This polynomial: x^4 - 48*x^3 - 12*x^2 - 33*x + 1613 has a root very near Pi. How can I know if this is unusual for the size of the coefficients? -- I think that it is. I suppose this problem is in the category you describe -- but sorry I don't know how to do it. Jim On Tue, Sep 16, 2008 at 8:42 AM, Christian Boyer <cboyer@club-internet.fr> wrote:
Let f(x , x , ..., x ) = k x + k x + ... + k x 1 2 n 1 1 2 2 n n
where k , k , ..., k are n REAL constants 1 2 n
and x , x , ..., x are n INTEGERS (positive or negative) 1 2 n
If we try to have the best possible approximations of
f(x , x , ... x ) = 0 1 2 n
do you know good algorithms giving best values of (x , x , ..., x )? 1 2 n
When n = 2, a good algorithm is to compute the continued fraction of k / k 2 1 At each step of the algorithm, the continued fraction will give more and more excellent (x , x ). 1 2
But which algorithm(s) for n > 2?
Christian.