Advertisement

Matrix Algebra pp 461-521 | Cite as

Numerical Methods

  • James E. Gentle
Chapter
  • 5.9k Downloads
Part of the Springer Texts in Statistics book series (STS)

Abstract

The computer is a tool for storage, manipulation, and presentation of data. The data may be numbers, text, or images, but no matter what the data are, they must be coded into a sequence of 0s and 1s because that is what the computer stores. For each type of data, there are several ways of coding. For any unique coding scheme, the primary considerations are efficiency in storage, retrieval, and computations. Each of these considerations may depend on the computing system to be used. Another important consideration is coding that can be shared or transported to other systems.

References

  1. Alefeld, Göltz, and Jürgen Herzberger. (1983). Introduction to Interval Computation. New York: Academic Press.zbMATHGoogle Scholar
  2. ANSI. 1989. American National Standard for Information Systems — Programming Language C, Document X3.159-1989. New York: American National Standards Institute.Google Scholar
  3. ANSI. 1992. American National Standard for Information Systems — Programming Language Fortran-90, Document X3.9-1992. New York: American National Standards Institute.Google Scholar
  4. ANSI. 1998. American National Standard for Information Systems — Programming Language C++, Document ISO/IEC 14882-1998. New York: American National Standards Institute.Google Scholar
  5. Bailey, David H. 1993. Algorithm 719: Multiprecision translation and execution of FORTRAN programs. ACM Transactions on Mathematical Software 19:288–319.CrossRefzbMATHGoogle Scholar
  6. Bailey, David H. 1995. A Fortran 90-based multiprecision system. ACM Transactions on Mathematical Software 21:379–387.CrossRefzbMATHGoogle Scholar
  7. Bickel, Peter J., and Joseph A. Yahav. 1988. Richardson extrapolation and the bootstrap. Journal of the American Statistical Association 83:387–393.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Blackford, L. S., J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley. 1997a. ScaLAPACK Users’ Guide. Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar
  9. Calvetti, Daniela. 1991. Roundoff error for floating point representation of real data. Communications in Statistics 20:2687–2695.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Chaitin-Chatelin, Françoise, and Valérie Frayssé. 1996. Lectures on Finite Precision Computations. Philadelphia: Society for Industrial and Applied Mathematics.CrossRefzbMATHGoogle Scholar
  11. Chan, T. F., G. H. Golub, and R. J. LeVeque. 1982. Updating formulae and a pairwise algorithm for computing sample variances. In Compstat 1982: Proceedings in Computational Statistics, ed. H. Caussinus, P. Ettinger, and R. Tomassone, 30–41. Vienna: Physica-Verlag.Google Scholar
  12. Chan, Tony F., Gene H. Golub, and Randall J. LeVeque. 1983. Algorithms for computing the sample variance: Analysis and recommendations. The American Statistician 37:242–247.MathSciNetzbMATHGoogle Scholar
  13. Cody, W. J. 1988. Algorithm 665: MACHAR: A subroutine to dynamically determine machine parameters. ACM Transactions on Mathematical Software 14:303–329.CrossRefzbMATHGoogle Scholar
  14. Cody, W. J., and Jerome T. Coonen. 1993. Algorithm 722: Functions to support the IEEE standard for binary floating-point arithmetic. ACM Transactions on Mathematical Software 19:443–451.CrossRefzbMATHGoogle Scholar
  15. Dempster, Arthur P., and Donald B. Rubin. 1983. Rounding error in regression: The appropriateness of Sheppard’s corrections. Journal of the Royal Statistical Society, Series B 39:1–38.zbMATHGoogle Scholar
  16. Gentle, James E. 2009. Computational Statistics. New York: Springer-Verlag.CrossRefzbMATHGoogle Scholar
  17. Gregory, R. T., and E. V. Krishnamurthy. 1984. Methods and Applications of Error-Free Computation. New York: Springer-Verlag.CrossRefzbMATHGoogle Scholar
  18. Grewal, Mohinder S., and Angus P. Andrews. 1993. Kalman Filtering Theory and Practice. Englewood Cliffs, New Jersey: Prentice-Hall.Google Scholar
  19. Gropp, William D. 2005. Issues in accurate and reliable use of parallel computing in numerical programs. In Accuracy and Reliability in Scientific Computing, ed. Bo Einarsson, 253–263. Philadelphia: Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
  20. Higham, Nicholas J. 2002. Accuracy and Stability of Numerical Algorithms, 2nd ed. Philadelphia: Society for Industrial and Applied Mathematics.CrossRefzbMATHGoogle Scholar
  21. IEEE. 2008. IEEE Standard for Floating-Point Arithmetic, Std 754-2008. New York: IEEE, Inc.Google Scholar
  22. Jansen, Paul, and Peter Weidner. 1986. High-accuracy arithmetic software — some tests of the ACRITH problem-solving routines. ACM Transactions on Mathematical Software 12:62–70.MathSciNetCrossRefzbMATHGoogle Scholar
  23. Jaulin, Luc, Michel Kieffer, Olivier Didrit, and Eric Walter. (2001). Applied Interval Analysis. New York: Springer.CrossRefzbMATHGoogle Scholar
  24. Kearfott, R. Baker. 1996. Interval_arithmetic: A Fortran 90 module for an interval data type. ACM Transactions on Mathematical Software 22:385–392.CrossRefzbMATHGoogle Scholar
  25. Kearfott, R. Baker, and Vladik Kreinovich (Editors). 1996. Applications of Interval Computations. Netherlands: Kluwer, Dordrecht.zbMATHGoogle Scholar
  26. Kearfott, R. B., M. Dawande, K. Du, and C. Hu. 1994. Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Transactions on Mathematical Software 20:447–459.CrossRefzbMATHGoogle Scholar
  27. Keller-McNulty, Sallie, and W. J. Kennedy. 1986. An error-free generalized matrix inversion and linear least squares method based on bordering. Communications in Statistics — Simulation and Computation 15:769–785.MathSciNetCrossRefzbMATHGoogle Scholar
  28. Kshemkalyani, Ajay D., and Mukesh Singhal. 2011. Distributed Computing: Principles, Algorithms, and Systems. Cambridge, United Kingdom: Cambridge University Press.zbMATHGoogle Scholar
  29. Kulisch, Ulrich. 2011. Very fast and exact accumulation of products. Computing 91:397–405.MathSciNetCrossRefzbMATHGoogle Scholar
  30. Lemmon, David R., and Joseph L. Schafer. 2005. Developing Statistical Software in Fortran 95. New York: Springer-Verlag.zbMATHGoogle Scholar
  31. Levesque, John, and Gene Wagenbreth. 2010. High Performance Computing: Programming and Applications. Boca Raton: Chapman and Hall/CRC Press.CrossRefGoogle Scholar
  32. Liem, C. B., T. Lü, and T. M. Shih. 1995. The Splitting Extrapolation Method. Singapore: World Scientific.CrossRefzbMATHGoogle Scholar
  33. Linnainmaa, Seppo. 1975. Towards accurate statistical estimation of rounding errors in floating-point computations. BIT 15:165–173.MathSciNetCrossRefzbMATHGoogle Scholar
  34. Metcalf, Michael, John Reid, and Malcolm Cohen. 2011. Modern Fortran Explained. Oxford, United Kingdom: Oxford University Press.zbMATHGoogle Scholar
  35. Moore, Ramon E. (1979). Methods and Applications of Interval Analysis. Philadelphia: Society for Industrial and Applied Mathematics.CrossRefzbMATHGoogle Scholar
  36. Nakano, Junji. 2012. Parallel computing techniques. In Handbook of Computational Statistics: Concepts and Methods, 2nd revised and updated ed., ed. James E. Gentle, Wolfgang Härdle, and Yuichi Mori, 243–272. Berlin: Springer.CrossRefGoogle Scholar
  37. Overton, Michael L. 2001. Numerical Computing with IEEE Floating Point Arithmetic. Philadelphia: Society for Industrial and Applied Mathematics.CrossRefzbMATHGoogle Scholar
  38. Parsian, Mahmoud. 2015. Data Algorithms. Sabastopol, California: O’Reilly Media, Inc.Google Scholar
  39. Stallings, W. T., and T. L. Boullion. 1972. Computation of pseudo-inverse using residue arithmetic. SIAM Review 14:152–163.MathSciNetCrossRefzbMATHGoogle Scholar
  40. Szabó, S., and R. Tanaka. 1967. Residue Arithmetic and Its Application to Computer Technology. New York: McGraw-Hill.zbMATHGoogle Scholar
  41. Unicode Consortium. 1990. The Unicode Standard, Worldwide Character Encoding, Version 1.0, Volume 1. Reading, Massachusetts: Addison-Wesley Publishing Company.Google Scholar
  42. Unicode Consortium. 1992. The Unicode Standard, Worldwide Character Encoding, Version 1.0, Volume 2. Reading, Massachusetts: Addison-Wesley Publishing Company.Google Scholar
  43. Walster, G. William. 1996. Stimulating hardware and software support for interval arithmetic. In Applications of Interval Computations, ed. R. Baker Kearfott and Vladik Kreinovich, 405–416. Dordrecht, Netherlands: Kluwer.CrossRefGoogle Scholar
  44. Walster, G. William. 2005. The use and implementation of interval data types. In Accuracy and Reliability in Scientific Computing, ed. Bo Einarsson, 173–194. Philadelphia: Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
  45. Wilkinson, J. H. 1959. The evaluation of the zeros of ill-conditioned polynomials. Numerische Mathematik 1:150–180.MathSciNetCrossRefzbMATHGoogle Scholar
  46. Wilkinson, J. H. 1963. Rounding Errors in Algebraic Processes. Englewood Cliffs, New Jersey: Prentice-Hall. (Reprinted by Dover Publications, Inc., New York, 1994).Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • James E. Gentle
    • 1
  1. 1.FairfaxUSA

Personalised recommendations