Calculating the first m+1 partial quotients of the root of a polynomial using Lagrange's method

f(x) = a[n]xn + ··· + a[0] is a polynomial with integer coefficients, with a[n] > 0, having no rational roots and having exactly one real positive root α, this being > 1. The method of Lagrange (1797) is used to find the first m + 1 partial quotients of α.


  1. Number Theory with Computer Applications, R. Kumanduri and C. Romero, Prentice Hall 1997, page 261.
  2. Elements of Computer Algebra with Applications, A.G. Akritas, Wiley 1989, 333-399.
  3. Art of computer programming, volume 2, D.E. Knuth, problem 13, Ch. 4.5.3.
  4. Continued fractions for some algebraic numbers, S. Lang and H. Trotter, J. für Math. 255 (1972) 112-134; Addendum 267 (1974) ibid. 219-220.
  5. A new proof of Vincent's theorem, A. Alesina, M. Galuzzi, L'Enseignement Math 44 (1988), 219-256.
  6. An explanation of some exotic continued fractions found by Brillart, H.M. Stark, Computers in number theory, Academic Press 1971, 21-35.
The input polynomial is checked to see that (a) it is squarefree and (b) has exactly one positive real root t and that t > 1 holds.
Our program will in fact output the correct answer when t is rational.
For a program that develops the continued fraction of all real roots of a general polynomial with integer coefficients, see the rootexp(f(X),m) function in CALC.

Enter the coefficients a[0],...,a[n] (n > 1) (separated by spaces):
Enter m (≥ 0):

Last modified 20th July 2006
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