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Basics of Numerical Analysis

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Computational Methods in Physics

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Abstract

An ever increasing amount of computational work is being relegated to computers, and often we almost blindly assume that the obtained results are correct. At the same time, we wish to accelerate individual computation steps and improve their accuracy. Numerical computations should therefore be approached with a good measure of skepticism.

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References

  1. T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, 2nd edn. (MIT Press/McGraw-Hill, Cambridge/New York, 2001)

    MATH  Google Scholar 

  2. D.E. Knuth, The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd edn. (Addison-Wesley, Reading, 1997)

    Google Scholar 

  3. IEEE Standard 754-2008 for Binary Floating-point Arithmetic (IEEE, 2008), http://grouper.ieee.org/groups/754/

  4. D. Goldberg, What every computer scientist should know about floating-point arithmetic. ACM Comput. Surv. 23, 5 (1991). An up-to-date version is accessible as Appendix D of the Sun Microsystems Numerical Computation Guide, https://docs.oracle.com/cd/E19957-01/806-3568/

  5. M.H. Holmes, Introduction to Numerical Methods in Differential Equations (Springer-Verlag, New York, 2007). (Example in Appendix A.3.1)

    Google Scholar 

  6. D. O’Connor, Floating Point Arithmetic, Dublin Area Mathematics Colloquium, 5 Mar 2005

    Google Scholar 

  7. GNU Multi Precision (GMP), Free Library for Arbitrary Precision Arithmetic, http://gmplib.org

  8. H.J. Wilkinson, Rounding Errors in Algebraic Processes (Dover Publications, Mineola, 1994)

    MATH  Google Scholar 

  9. D.E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd edn. (Addison-Wesley, Reading, 1980)

    Google Scholar 

  10. J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, 3rd edn., Texts in Applied Mathematics, vol. 12 (Springer, Berlin, 2002)

    Chapter  Google Scholar 

  11. M.J.D. Powell, Approximation Theory and Methods (Cambridge University Press, Cambridge, 1981); See also C. Hastings, Approximations for Digital Computers (Princeton University Press, Princeton, 1955). Which is a pedagogical jewel; a seemingly simplistic outward appearance hides a true treasure-trove of ideas yielding one insight followed by another

    Google Scholar 

  12. W. Fraser, A survey of methods of computing minimax and near-minimax polynomial approximations for functions of a single variable. J. Assoc. Comput. Mach. 12, 295 (1965)

    Article  MathSciNet  Google Scholar 

  13. H.M. Antia, Numerical Methods for Scientists and Engineers, 2nd edn. (Birkhäuser, Basel, 2002). See Sections 9.11 and 9.12

    Google Scholar 

  14. R. Pachón, L.N. Trefethen, Barycentric-Remez algorithms for best polynomial approximation in the chebfun system, Oxford University Computing Laboratory, NAG Report No. 08/20

    Google Scholar 

  15. G.A. Baker, P. Graves-Morris, Padé Approximants, 2nd edn., Encyclopedia of Mathematics and its Applications, vol. 59 (Cambridge University Press, Cambridge, 1996)

    Google Scholar 

  16. P. Gonnet, S. Güttel, L.N. Trefethen, Robust Padé approximation via SVD. SIAM Rev. 55, 101 (2013)

    Article  MathSciNet  Google Scholar 

  17. P. Wynn, On the convergence and stability of the epsilon algorithm. SIAM J. Numer. Anal. 3, 91 (1966)

    Article  MathSciNet  ADS  Google Scholar 

  18. P.R. Graves-Morris, D.E. Roberts, A. Salam, The epsilon algorithm and related topics. J. Comput. Appl. Math. 12, 51 (2000)

    Article  MathSciNet  Google Scholar 

  19. W. van Dijk, F.M. Toyama, Accurate numerical solutions of the time-dependent Schrödinger equation. Phys. Rev. E 75, 036707 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  20. K. Kormann, S. Holmgren, O. Karlsson, J. Chem. Phys. 128, 184101 (2008); See also C. Lubich, Quantum Simulations of Complex Many-Body Systems: from Theory to Algorithms, J. Grotendorst, D. Marx, A. Muramatsu (eds.), John von Neumann Institute for Computing, Jülich, NIC Series, vol. 10 (2002), p. 459

    Google Scholar 

  21. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, 10th edn. (Dover Publications, Mineola, 1972)

    MATH  Google Scholar 

  22. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1980)

    MATH  Google Scholar 

  23. P.C. Abbott, Asymptotic expansion of the Keesom integral. J. Phys. A: Math. Theor. 40, 8599 (2007); M. Battezzati, V. Magnasco, Asymptotic evaluation of the Keesom integral. J. Phys. A: Math. Theor. 37, 9677 (2004)

    Google Scholar 

  24. T.M. Apostol, Mathematical Analysis, 2nd edn. (Addison-Wesley, Reading, 1974)

    MATH  Google Scholar 

  25. P.D. Miller, Applied Asymptotic Analysis, Graduate Studies in Mathematics, vol. 75 (AMS, Providence, 2006)

    Google Scholar 

  26. A. Erdélyi, Asymptotic Expansions (Dover, New York, 1987)

    MATH  Google Scholar 

  27. R. Wong, Asymptotic Approximations of Integrals (SIAM, Philadelphia, 2001)

    Book  Google Scholar 

  28. J. Wojdylo, Computing the coefficients in Laplace’s method. SIAM Rev. 48, 76 (2006). While [27] in Section II.1 describes the classical way of computing the coefficients \(c_s\) by series inversion, this article discusses a modern explicit method

    Google Scholar 

  29. F.J.W. Olver, Asymptotics and Special Functions (A. K. Peters, Wellesley, 1997)

    MATH  Google Scholar 

  30. S. Wolfram, Wolfram Mathematica, http://www.wolfram.com

  31. N. Bleistein, R.A. Handelsman, Asymptotic Expansion of Integrals (Holt, Reinhart and Winston, New York, 1975)

    MATH  Google Scholar 

  32. L.D. Landau, E.M. Lifshitz, Course in Theoretical Physics, Vol. 3: Quantum Mechanics, 3rd edn. (Pergamon Press, Oxford, 1991)

    Google Scholar 

  33. P.M. Morse, H. Feshbach, Methods of Theoretical Physics, vol. 1 (McGraw-Hill, Reading/New York, 1953)

    MATH  Google Scholar 

  34. A.D. Polyanin, V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. (Chapman & Hall/CRC, Boca Raton, 2003), p. 215

    MATH  Google Scholar 

  35. J. Boos, Classical and Modern Methods in Summability (Oxford University Press, Oxford, 2000). The collection of formulas and hints contained in the book T.J.I’a. Bromwich, An Introduction to the Theory of Infinite Series (Macmillan, London, 1955), also remains indispensable

    Google Scholar 

  36. A. Sofo, Computational Techniques for the Summation of Series (Kluwer Academic/Plenum Publishing, New York, 2003)

    Book  Google Scholar 

  37. M. Petkovšek, H. Wilf, D. Zeilberger, A=B (A. K. Peters Ltd., Wellesley, 1996)

    MATH  Google Scholar 

  38. D.M. Priest, On properties of floating-point arithmetics: Numerical stability and the cost of accurate computations, Ph.D. thesis, University of California at Berkeley (1992)

    Google Scholar 

  39. N.J. Higham, The accuracy of floating point summation. SIAM J. Sci. Comput. 14, 783 (1993)

    Article  MathSciNet  Google Scholar 

  40. J. Demmel, Y. Hida, Accurate and efficient floating point summation. SIAM J. Sci. Comput. 25, 1214 (2003)

    Article  MathSciNet  Google Scholar 

  41. W. Kahan, Further remarks on reducing truncation errors. Commun. ACM 8, 40 (1965)

    Article  Google Scholar 

  42. P. Linz, Accurate floating-point summation. Commun. ACM 13, 361 (1970)

    Article  MathSciNet  Google Scholar 

  43. T.O. Espelid, On floating-point summation. SIAM Rev. 37, 603 (1995)

    Article  MathSciNet  Google Scholar 

  44. I.J. Anderson, A distillation algorithm for floating-point summation. SIAM J. Sci. Comput. 20, 1797 (1999)

    Article  MathSciNet  Google Scholar 

  45. G. Bohlender, Floating point computation of functions with maximum accuracy. IEEE Trans. Comput. 26, 621 (1977)

    Article  MathSciNet  Google Scholar 

  46. D. Laurie, Convergence Acceleration, Appendix A (pp. 227–261) in the fascinating book F. Bornemann, D. Laurie, S. Wagon, J. Waldvögel, The SIAM 100-Digit Challenge. A Study in High-Accuracy Numerical Computing (SIAM, Philadelphia, 2004)

    Google Scholar 

  47. C. Brezinski, M.R. Zaglia, Extrapolation Methods (North-Holland, Amsterdam, 1991). A comprehensive historical review is offered by C. Brezinski, Convergence acceleration during the 20th century. J. Comput. Appl. Math. 122, 1 (2000)

    Google Scholar 

  48. E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep. 10, 189 (1989)

    Article  ADS  Google Scholar 

  49. K. Knopp, Theory and Application of Infinite Series (Blackie & Son, London, 1951)

    MATH  Google Scholar 

  50. http://www.nr.com/webnotes?5. The implementation described here is particularly attractive, as it automatically changes \(N\) and \(J\) such that the required maximum error is achieved most rapidly

  51. H. Cohen, F.R. Villegas, D. Zagier, Convergence acceleration of alternating series. Exp. Math. 9, 4 (2000)

    Article  MathSciNet  Google Scholar 

  52. H.H.H. Homeier, Scalar Levin-type sequence transformations. J. Comput. Appl. Math. 122, 81 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  53. GSL (GNU Scientific Library), http://www.gnu.org/software/gsl

  54. E.T. Whittaker, On the reversion of series. Gaz. Mat. 12(50), 1 (1951)

    MathSciNet  MATH  Google Scholar 

  55. R.P. Brent, T.H. Kung, Fast algorithms for manipulating formal power series. J. Assoc. Comput. Mach. 25, 581 (1978)

    Article  MathSciNet  Google Scholar 

  56. F. Johansson, A fast algorithm for reversion of power series. Math. Comput. 84, 475 (2015)

    Article  MathSciNet  Google Scholar 

  57. D.A. Leahy, J.C. Leahy, A calculator for Roche lobe properties. Comput. Astrophys. Cosmol. (2015) 2:4

    Google Scholar 

  58. G. Marsaglia, Evaluation of the normal distribution. J. Stat. Soft. 11, 1 (2004)

    Google Scholar 

  59. J.F. Hart et al., Computer Approximations (Wiley, New York, 1968); See also W.J. Cody, Rational Chebyshev approximations for the error function. Math. Comput. 23, 631 (1969)

    Google Scholar 

  60. J.R. Philip, The function inverfc \(\theta \). Austral. J. Phys. 13, 13 (1960). See also L. Carlitz, The inverse of the error function. Pac. J. Math. 13, 459 (1962)

    Google Scholar 

  61. O. Vallée, M. Soares, Airy Functions and Applications to Physics (Imperial College Press, London, 2004)

    Book  Google Scholar 

  62. G. Szegö, Orthogonal Polynomials (AMS, Providence, 1939)

    Book  Google Scholar 

  63. G.N. Watson, Theory of Bessel Functions (Cambridge University Press, Cambridge, 1922)

    MATH  Google Scholar 

  64. W.R. Gibbs, Computation in Modern Physics (World Scientific, Singapore, 1994). (See Sections 12.4 and 10.4)

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Simon Širca & Martin Horvat

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  1. Simon Širca
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Correspondence to Simon Širca .

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Širca, S., Horvat, M. (2018). Basics of Numerical Analysis. In: Computational Methods in Physics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-78619-3_1

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