Using Variables

After entering a few starting letters of a variable name or built-in command/function name in the Command Window, you can use the TAB key from the keyboard to access matching commands/functions, including your developed function files. For example, if you typed >> AB and then pressed TAB, the rest of ABCD variable calculation expression would appear as an option.

Another useful feature of the Command Window is using the keyboard’s up-arrow (↑) to recall previously typed variables or commands. You simply type a few starting letters of any previously typed commands or function names and then press the up-arrow (↑). For example, >> f↑ recalls the previously typed-in command, >> format long. Moreover, the up-arrow (↑) can be associated with the TAB key to recall previously entered commands in the Command Window.

The values and attributes of all variables entered in the Command Window will be saved in the workspace until you clean up the workspace by deleting the variables using the clear, clearvars, or clear all command or by using the right and left mouse button options to select the variables and delete them. In addition, all of the variables and their attributes are saved in the workspace until the MATLAB package is closed.

From these examples, it is clear that MATLAB reads every entry as an array/matrix. For example, scalar is read by MATLAB as an array of size 1-by-1. This attribute of MATLAB is logically linked to its name, MATrix LABoratory. MATLAB’s default storage (memory allocation) is double precision, which is the maximum available space allocated. However, for memory efficiency and faster calculation purposes, other storage formats can also be used. MATLAB supports single precision or integer type and int8 … 64, uint8 …64 storage format types. Table 1-1 shows how data can be saved in every storage class type and the conversion function used in MATLAB for each type.

In calculations of the Fnew1 and Fnew2 variables from the int8 and uint16 formatted variable values, the allocated storage space cannot accommodate any more values (i.e., for int8 maximum allocated storage space is 127 = 2⁷ − 1 and for uint16 it is 65535 = 2¹⁶ − 1). Therefore, errors in the calculations have taken place. Note that in such cases, MATLAB does not show an error. Figure 1-9 shows all the data storage types and data formats supported in MATLAB.

The Command Window’s history pane is a good way to review all of the entries that will be kept unless you delete them. You can also change the setting in the Preferences window to clear the history of entries after ending the session.

Or use Ctrl+Q from the keyboard to quit.

An alternative way to exit is by clicking close (the x) in the upper-right corner of the main window. This will close the whole package.

When to Use the Command Window

Many of the operations performed in the Command Window, such as performing calculations and analyses and viewing variables or file contents, can also be done other ways. For example, most of the operations carried out in the Command Window can also be done via GUI tools. You can create new variables , delete them , or open them .

Similarly, you can create a MATLAB file with or an M-file with or open an existing MATLAB file with . You can delete the files via the right and left mouse button options. You can also view the current directory or change it using .

One of the most essential functions of the Command Window that cannot be done easily with GUI tools is viewing the error and warning messages obtained while and after executing M-files, MLX-files, and MDL/SLX-files. This is essential for good programming. Another good use of the Command Window is to obtain quick help with MATLAB commands/functions.

Numerical Data/Arrays

Note that I have set up the seed value of the random number generator in order to generate the permanent values from the random number generators rand(), randn(), and randi().

Now it is time to check the attributes/properties of the numerical entries created in the Command Window and saved in the workspace. You do this by using the whos command, as shown in Figure 1-11.

Note in these examples that the colon is one of the essential operators in managing and manipulating matrix and array elements. For example, NewDr(2, :) is equivalent to NewDr(2, 1:end). Both select all elements along row 2. Likewise, NewDr(:,:) is equal to NewDr(1:end, 1:end). They both select all elements starting from the first one, up to the last one.

These are a few examples of creating arrays in the Command Window. As stated, numerical arrays can be imported from other data files, such as *.dat, *.txt, *.xls, *.xlsx, and *.csv, as well as image files, such as *.jpg/jpeg, *.tiff, *.png, etc.

NaN (Not-a-Number)

Sometimes, you may need to remove NaN from your data. For instance, say you are analyzing measured data with some missing points (NaN). You need to remove the NaN from the data. How do you address this problem?

This answer is correct. This approach is quite straightforward, but with very large data sets, it will become tedious and time consuming or might even be impossible.

Moreover, there are several other MATLAB functions that compute mean values, standard deviations, covariance values , etc., of numerical data arrays with NaN elements.

Here, the isnan() function identifies which elements of A_var are NaN and which ones are not. The new logical array called Index contains 1s and 0s. The 1s represent NaN elements and the 0s represent all other numerical elements.

Character Types of Variables

Note that these variables are kept and used in the coming sections to generate logical, table, cell, and structure types of array variables.

Function Handle

Where MY_function is a function file, function expression, or another function handle.

Here the function handle called F1 calls the function file called MY_function.m and executes it with the user-specified input data for the a, b, and c variables.

Figure 1-12 shows the list of variables and function handles created in this section that reside in the workspace.

Broader applications and uses of function handles are discussed in other chapters of the book—in Chapter 2, “Programming Essentials,” and Chapter 8, “Ordinary Differential Equations”.

Logical Arrays

Note that we set the seed value of the random number generator rng() in order to generate permanent element values with the uniform random number generator rand().

In this section, we generated the numerical and logic matrices and character strings, as shown in Figure 1-13.

More uses/examples of logical arrays and indexing are discussed in other sections of the book.

Table Arrays

It is also possible to convert Table Arrays into arrays and cell arrays by using the tabel2array() and table2cell() commands , respectively. Understanding and working with Table Arrays will be of great help not only when you’re preparing reports but also when you’re importing data using the Import wizard and manipulating various data sets from external files (e.g. .txt, .xls, .dat, etc.) into the MATLAB workspace.

Cell Arrays

Cell arrays are useful for accommodating various types of arrays (numerical, character, logical, table, and function handle) in different cells of one cell type variable by preserving all attributes of each variable, unchanged. They might be very handy to carry or pass various data sets inside one variable. The cell arrays contain indexed data containers—cells accommodating lists of text, character strings, combinations of text and numerical data, and numerical arrays, function handles, structure arrays, and tables. One of the most essential features of cell arrays is that they require curly brackets to be used in specifying cell addresses. Another important feature of cell arrays is that data read by MATLAB will be in cell array mode.

By double-clicking on the cell variable name in the workspace, the contents of the cell can be viewed, as shown in Figure 1-14.

You can change the contents of the cell array by double-clicking in each cell and entering the new values.

Structure Arrays

Structure arrays can accommodate all of the previously created array and entry (variable) types, namely, all types of numeric, logical, character, table, cell, and function handles. They can store data not only of different types but also of different sizes. One of the more important aspects of structure arrays is that they are suitable to code generation. Moreover, they are very useful in programming, data processing, data acquisition, and reading from Simulink models.

Figure 1-15 shows the list of variables saved in the workspace and the contents of the Astr and Bstr structure arrays .

From the attributes of the variables in the workspace (see Figure 1-12), you can determine the variable type (scalar, array, logical, table, cell, structure, character, or function handle) and its storage type (double, single, uint8, or int8). Moreover, the symbols representing each variable shown in Figure 1-15 demonstrate some of the MATLAB supported data (array) sets shown in Figure 1-9.

It must be noted that many of these arrays can be converted from one type to another, as you have seen in some of the examples . For example, a cell array can be converted into a table array via the cell2table() function or similarly, a structure array can be converted into a table array via struct2table(). A table array can be converted into a cell and table array via table2cell() and table2struct(), respectively.

Complex Numbers

Two letters—i and j or 1i and 1j—are reserved for notating imaginary numbers. Therefore, it is advised not to use these letters to assign variable names. An alternative approach to assigning numbers is to multiply them by sqrt(-1).

Precision

MATLAB’s precision is not absolute.

In these expressions, t is a time vector containing a row of elements, such as [0, π/50, … 2π]. Some values of F are zero and some are non-zero, even though they are very small numbers. The reason for this is that all of the trigonometric functions, including exponential and logarithmic functions, are approximated by a polynomial of degree 13 with only odd powers of the argument variable (in this example, t). For instance, sin(t) ≈ t − c₁t³ + c₂t⁵ + … + c₆t¹³ = p(t). The computation algorithm for all of these functions is implemented based on fdlibm, a “Freely Distributable Math Library” developed by at Sun Microsystems by K. C. Ng and others (for more information, see www.netlib.org/fdlibm).

The allocated data storage int8 can hold up to 2⁸ − 1 integer numbers. All MATLAB supported data storage types are shown in Figure 1-15.

M-File Editor

The M-file editor window menu and GUI tools are sub-grouped in three tabs—EDITOR, PUBLISH, and VIEW, as shown in Figures 1-16, 1-17, and 1-18. Note that there are three main menu sub-groupings—HOME, PLOTS, and APPS—which belong to the main MATLAB window and were shown in initial sections.>

In the M-file editor’s main tools menu (see Figure 1-16), there are five sub-sections—FILE, NAVIGATE, EDIT, BREAKPOINTS, and RUN. All of the tools in each subsection are quite intuitive. For example, the FILE sub-section has GUI tools used to create a new file or open existing files, save the current file, find M-files, compare different versions of the M-files with the same names, and print the current M-file. Similarly, the NAVIGATE subsection tools help a user move the cursor within the current file and find keywords and, if necessary, substitute them with other words. The EDIT subsection tools allow you to insert a new section into the current file, add or remove a comment line or wrap comments and add indents to make the file more readable. The BREAKPOINTS subsection enables you to debug/edit the current M-file code. Finally, the RUN subsection has GUI tools that run different cell sections of the current M-file step by step and run the current M-file and measure the evaluation time in different sections of the file. It should be noted that when writing M-files, the EDITOR window tools are mainly used.

The PUBLISH tools, shown in Figure 1-17, are used to generate report files in different file formats, such as HTML, DOC, PPT, etc. The PUBLISH window’s GUI tools are very intuitive and similar to many document editing software applications.

The VIEW tools, shown in Figure 1-18, are used to display several windows of M-files and MLX-files (documents) within one window area. You can split the view side by side or top to bottom by using the GUI tools called TILES, DOCUMENT TABS, or SPLIT DOCUMENT. The tickmark options in the DISPLAY subsection are very handy to display data-tips, show line numbers, and highlight the current line while editing M-files.

MLX-File Editor

The MLX-file (live editor) editor tab (shown in Figures 1-19, 1-20, and 1-21) contains many of the M-file editor tools along with several other different GUI tools. The LIVE EDITOR tab (see Figure 1-19) has one main different sub-section from the M-file editor, called TEXT, and it contains most of the functions of the PUBLISH tools of the M-file editor.

Moreover, the MLX-editor’s INSERT tab (see Figure 1-20) has a few very handy tools to write/edit equations directly in the MLX-file and insert subsections with comments. You can insert control GUI tools, such as slider and drop-down boxes, within the current MLX-file content.

The VIEW window (shown in Figure 1-21) of the MLX-editor has some similar tools of the M-file editor’s VIEW and some other tools, such as DOCUMENT TABS, DISPLAY, OUTPUT, and LAYOUT, by which a user can display script (code) line numbers (excluding the comment lines), data-tips, full screen view, clean up the output, and show or hide output.

M-files and MLX-files can be created in several different ways—by using GUI buttons or or by typing the command in the Command Window.

In order to demonstrate some of the highlighted tools and options of the M-file and MLX-file editors, let’s look at the following example to demonstrate that MATLAB’s precision is not absolute via the Pythagorean Theorem.

Example:

First, the solution script of this simple example will created in the M-file editor and the results will be published. Subsequently, all of the simulations will be carried out within the MLX-editor to demonstrate similar and different features of the both editors.

Note that in this script, we used % to insert comments (non-executable information) and to insert and display Greek letters.

% Comments

Comments are not executable and contain additional information for the users. The % sign is used to place comments and remarks or any additional information within M-files and MLX-files. The comments can be added on a separate line or after the command syntax as long as they start with %. Using a double sign (%%) followed by a space automatically makes the following comments bold. Moreover, inserting %% at the beginning of a line and leaving a blank space after it creates a cell mode in the script. We discuss in detail the cell mode options and advantages in Chapter 2. Note that there are several other functionalities of the % sign. It is used to format specifications for write, display, read purposes, which we discuss in Chapter 2.

This is slightly edited with the PUBLISH tools, such as bold for Steps 1, 2, ..6) under INSERT INLINE MARKUP and Publish (PUBLISH); see Figure 1-17. Note that to make selected lines of comments bold, you first select the line and then press the button. Note that in this script, we used the LaTEX commands to insert the Greek letters (α, π) and the equation F = 1 − (sin²α+cos² α).

The M/MLX editors are compatible with most common LaTEX mathematical mode commands. The LaTEX compatible mathematical commands and symbols can be inserted for plot titles, axis labels, graphic notes, and so forth, which are discussed in examples in the following chapters (programming, plots, and ODEs). For example, to insert the expression: x² + y² = R², you enter the expression: x^2+y^2=R^2. Or to insert α, β, Ω, Ψ, you type alpha, eta, Omega, Psi. Moreover, to insert the equations with Greek letters, the notations need to start with $$ signs and end with $ (See Step 1. Problem statement and Step 5. Plot …). For more information and help with various mathematical expressions and symbols to write in LaTEX math, type the following in the command window: >> doc latex

See [6] also. Note that %% at the beginning initializes automatic recognition of the LaTEX compatible mathematical commands within Step 1.

%%

%% *% Step 1. Problem statement*

% MATLAB's precision is not absolute.

% Pythagorean Theorem: F = 1 - (sin^2 $$alpha$ + cos^2 $$alpha$ );

% Input variable: $$alpha$ = -0.5* $$pi$ ...0.5* $$pi$

%% *% Step 2. Define the input variable*

alpha = -pi/2:pi/100:pi/2;

%% *% Step 3. Perform the computation*

F = 1-(sin(alpha).^2+cos(alpha).^2);

%% *% Step 4. Define for which values of $$alpha$ $$F(alpha) =0$*

Findex=find(F==0); F0=F(Findex);

%% *% Step 5. Plot the simulation results: $$alpha$ vs. $$F(alpha)$*

plot(alpha, F, 'b-', alpha(Findex), F0, 'ro'),

legend('alpha vs F(alpha)', 'F(alpha) = 0')

title('The Pythagorean Theorem '), grid on

xlabel('alpha'), ylabel('F(alpha) = 1-(sin^2(alpha)+cos^2(alpha))')

%% *%Step 6. Publish all of the obtained results including the whole script*

After writing the script in the editor, save the M-file with a valid filename (e.g., P1.m). Now, to publish it as an HTML format, press the Publish button under the PUBLISH tab (see Figure 1-17). After pressing the button, the script is executed automatically and its HTML-formatted report will be generated, as shown in Figure 1-19.22

3

Note that the bold lines (Figure 1-19), which are the starting lines of cell modes preceded with %%, have been recognized by the M-file editor automatically and put in contents and hyperlinked, such as % Step 1 ... Step 2 ... Step 6.

Now let’s try the same procedure with the MLX-file editor. Note that the comments are edited using the TEXT tab tools of the MLX-editor (shown in Figure 1-19). The parts of the script are entered as text (comments) by using . The Step 1 … Step 2 … Step 3… Step 6 lines are made bold with .

Step 1. Problem statement

MATLAB's precision is not absolute.

Pythagorean Theorem: F = 1 - (sin^2 (alpha) + cos^2(alpha));

Input variable: alpha = -0.5*pi ... 0.5*pi

Step 2. Define the input variable

Step 3. Perform the computation

Step 4. Define for which values of alpha, F(alpha) = 0.

Step 5. Plot the simulation results: alpha vs. F(alpha)

Step 6. Publish all of the obtained results including the whole script

Note that there is an alternative way to make the chosen lines bold. That is to use %% followed by a blank space, as in the M-file editor. In this case, such editing is automatically detected as a header of the following section of the script. If you enter this in the Code section of MLX-editor:

%% Step 2. Define the input variable.

and press Enter, MATLAB automatically creates this bolded text header:

Step 2. Define the input variable.

Now, in between Step 2, Step 3, Step 5, and Step 6, the following executable commands are inserted by placing the cursor on the desired line and pressing the button from CODE sub-tab (see Figure 1-19). Finally, the complete code is obtained.

Step 1. Problem statement

MATLAB's precision is not absolute.

Pythagorean Theorem: F = 1 - (sin^2 (alpha) + cos^2(alpha));

Input variable: alpha = -0.5*pi ... 0.5*pi

Step 2. Define the input variable

alpha = -pi/2:pi/100:pi/2;

Step 3. Perform the computation

F = 1-(sin(alpha).^2+cos(alpha).^2);

Step 4. Define for which values of alpha, F(alpha) = 0.

Findex=find(F==0);F0=F(Findex);

Step 5. Plot the simulation results: alpha vs. F(alpha)

plot(alpha, F, 'b-', alpha(Findex), F0, 'ro'),

legend('alpha vs F(alpha)', 'F(alpha) = 0')

title('The Pythagorean Theorem'), grid on

xlabel('alpha'), ylabel('F(alpha) = 1-(sin(alpha)^2+cos(alpha)^2)')

Step 6. Publish all of the obtained results including the whole script

Save the file as an *.mlx file (e.g., P2.mlx). Insert the mathematical expressions with the Equation tools under the INSERT sub-tab (shown in Figure 1-20) and by using . When you press the button, the whole range of symbols, structures, and matrices will be opened as shown in the following image.

Now you bring the cursor where you want to insert the mathematical expressions and insert the symbols by selecting the necessary symbols, or by using LaTEX expressions. For example, alpha gives α. Enter the expressions of the Pythagorean formulation and input the variable range, then finalize the script.

Step 1. Problem statement

MATLAB's precision is not absolute

The Pythagorean Theorem: F = 1 − (sin²α + cos²α)

Input variable: α =  − 0.5α...0.5α

Step 2. Define the input variable

alpha = -pi/2:pi/100:pi/2;

Step 3. Perform the computation

F = 1-(sin(alpha).^2+cos(alpha).^2);

Step 4. Define for which values of α, F(α) = 0

Findex=find(F==0);

F0=F(Findex);

Step 5. Plot the simulation results: α vs. F(α)

plot(alpha, F, 'b-', alpha(Findex), F0, 'ro'),

legend('alpha vs F(alpha)', 'F(alpha) = 0')

title('The Pythagorean Theorem '), grid on

xlabel('alpha'), ylabel('F(alpha) = 1-(sin^2(alpha) +cos^2(alpha))')

Step 6. Publish all of the obtained results including the whole script

After executing the saved MLX-file (by pressing the Run button shown in Figure 1-19), you’ll get the script outputs via Output Inline in LAYOUT. See Figure 1-24 and Figure 1-25.

Note that in the MLX editor, all of the executable and non-executable lines of the script are identified automatically and placed in separate sections. There are other salient issues about hints, warnings, and error messages from the M/MLX editors that we discuss in Chapter 2.

Exercise 1

Exercise 2

Exercise 3

Get help with the MATLAB exp (exponential) function using the Command Window. Use the help, lookfor, doc, and help browser commands and then compare the results of the four help options.

Exercise 4

Create and open an *.m file called learn.m in M-file editor window using the Command Window. (Hint: >> edit ...). Insert two commands in it that display the current date and time. Hint: date, clock.

Exercise 5

Create a shortcut (a set of favorite commands) that opens a new M-file named My_Shortcut and simultaneously closes all figure windows and clears the Command Window and Workspace from all previously entered data and commands.

Exercise 6

Change from the Preferences window a display of data formats using the Command Window. Make it hexadecimal format. Hint: format.

Exercise 7

Which commands are used to clean up the Workspace, Command Window, and History?

Exercise 8

Given x = 2.25, y = 3.1 and z = 13.20, use MATLAB to evaluate the following expressions:

Use the Array Editor to change the assigned values for x, y and z and re-execute the expressions to compute A and B.

Exercise 9

Exercise 10

Exercise 11

Save all computation results from Exercises 9 and 10 in a MAT file called my_FIRSTwork.mat and clean up your MATLAB workspace, removing all variables except for A and B from Exercise 8.

Exercise 12

Exercise 13

Exercise 14

Exercise 15

Why is day_DUE numeric and equal to a 10-element row matrix?

What do these numbers represent?

Exercise 16

Given: >> A=[1,2; -12.0, 3]; mat2str(A); SNA=ans+0; char(SNA)

What is hidden behind the variable SNA?

Exercise 17

Exercise 18

Create a function handle and inline function of the following mathematical expression:

h(θ, t) = 1.3 * e^−tiθ. Note that i represents an imaginary number.

Exercise 19

It is analytically proven that cos2α = 2 cos²α − 1. Use MATLAB to compute the equality for different values [] of α and define the values of α in which the accuracy of MATLAB calculations does not represent equality.

Exercise 20

Use MATLAB to compute the expression in a most accurate way. Note that it is in a fifth root.

Exercise 21

How do you fix this problem and make the results legible?

Exercise 22

Create a 5-by-5 matrix called A by using randi() within [1, 20] divided by 3. Display A as rational numbers, as shown here. Note that your array numbers (in the numerator) of A will differ from the ones shown here. Why does your answer differ from the one shown here?

Exercise 23

Create the following array in the most efficient way (at least two different ways). Note the display format of the A2 elements.

Exercise 24

Using randi(), create a 15 by 15 array with elements ranging from -125 to 127 and save it in the most memory-efficient way with the name A3.

Exercise 25

Create the following HTML report using M-file editor tools.

Exercise 26

Exercise 27

Create three numerical (row matrix) arrays (variables called AJ, IS, and LJ) so that, when you subtract three from each and then perform one conversion operation, you obtain Al-Khwarizmi, Ibn Sina, and Lennart Johansson.

Exercise 28

Save all of your created variables (A, A2, A3, A4, A5, and A6) from Exercises 22-27 in a *.mat file with your last name, e.g. Jones_HW2.mat.

Exercise 29

Create a cell array (called A) containing three variables: a=4/5, b='matlab'+0, and c=sin(π), and create a structure (called B) containing four variables: a, b, c, and A. Show how you access the a, b, and c variables residing inside A and B.

Exercise 30

Create the following variables and entries in the MLX File Editor:

Function handle F: F(ω₁, ω₂, θ) =  cos (ω₁θ) −  sin (ω₂θ)

Identity matrix:

Magic numbers:

Multiply the matrix by 2 and subtract it from the M matrix. Call the new matrix MI:

Locate array L by locating/comparing the elements of MI that are greater than 1 and less than 6:

Table array TM from M:

Cell Array CA containing F, I, M, MI, L, TM

Structure SA containing F, I, M, MI, L, TM, CA.

Exercise 31

Explain how to obtain the following Layout (1), Preferences (2), and Quick Access (3) windows, as shown in the screenshots. Note: Fonts and Colors in (2).

Exercise 32

Create these files (MLX/M-files) and explain how to display the results as shown in the figure in the Live Editor window.

Note that there are four windows displaying MLX and M-files; Equations, Greek letters, Plot figures, hyperlinks, data tips, and how to insert an image.

Exercise 33

Exercise 34

Exercise 35

How do you get a logical array c = (1 1 1 1 1 0 0 0 0 0 0) from the array a = (-5, -4, -3 ... 3, 4, 5) whose elements are linear (equally spaced)?

Exercise 36

There are errors. Find errors and fix them. What is the size of x and F now?

Exercise 37

Exercise 38

Exercise 39

Exercise 40

Exercise 41

And by applying logical indexing (logical array) and element-wise matrix multiplication operations, generate the following array:

1. 5.

   The command that creates a structure array called S5 and the command that writes its Q1, H, and HLG.

    
2. 6.

   Explain why the converted numbers (in Step 3) of the binary representations (of 12321 and 987654321) differ from the original decimal numbers, i.e. 12321 and 987654321?

    

Exercise 42

Exercise 43

Exercise 44