Convergence

Tags

math

Diverging is when a numerical sequence diverges away from a common solution and continually gives bigger and less accurate solutions. Converging is when a sequence stabilises to a fixed number (root value) after a misc number of iterations. Continuing the sequence only provides a better approximation of the root, and we often stop once we reach an accurate value.

For example consider the first few values of the recurrence relation $x_{n+1}=\frac{x_n^5+3}{5}$ beginning with $x_0=-1.5$.

$$\begin{array}{rlrlrlrlrl}{x}_0& =-1.5x_1& =-0.91875x_2& =0.46907x_3& =0.60454x_4& =0.61614x_5& =0.61776x_6& =0.61799x_7& =0.61802x_8& =0.61803\end{array}$$

After 8 iterations the value of the sequence converges/stabilises to $0.8180$ to 4 decimal places.

For another example in gradient-descent when the change in the gradient of some value at some point is $0$ we say it's converged to a local-minimum or maximum.