Math 132A Assignment 4
Due: Thursday, February 8th at Midnight on Gradescope.
Don’t forget you will not have a computer on the midterm so it ’s important you know how to do these by hand and calculator.
1. Recall from class that we discussed the general quadratic function f : Rn → Rn defined by
for an n × n symmetric matrix Q and b ∈ Rn.
(a) Prove that ∇f (x) = Qx − b and ∇2 f (x) = Q.
(b) Starting from x(0) = (1, 1.5)T, determine the the first three iterates in the method of steepest descent applied to such an f with
2. Apply three iterations of the method of steepest descent to the function
starting at x(0) = (0, −2).
3. The function
is known as Rosenbrock ’s function or the banana function. This function is considered “nasty” and is often used to test algorithms.
(a) Prove that (1, 1) is the unique global minimizer of f.
(b) With a starting point of (0, 0)T, apply two iterations of Newton’s method.
(c) Repeat part (b) with the method of steepest descent but with fixed step size α = 0.05.