EXPLANATION FILE OF PROGRAM BERNOU ================================== 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]. ---------------------------------------- End of file Bernou.txt