### Bisection

Closed.

#### Requirements

A function f(x) and an interval [a, b] (in which f(x) is continuous) containing one and only one root.

#### Risks

If the function is discontinuous, the program can have an error when evaluating. Also, if the interval has more or less than one roots, unexpected behavior can happen.

#### Convergence rate

Linear

If the criteria for the interval is satisfied, it always converges. Also, this is the easiest method to program and the easier to understand. The smaller the interval given, the faster it converges.

This is the method with the lowest converge rate.

#### Fault tolerance

As long as the criteria are satisfied, this method doesn’t run.

One.

### Newton-Raphson

Open.

#### Requirements

A function f(x), the derivative df(x) of f(x) and a starting point.

#### Risks

If the function presents small slopes in any iteration, the method diverges. On the other hand, if the function presents really big slopes in any iteration, the method converges really slow.

#### Convergence rate

This method has the quickest converging rate. Also, it just requires a starting point instead of two.

This method requires the derivative, which can be hard to calculate and/or evaluate. Also, small slope functions (such as log(x)) diverge.

#### Fault tolerance

Some functions, such as periodic functions, can lead to unexpected behaviors with certain values. Also, small slopes fail.

One.

### Secant

Open.

#### Requirements

A function f(x) and two starting points (different).

#### Risks

Similar as Newton Raphson. Also, this method can fail if both starting values are similar.

#### Convergence rate

Aureal number.

It runs quicker than bisection and does not require a derivative.

Can fall in the same errors as Newton-Raphson.

#### Fault tolerance

Fails less than Newton-Raphson, but still can have unexpected behaviours.

One.

### Bairstow

Open.

#### Requirements

A polynomic function f(x) and two starting values.

#### Risks

The farther the starting values from the roots, the longer it takes to converge.

#### Convergence rate

This method allows us to find complex roots, and it may take a long time, but it doesn’t fail.