EXPLANATION FILE OF PROGRAM BERNOU
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Bernouilli's Method
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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