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hw4.dvi

ECS130 Homework Assignment #4 Due: 4:00pm, March 13, 2017

1. Consider the following cubic polynomial

p(x) = 816×3 − 3835×2 + 6000x− 3125.

It has three three closely spaced roots: 25/15, 25/16, 25/17

(a) Plot p(x) for 1.43 ≤ x ≤ 1.71. Show the location of the three roots.

(b) Starting with the interval [1,2], what does the bisection method do?

(b) Starting with x0 = 1.5, what does Newton’s method do?

(c) Starting with x0 = 1 and x1 = 2, what does the secant metod do?

2. Investigate the behavior of the secant method on the function

f(x) = sign(x− 2)

|x− 2|

Hint: start from “Example Newton 2” from our class website.

3. Let

f(x1, x2) =
1

2
(x2

1
− x2)

2 +
1

2
(1− x1)

2

(a) What is the minimizer of f(x1, x2)?

(b) Compute one iteration of Newton’s method for minimizing f(x1, x2) starting from the
point (2, 2). Is this a good step?

4. Let

f(x1, x2) =
1

2
x2
1
+

9

2
x2
2
.

It’s easy to see that the minimizer is x∗ = (0, 0)

(a) Derive the steepest descent method for finding the minimzer of of f(x).

(b) Compute the first four iterations starting from the point (9, 1).

1

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