Gradient Descent and Cauchy Method in Differential Equations

[kidsasd987]

http://www.math.uiuc.edu/documenta/vol-ismp/40_lemarechal-claude.pdf



I don't understand why we use theta for equation (1)

Θ>0

but why α=-θX?



Thanks.

 

[HallsofIvy]

It was already given that X is the derivative with respect to x and so $\theta X$ is gives a slight change along the tangent plane in the x direction. I presume it is negative because they want to think of $\theta$ going to 0 as the point, $(x_0- \theta X, y_0- \theta Y, z_0-\theta Z)$, moving toward $(x_0, y_0, z_0)$ along the path defined by the differential equation.