag (2000)

[QV94] Quarteroni A. and A. Valli. Numerical Approximation of Partial Differential Equations. Springer-Verlag (1994)

[QT90] Quinn T. and S. Tremaine. "Roundoff Error in Long-Term Planetary Orbit Integrations" Astron. J. 99, no. 3 (1990): 1016-1023

[R93] Reich S. "Numerical Integration of the Generalized Euler Equations" Tech. Rep. 93-20, Dept. Comput. Sci. Univ. of British Columbia (1993)

[R99] Reich S. "Backward Error Analysis for Numerical Integrators" SIAM J. Num. Anal. 36 (1999): 1549-1570

[R98] Rubinstein B. "Numerical Solution of Linear Boundary Value Problems" Mathematica MathSource package, http://library.wolfram.com/database/MathSource/2127/

[RM57] Richtmeyer R. and K. Morton. Difference Methods for Initial Value Problems. Krieger Publishing Company (1994) (original edition 1957)

[R87] Robertson B. C. "Detecting Stiffness with Explicit Runge-Kutta Formulas" Report 193/87, Dept. Comp. Sci. University of Toronto (1987)

[SC94] Sanz-Serna J. M. and M. P. Calvo. Numerical Hamiltonian Problems: Applied Mathematics and Mathematical Computation, no. 7. Chapman and Hall (1994)

[S91] Schiesser W. The Numerical Method of Lines. Academic Press (1991)

[S86] Shampine L. F. "Conservation Laws and the Numerical Solution of ODEs" Comp. Maths. Appl. 12B (1986): 1287-1296

[S87] Shampine L. F. "Control of Step Size and Order in Extrapolation Codes" J. Comp. Appl. Math. 18 (1987): 3-16

[S94] Shampine L. F. Numerical Solution of Ordinary Differential Equations. Chapman and Hall (1994)

[SB83] Shampine L. F. and L. S. Baca. "Smoothing the Extrapolated Midpoint Rule" Numer. Math. 41 (1983): 165-175

[SG75] Shampine L. and M. Gordon. Computer Solutions of Ordinary Differential Equations. W. H. Freeman (1975)

[SGT03] Shampine L., I. Gladwell, and S. Thompson. Solving ODEs with MATLAB. Cambridge University Press (2003)

[SBB83] Shampine L. F., L. S. Baca, and H. J. Bauer. "Output in Extrapolation Codes" Comp. and Maths. with Appl. 9 (1983): 245-255

[SS03] Sofroniou M. and G. Spaletta. "Increment Formulations for Rounding Error Reduction in the Numerical Solution of Structured Differential Systems" Future Generation Computer Systems 19, no. 3 (2003): 375-383

[SS04] Sofroniou M. and G. Spaletta. "Construction of Explicit Runge-Kutta Pairs with Stiffness Detection" Mathematical and Computer Modelling, special issue on The Numerical Analysis of Ordinary Differential Equations, 40, no. 11-12 (2004): 1157-1169

[SS05] Sofroniou M. and G. Spaletta. "Derivation of Symmetric Composition Constants for Symmetric Integrators" Optimization Methods and Software 20, no. 4-5 (2005): 597-613

[SS06] Sofroniou M. and G. Spaletta. "Hybrid Solvers for Splitting and Composition Methods" J. Comp. Appl. Math., special issue from the International Workshop on the Technological Aspects of Mathematics, 185, no. 2 (2006): 278-291

[S84] Sottas G. "Dynamic Adaptive Selection Between Explicit and Implicit Methods When Solving ODEs" Report, Sect. de math, University of Genève, 1984

[S68] Strang G. "On the Construction of Difference Schemes" SIAM J. Num. Anal. 5 (1968): 506-517

[S70] Stetter H. J. "Symmetric Two-Step Algorithms for Ordinary Differential Equations" Computing 5 (1970): 267-280

[S90] Suzuki M. "Fractal Decomposition of Exponential Operators with Applications to Many-Body Theories and Monte Carlo Simulations" Phys. Lett. A 146 (1990): 319-323

[T59] Trotter H. F. "On the Product of Semi-Group Operators" Proc. Am. Math. Soc. 10 (1959): 545-551

[V78] Verner J. H. "Explicit Runge-Kutta Methods with Estimates of The Local Truncation Error" SIAM J. Num. Anal. 15 (1978): 772-790.

[V79] Vitasek E. "A-Stability and Numerical Solution of Evolution Problems" IAC 'Mauro Picone', Series III 186 (1979): 42

[W76] Whitham G. B. Linear and Nonlinear Waves. John Wiley and Sons (1976)

[WH91] Wisdom J. and M. Holman. "Symplectic Maps for the N-Body Problem" Astron. J. 102 (1991): 1528-1538

[Y90] Yoshida H. "Construction of High Order Symplectic Integrators" Phys. Lett. A. 150 (1990): 262-268

[Z98] Zanna A. "On the Numerical Solution of Isospectral Flows" Ph.D. Thesis, Cambridge University, 1998