4. Gradient descent.Gradient descent is one of the most popular algorithms in data science and by far the mostcommon way to optimise neural networks. A function is minimised by iteratively movinga little bit in the direction of negative gradient. For the two-dimensional case, the step ofiteration is given by the formulaXn+1Yn+1n+¹) = (xn) - € V ƒ (Xxnx Y₁).EYnIn general, & does not have to be a constant, but in this question, for demonstrative purposes,we set = 0.1. Let f(x, y) = 3.5x² - 4xy +6.5y² and xo and yo be any real numbers.(a) For all x, y ER compute Vf(x, y) and find a matrix A such thatΤηYnA¹ (*) = (*)in terms of xo and yo and powers of A.- ε ▼ f(x, y).Write an expression for(b) Find the eigenvalues of A.(c) Find one eigenvector corresponding to each eigenvalue.(d) Find matrices P and D such that D is diagonal and A = PDP-¹.(e) Find matrices Dn, P-1 and An. Find formulas for an and yn.(f) Suppose to = yo = 1. Find the smallest N EN such that |(YN< 0.05,)IN-1YN-1,N(g) Sketch the region R consisting of those (ro, yo) such that N ≥ 0, YN ≥ 0 and|(x)| ≤> 0.05,where N is the number found in part (f). Write an equation for the boundary of R.Which points of the boundary belongs to R and which do not?≤0.05.

Note: According to the guidelines we are allowed to solve three subparts at a time. Please post rest…

Answered: 4. Gradient descent. Gradient descent…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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