4.2.1. Examples

Consider the system of equations:

x + 2y + 3z = 1 3x + 3y + 4z = 1 2x + 3y + 3z = 2

Find the solution x to the system of equations.

Solution: Use the MATLAB editor to create a file: File > New > M-file.

Enter the following statements in the file:

A = [1 2 3; 3 3 4; 2 3 3]; b = [1; 1; 2]; x = A\b

Save the file, for example, example1.m.

Run the file, in the command line, by typing:

example1

x = -0.5000 1.5000 -0.5000

When execution completes, the variables (A, b, and x) remain in the workspace. To see a listing of them, enter 'whos' at the command prompt.

Note: The MATLAB editor is both a text editor specialized for creating M-files and a graphical MATLAB debugger. It has numerous menus for tasks such as saving, viewing, and debugging.

In this example, we are tasked with solving a system of linear equations using MATLAB. The equations are first represented in matrix form, where matrix A contains the coefficients and vector b contains the constants from the equations. We create a new M-file in the MATLAB editor, where we enter the commands necessary to perform the calculation using matrix division (A\b). By running the file, MATLAB computes the values of x, y, and z and displays the results. The variables remain in the MATLAB workspace for further analysis.

1. Open the MATLAB editor to create a new M-file.
2. Enter the matrices A and b, and use the matrix division command to obtain the solution.
3. Save the file and execute it from the command window, allowing us to view the solution.

Think of the system of equations as a puzzle where each equation represents a different constraint. Your job is to find the piece (values of x, y, z) that fits all the constraints together. Just like in a puzzle, you have to make sure each piece interacts correctly with the others to complete the picture.

Plot the following cosine functions, y = 2cos(x), y = cos(x), and y = 0.5cos(x), in the interval 0 ≤ x ≤ 2π.

Create a file, say example2.m, which contains the following commands:

x = 0:pi/100:2pi; y1 = 2cos(x); y2 = cos(x); y3 = 0.5cos(x); plot(x, y1, '--', x, y2, '-', x, y3, ':') xlabel('0 ≤ x ≤ 2π') ylabel('Cosine functions') legend('2cos(x)', 'cos(x)', '0.5cos(x)') title('Typical example of multiple plots') axis([0 2pi -3 3])

Run the file by typing example2 in the Command Window.

In this example, the goal is to plot three different cosine functions that vary in amplitude. The commands specify the range for x and calculate corresponding y values for each cosine function. The 'plot' command is then used to visualize all three functions on the same graph, where each function is represented by a different line style. Additionally, labels for the x and y axes, a legend for the curves, and a title for the plot are added. This example illustrates how to create visual representations of data in MATLAB effectively.

1. Define the range for x using a vector.
2. Calculate the y values for each cosine function based on this x.
3. Use the plot function to visually represent these values in one graph with appropriate labels and styles.

Imagine you are an artist creating a series of paintings. Each painting (cosine function) has a different color (amplitude), but they are all based on the same theme (the cosine function itself). By displaying them all together in one exhibition (the plot), viewers can appreciate the differences and variations while still relating them back to the same underlying concept.

--