When you do some work for someone, you expect to get paid. The same is true if you loan money to someone. You expect them to pay you interest for the use of your money. On the other side of the coin (no pun intended), if you need to borrow money, you must pay someone else interest for the use of their money. Rather than doing this transaction individual by individual, society has banks that serve as intermediaries who pay interest for money given to them and who charge interest for money that they loan. Thus in the financial world, money is like labor or goods and services. Businesses that need money must pay for it. This is the way the business, and thus the engineering, world works.
A general principle that applies to all financial transactions is called the time value of money (or future value of money). A dollar that you have today is worth more in the future because you don't hide it under your mattress. You invest it in a savings account, certificate of deposit, stock or bond or some other "financial instrument" and your money earns interest. This works because the institution that takes your money immediately loans it to someone else for their use and charges them interest.
The time value of money is computed using the formula: F = P(1 + a)^{n} where:
P = the present value of your money,
F = the future value of your money after n periods (months or years),
a = the percent interest paid for your money per period (month or year), and
n = the number of periods (months or years) over which the interest is computed (or compounded)
MATLAB is short for Matrix Laboratory. It is a computing environment that makes it easy to perform very complex numeric computations. In numeric computing, we compute initial, intermediate, and final numeric values (numbers) in order to solve problems. This is in contrast with symbolic computing where symbols are used to create expressions and equations and then the equations are manipulated without much thought to the specific underlying value for each symbol.
In numeric computing, the values themselves are needed to find the next step in the solution and the solution is one or more values instead of a symbol or symbolic expression. Note: Variable names are used in MATLAB, but they are not abstract symbolic variables as you might see in a math equation. Rather they are more like nicknames for values that have been computed.
Let's get started!
1. Use MATLAB's
to answer the following questions or perform the described calculationsType the following into the
(don't type the >> -- that's the command prompt):>> F = 10000*(1+0.04)^5
Hopefully, you are considering how best to do this many calculations without retyping all of the values needed. Hint: First, enter the interest rates into a single vector of interest rates and save it with the name a and then perform an element-wise calculation using all elements in the vector. Type the following to create the vector using the colon operator:
>> a = 0.04 : 0.005 : 0.07
Repeat an earlier computation by using the up arrow until the calculation is at the command prompt and then editing as needed for the new calculation. Find your calculation from part a and change the 0.04 to the name of your vector of interest rates, a.
>> F = 10000*(1+a)^5
Before we explain the error, let's just fix it. Type a period (.) before the ^ symbol in the expression and press the
key. The (.^) is an element-wise operator. It indicates that the power operator needs to occur on each value (element) in the matrix, instead of performing matrix algebra.2. Matrix algebra in MATLAB
The power operator by itself (^) performs the exponential multiplication as defined by matrix algebra. Let's review matrix multiplication first.
SHOW YOUR COMPLETED MATRIX TO A LAB LEADER
Check your work using MATLAB:
>> A = [ 1 , 2 ; 3 , 4 ; 5 , 6 ; 7 , 8 ]
>> B = [ 1, 2, 3 ; 4, 5, 6 ]
>> C = A * B
>> D = B * A
Now what happened?
The error MATLAB produced is due to the fact that MATLAB tried to multiply two matrices with incompatible dimensions. A 4×2 matrix times a 2×3 matrix is defined, but multiplying a 2×3 matrix with a 4×2 matrix is undefined in matrix algebra, so the B*A operation failed.
>> one_plus_a = 1 + a
Now, one_plus_a contains the quantities of each interest rate plus the value 1. The addition of the scalar value to the matrix of interest rate values (1 + a) works fine. This is because the matrix algebra interpretation of a scalar added to a matrix is to add the scalar to each element of the matrix. Try adding two non-scalar matrices with different dimensions and again an error is the result because that operation is undefined in matrix algebra.
However, when we tried to raise the quantity (1 + a) to the fifth power, MATLAB produced the error because the matrix dimensions were incompatible for the multiplication operation. Matrix algebra does not allow a 1×N matrix to be multiplied by another 1×N matrix. What we really wanted was each element of the matrix to be raised to the fifth power. This is what element-wise operators do. Thus, replacing the ^ with .^ tells MATLAB to perform the calculation on each element.
Define a square matrix (4×4) and name it x. Then, compute x^2 and x.^2 and store the results in the variables x2_matrix_algebra_mult and x2_elementwise_mult, respectively.
3. Future Value of Money (Part 2)
The up arrow will work, but the command we want now is several commands away. Another way to repeat a previously entered command is to drag it from the
window into the . If the window is not visible, under the tab in the main menu select the drop-down menu. Then under the heading, select .Which of the following investment options is the better investment?
Caution: Do not reenter commands if you can easily drag them from the
or repeat them with the up arrow key. Be sure to edit as necessary before pressing .4. Publishing your work (to HTML)
You have solved some problems, but need to present your work and the results to your boss or instructor. MATLAB has several tools to help publish your solutions to different forms. In this class, we ask that you publish your homework solutions to HTML or PDF.
Note: HTML is currently the most common type of web page found on the internet. If you copy your HTML pages to a computer folder that is available via the internet (on a web server), then you will be able to see your work from anywhere you have internet access.
Here's how to save your work in an m-file:
Here's how to publish your work (to HTML) using MATLAB:
%% Team Lab 1
% <<../tvom_formula.gif>>
Now, let's make our work more user-friendly by adding some cell headers and comments. Be sure to edit, remove, or comment out commands that were errors or duplicates from the original command (code) list.
%% 1.a Numeric Calculations
%% 2.a Matrix Multiplication
%% 2.b Add a scalar to a matrix
%% 2.c Matrix Squared vs Element-wise operator
%% 3.a Plot the Future Value of Money vs Interest Rate
%% 3.b Investment Option
5. Publishing multiple plots
When your work requires multiple plots, make sure that each plot is in its own cell division before publishing to HTML. Otherwise, only the last plot will be displayed on the final html page. Try getting two or more plot figures to display on your HTML page.
Add a new section to your program script and add this command to plot the first row of your squared x matrix. Note: this command will only work if you named matrices as directed in earlier steps.
>> plot(x(1,:),x2_elementwise_mult(1,:));
Save, publish and view the results to confirm that two separate plots are visible.
6. Displaying equations in the published document page
There are lots of things you can do to make your final document a better presentation of your work. One cool thing MATLAB can do is display professionally formatted equations in your page. Even though MATLAB does not allow you to enter equations and solve them, you can display the equations that your work is based on in the final document.
%
% $$F = P(1+a)^n$$
%
%
% $$y = \frac{\sin(x)}{\cos( \frac{\pi}{4} ) }$$
%
x = 1:0.1:10;
y = sin(x)/cos(pi/4);
plot(x,y)
title('sin(x)/cos(\pi/4) vs x');
xlabel('x');
ylabel('sin(x)/cos(\pi/4)');
Note: This syntax is based on LaTex syntax for entering equations to be formatted. While there is no guarantee that all LaTex commands will work the same in MATLAB as in LaTex, many will and it's worth a try.
7. Displaying lists in the published document
It is easy to display a list of items in a cell comment section. MATLAB can produce bulleted lists and numbered lists.
%
% * Name 1
% * Name 2
% * Name 3
%
%% Numbered list example
% # Launch MATLAB
% # Create script file
% # Write MATLAB code
% # Run script
% # Save and publish
% # Print or upload as needed
8. Publish your work to PDF
Show your published PDF to a Lab Leader.
9. Find the published files you created
The published files you created are automatically saved for you according to the default configuration settings or those that you edited. If you are asked to print your work, use
while your published document is showing. If we ask you to upload a PDF, then you will need to know where the PDF file is located.10. Getting HELP!
We have presented a lot of different things to try in MATLAB, but what if you need help while working in MATLAB? Help is just a click or a few typed characters away.
If you know the name of the command you need:
Type help plot to see help displayed in the .
Type doc plot to launch the browser.
If you don't know the name of the command:
Type doc or click the blue question mark near the top of your screen to launch the browser and then search for a command or topic that will help you solve the problem.
The solution for this lab will be available shortly after 6 pm on Thursday. Please review and ask questions before starting the quiz. Be sure to complete Quiz 1.