I'm trying to numerically find the smallest real root of a polynomial in a given range. My initial plan was to shift the polynomial so that the bottom of the range was 0, expand the resulting expressions to find the new coefficients, then use the Jenkins-Traub algorithm until it finds the first (smallest) root or increases out of the range. However, the Wiki page says that it only generally finds the roots in increasing order, so it sounds like that's not guaranteed. Is there a way to guarantee finding the first root? If not, what is the most efficient way to solve the problem?
Finding all the roots then sorting is possible, but is inefficient as the degree of the polynomial gets larger, and, I hope, unnecessary. The bisection method is another common algorithm, but consider the polynomial: $-5x^2+5x-1$ with a range of (0,2), which would cause the angle bisection to fail. The best answer is one that guarantees success with the fastest time for a reasonable degree polynomial (less than say 10).