A Short Introduction to Scilab

Terence Leung Ho Yin, Tsing Nam Kiu

Table of Contents

  1. About Scilab
  2. Installing and Running Scilab
  3. Documentation and Help
  4. Scilab Basics
    • Common Operators
    • Common Functions
    • Special Constants
    • The Command Line
    • Data Structures
    • Strings
    • Saving and Loading Variables
  5. Dealing with Matrices
    • Entering Matrices
    • Calculating Sums
    • Subscripts
    • The Colon Operator
    • Simple Matrix Generation
    • Concatenation
    • Deleting Rows and Columns
    • Matrix Inverse and Solving Linear Systems
    • Entry-wise operations, Matrix Size
  6. The Programming Environment
    • Creating Functions
    • Flow Control
    • Some Programming Tips
    • Debugging
  7. Plotting Graphs
    • 2D Graphs
    • 3D Surfaces
  8. Scilab versus Matlab
  9. References

Top1. About Scilab

Scilab is a freely distributed open source scientific software package, first developed by researchers from INRIA and ENPC, and now by the Scilab Consortium. It is similar to Matlab, which is a commercial product. Yet it is almost as powerful as Matlab. Scilab consists of three main components:

Scilab is specialized in handling matrices (basic matrix manipulation, concatenation, transpose, inverse, etc.) and numerical computations. Also it has an open programming environment that allows users to create their own functions and libraries.

For further information and documentation, visit the Scilab homepage:

www.scilab.org

Top2. Installing and Running Scilab

First, you must have the software. Go to the download section in the Scilab homepage, find a right version for your operating system (platform), and then click to download. For easy installation, it is advisable to download the installer (for binary version). Then double click the downloaded file and follow the instructions to complete the installation.

To run Scilab, type

scilex

in the command prompt in the folder bin under the installation directory, or click the shortcut in the start menu if you use Windows. To exit the program, type

exit

or close the window of the main program.

Top3. Documentation and Help

To find the usage of any function, type

help function_name

For example:

help sum

If you want to find functions that you do not know, you may just type

help

and search for the keywords of the functions. Finally, if you want more information, you may visit the Scilab homepage. There is a section called documentation. It is very resourceful.

Top4. Scilab Basics

Common Operators

Here is a list of common operators in Scilab:

+ addition
- subtraction
* multiplication
/ division
^ power
' conjugate transpose

Common Functions

Some common functions in Scilab are:

sin, cos, tan, asin, acos, atan, abs, min, max, sqrt, sum

E.g., when we enter:

sin(0.5) 

then it displays:

ans =
   0.4794255

Another example:

max(2, 3, abs(-5), sin(1))
ans = 
   5

Special Constants

We may wish to enter some special constants like, i (sqrt(-1)) and e. It is done by entering

%pi  %i  %e

respectively. There are also constants

%t  %f

which are Boolean constants representing true and false, respectively. Boolean variables would be introduced later.

The Command Line

Multiple commands, separated by commas, can be put on the same command line:

A = [1 2 3],  s = tan(%pi/4) + %e
 A  =
    1.   2.   3.
 s  =
    3.7182818

Entering a semi-colon at the end of a command line suppresses showing the result (the answer of the expression):

A = [1 2 3];  s = tan(%pi/4) + %e;

Here the vector [1 2 3] is stored in the variable A, and the expression tan(%pi/4) + %e is evaluated and stored in s, but the results are not shown on the screen.

A long command instruction can be broken with line-wraps by using the ellipsis (...) at the end of each line to indicate that the command actually continues on the next line:

s =  1 -1/2 + 1/3 -1/4 + 1/5 - 1/6 + 1/7 ...
     - 1/8 + 1/9 - 1/10 + 1/11 - 1/12;

Anything that is typed after a pair of slashes // will be ignored, and hence can serve as comments or code annotations:

%e^(%pi * %i) + 1  // should equal 0, as in the Euler's identity
ans  =
   1.225D-16i 

Using the up and down arrow in the command line can recall previous commands.

Data Structures

Scilab supports many data structures. Examples are: (real or complex) matrix, polynomial, Boolean, string, function, list, tlist, sparse, library. Please read the Scilab documentation for details. To query for the type of an object, type:

typeof(object)

Strings

To enter a string, enclose it with either single or double quotations. For example:

'This is a string'

or

"this is also a string"

To concatenate strings, use the operator + :

"Welcome " + "to " + "Scilab!"
ans =
   Welcome to Scilab!

There are some basic string handling functions such as

strindex   strsplit   strsubst   part

Please refer to Scilab's documentation for details.

Saving and Loading Variables

To save and to load variables, we use the save and load functions:

save('file_name', var1, var2, ...);
load('file_name', 'var1', 'var2', ...);

where file_name is the name of the file to be saved or loaded, and var1, var2, ..., are variable names.

Notice that the variable name has to match when it is to be saved. Here are some illustrations.

a = 3; b = %f; s = 'scilab';
save('save.dat', a, b, s);
clear a;  // delete the variable a
clear b;
clear s;
load('save.dat', 'a', 'b', 's');
// load all the saved variables

load('save.dat','b');
// It loads only variable b, but not
// variable a in the name of b

load('save.dat','d');
// It will not show any error messages.
// Variable d is undefined, not empty.

listvarinfile('save.dat');
// list variables in a file saved by 
// the function save

Name              Type         Size          Bytes
----------------------------------------------------
a                 constant     1 by 1        24
b                 boolean      1 by 1        20
s                 string       1 by 1        44

Programming Environment

Top5. Dealing with Matrices

Entering Matrices

There are many ways to enter a matrix. Here is the simplest method:

  1. Separate each element in a row by a blank space or a comma,
  2. Separate each row of elements with a semi-colon, and
  3. Put the whole list of elements in a pair of square brackets.

For example, to enter a 3 x 3 magic square and assign to the variable M :

M = [8 1 6; 3 5 7; 4 9 2]
M = 
   8.   1.   6.
   3.   5.   7.
   4.   9.   2.

Calculating Sums

For a magic square, we may wish to check for its column sums and row sums and the sum of diagonals. This is done by entering:

sum(M,'c') // column sums
ans = 
   15.
   15.
   15.
sum(M,'r') // row sums
ans =
   15.  15.  15.

The sum of the main diagonal is easily done with the help of the function diag.

diag(M)
ans =
   8.
   5.
   2.

Subscripts

It is a bit more difficult to find the sum of the other diagonal. We will show two ways to accomplish it. One method is to find the sum manually, i.e., to read the specified elements and then to sum them up.

M(1,3) + M(2,2) + M(3,1)
ans =
   15.

It is possible to access elements in a matrix using a single index. This by treating a matrix as a long vector formed by stacking up the columns of the matrix. E.g., the values of M(1), M(2), M(3), M(4), M(5) are 8, 3, 4, 1, 5, respectively.

Accessing out-of-bound elements will result in an error, like entering:

M(3,4)
    !--error 21
invalid index

A smarter way to get the sum of the other diagonal is to use the function mtlb_fliplr, where mtlb stands for Matlab. This is to flip a matrix left-to-right (lr):

mtlb_fliplr(M)
ans =
   6.   1.   8.
   7.   5.   3.
   2.   9.   4.

The desired result would then be obtained by typing sum(diag(mtlb_fliplr(M))).

The Colon Operator

The colon operator is one of the most important operators in Scilab. The expression 1:10 results in a row matrix with elements 1, 2, ..., 10, i.e.:

1:10
ans =
   1.  2.  3.  4.  5.  6.  7.  8.  9.  10.

To have non-unit spacing we specify the increment:

10 : -2 : 2
ans =
   10.  8.  6.  4.  2.

Notice that expressions like 10:-2:1, 10:-2:0.3 would produce the same result while 11:-2:2 would not.

Subscript expressions involving colons refer to parts of a matrix. M(i:j, k) shows the i-th row to j-th row of column k. Similarly,

M(3,2:3)
ans =
   9.  2.

Some more examples:

M(3,[3,2])
ans =
   2.    9.

M([2,1], 3:-1:1)
ans =
   7.   5.   3.
   6.   1.   8.

The operator $, which gives the largest value of an index, is handy for getting the last entry of a vector or a matrix. For example, to access all elements except the last of the last column, we type:

M(1:$-1, $)

We sometimes want a whole row or a column. For example, to obtain all the elements of the second row of M, enter:

M(2,:)
ans =
   3.   5.   7.

Now we have a new way to perform operations like mtlb_fliplr(M). It is done by entering M(:, $:-1:1). However the function mtlb_fliplr(M) would obtain result faster (in computation time) than using the subscript expression.

Simple Matrix Generation

Some basic matrices can be generated with a single command:

zeros all zeros
ones all ones
eye dentity matrix (having 1 in the
main diagonal and 0 elsewhere)
rand random elements (follows either
normal or uniform distribution)

Some illustrations:

zeros(2,3)
ans =
   0.   0.   0.
   0.   0.   0.

8 * ones(2,2)
ans = 
   8.   8.
   8.   8.

eye(2,3)
ans =
   1.   0.   0.
   0.   1.   0.

rand(1,3,'uniform') // same as rand(1,3)
ans =
   0.2113249   0.7560439   0.0002211

Concatenation

Concatenation is the process of joining smaller size matrices to form bigger ones. This is done by putting matrices as elements in the bigger matrix:

a = [1 2 3]; b = [4 5 6]; c = [7 8 9];
d = [a b c]
d =
   1.  2.  3.  4.  5.  6.  7.  8.  9.

e = [a; b; c]
e =
   1.   2.   3.
   4.   5.   6.
   7.   8.   9.

Concatenation must be row/column consistent:

x = [1 2]; y = [1 2 3];
z = [x; y]
       !-error 6
inconsistent row/column dimensions

We can also concatenate block matrices, e.g.:

[eye(2,2) 5*ones(2,3); zeros(1,3) rand(1,2)]
ans =
   1.   0.   5.   5.          5.
   0.   1.   5.   5.          5.
   0.   0.   0.   0.6525135   0.3076091

Remember that it is an error to access out-of-bound element of a matrix. However, it is okay to assign values to out-of-bound elements. The result is a larger size matrix with all unspecified entries 0:

M = matrix(1:6, 2, 3); M(3,1) = 10
M =
   1.    3.   5.
   2.    4.   6.
   10.   0.   0.

It is remarked that this method is slow. If the size of the matrix is known beforehand, we should use pre-allocation:

M = zeros(3,3); // pre-allocation
M([1 2], :) = matrix(1:6, 2, 3);
M(3,1) = 10;

Deleting Rows and Columns

A pair of square brackets with nothing in between represents the empty matrix. This can be used to delete rows or columns of a matrix. To delete the 1st and the 3rd rows of a 4x4 identity matrix, we type:

A = eye(4,4);
A([1 3],:) = []
A =
   0.   1.   0.   0.
   0.   0.   0.   1.

If we delete a single element from a matrix, it results in an error, e.g.:

A(1,2) = []
         !-error 15
submatrix incorrectly defined

If we delete elements using single index expression, the result would be a column vector:

B=[1 2 3; 4 5 6]; 
B(1:2:5)=[]
B =
   4.
   5.
   6.

Matrix Inverse and Solving Linear Systems

The command inv(M) gives the inverse of the matrix M. If the matrix is badly scaled or nearly singular, a warning message will be displayed:

inv([1 2;2 4.0000001])
 warning
 matrix is close to singular or badly scaled. rcond =    2.7778D-09
 ans  =
    40000001.  - 20000000.
  - 20000000.   10000000.

Solving a system of linear equations Ax = b, i.e., to find x that satisfies the equation, when A is a square, invertible matrix and b is a vector, is done in Scilab by entering A\b :

A = rand(3,3), b = rand(3,1)
A =
   0.2113249   0.3303271   0.8497452
   0.7560439   0.6653811   0.6857310
   0.0002211   0.6283918   0.8782165
b =
   0.0683740
   0.5608486
   0.6623569

x = A \ b
x = 
   - 0.3561912
     1.7908789
   - 0.5271342

Another method is to type inv(A) * b. Although it gives the same result, it is slower than A\b because the first method mainly uses Gaussian Elimination which saves some computation effort. Please read the Scilab help file for more details about the slash operator when A is non-square.

A / b solves for x in the equation xb = A.

Entry-wise operations, Matrix Size

To add 4 to each entry of a matrix M, using M + 4 * ones(M) is correct but troublesome. Indeed this can be done easily by M + 4. Subtraction of a scalar from a matrix entry-wise is done similarly.

Multiplying 2 to the second column and 3 to the third column of M can be achieved by using the entry-wise multiplication operator .* :

M .* [1:3; 1:3]

Entry-wise arithmetic operations for arrays are:

+ addition
- subtraction
.* multiplication
.^ power
./ right division
.\ left division

Thus, instead of typing

s =  1 -1/2 + 1/3 -1/4 + 1/5 - 1/6 + 1/7 ...
     - 1/8 + 1/9 - 1/10 + 1/11 - 1/12;

one may simply type

s = sum((1:2:12) .\ 1) - sum((2:2:12) .\ 1)

or

s = sum(1 ./ (1:2:12)) - sum(1 ./ (2:2:12))

Note that 1./1:2:12 is interpreted as (1./1):2:12. Similarly, 1:2:12.\1 is interpreted as 1:2:(12.)\1.

To enter the matrix M = [1 2 3 4 5; 6 7 8 9 10], one may use:

M = zeros(2,5);
M(:) = 1:10;

Yet an almost effortless method is to use the function matrix, which reshapes a matrix to a desired size:

M = matrix(1:10,5,2)'

How to enter N = [1 2; 3 4; 5 6; 7 8; 9 10] easily? Hint: think of some simple operations on a matrix.

A handy function in Scilab called size returns the dimensions of the matrix in query:

size(M)
ans =
   2.   5.

while size(M,1) returns 2 (number of rows) and size(M,2) returns 5 (number of columns).

Dealing with Matrices

Top6. The Programming Environment

Creating Functions

Scilab has an open programming environment that enables users to build their own functions and libraries. It is done by using the built-in editor SciPad. To call the editor, type scipad() or editor(), or click Editor at the menu bar.

The file extensions used by scilab are .sce and .sci. To save a file, click for the menu File and choose Save. To load a file, choose Load under the same menu. To execute a file, type

exec('function_file_name');

in the command line, or click for load into Scilab under the menu Execute.

To begin writing a function, type:

function [out1, out2, ...] = name(in1, in2, ...)

where function is a keyword that indicates the start of a function, out1, out2 and in1, in2, ..., are variables that are outputs and inputs, respectively, of the function; the variables can be Boolean, numbers, matrices, etc., and name is the name of the function. Then we can enter the body of the function. At the end, type:

endfunction

to indicate the end of the function.

Comment lines begin with two slashes //. A sample function is given as below:

function [d] = distance(x, y)
// this function computes the distance
// between the origin and the point (x, y)
d = sqrt(x^2 + y^2);
endfunction

A Scilab function can call upon itself recursively. Here is an example.

Unlike Matlab, Scilab allows multiple function declaration (with different function names) within a single file. Also Scilab allows overloading (it is not recommended for beginners). Please refer to the chapter overloading in its help file for details.

Flow Control

A table of logical expressions is given below:

== equal
~= not equal
>= greater than or equal to
<= less than or equal to
> greater than
< less than
~ not

If a logical expression is true, it returns a Boolean variable T (true), otherwise F (false).

The if statement

It has the basic structure:

if condition
   body
end

The body will be executed only when the condition statement is true. Nested if statements have the structure:

if condition_1
   body_1
elseif condition_2
   body_2
elseif condition_3
   body_3
elseif ...
   ...
end

As an example, we make use of the fact that a Scilab function can call upon itself recursively to write a function which gives the n-th Fibonacci number in the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... :

function [K] = fibonacci(n)
//function [K] = fibonacci(n)
//Gives the n-th term of the Fibonacci sequence 0,1,1,2,3,5,8,13,...
if n==1
  K = 0;
elseif n==2
  K = 1;
elseif n>2 & int(n)==n  // check if n is an integer greater than 2
  K = fibonacci(n-1) + fibonacci(n-2);
else
  disp('error! -- input is not a positive integer');
end
endfunction

Note that this is not an efficient way to find the n-th Fibonacci number when n is large.

The for loop

It has the basic structure:

for variable = initial_value : step : final_value
   body
end

The loop will be executed a fixed number of times specified by the number of elements in the array variable. A slightly modified version is:

str = 'abcdr';
s = ''; // an empty string
for i = [1 2 5 1 3 1 4 1 2 5 1]
   s = s + part(str, i);
end
disp(s);
s = abracadabra

The while loop

It has the basic structure:

while condition
   body
end

The loop will go on as long as the condition statement is true. Here we give an example of the Euclidean Algorithm.

function [n1] = hcf(n1, n2)
// n1 and n2 are positive integers
if n2 > n1
   tem = n2; n2 = n1; n1 = tem; // to ensure n1>=n2
end

r = pmodulo(n1, n2); // remainder when n2 divides n1
n1 = n2;  n2 = r;

while r ~= 0
   r = pmodulo(n1, n2);
   n1 = n2;  n2 = r;
end

endfunction

The break and continue commands

They are for ending a loop and to immediately start the next iteration, respectively. For example:

// user has to input 10 numbers and for those which
// are integers are summed up, the program are
// prematurely once a negative number is entered.

// It is not well written but just to illustrate the
// use of the "break" and "continue" commands
result = 0;
for i = 1:10
   tem = input('please input a number');
   if tem < 0
      break;
   end
   if tem ~= int(tem) //integral part
      continue;
   end
   result = result + tem;
end
disp(result);

Some Programming Tips

The concept of Boolean vectors and matrices is important. The function find is useful too. It reports the indices of true Boolean vectors or matrices. For example:

M = [-1 2; 4 9]; M > 0
ans =
   F  T
   T  T

M(M > 0)'
ans =
   4.   2.   9.
end

In contrast,

find(M > 0)
ans = 
   2.   3.   4.
   
M(find(M>0))'
ans =
   4.   2.   9.

We remark that M(M>0) gives results quicker than M(find(M>0) because the find function is not necessary here.

It is important to distinguish & and and, | and or. The first one of each pair is entry-wise operation and the other one reports truth value based on all entries of a Boolean matrix.

M = [0 -2; 1 0]; M==0 | M==1
ans =
   T  F
   T  T

and(M >= 0) // true iff all entries are true
ans =
   F

or(M == -2) // false iff all entries are false
ans =
   T

Debugging

The most tedious work in programming is to debug. It can be done in two ways: either using Scilab's built-in debugger, or modifying the program so that it serves the same purpose as a debugger.

The Scilab debugger is similar to those debuggers in other programming languages and is simple to use. We present the second method to offer programmers greater flexibilities when debugging.

To insert breakpoints we use

pause

To end pause we use

abort

To set the output of the function we may use

return

To display variables we use

disp (variable)

For details please read the Scilab documentation.

Programming Environment

Top7. Plotting Graphs

2D Graphs

The plot function has different forms, depending on the input arguments. If y is a vector, plot(y) produces a piecewise linear graph of the elements of y versus the index of the elements of y. When x and y are vectors of the same length, plot(x,y) produces a graph of y versus x. E.g., to plot the value of the sine function from zero to 2 pi:

t = (0:1/100:2) * %pi;
y = sin(t);
plot(t,y);
2D graph plot

3D Surfaces

The command plot3d(x,y,z) plots 3D surfaces. Here x and y (x-axis and y-axis coordinates) are row vectors of sizes n1 and n2, and the coordinates must be monotone, and z is a matrix of size n1xn2 with z(i,j) being the value (height) of the surface at the point (x(i),y(j)).

// A simple plot of z = f(x,y)
t=[0:0.3:2*%pi]'; 
z=sin(t)*cos(t');
plot3d(t,t,z)
3D surface plot

Top8. Scilab versus Matlab

This section is based on some user comments found in the internet, thus not necessarily all true. It is intended to give readers a general image about their differences besides those in syntax.

Top9. References

  1. Scilab help file (its own documentation)
  2. Scilab for dummies
  3. Matlab primer
  4. A pratical introduction to Matlab

19 December 2006, revised 5 January 2009.

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