Machine Learning II
Homework #5
Homework 5:
• Show ALL Work, Neatly and in Order.
• No credit for Answers Without Work.
• Submit a single pdf file includes all of your solutions.
• DO NOT submit individual files or images.
• For coding questions, submit ONE .py file and include your comments.
E.1:
Consider the following function:
F(x) = [1+ (x1 +x2 −5)
2
][1+ (3x1 −2x2)
2
]
i. Perform one iteration of Newton’s method, starting from the initial guess
10 10T
ii. Repeat part (i), starting from the initial guess
2 2T
iii. Find the minimum of the function, and compare with your results from the previous two
parts.
E.2:
For the following functions find the first and second directional derivatives from the point
1 1T
in the direction
−1 1T
.
i. F(x) = 7
2
x
2
1 −6x1x2 −x
2
2
ii. F(x) = 5x
2
1 −6x1x2 +5x
2
2 +4x1 +4x2
iii. F(x) = 9
2
x
2
1 −2x1x2 +3x
2
2 +2x1 −x2
iv. F(x) = −1
2
(7x
2
1 +12x1x2 −2x
2
2
)
1
E.3:
For the functions of Exercise E2:
i. Find the stationary points.
ii. Test the stationary points to find minima, maxima or saddle points
iii. Provide rough sketches of the contour plots, using the eigenvalues
2
