By now, you should have learned the basics of plotting in Matlab using previous post. In this session we want to look closer to how we can plot a conditional plot using Matlab Here is an easy example of a conditional plot

plot f

Conditional plotting in Matlab

One of the way you can walk to easily plot f is the following Matlab code

x=-5:0.1:5;
for n=1:length(x)
if x(n)<1
f(n)=x(n)^3;
else
f(n)=x(n)^2;
end
end
plot(x,f,'--r','linewidth',2)

which returns

If you were just to plot half of the graph each time and add up plots, the code would look like

% Part 1
x=-5:0.1:1;
f=x.^3;
plot(x,f,'--b','linewidth',2);
% Part 2
x=1:0.1:5;
f=x.^2;
hold on;
plot(x,f,'--r','linewidth',2)

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There is a function behind every Matlab code that performs a computational task, and here is how you can create your own function that will help you go faster while working with Matlab.

We will be creating a simple one-line code function in this post, just to help you have a glimpse into how creating functions in Matlab

Write a function in Matlab

Here is what you do when writing your own function in Matlab.

Open a new function file

Here is what appear on the next window

In Matlab, the section after the % is considered as a comment (the section that appears in green in the window above)

Here is how we need to structure our Matlab function

[php]function [v] = vol(L,W,H)
% L is the length of the 3D rectangle
% W is the width of the 3D rectangle
% H is the height of the 3D rectangle
v= L*W*H;
end[/php]

You need to copy the whole code above and paste it in the file, Save the file with the name vol.m and make sure not to change the directory Matlab will guide you to.

Here is how it will look

How to use the Matlab function we have just created

Every time you will need to compute the such a volume, simply type the following code

vol(L,W,H)

where L, W, And H will be real numbers.

Example

Let’s say, we need the volume of a 3D rectangle with L, W and H being respectively 10, 3, 15, we can simply type:

vol(10,3,15)

And Matlab will call the function vol, compute the volume of the 3D rectangle and display the following.

If you are working in an exercise where you will often need this volume, you can just use this function instead of actually computing manually the volume each time you need it. It might not look very important in this exercise, but when you have a complex operation you need to perform often in Matlab, creating a function doing it automatically will be time-efficient and necessary.

Using this technique, you can create more complex functions in Matlab and have an easier life while using them. Learn more about functions in Matlab.

In the world of tech devices, the Apple brand is the Holy Grail and wherever it is mentioned other tech brands usually bow their heads in shame. And same goes for Matlab the undisputed king of the numerical computations niche. This software is loved by both experienced engineers and students alike, saddled with the responsibility of computing numbers, processing images, and data acquisition. Therefore, it’s widespread fame, features and abilities are well known but little do we talk about the cost of purchasing the original software as well as its add-ons and the effect they have on our pockets.

For anyone who does not have the resources to purchase Matlab’s software or the corresponding add-ons you need to handle certain projects, here is some god news for you. There are a couple of matrix language software applications out there on the market that you can easily purchase without punching a hole in your pockets and we will be reviewing three of such alternatives in the coming paragraphs. And here is our list of criteria to consider for the alternatives provided in the list below.

It runs on Windows: the popularity of Windows OS means that most students or people in general will have to run the software on a Windows powered PC sooner or later. So our alternative must run on Windows.

It should be Compatible with Matlab: since Matlab has the first-mover advantage over its competitors, it is important for others to ensure imported and exported files can be used on Matlab and vice versa.
It should have a good User Interface: A seamless interface which is easy to use coupled with a basic command line and other simplistic features will also be considered.

Matlab alternatives

Octave

Octave is a free matrix language software application built by GNU engineers with the aim of simplifying computations as well as to provide students with a non-commercial software option to work with. This software is spectacular in many ways because it was basically developed by engineers to be used by engineers; therefore, it cuts out unnecessary fluff and procedures during its use. Using our criteria to review Octave we have:

Octave was built to run on both Windows and Linux OS’s seamlessly and to top this off, the software is totally free on both platforms and can be downloaded from the GNU website or at some engineering online forums.

Also important to this discussion is the fact that Octave is 95% compatible with MATLAB which makes it a great resource for students. Octave functions are declared in the same way as MATLABs’ and it has simple commands for saving and loading matrices and commands that were built for MATLAB. On an unrelated note, Octave works simultaneously with other GNU software applications such as Gnuplot and plplot which makes it capable of plotting diverse graphs and also display images.

Octave’s fast matrix routines, syntax, inclusion of an iterative history, numerous mathematical solutions utilities and its soft learning curve makes it a great alternative to MATLAB

Scilab

Another contender to the MATLAB throne, known for its high-level computation abilities is Scilab. Scilab is an open source software for numerical computation and provides a solid computing ecosystem for engineering and other scientific computation.

Some long time users of this software application claim that it is quite better than both Octave and MATLAB due to the advanced math operations and engineering analysis it can handle but whether this is true or not, it stands as a testament to its computational abilities as a good alternative to MATLAB both students and working professionals should consider. Reviewing Scilab using our criteria we get:

Scilab takes the compatibility issue a step further for it is compatible on more platform operating systems than the Octave Scilab runs on all Windows operating systems, Linux as well as OS X. The software application can be found and downloaded online from Scilab Enterprise’s official website.

Unlike the Octave which is closely modelled after MATLAB in terms of syntax, Scilab is a little bit different and its functions and commands are declared by using different but simple formats which anyone can easily learn. Scilab takes simulation, design and analysis a step further by including a ‘hybrid dynamic systems modeller and simulator’ for modelling mechanical systems, control systems and other high-level mathematical operations.

Scilab may have some differences from MATLAB but any active user of the software application can quickly learn how to use it as a powerful tool for numerical computation and otherwise which makes it a powerful alternative to MATLAB. Do not forget that it’s also free.

FreeMat

This is another open source environment for rapid engineering and scientific processing that is quite similar to MATLAB in terms of syntax which makes it a perfect alternative to MATLAB. The FreeMat platform was built in MIT as a cheaper alternative to the more commercial numerical computation languages out there and it’s completely free package ensures it comes cheap to students.

Like Scilab, FreeMat works on multiple operating systems which include Windows, OS X, and Linux. As stated earlier, the application is totally free and can be downloaded from the FreeMat official website and other student design and simulation forums.

In terms of compatibility, FreeMat can easily be called the closest clone to MATLAB as it supports approximately 97% of the features in MATLAB which is quite good for anyone looking for something close to MATLAB but cheaper. Its cloned features include support for; all the integer types Matlab supports, dynamic array types, N-dimensional array manipulation , built in arithmetic manipulation of data types, command lines etc. A good knowledge of MATLAB is all you need to start using FreeMat.

The multiple similarities between FreeMat and MATLAB may make it the best alternative software on this list. But there is no denying that it is one of the favourites anyone looking for something really similar should consider using.

Final thought

In conclusion, despite the differences between the above-listed languages and MATLAB, each and every one of these applications has its own merit which makes them viable options for your numerical computations at no cost whatsoever. Students and hobbyists should consider them as perfect alternatives.

We have recently learned how to solve polynomial equations in Matlab, now we are going to visit a trick that will help you learn how to do Division and Multiplication of polynomial using Matlab.

Polynomial Division example

Polynomial Multiplication example

Matlab Polynomial

Multiplication of polynomial can be a very dreary task, so do the division of polynomial. Matlab uses the functions conv and deconv to help you do these tasks with the least commotion possible, and most importantly with the assurance to find the right result the quickest way possible.

Let get on some examples, those will help easily learn how to make use of conv (for multiplication) and deconv (for division).

Example 1

The code

s=[1 4];
t=[1 4 -3];
conv(s,t)

Which returns

Rewritten, it gives the following

Example 2

The code

c=[1 0 4];
u=[1 3];
[p,r]=deconv(c,u)% r is the remainder of the division

Which returns

Rewritten, it gives

A bit of clarity

Top-down conversion. Here we simply need to indicate the coefficient and their position in the polynomial equation.

Bottom-up conversion. Here we need to read Matlab outputs and rewrite it the way it is supposed to be, and it goes like this.

Remember you need to put a zero, in case an order is not represented in the equation. See the example below

There are two operators allowing to divide in Matlab:

• The right division represented by the symbol / (slash)
• The left division represented by the symbol \ (Backslash)

These two operators differ from each other

Using numbers, the right division will be the conventional division we all day make use of, so

Contrary to the right division, the left division reverse the division, meaning

Matrix division in Matlab

The right Matrix divide

Let’s consider two Matrices A and B

Using the right division

Matlab code

A=[1 2 ; 2 2];
B=[3 2 ; 1 1];
A/B % You can also use A*inv(B)

which returns

rewritten, it will look like this

The left Matrix divide

The right matrix divide is roughly the same as

Which leads to a complete different result from the preceding operator.

This technique can be used to quickly compute the solution of the equation

So, using our early defined matrices

will be written

A=[1 2 ; 2 2];
B=[3 2 ; 1 1];
A\B % You can also use inv(A)*B

which returns

Which rewritten will look like

The Matrix division, element by element

We thought it will be also necessary you have a grip on the element-by-element Matrix division in Matlab

To divide Matrices, element-by-element, the following formula is useful

Where

The code

A./B

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Plotting is a very important step in the study of a system since it helps understand the behavior and the restrictions of the system with less effort possible.

Plotting an equation is not always an easy task especially if you have to work out all the parameters manually, luckily nowadays there are tons of tools that help plot equations.

Matlab is the most popular tool out there when it comes to the study of the system, and in this post, we are going to work with some basic equations, Matlab, and most importantly make use of EZPLOT for plotting equations.

Plotting equations in Matlab using EZPLOT

 Ezplot Matlab example

To plot this equation in Matlab using the EZPLOT, We will write the equation the following way

Now type the following code and press ENTER

ezplot('y-x^3-x^2+x-1',[-5 5 -5 5])

And you will obtain the following figure

If you would like to have more information about the command we just used, you can find it by using the following code

doc ezplot

While using the EZPLOT, the structure of the code is the following

ezplot(fun,[xmin,xmax,ymin,ymax])

Where fun is the equation, xmin and xmax are respectively the lower and the highest value of x, ymin and ymax the lowest and highest value of y. If the most interesting part of the graph your are plotting is happening in a larger interval than the one we just plotted, you may need to increase the value of xmin, xmax, ymin, and ymax.

These are a few basic tools you will need to know if you are just starting learning Matlab.

Scalar functions in Matlab

Before fully diving into trigonometric functions, it is necessary to state that Matlab takes as input radians while working with trigonometric function meaning and uses the symbol pi for π

Trigonometric sine

Matlab code

sin(pi)

Trigonometric cosine

Matlab code

cos(pi)

Trigonometric tangent

Matlab code

tan(3*pi/2)

Trigonometric inverse sine (arcsine)

Matlab code

asin(1)

Trigonometric inverse cosine (arccos)

Matlab code

acos(0.5)

Trigonometric inverse tangent (arctan)

Matlab code

atan(3/2)

e Matlab (exponential)

Matlab code

exp(x)

Natural logarithm

Matlab code

log(5/2)

Absolute value

Matlab code

abs(a*x+b)

Square root

Matlab code

sqrt(a*b)

Remainder after division

This function helps you find the remainder after a division. We will use as an example the remainder of the division of 5 by 2, which is one.

Matlab code

rem(5,2)

which returns

Round towards nearest integer

This function will allow you to round values towards the nearest integer if you are not interested in working with decimal numbers. It is most useful while you want to round the result of an operation.

Matlab code

round(2.9)

Which returns

Round towards negative infinity

floor(x) rounds the elements of x to the nearest integers towards minus infinity.

Round towards positive infinity

As opposed to the floor function, ceil(x) rounds the elements of x to the nearest integers towards infinity.

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Solving a system of equations with two unknowns is a very easy cake to bite but when the number of unknown exceed two, solving the system of equations becomes complicated and time-consuming.

In this post, we are going to show you how you can use your computer and Matlab to solve a system of many equations.

Caution: the following technique works only when the number of equations and the number of unknowns are the same. We will solely work will linear equations

Can Matlab solve a system of equations?

Yes, Matlab can help you easily solve a system of equations. Let’s jump into some examples and show you how you go about it.

Using Matlab to Solve a system of equation with two unknowns

Let’s consider the following system of equations

The above equation can be written in the matrix form

The equation can be rewritten as

Now we are set to use Matlab!

A=[2 3;1 1];
B=[8;3];
X=inv(A)*B

The last line does not have; at the end.

You will see an answer like this

Which simply means that

Let’s elaborate a little more in detail here before moving forward. What we have done above is the following.

We have written the equation in the form

Where

So to have at the end the equation on the form

Example 2: System of the equation with three unknowns

let’s consider the following system of equations

Using the same technique we used above we can write the system in the following form

Giving commands to Matlab will look like the following

A=[1 3 -2;3 2 -1;2 -1 -3];
B=[1;2;13];
X=inv(A)*B

To end up with the following result

Final thought

If you are working on a system of equations where the number of unknowns is equal to the number of equations, this method is a good way to go. This restriction is due to the fact that using the system of matrices to solve such system of equations requires that the matrix A is invertible. If there are no possibilities of finding the inv(A) then this method is totally useless.

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Lately we have learned some basics about Matlab matrix operations. Here come the part 2 of that post, where we will test our abilities of manipulating matrices in Matlab

Matlab matrix operation exercise

Exercise 1

Create a 1 x 5 vector A with all elements equal to 0

Exercise 2

Create a 3 x 1 vector B with with all elements equal to 1

Exercise 3

Create a 1 x 5 vector C with elements equal to 1, 2, 3, 4, π respectively.

Exercise 4

Create 1-row vector D with element’s from 3 to 27 with step 3 by using the appropriate operator

Exercise 5

Create a 3 x 3 matrix E with elements with random values.

Exercise 6

Create the following matrix

Exercise 7

Check the size of F

Exercise 8

Create the following Matrix

1. find the transpose of G
2. print on display the element G₁₃ of the transpose of G
3. print on display the second row of the transpose of G
4. print on display the third column of the transpose of G
5. find the determinant of G

Exercise 9

Create the following Matrices

1. Find the Matrix H+J
2. Find the Matrix H * J
3. Multiply H by J, element-by-element

Exercise 10

Solve the linear system of equations ax=b if

Exercise 11

1. Create a 1 x 5 vector with elements with random values
2. Use the “for” loop to find the greatest element of the matrix
3. Use the “while” loop to find the smallest element of the matrix

Exercise 12

Use the diag operator to generate the matrix

Exercise 13

Let’s consider the following vector

1. Define a vector K, K being the vector A sorted in ascending order
2. Define a vector L, L being the vector A sorted in Descending order

We have so far been working with numbers, totally ignoring one of the most rewarding ability Matlab put to our use: the possibility to work with symbolic expressions.

In many college algebraic courses, it is taught how to use and simplify equations with symbolic expressions, where one of the major task is to be learn how to express one symbol with respect to others. Here we will attempt to use Matlab to solve some problems we lately worked on using real numbers.

Example 1

Let’s consider the following equation

We all know that this is second order polynomial equation and we know how to solve it. Let’s try using Matlab to solve this very equation as it is, assuming we don’t know what the value of the coefficients are.

The code

syms a b c x
f = a*x^2 + b*x + c
solve(f)

Which returns

Let’s ask Matlab to give us less difficulties reading the answer

The code

pretty(ans)

Which returns

Which we all remember from basic algebra.

If you would like to solve the equation with respect to a, you can state it like this

solve(f,a)

Which returns

Example 2

Let’s now use the following equation

Just like the second order polynomial equation in example 1, this will be like it

The code

syms a b c d x;
f = a*x^3 + b*x^2 + c*x + d;
l = solve(f);
pretty(l)

Which returns

Example 3

This is also a nice method you can make use of, to help you remember a formula. Let’s consider the following Matrix

Let’s find the determinant of A

The code

syms A11 A12 A13 A21 A22 A23 A31 A32 A33
A=[A11 A12 A13 ; A21 A22 A23 ; A31 A32 A33]
l = det(A)

Which returns

Example 4

Let’s do some Matrix operation with the following matrices: Addition and Multiplication.

the code

syms B11 B12 B21 B22 C11 C12 C21 C22;
B = [B11 B12 ; B21 B22];
C = [C11 C12 ; C21 C22];
Add = B + C;
Mul = B*C;
Add
Mul

Which returns

Example 5

Let’s end this session with solving the following equation.

The code

syms a b x
g = a^x + b;
l = solve(g);
pretty(l)

Which returns