A Test Set for Molecular Dynamics Algorithms

  • Eric Barth
  • Benedict Leimkuhler
  • Sebastian Reich
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 24)


This article describes a collection of model problems for aiding numerical analysts, code developers and others in the design of computational methods for molecular dynamics (MD) simulation. Common types of calculations and desirable features of algorithms are surveyed, and these are used to guide selection of representative models. By including essential features of certain classes of molecular systems, but otherwise limiting the physical and quantitative details, it is hoped that the test set can help to facilitate cross-disciplinary algorithm and code development efforts.


Molecular Simulation Pair Correlation Function Liquid Argon Langevin Dynamic Velocity Autocorrelation Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comp. Phys., 73 (1987), pp. 325–348CrossRefGoogle Scholar
  2. 2.
    R. Krasny and Z.-H. Duan, Treecode algorithms for computing nonbonded particle interactions, this volume.Google Scholar
  3. 3.
    M. Tuckerman, G. Martyna and B.J. Berne, Molecular Dynamics algorithms for multiple time scales: Systems with Long Range Forces, J. Chem. Phys., 94 (1991), pp. 6811–6815CrossRefGoogle Scholar
  4. 4.
    R.D. S Keel and J. Izaguirre, The Five Femtosecond Time Step Barrier, in P. Deuflhard, J. Hermans, B. Leimkuhler, A. Mark, S. Reich, R. D. Skeel, Computational Molecular Dynamics, Challenges, Methods, Ideas (Springer-Verlag), pp. 303–318, (1998)Google Scholar
  5. 5.
    T. Schlick, R. D. Skeel, A. T. Brunger, L. V. Kalé, J. Hermans, K. Schulten and J.A. Board Jr. , Algorithmic Challenges in Computational Molecular Biophysics, J. Comp. Phys., 151 (1999), pp. 9–48CrossRefGoogle Scholar
  6. 6.
    S. Nosé, A unified formulation of the constant temperature molecular dynamics methods, J. Chem. Phys., 81 (1984), pp. 511–519CrossRefGoogle Scholar
  7. 7.
    G.J. Martyna, M.E. Tuckerman, D.J. Tobias, and M.L. Klein, Explicit reversible integration algorithms for extended systems, Mol. Phys. 87 (1996), pp. 1117CrossRefGoogle Scholar
  8. 8.
    S.D. Bond, B.J. Leimkuhler and B.B. Laird, The Nose-Poincare method for constant temperature molecular dynamics, J. Comp. Phys. 151 (1999), pp. 114–134CrossRefGoogle Scholar
  9. 9.
    J.J. Moré, B.S. Garbow and K.E. Hillstrom, Testing Unconstrained Optimization Software, ACM TOMS, 7 (1981), 17–41CrossRefGoogle Scholar
  10. 10.
    E.D. Dolan and J.J. Moré, Benchmarking optimization software with COPS, Mathematics and Computer Science Division, Argonne National Laboratory, Technical Report ANL/MCS-246, November 2000 (Revised November 30), http: //www-unix.mcs.any.gov/~more/cops/Google Scholar
  11. 11.
    C.A. Floudas, P.M. Pardalos, C. Adjiman, W.R. Esposito, Z.H. Gümüs, S.T. Harding, J.L. Klepeis, C.A. Meyer, C.A. Schweiger, Handbook of test problems in local and global optimization, Volume 33 of Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, 1999Google Scholar
  12. 12.
    W.M. Lioen and J.J.B. De Swart, Test set for initial value problems, Report MAS-R 9832, Centrum voor Wiskunde en Informatica, Amsterdam http://www.cwi.nl/cwi/projects/IVPtestsetGoogle Scholar
  13. 13.
    E. Hairer and G. Wanner, Solving ordinary differential equations. Volume II, Springer Series in Comput. Mathematics, Vol. 14, Springer-Verlag 1996, see also http://www.zib.de/uwe.poehle/ode.htmlGoogle Scholar
  14. 14.
    Proceedings of the first meeting on the critical assessment of techniques for protein structure prediction, Proteins: Structure, Function and Genetics, 23 (1995), see also http://predictioncenter.llnl.govGoogle Scholar
  15. 15.
    B.J. Alder and T.E. Wainwright, Phase transition for a hard sphere system, J. Chem. Phys., 27 (1957), pp. 1208–1209CrossRefGoogle Scholar
  16. 16.
    J.D. Weeks, D. Chandler, and H.C. Andersen, Role of repulsive forces in determining the equilibrium structure of simple liquids, J. Chem. Phys., 54 (1971), pp. 5237–5247CrossRefGoogle Scholar
  17. 17.
    R.M. Stratt, S.L. Holmgren, and D. Chandler, Constrained impulsive molecular dynamics, Mol. Phys., 42 (1981), pp. 1233–1243CrossRefGoogle Scholar
  18. 18.
    S.-H. Suh, L. Mieryteran, H.S. White, and H.T. Davis, Molecular dynamics study of the primitive model of 1–3 electrolyte solutions, Chem. Phys., 142 (1990), pp. 203–211CrossRefGoogle Scholar
  19. 19.
    Y.A. Houndonougbo, B.B. Laird and B.J. Leimkuhler, Molecular dynamics algorithms for mixed hard-core/continuous potentials, Mol. Phys., 98 (1999), pp. 309–316CrossRefGoogle Scholar
  20. 20.
    A. Rahman, Correlations in the motion of atoms in liquid argon, Phys. Rev. A, 136 (1964), pp. 405–411Google Scholar
  21. 21.
    L. Verlet, Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard-Jones Molecules, Phys. Rev., 159 (1967), pp. 98–103CrossRefGoogle Scholar
  22. 22.
    R.D. Ruth, A canonical integration technique, IEEE Trans. Nucl. Sci., 30 (1983), p. 2669–2671CrossRefGoogle Scholar
  23. 23.
    F.M. Lasagni, Integration methods for Hamiltonian differential equations, Unpublished manuscript, (1990).Google Scholar
  24. 24.
    E. Hairer Backward analysis of numerical integrators and symplectic methods, in K. Burrage, C. Baker, P. v.d. Houwen, Z. Jackiewicz, and P. Sharp, editors, Scientific Computation and Differential Equations, volume 1 of Annals of Numer. Math., pp. 107–132, Amsterdam, J.C. Baltzer. 1994, Proceedings of the SCADE′93 conference, Auckland, New-Zealand, January 1993Google Scholar
  25. 25.
    G. Benettin and A. Giorgilli, On the Hamiltonian Interpolation of Near to the Identity Symplectic Mappings, J. Statist. Phys., 74 (1994), pp. 1117–1143CrossRefGoogle Scholar
  26. 26.
    A. Rahman and F.H. Stillinger, Molecular dynamics study of liquid water, J. Chem. Phys., 55 (1971), pp. 3336–3359CrossRefGoogle Scholar
  27. 27.
    J.P. Ryckaert, G. Ciccotti and H. J.C. Berendsen, Numerical integration of the Cartesian equations of motion of a system with constraints: molecular dynamics of n-alkanes, J. Comp. Phys., 23 (1977), pp. 327–341CrossRefGoogle Scholar
  28. 28.
    J.A. Mccammon, B.R. Gelin and M. Karplus, Dynamics of folded proteins, Nature, 267 (1977), pp. 585–590CrossRefPubMedGoogle Scholar
  29. 29.
    A.K. Mazur, V.E. Dorofeev and R.A. Abagyan, Derivation and testing of explicit equations of motion for polymers described by internal coordinates, J. Comp. Phys., 92 (1991), pp. 261–272CrossRefGoogle Scholar
  30. 30.
    S. He and H.A. Scheraga, Macromolecular conformational dynamics in torsional angle space, J. Chem. Phys., 108 (1998), pp. 271–286CrossRefGoogle Scholar
  31. 31.
    S.-H. Lee, K. Palmo and S. Krimm, A new formalism for molecular dynamics in internal coordinates, J. Chem. Phys., to appearGoogle Scholar
  32. 32.
    B.R. Brooks, R.E. Bruccoleri, B.D. Olafson, D.J. States, S. Swaminathan and M. Karplus, CHARMM: A program for macromolecular energy, minimization, and dynamics calculations, J. Comp. Chem, 4 (1983), pp. 187–217CrossRefGoogle Scholar
  33. 33.
    A.D. Mackerell Jr., D. Bashford, M. Bellott, R.L. Dunbrack Jr., J. Evanseck, M.J. Field, S. Fischer, J. Gao, H. Guo, S. Ha, D. Joseph, L. Kuchnir, K. Kuczera, F.T.K. Lau, C. Mattos, S. Michnick, T. Ngo, D.T. Nguyen, B. Prodhom, W.E. Reiher Iii, B. Roux, M. Schlenkrich, J. Smith, R. Stote, J. Straub, M. Watanabe, J. Wiorkiewicz-Kuczera, D. Yin and M. Karplus, An all-atom empirical potential for molecular modeling and dynamics of proteins, J. Phys. Chem., 102 (1998), pp. 3586–3616CrossRefGoogle Scholar
  34. 34.
    S.J. Weiner, P.A. Kollman, D.T. Nguyen, and D.A. Case, An all atom force field for simulations of proteins and nucleic acids, J. Comp. Chem., 7 (1986), pp. 230–252 http://www.amber.ucsf.edu/amber/CrossRefGoogle Scholar
  35. 35.
    W.D. Cornell, P. Cieplak, C.I. Bayly, I.R. Gould, K.M. Merz, Jr, D.M. Ferguson, D.C. Spellmeyer, T. Fox, J.W. Caldwell and P.A. Kollman, A second generation force field for the simulation of proteins and nucleic acids, J. Am. Chem. Soc, 117 (1995), pp. 5179–5197CrossRefGoogle Scholar
  36. 36.
    W.L. Jorgensen and J. Tirado-Rives, The OPLS potential functions for proteins. Energy minimization for crystals of cyclic peptides and crambin, J. Am. Chem. Soc., 110 (1988), pp. 1657–1666CrossRefGoogle Scholar
  37. 37.
    http://bmbiris.bmb.uga.edu/wampler/8200/using-ff/mmrefs.html.Google Scholar
  38. 38.
    T. Schlick, Optimization methods in computational chemistry, in Reviews in Computational Chemistry, Volume 3, Chapter 1, pages 1–71, K. B. Lipkowitz and D. B. Boyd eds., VCH Publishers, New York (1992)CrossRefGoogle Scholar
  39. 39.
    N.B. Slater, Classical motion under a Morse potential, Nature, 180 (1957), pp. 1352–1353CrossRefGoogle Scholar
  40. 40.
    M. Mandziuk and T. Schlick, Resonance in chemal systems simulated by the implicit midpoint method, Chem. Phys. Lett., 237 (1995), pp. 525–535CrossRefGoogle Scholar
  41. 41.
    J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian problems, Chapman and Hall, 1994.Google Scholar
  42. 42.
    J. Frank, W. Huang and B. Leimkuhler, Geometric integrators for classical spin systems, J. Comp. Phys., 133 (1997), pp. 160–172.CrossRefGoogle Scholar
  43. 43.
    H.C. Andersen, Rattle: a ‘velocity’ version of the shake algorithm for molecular dynamics calculations, J. Comp. Phys., 52 (1983), pp. 24–34CrossRefGoogle Scholar
  44. 44.
    B. Leimkuhler and R.D. Skeel, Symplectic numerical integrators in constrained Hamiltonian systems, J. Comp. Phys., 112 (1994), pp. 117–125CrossRefGoogle Scholar
  45. 45.
    E. Barth, K. Kuczera, B. Leimkuhler and R.D. Skeel, Algorithms for constrained molecular dynamics, J. Comp. Chem., 16 (1995), pp. 1192–1209CrossRefGoogle Scholar
  46. 46.
    D.J. Evans, Computer “experiment” for nonlinear thermodynamics of Couette flow, J. Chem. Phys., 78 (1983), pp. 3297–3302CrossRefGoogle Scholar
  47. 47.
    R.J. Loncharich, B.R. Brooks and R.W. Pastor, Langevin dynamics of peptides: The frictional dependence of isomerization rates of N-Acetylalanyl-N′-Methylamide, Biopolymers, 32 (1992), pp. 523–535CrossRefPubMedGoogle Scholar
  48. 48.
    S. Nosé, A molecular dynamics method for simulations in the canonical ensemble, Mol. Phys., 52 (1984), pp. 255–268CrossRefGoogle Scholar
  49. 49.
    J.B. Sturgeon and B.B. Laird, Symplectic algorithm for constant-pressure molecular-dynamics using a Nose-Poincare thermostat, J. Chem. Phys. 112 (2000), 3474CrossRefGoogle Scholar
  50. 50.
    G.J. Martyna, M.L. Klein, and M.E. Tuckerman, Nose-Hoover chains: The canonical ensemble via continuous dynamics, J. Chem. Phys., 97 (1992), pp. 2635–2643CrossRefGoogle Scholar
  51. 51.
    W.G. Hoover, C.G. Hoover, and D.J. Isbister, Chaos, ergodic convergence, and fractal instability for a thermostatted canonical harmonic oscillator, Phys. Rev. E, 63 (2001), 026029CrossRefGoogle Scholar
  52. 52.
    D. Okunbor and R.D. Skeel, Canonical numerical methods for molecular dynamics simulations, J. Comp. Chem., 15 (1994), pp. 72–79CrossRefGoogle Scholar
  53. 53.
    J.L. Yarnell, M.J. Katz, R.G. Wenzel and S.H. Koenig, Structure factor and radial distribution function for liquid argon at 85°K, Phys. Rev. A, 7 (1973), pp. 2130–2144CrossRefGoogle Scholar
  54. 54.
    A.K. Soper, On the determination of the pair correlation function from liquid structure factor measurements, Chem. Phys., 107 (1986), pp. 61–74CrossRefGoogle Scholar
  55. 55.
    M.P. Allen and D.J. Tildesley, Computer simulation of liquids, Oxford Science Publications, 1987Google Scholar
  56. 56.
    D.C. Rapaport, The art of molecular dynamics simulation, Cambridge University Press, 1995, http://uk.cambridge.org/physics/resourceGoogle Scholar
  57. 57.
    D. Frenkel and B. Smit, Understanding molecular simulation. From algorithms to applications, Academic Press, 1996Google Scholar
  58. 58.
    H. Gould and J. Tobochnik, An introduction to computer simulation methods: Applications to physical systems, Addison-Wesley, 1988Google Scholar
  59. 59.
    J.A. Izaguirre, S. Reich and R.D. Skeel, Longer time steps for molecular dynamics, J. Chem. Phys, 110 (1999), pp. 9853–9864CrossRefGoogle Scholar
  60. 60.
    F.H. Stillinger, Theory and molecular models for water, Adv. Chem. Phys., 31 (1975), pp. 1–101CrossRefGoogle Scholar
  61. 61.
    F.H. Stillinger, Water revisited, Science, 209 (1980), pp. 451–457CrossRefPubMedGoogle Scholar
  62. 62.
    W. Jorgensen, J. Chandrasekar, J. Madura and R. Impey and M. Klein, Comparison of simple potential functions for simulating liquid water, J. Chem. Phys., 79 (1983), pp. 926–935CrossRefGoogle Scholar
  63. 63.
    H.J.C. Berendsen, J.P.M. Postma, W.F. Van Gunsteren and J. Hermans, in Intermolecular Forces, B. Pullman, Editor, Reidel, Dordrecht, 1981Google Scholar
  64. 64.
    F.H. Stillinger and A. Rahman, Improved simulation of liquid water by molecular dynamics, J. Chem. Phys., 60 (1974), pp. 1545–1557CrossRefGoogle Scholar
  65. 65.
    M.W. Mahoney and W.L. Jorgensen, A five-site model for liquid water and the reproduction of the density anomaly by rigid, nonpolarizable potential functions, J. Chem. Phys., 112 (2000), pp. 8910–8922CrossRefGoogle Scholar
  66. 66.
    K. Toukan and A. Rahman, Molecular dynamics study of atomic motions in water, Phys. Rev. B., 32 (1985), pp. 2643–2648CrossRefGoogle Scholar
  67. 67.
    H.J.C. Berendsen, J.R. Grigera, and T.P. Straatsma, The missing term in effective pair potentials, J. Phys. Chem., 91 (1987), pp. 6269–6271CrossRefGoogle Scholar
  68. 68.
    S.J. Stuart and B.J. Berne, Effects of Polarizability on the Hydration of the Chloride Ion, J. Phys. Chem., 100 (1996), pp. 11934–11943CrossRefGoogle Scholar
  69. 69.
    J. Anderson, J.J. Ullo and S. Yip, Molecular dynamics simulation of dielectric properties of water, J. Chem. Phys., 87 (1987), pp. 1726–1732CrossRefGoogle Scholar
  70. 70.
    R.R. Gabdoulline and Chong Zheng, Effects of the cutoff center on the mean potential and pair distribution functions in liquid water, J. Comp. Chem., 16 (1995), pp. 1428–1433CrossRefGoogle Scholar
  71. 71.
    M. Saito, Molecular dynamics simulations of proteins in solution: Artifacts caused by the cutoff approximation, J. Chem. Phys., 101 (1994), pp. 4055–4061CrossRefGoogle Scholar
  72. 72.
    R.M. Levy and E. Gallicchio, Computer simulations with explicit solvent: Recent progress in the thermodynamic decomposition of free energies and in modeling electrostatic effects, Annu. Rev. Phys. Chem., 49 (1998), pp. 531–567CrossRefPubMedGoogle Scholar
  73. 73.
    A.W. Appel, An efficient program for many-body simulations, SIAM J. Sci Stat. Comput., 6 (1985), pp. 85–103CrossRefGoogle Scholar
  74. 74.
    J. Barnes and P. Hut, A hierarchical O(N log N) force calculation algorithm, Nature, 324 (1986), pp. 446–449CrossRefGoogle Scholar
  75. 75.
    Z.-H. Duan and R. Krasny, An adaptive treecode for computing nonbonded potential energy in classical molecular systems, J. Comp. Chem., 21 (2000), pp. 1–12CrossRefGoogle Scholar
  76. 76.
    R.W. Hockney and J.W. Eastwood, Computer simulation using particles, McGraw-Hill, New York, 1981Google Scholar
  77. 77.
    T. Darden, D. York and L. Pedersen, Particle mesh Ewald: an N*log(N) method for computing Ewald sums, J. Chem. Phys., 98 (1993), pp. 10089–10092CrossRefGoogle Scholar
  78. 78.
    Z.-H. Duan and R. Krasny, An Ewald summation based multipole method, J. Chem. Phys., 113 (2000), pp. 3492–3495 http://www. math. lsa. umich. edu/~zduan/math/CrossRefGoogle Scholar
  79. 79.
    E. Barth and T. Schlick, Overcoming stability limitations in biomolecular dynamics: Combining force splitting via extrapolation with Langevin dynamics in LN, J. Chem. Phys., 109 (1998), pp. 1617–1632CrossRefGoogle Scholar
  80. 80.
    T. Bishop, R. Skeel and K. Schulten, Difficulties with multiple timestep-ping and the fast multipole algorithm in molecular dynamics, J. Comp. Chem., 18 (1997), pp. 1785–1791CrossRefGoogle Scholar
  81. 81.
    P. Procacci, M. Marchi and G. Martyna, Electrostatic calculations and multiple time scales in molecular dynamics simulation of flexible molecular systems, J. Chem. Phys., 108 (1998), pp. 8799–8803CrossRefGoogle Scholar
  82. 82.
    A.K. Soper and M.G. Phillips, A new determination of the structure of water at 25C, Chem. Phys., 107 (1986), pp. 47–60CrossRefGoogle Scholar
  83. 83.
    J. Zhou, S. Reich and B.R. Brooks, Elastic molecular dynamics with self-consistent flexible constraints, J. Chem. Phys., 112 (2000), pp. 7919–7929CrossRefGoogle Scholar
  84. 84.
    M. Tuckerman, B.J. Berne and G.J. Martyna, Reversible multiple time scale molecular dynamics, J. Chem. Phys., 97 (1992) pp. 1990–2001CrossRefGoogle Scholar
  85. 85.
    M. Tuckerman and B.J. Berne, Molecular dynamics in systems with multiple timescales — systems with stiff and soft degrees of freedom and with short and long-range forces, J. Chem. Phys., 95 (1991), pp. 8362–8364CrossRefGoogle Scholar
  86. 86.
    H. Grubmüller, H. Heller, A. Windemuth and K. Schulten, Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions, Mol. Sim., 6 (1991), pp. 121–142CrossRefGoogle Scholar
  87. 87.
    R. Zhou and B.J. Berne, A new molecular dynamics method combining the reference system propagator algorithm with a fast multipole method for simulating proteins and other complex systems, J. Chem. Phys., 103 (1995), pp. 9444–9459CrossRefGoogle Scholar
  88. 88.
    P. Procacci and M. Marchi, Taming the Ewald sum in molecular dynamics simulations of solvated proteins via a multiple time step algorithm, J. Chem. Phys., 104 (1996), pp. 3003–3012CrossRefGoogle Scholar
  89. 89.
    P. Procacci and B. Berne, Computer simulation of solid C 60 using multiple time-step algorithms, J. Chem. Phys., 101 (1994), pp. 2421–2431CrossRefGoogle Scholar
  90. 90.
    W.C. Still, A. Tempczyk R.C. Hawley and T. Hendrickson, Semian-alytical treatment of solvation for molecular mechanics and dynamics, J. Am. Chem. Soc., 112 (1990), pp. 6127–6129CrossRefGoogle Scholar
  91. 91.
    D. Qiu, P.S. Shenkin, F.P. Hollinger and W.C. Still, The GB/SA continuum model for solvations. A fast analytical method for the calculation of approximate Born radii, J. Phys. Chem. A, 101 (1997), pp. 3005–3014CrossRefGoogle Scholar
  92. 92.
    P.J. Kraulis, MOLSCRIPT: A program to produce both detailed and schematic plots of protein structures, J. of Appl. Cryst., 24 (1991), pp. 946–950. http:/ /www. avatar.se/molscript/CrossRefGoogle Scholar
  93. 93.
    H.B. Thompson, Calculation of Cartesian coordinates and their derivatives from internal molecular coordinates, J. Chem. Phys., 47 (1967), pp. 3407–3410CrossRefGoogle Scholar
  94. 94.
    J. Hermans, Rationalization of Molecular Models, Methods in Enzymology, 115 (1985), pp. 171–189CrossRefPubMedGoogle Scholar
  95. 95.
    H. Bekker, H. J.C. Berendsen and W.F. Van Gunsteren, Force and virial of torsional-angle dependent potentials, J. Comput. Chem., 16 (1995), 527–533CrossRefGoogle Scholar
  96. 96.
    G. Zhang and T. Schlick, LIN: A new algorithm combining implicit integration and normal mode techniques for molecular dynamics, J. Comp. Chem., 14 (1993), pp. 1212–1233CrossRefGoogle Scholar
  97. 97.
    D.J. Tobias and C.L. Brooks III, Molecular dynamics with internal coordinate constraints, J. Chem. Phys., 89 (1988), pp. 5115–5127CrossRefGoogle Scholar
  98. 98.
    E. Barth, M. Mandziuk and T. Schlick, A separating framework for increasing the timestep in molecular dynamics, in Computer Simulation of Biomolecular Systems: Theoretical and Experimental Applications, Volume 3, chapter 4, W.F. van Gunsteren, P.K. Weiner and A.J. Wilkinson, ESCOM, Leiden, The Netherlands, 1996Google Scholar
  99. 99.
    H.B. Grubmüller and Paul Tavan, Molecular dynamics of conformational sub states for a simplified protein model, J. Chem. Phys., 101 (1994), pp. 5047–5057CrossRefGoogle Scholar
  100. 100.
    J.D. Honeycutt and D. Thirumalai, Metastability of the folded states of globular proteins, Proc. Natl. Acad. Sci. USA, 87 (1990), pp. 3526–3529CrossRefPubMedGoogle Scholar
  101. 101.
    J.-E. Shea, Y.D. Nochomovitz, Z. Guo and C.L Brooks, III, Exploring the space of protein folding Hamiltonians: The balance of forces in a minimalist ß-barrel model, J. Chem. Phys, 109 (1998), pp. 2895–2903CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Eric Barth
    • 1
  • Benedict Leimkuhler
    • 2
  • Sebastian Reich
    • 3
  1. 1.Department of Mathematics and Computer ScienceKalamazoo CollegeKalamazooUSA
  2. 2.Department of Mathematics and Computer ScienceUniversity of LeicesterLeicesterUK
  3. 3.Department of MathematicsImperial CollegeLondonUK

Personalised recommendations