EXPLANATION FILE OF PROGRAM BERNOU
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  Bernouilli's Method
  -------------------

    This method allows calculating the greatest root of the polynomial, alpha;
  by dividing the polynomial by (x-alpha), we can apply again the method and so
  on to determine all the roots.

    Let us consider the equation phi, associated to the polynomial

                      n      n-1
           Pn(x) = a0x  + a1x    + ... + an-1x + an:

           phi = a  y  + a  y   + ... + a y    = 0
                  0  k    1  k-1         n

    The particular solutions of this equation (looking like a linear differen-
  tial equation) have the form yk = beta^k where the beta's are the roots of the
  equation:

                 k         k-1               k-n
           a beta  + a beta    + ... + a beta    = 0
            0         1                 n

  of the same kind than Pn(x) = 0 the roots of which, supposed distinct, are
  x1, x2, ... xn. The equation phi being linear, the general solution is a lin-
  ear combination:

                   k      k            k
           y  = c x  + c x  + ... + c x
            k    1 1    2 2          n n

    If the roots are such than |x1| > |x2| > |x3| > ... > |xn| then

                   k                     k
           y  = c x  [ 1 + (c2/c1)(x2/x1)  + ...]
            k    1 1

                     k+1                    k+1
    and    y    = c x    [1 + (c2/c1)(x2/x1)    + ...]
            k+1    1 1

                                             k+1
                          [1 + (c2/c1)(x2/x1)   + ...]
    hence  y   / y  =  x1 ----------------------------  
            k+1   k                          k
                          [1 + (c2/c1)(x2/x1)  + ...]


    as x1 > x2, (x2/x1)^k, (x3/x1)^k ... --> 0 when k --> infinity

    so y   / y  --> x1 when k --> inf and  x1 = Lim    (y   / y )
        k+1   k                                k-->inf.  k+1   k

    If equation phi has n+1 coefficients, a particular solution, yk can be de
  termined from the n values, y   , y   ,...,y     y   , this allows recursi-
                                   k-1   k-2      n-2,  n-1
  vely calculating particular solutions of equation phi.

    To calculate the greatest root of the polynomial, we start from particular
  solutions: y0=0, y1=0, y2=0,...,yn-1=1 to calculate yn, also a particular
  solution. The same process is used to calculate yn+1 from y1=0, y2=0...yn-1=1,
  yn and so on until yK+1/yk almost does not vary.

  From [BIBLI 04].
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End of file Bernou.txt