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.