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

f(x) = a[n]x^{n} + ··· + 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.
### References

*Number Theory with Computer Applications*, R. Kumanduri and C. Romero, Prentice Hall 1997, page 261.
*Elements of Computer Algebra with Applications*, A.G. Akritas, Wiley 1989, 333-399.
*Art of computer programming*, volume 2, D.E. Knuth, problem 13, Ch. 4.5.3.
*Continued fractions for some algebraic numbers*, S. Lang and H. Trotter, J. für Math. 255 (1972) 112-134; Addendum 267 (1974) ibid. 219-220.
*A new proof of Vincent's theorem*, A. Alesina, M. Galuzzi, L'Enseignement Math **44** (1988), 219-256.
*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.

*Last modified 20th July 2006*

Return to main page