ECS130 Homework Assignment #4 Due: 4:00pm, March 13, 2017
- 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? - 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
(a) What is the minimizer of f(x1,x2)?
f ( x 1 , x 2 ) = 1 ( x 21 − x 2 ) 2 + 1 ( 1 − x 1 ) 2 22
(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 ( x 1 , x 2 ) = 1 x 21 + 9 x 2 2 . 22
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).
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