obtaining all numerical roots of a function in an interval

edit

def find_root_recursive(func,a,b,tol=0.000000000001):
    L = []
    try:
        x0 = find_root(func,a,b)
        L.append(x0)
        L += find_root_recursive(func,a,x0-tol,tol)       
        L += find_root_recursive(func,x0+tol,b,tol)       
    except:
        pass
    return L


L = find_root_recursive(sin,-50,50)
L.sort()
L

Formally, it can't be done. For example, $\sin\displaystyle\frac{1}{x}$ has an infinity of roots between -1 and 1...

If you run the following code you get [-0.973248989467730, 0.973248989467730, -0.229752920547361, 0.229752920547361], but if you uncomment assume(x>0) you get [0.973248989467730, 0.229752920547361].

x=var('x')
#assume(x>0)
f(x) = -x*x+x**4+0.05
roots = solve(f(x),x,solution_dict=True)
roots = [s[x] for s in roots]
num_roots = map(n, roots)
print num_roots