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I've encountered the Del operator while reading about gradient descent methods and I simply don't understand it. Here is an example of it's usage in gradient descent:

$b = a - \gamma \nabla F(a)$

I've seen some examples of the formula itself but for the whole understanding I need to know the meaning of the $\nabla$ operator.

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    What do you men by "meaning"?2011-04-27
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    Oh ok, want to now what kind of computation it does and of what cause.2011-04-27
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    Can you clarify what $a$,$b$,$\gamma$ and $F$ are? (i.e. are they scalars or vectors)2011-04-27
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    $b$ is the starting point of computation (which should be minimized within a matrix I guess), $a$ is the neighbour point of $b$, $\gamma$ is the step width of the descendence and $F(a)$ is, as far as I know, the gradient function of descendence2011-04-27
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    If $F(a)$ is actually some multivariate function $F(a_1,a_2,\dots)$, then $\nabla F(a)$ is the vector of partial derivatives with respect to each of the components of $a$.2011-04-27

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The Del symbol (more properly, nabla) $\nabla$ indicates taking the gradient. That is, for a scalar function $F$ the gradient $\nabla{F}$ is the vector of first partial derivatives. The direction given by the gradient vector is the steepest increase (ascent) of the function's value. Thus the formula for root-finding involves adjust a current estimate of the root location by an amount proportional to the gradient, taking into account the steepness of the slope there.