Slope-based root-finding method

Definition

The slope-based root-finding method is a first-order root-finding method for a function of one or more variables. Explicitly, given a vector-valued function ${\vec {F}}:\mathbb {R} ^{n}\to \mathbb {R} ^{m}$, the slope-based root-finding method does the following to compute the $(k+1)^{th}$ iterate ${\vec {x}}^{(k+1)}$ from the $k^{th}$ iterate ${\vec {x}}^{(k)}$:

${\vec {x}}^{(k+1)}={\vec {x}}^{(k)}-\alpha ^{(k)}{\vec {F}}({\vec {x}}^{(k)})$

where the learning rate $\alpha ^{(k)}$ is computed based on the iteration. It may be chosen to be constant.

The corresponding optimization method is termed gradient descent.