TrickiQuick description
Nonlinear equations are ubiquitous in mathematics, and we need practical methods to solve such equations. Methods vary in their reliability and robustness as well as their speed and ease of implementation, as well as applicability. For example, the Newton-Raphson method can be applied to systems of nonlinear equations
and can converge very fast, while bisection is extremely reliable, but is much slower and can only be used for a single scalar equation in one variable.
Prerequisites
Calculus.
Example 1
Bisection is extremely robust but relatively slow for solving 
once we have 
where 
and 
have opposite signs.
One simply computes 
and determines the sign of 
. One then either updates 
or 
and repeats the process depending on the sign of to ensure that we again have and of opposite signs. Of course, if 
, we stop. Normally we stop when 
, a pre-determined tolerance.
As long as 
is continuous, this method will converge thanks to the intermediate value theorem.
Speed can be improved using so-called bracketing methods such as Dekker's or Brent's method.
Example 2
The so-called Newton-Raphson method is based on the linear approximation 
for 
.