2 (First order) Descent methods, rates 2 2.1 Gradient descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 What can we achieve? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Second order methods: Newton’s method . . . . . . . . . . . . . . . . . 7 2.4 Multistep first order methods . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.1 Heavy ball method . ...

In this thesis I will describe an explicit method for performing an 8-descent on elliptic curves. First I will present some basics on descent, in particular I will give a generalization of the definition of n-coverings, which suits the needs of higher descent. Then I will sketch the classical method of 2-descent, and the two methods that are known for doing a second 2-descent, also called 4-des...

‎Based on an eigenvalue analysis‎, ‎a new proof for the sufficient‎ ‎descent property of the modified Polak-Ribière-Polyak conjugate‎ ‎gradient method proposed by Yu et al‎. ‎is presented‎.