[answered] MATH125 FALL 2016 COMPUTER ASSIGNMENT DUE: In particular, y

I have done problem one, and I don't know how to move on. Please help.

MATH125

 

FALL 2016

 

COMPUTER ASSIGNMENT

 

DUE: NOV 17 In this assignment we use MATLAB to perform numerical computations and generate graphics. In order to complete this assignment, you need to know the basic

 

syntax of MATLAB (to be covered in the discussion class on Nov 3). In particular,

 

you need to know how to define and plot functions and to write simple loops. Your

 

submitted assignment should contain all codes and outputs of the programs, as well

 

as necessary explanation. While you are encouraged to discuss with fellow students,

 

your must write your codes yourself and write up your own work.

 

1. (Warm up) Consider the function f (x) = sin x. Also consider the following

 

polynomials:

 

p1 (x) = x

 

x3

 

6

 

x5

 

x3

 

+

 

p5 (x) = x ?

 

6

 

120

 

p3 (x) = x ? These polynomials are approximations of the sine curve. Plot f , p1 , p3 and

 

p5 on the same graph over the interval [??, ?].

 

2. (Simple loop, complicated behaviors) The computer allows us to simulate

 

the dynamics of many systems (e.g. weather). As an example, write a loop

 

that performs the following computation:

 

(i) Start with a number x0 in (0, 1).

 

(ii) Given xn , define xn+1 by xn+1 = 3.6xn (1 ? xn ). For example, x1 =

 

3.6x0 (1 ? x0 ), x2 = 3.6x1 (1 ? x1 ), and so on.

 

(iii) Repeat (ii) until we get x100 .

 

Plot a single output sequence (xn versus n for n = 0, 1, 2, . . . , 100) for your

 

choice of x0 in (0, 1). You will see that the output changes a lot even if

 

you change the condition x0 just a little bit. This is a classic example of

 

chaotic behavior.

 

3. (Method of bisection) The method of bisection is a method to find a root

 

of the equation f (x) = 0, where f is continuous. It is based on the intermediate value theorem. We will use the method as described here:

 

https://en.wikipedia.org/wiki/Bisection method#The method

 

The input of the method is the following: a continuous function f , and an

 

interval [a, b] for each f (a) and f (b) have opposite signs.

 

(a) Consider the cubic polynomial f (x) = x3 ? x ? 2 and [a, b] = [1, 2]

 

(this is the example in Wikipedia). Try to reproduce the table there,

 

and find an approximate root x? of f (x) = 0 in [1, 2] which is accurate

 

up to 2?10 .

 

Date: October 25, 2016.

 

1 2 MATH125 (b) Consider the function f (x) = sin x?x?0.5, and [a, b] = [?2, ?1]. Run

 

the method of bisection with 20 iterations. For each iteration, record

 

the value of the midpoint as well as the value of f at the midpoint.

 

4. (Numerical integration) In this problem we compute the Riemann sums

 

corresponding to the definite integral

 

Z 1p

 

1 ? x2 dx.

 

0 ?

 

The exact value of this integral is ?4 . Write f (x) = 1 ? x2 .

 

(a) Write a script that performs the following:

 

? Input: a positive integer n, such as n = 100.

 

Pn

 

? Output: the value of the Riemann sum i=1 f (xi ) n1 , where xi

 

is chosen according to the midpoint rule.

 

(b) Compute the Riemann sum for n = 5, 10, 25, 50, 100, 200, 500, 1000.

 

Arrange your results in a table form.

 

(c) Recall that the exact value of the integral is ?4 . For each n, let In be

 

the Riemann sum with n terms and ? En = ? In 4

 

be the error of the approximation. Plot En against n, and log10 En

 

against log10 n. Briefly comment on the results.

 

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