Glasnik Matematicki, Vol. 48, No. 2 (2013), 231-247.

HOUSEHOLDER'S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS

Vinko Petričević

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: vpetrice@math.hr


Abstract.   There are numerous methods for rational approximation of real numbers. Continued fraction convergent is one of them and Newton's iterative method is another one. Connections between these two approximation methods were discussed by several authors. Householder's methods are generalisation of Newton's method. In this paper, we will show that for these methods analogous connection with continued fractions hold.

2010 Mathematics Subject Classification.   11A55.

Key words and phrases.   Continued fractions, Householder's iterative methods.


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DOI: 10.3336/gm.48.2.02


References:

  1. G. Chrystal, Algebra, Part II, Chelsea, New York, 1964.
    MathSciNet    

  2. A. Dujella, Newton's formula and continued fraction expansion of √d, Experiment. Math. 10 (2001), 125-131.
    MathSciNet     CrossRef

  3. A. Dujella and V. Petričević, Square roots with many good approximants, Integers 5(3) (2005), #A6. (electronic)
    MathSciNet    

  4. N. Elezović, A note on continued fractions of quadratic irrationals, Math. Commun. 2 (1997), 27-33.
    MathSciNet    

  5. E. Frank, On continued fraction expansions for binomial quadratic surds, Numer. Math. 4 (1962), 85-95.
    MathSciNet     CrossRef

  6. E. Frank and A. Sharma, Continued fraction expansions and iterations of Newton's formula, J. Reine Angew. Math. 219 (1965), 62-66.
    MathSciNet    

  7. A. S. Householder, The Numerical Treatment of a Single Nonlinear Equation, McGraw-Hill, New York, 1970.
    MathSciNet    

  8. T. Komatsu, Continued fractions and Newton's approximants, Math. Commun. 4 (1999), 167-176.
    MathSciNet    

  9. J. Mikusiński, Sur la méthode d'approximation de Newton, Ann. Polon. Math. 1 (1954), 184-194.
    MathSciNet    

  10. R. A. Mollin, Infinite Families of Pellian Polynomials and their Continued Fraction Expansions, Results Math. 43 (2003), 300-317.
    MathSciNet     CrossRef

  11. R. A. Mollin and K. Cheng, Continued Fraction Beepers and Fibonacci Numbers, C. R. Math. Rep. Acad. Sci. Canada 24 (2002), 102-108.
    MathSciNet    

  12. O. Perron, Die Lehre von den Kettenbrüchen I, Dritte ed., B. G. Teubner Verlagsgesellschaft m.b.H., Stuttgart, 1954.
    MathSciNet    

  13. V. Petričević, Newton's approximants and continued fraction expansion of (1+√d)/2, Math. Commun. 17 (2012), 389-409.
    MathSciNet    

  14. P. Sebah and X. Gourdon, Newton's method and high order iterations, preprint, 2001, http://numbers.computation.free.fr/Constants/Algorithms/newton.ps

  15. A. Sharma, On Newton's method of approximation, Ann. Polon. Math. 6 (1959), 295-300.
    MathSciNet    

  16. K. S. Williams and N. Buck, Comparision of the lengths of the continued fractions of √D and (1+ √D)/2, Proc. Amer. Math. Soc. 120 (1994), 995-1002.
    MathSciNet     CrossRef

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