Math 1070 Matlab Assignment 4

Part 0: Getting started

To access the files needed for this assignment, in the class web page

http://www.pitt.edu/~trenchea/Math1070_Fall_Semester_2016.html

click on

A collection of matlab codes accompanying the text 

and then on Chapter5

Copy the files simpson.m and trapezoidal.m into your
working directory.

NOTE: type

>> format long

in matlab at beginning of your work on this assignment.

The function 

function [integral,difference,ratio]=trapezoidal(a,b,n0,index_f)

uses the composite trapezoidal rule with n subdivisions to  
integrate the function f over the interval [a,b]. The 
values of n used are

n = n0,2*n0,4*n0,...,256*n0

Read the comments at the beginning to understand what it is doing.
Note that the last parameter index_f allows for different functions
to be integrated. The index is given at the end of the file.

For example,

>> [integral,difference,ratio]=trapezoidal(-1,1,1,4)

will integrate exp(cos(x)) on [-1,1] with 
n = 1,2,4,...,256 subintervals. On output 
the vector integral stores the corresponding numerical integral,
The differences of successive numerical 
integrals are returned in the vector difference:

difference(i) = integral(i)-integral(i-1),  i=2,...,9

The entries in ratio give the rate of decrease in these
differences.

Call the function with various parameters.

Do the same with the function simpson in simpson.m

No need to submit anything up to this point.

Part 1: Submit the results for all you are asked to do.

For BOTH TRAPEZOIDAL AND SIMPSON, do the following:

1. Add the function f(x) = sin(x) as case 5 in the index.

2. Run the program to compute the integral of sin(x) on [0,pi/2].
   (Note that the exact value is I = 1.) Submit the output vectors
   integral, difference, and ratio.

3. Use the vector ratio to determine the order of convergence, i.e.,
   the constant "p" in the estimate

   I - I_n = c/(n^p)

   Comment if the results confirm the theory.

4. Use Richardson extrapolation and error estimate to obtain an improved
   integral value and an error estimate for n = 8. 

5. For n= 8: Compare the true error with
    a) the asymptotic error estimate
    b) the Richardson error estimate

6. For n= 8: Compare the true integral with
    a) the trapezoidal rule answer I_n
    b) the Richardson extrapolation value

ONLY FOR TRAPEZOIDAL:

7. Repeat steps 2. and 3. for the integral of sin(x) on [0,2*pi].
   (Note that the exact value is I = 0.) Explain the accelerated 
   rate of convergence.

Part 2: Submit the results for all you are asked to do.

DO THE FOLLOWING ONLY FOR THE TRAPEZOIDAL RULE.

8. Write a new function corrected_trapezoidal that implements the
   corrected trapezoidal rule. To do this just copy trapezoidal.m into
   corrected_trapezoidal.m and modify the latter file. Submit your code
   in  corrected_trapezoidal.m.

9. Run corrected_trapezoidal to compute the integral of sin(x) on [0,pi/2].
   Submit the output vectors integral, difference, and ratio.

10. Use the vector ratio to determine the order of convergence, i.e.,
   the constant "p" in the estimate

    I - I_n = c/(n^p)

11. Compare your estimate with the estimate for the trapezoidal rule from 3.