{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Fall2011-HW2-soln

# Fall2011-HW2-soln - [4 6 Table Approximation p n and...

This preview shows pages 1–2. Sign up to view the full content.

Homework 2 Solutions Math 371, Fall 2011 Page numbers in the International Edition are +12. Here is an example of two MATLAB scripts which, together, allow you to pass a pass a function into your code. This allows you to reuse your code for other functions. Program 1 function p=myfun(x), p=1/x-cos(x); return Program 2 function p=bisection(myfun, a, b, nmax), for n = 1 : nmax, p(n) = (a+b)/2; if(feval(myfun,p(n))*feval(myfun,a) < 0 ), b=p(n); else a=p(n); end end 1. (Rootfinding and Optimization) (a) Suppose f ( x ) is differentiable on [ a, b ] . Discuss how you might use a rootfinding method to identify a local extremum of f ( x ) inside [ a, b ] . Roots of the derivative give possible locations of local extrema. (b) Let f ( x ) = log x - sin x . Prove that f ( x ) has a unique maximum in the interval [4 , 6] . (Note that log means natural logarithm.) f (4) > 0 , f (6) < 0 , and the continuity of f ( x ) give the existence of at least one maximum. f ( x ) < 0 for x [4 , 6] shows the maximum is unique. (c) Approximate this local maximum using six iterations of the Bisection method with starting interval

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: [4 , 6] . Table: Approximation p n and absolute error | p n-p | versus iteration number n in Bisection method 1 n p n | p n-p | 5.0000000 0.0828141 1 4.5000000 0.4171859 2 4.7500000 0.1671859 3 4.8750000 0.0421859 4 4.9375000 0.0203141 5 4.9062500 0.0109359 6 4.9218750 0.0046891 2. (Order of Convergence) Apply the theorems on pp. 90-91 to the following problems: (a) Section 2.3 #8 g ( √ a ) = √ a . The conditions of the theorem on page 91 hold. g Í ( √ a ) = 0 but g Í ( √ a ) Ó = 0, so α = 2 and the asymptotic error constant is 1 / 2 √ a . (b) Section 2.3 #9 g ( √ a ) = √ a . The conditions of the theorem on page 91 hold. g Í ( √ a ) = g Í ( √ a ) = 0, but g Í ( √ a ) Ó = 0. Thus α = 3 and the asymptotic error constant is 1 / 4 a . 2...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern