Starting from:
$30
Computational Project 4
Computational Project 4
1. Integration
The overall goal of this problem is to find the area of the bat sign using numerical
integration. The approach for finding the area of the bat sign will be to integrate
two functions, b1(x) and b2(x), which evaluate the outline of the top and bottom
halves of the sign. The total area is thus
I = b1 dx
−7
7
∫ − b2 dx
−7
7
∫
Code for calculating b1 and b2 are provided on D2L (bat_func_1 and bat_func_2,
respectively) where the input for each function is x and the output is f(x). Note that
the output will symbolic. Try the following commands in the command window:
bat_func_1(1)
double(bat_func_1(1)) %casts output as double—use to show numeric values
Treat these codes like “black boxes”, i.e., you can evaluate them for any x, but
pretend you do not know the analytical expressions contained in b1 and b2.
However, you may note the following: the symbol is symmetric about x = 0, the x
coordinate of the discontinuities in b1(x) are located at x = ±0.5, ±0.75, ±1.0 and
±3.0, and the x coordinate of the discontinuities in b2(x) are located at x = 0, ±2
and ±4.
Find the area of the bat sign accurate to within an absolute error of 0.005 for each
integral. Report the number of times you needed to evaluate b1(x) and b2(x) to
find your final integral.
Your code should try more and more points (smaller spacing) until the desired
error tolerance is reached. At this error, it should stop automatically and output
your answer. For each attempt, display the number of points, the integral estimate,
and the estimate of the error.
Part of your grade for this project will be for efficiency. It is easy to write a very
simple integration scheme that uses many points with the rectangle method, but it
will be quite slow. Part of your grade will also be for program style, so think about
how you can write a program that uses separate functions in a clean and clear
manner.
2. Derivative of right upper edge
Write a code the makes a plot of the derivative of b1(x) over the range 3 ≤ x ≤ 7,
calculated such that the average error in the derivative across all points of the plot
is less than 0.01. Your code should use finite difference methods to find the
derivative, adapting the step size such that this error tolerance is met. Display the
step size and estimated error each time the derivative is estimated. Similar to
problem 1, the efficiency and style of your program will factor into the overall
grade.
Hint: to determine the error in an estimate of the derivative, you may need to
compare to a previous estimate with a smaller interval size, h. This can be a bit
tricky, since changing h changes the number of point for which you calculate the
derivative. So be careful with the alignment when making this comparison. You
could use the following to compare the previous derivative, Dold, calculated using
h = h1 to Dnew, calculated using h = h1/2 with:
difference = Dnew(1:2:end) - Dold
Remember that you should submit only one m file and your code should be
commented.
1. Integration
The overall goal of this problem is to find the area of the bat sign using numerical
integration. The approach for finding the area of the bat sign will be to integrate
two functions, b1(x) and b2(x), which evaluate the outline of the top and bottom
halves of the sign. The total area is thus
I = b1 dx
−7
7
∫ − b2 dx
−7
7
∫
Code for calculating b1 and b2 are provided on D2L (bat_func_1 and bat_func_2,
respectively) where the input for each function is x and the output is f(x). Note that
the output will symbolic. Try the following commands in the command window:
bat_func_1(1)
double(bat_func_1(1)) %casts output as double—use to show numeric values
Treat these codes like “black boxes”, i.e., you can evaluate them for any x, but
pretend you do not know the analytical expressions contained in b1 and b2.
However, you may note the following: the symbol is symmetric about x = 0, the x
coordinate of the discontinuities in b1(x) are located at x = ±0.5, ±0.75, ±1.0 and
±3.0, and the x coordinate of the discontinuities in b2(x) are located at x = 0, ±2
and ±4.
Find the area of the bat sign accurate to within an absolute error of 0.005 for each
integral. Report the number of times you needed to evaluate b1(x) and b2(x) to
find your final integral.
Your code should try more and more points (smaller spacing) until the desired
error tolerance is reached. At this error, it should stop automatically and output
your answer. For each attempt, display the number of points, the integral estimate,
and the estimate of the error.
Part of your grade for this project will be for efficiency. It is easy to write a very
simple integration scheme that uses many points with the rectangle method, but it
will be quite slow. Part of your grade will also be for program style, so think about
how you can write a program that uses separate functions in a clean and clear
manner.
2. Derivative of right upper edge
Write a code the makes a plot of the derivative of b1(x) over the range 3 ≤ x ≤ 7,
calculated such that the average error in the derivative across all points of the plot
is less than 0.01. Your code should use finite difference methods to find the
derivative, adapting the step size such that this error tolerance is met. Display the
step size and estimated error each time the derivative is estimated. Similar to
problem 1, the efficiency and style of your program will factor into the overall
grade.
Hint: to determine the error in an estimate of the derivative, you may need to
compare to a previous estimate with a smaller interval size, h. This can be a bit
tricky, since changing h changes the number of point for which you calculate the
derivative. So be careful with the alignment when making this comparison. You
could use the following to compare the previous derivative, Dold, calculated using
h = h1 to Dnew, calculated using h = h1/2 with:
difference = Dnew(1:2:end) - Dold
Remember that you should submit only one m file and your code should be
commented.
1 file (60.6KB)
