Numerically find all roots in an interval

edit

I don't think that there's anything in Sage, but you can use something like the following function:

def find_all_roots(f, a, b, eps=0.0000000001):
    roots = []
    intervals_to_check = [(a,b)]
    while intervals_to_check:
        start, end = intervals_to_check.pop()
        try:
            root = find_root(f, start, end)
        except RuntimeError:
            continue
        if root in roots:
            continue
        if abs(f(root)) < 1:
            roots.append(root)
        intervals_to_check.extend([(start, root-eps), (root+eps, end)])
    roots.sort()
    return roots

It then behaves like the following:

sage: f =  tan(z)+z/sqrt(9*pi^2-z^2)
sage: find_all_roots(f, 0, 9.4)
[0.0, 2.8359523267114892, 5.6414610103726988, 8.3387745764121703]

Note that I had to change the endpoint since find_root does not like it when the function is not defined.