Determining the properties of the Bisection method.

Question

If we use bisection to determine the roots of a function, can we say that the closer the function values of the two starting points are to zero, the faster the process converges? Also what happens if we have more than $2$ roots in the start interval? Will bisection then not converge?

Answer 1

1) No, not necessarily. For instance the endpoints of the interval might be close to nearby roots, outside.

2) If the function is continuous and you have a change of sign, you will converge to one of the roots (unpredictably which one).

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It is not a simple matter to check convergence based on the function values, because a test to zero is not meaningful as long as you don't have a reference order of magnitude. To illustrate, notice that $f(x)$ and $10000 f(x)$ have precisely the same roots, but the function values differ by a factor $10000$.

In the case of bissection, you know exactly where you are going on the $x$'s, because every iteration brings one more exact bit. Note that when you are getting real close to a root, chances are high that the evaluations of $f$ become very inaccurate and even wrong in sign. It is useless to continue the iterations past that point.