I need you help for solving this problem, and thanks in advantage for your help.
I have a polynom like P =x^6-4*x^3+x-2; and i would like to find an approximate value of the roots in some interval [a,b] =[-2,2] using sturm sequence.
The method is based on:
1) first construct the sturm sequence:
For given polynom P =x^6-4*x^3+x-2;
S[k] is the sturm sequence.
2) let f(a)= number of change of sign in the sturm sequence and f(b) the same . so f(b)-f(a) give the number of roots in the interval [a,b].
3) If f(b)-f(a) =0 so there are no roots
and if f(a)-f(b)=1 one can find the root
4) if f(a) -f(b) >2 :
given toterance tol=0.001; for example
if the abs(a-b)<2*epsilon we display a message that there are k roots at (b+a)/2
with our error tolerance
5) otherwise if c=(b+a)/2 is not a root of P_k(x) for any k, ( where p_k is an element of the sturm sequence )
we divide the interval into equal halves [a,c] and [x,b] and we run step 2 on each interval
else if c is a root of one of these p_k(x) add any time account to c so that c lies close the middle of [a,b] and not a root
6) Give all the roots ( approximate the rrots with small error epsilon).
I kindly appreciate your help