Finding  Zeros:

f(x)= F'(x)

Initial step f(a) f(b) <0

Methods:
Bisection: no additional assumptions. x{1}= 1/2 (a+ b) + a, then x1 replaces either  a  or  b.
Convergence: reducing the interval: rate 2^N.
Negative: the values of functions are not count.

Newton method. Assume that f(x) is close to linear. Then f(x{k+1}) = f(x{k}) + f'(x{k}) (x{k+1} - x{k})=0
and we obtain: x{k+1} = x{k} - f(x{k}/f'(x{k}).
f(x)= x^2- a then 1/2(x{k} + a/x{k})
Difficulty: iteration may diverge.

Secant method: f'(x{k} is replaced by (f{k} - f{k-1})/x{k} - x{k-1})
Pro: do not use derivative, use already computed function.
Negative: divergence.

Method of false position or "regula falsi"
Compute the sign f{k+1} f{k} and f{k} f{k-1}
Replace either x{k} or x{k-1} depending on the sign.
Convergence rate may be low Ex: f=-3 + x^(1/4).

Saveguarded algorithms:
Combination of bisection and Newton or Secant methods.
Use Newton or Secant, if the step is "reasonable":
x{k+1} in [a, b]
if the step is too large, replace it with bisection
If the step is too small, enlarge it to a fixed number delta.

Optimum of  one-dimensional function.

Optimal (Fibonacci) search.

Golden mean: an asymptotics.

Gradient methods. Choice of the length of the step.
Newton method.
Approximation method. (Secant method)
Modifications. Safeguard measures.

Global search. Lower estimate by piece wise linear function