Introduction to Scilab1

1.1 What is ScilabIf you are already familiar with the software package known as Matlab and you are also aware of the free software movement, then it is easy to answer the question “ What is Scilab? ” – Scilab is a free software alternative to Matlab. For the sake of readers who may be unfamiliar with either or both of Matlab and free softwares, we present in this chapter a brief introduction of the same. However, the reader may also skip this chapter in the first reading and begin with the second chapter, and gain a first hand understanding of Scilab. 1.2 Matlab®Matlab, believed to be used by over a million users in industry and academia, appeared in late 1970s and has become a hot favourite of users who are either not skilled or simply uninterested in getting entangled with syntax of languages like C or Pascal, even to solve simple problems. Matlab is hailed as the language of technical computing and often described as a “Quick and Dirty” programming language. The interpreted nature makes it “quick” and the “dirty” part indicates its flexibility of syntax, which programming puritans might not agree with. Formally, Matlab is a numerical computing environment and fourth generation programming language. It is developed by The MathWorks. Among Mathlab’s capabilities, the key is matrix manipulation (the name Matlab itself is an acronymn of Matrix Laboratory), handling of functions and data, implementation of algorithms and creation of user interfaces. Matlab is well known for the many tool boxes (specialized collection of commands and features which can be added on to basic Matlab), in specialized areas of science, technology, mathematics, statistics etc. Release 7.8 is in vogue in 2009.

Matlab is a proprietary software and a licence fee is mandatory for its use. The tool boxes are optional and each comes under its own licence fee. As we will see in the next section, the free software movement attempts to offer technically, ethically superior and free alternatives to proprietary softwares. There are many free open source alternatives to Matlab, in particular GNU Octave, FreeMat, and Scilab. They attempt to be compatible with the Matlab language. The statistical language S also can be considered in this category as it treats arrays as basic entities. The open source language R is an implementation of S.

1.3 Free Software

Wikipaedia defines free software as follows; “Free software or software libre is software that can be used, studied, and modified without restriction, and which can be copied and redistributed in modified or unmodified form

1 These notes are extracted from a draft of a book under preparation and should not be circulated. Feedback on errors, readability etc will be greatly appreciated. Pls email hemaachuth@gmail.com

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either without restriction, or with minimal restrictions only to ensure that further recipients can also do these things and that manufacturers of consumer-facing hardware allow user modifications to their hardware. Free software is available gratis (free of charge) in most cases”. Free software ideas originated from Richard Stallman who conceived the free software movement in 1983. The Free Software Foundation was founded soon after to advance Stallman’s free software ideas. It may be noted that there are currently alternative terms for free software such as "software libre", "Free and Open Source Software" ("FOSS") and "Free, Libre and Open Source Software" ("FLOSS").

A free software generally permits 4 kinds of freedom to its users:

The freedom to run the software anywhere, anyway The freedom to copy and distribute The freedom of access to the source code of the software The freedom to modify and redistribute

The flagship of the free software movement is Linux, the free operating system that has made giant strides in the recent past. There are of course a large number of softwares ranging from server softwares to tiny utilities that have proven the strength of free softwares, such as:

MySQL relational database; Apache web server GIMP raster drawing and image editor; OpenOffice.org office suite TeX and LaTeX scientific typesetting packages Mozilla Web Browser

You can see an exhaustive and growing list of free softwares at the Free Software Directory at directory.fsf.org.

1.4 History of Scilab

The following is the brief history of Scilab, as summarized in the Scilab’s official website:

In the 80’s, a CACSD (Computer Aided Control System Design) software was created at the IRIA (“Institut de Recherche en Informatique”) at Rocquencourt in France. It was inspired by the Matlab fortran software from Cleve Moler who founded “The MathWorks” company later with John Little. Blaise was mainly developed by François Delebecque and Serge Steer and its purpose was to have a tool for Researchers in Automatic Control.

Then Blaise became Basile and was sold during a few years by

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Simulog, the first subsidiary company of INRIA (“Institut National de Recherche en Informatique et Automatique”).

At the beginning of the 90’s, Simulog stopped selling Basile, and the software name changed another time to become Scilab. It was then developed by the Scilab Group composed by six researchers: Jean-Philippe Chancelier from the ENPC (“École Nationale des Ponts et Chaussées”), and François Delebecque, Claude Gomez, Maurice Goursat, Ramine Nikoukhah and Serge Steer from INRIA.

It was then decided to distribute Scilab as free Open Source software. Scilab 1.1, the first released version of Scilab, was put on anonymous ftp site on January, 2nd, 1994.

Then the Scilab Group, with the active collaboration of external developers, developed Scilab until the end of 2002, distributing source and binary versions of Scilab on the Internet up to Scilab 2.7 version.

At the beginning of 2003, to take in consideration the increased number of people downloading and using Scilab, and to ensure its future, development, maintenance, support and promotion, INRIA decided to create the Scilab Consortium with the support of companies and academic organizations.

To provide an appropriate response to the sustained growth of the operation, the Scilab Consortium integrates the Digiteo research network (2008)

The new organization anticipates an intensification of the resources of the Consortium with for major objectives to strive for excellence in the areas considered strategic by its members, to integrate recent research outcomes, to expand the community of contributors, to develop a powerful open source ecosystem, and to consolidate the operation at the European and international level.

However, the Scilab Consortium keeps a privileged link with INRIA as the Institute remains one of its members, continues to support actively its development and is also a founder member of Digiteo.

1.5 Downloading & Installing ScilabScilab is available for different platforms such as GNU/Linux platforms, Windows 2003/XP/VISTA platforms, MacOSX 10.5 Intel platforms

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The home page of Scilab is www.scilab.org. At the time of writing this book, the latest stable version of Scilab is 5.1.1. You can get the installation file from www.scilab.org/download/5.1.1/scilab-5.1.1.exe.

A quick taste of ScilabTo enable the reader to get a quick taste of the features of Scilab, we present in this chapter an informal introduction to a set of hand-picked commands. Obviously, the treatment is not comprehensive or deep, as the later chapters attempt to do that. The sequencing of the commands are also not necessarily logical.

Double click on the Silab icon scilab-4.0.lnk

and launch the Scilab window, as shown below. We can now start giving Scilab commands.

Fig 2.4 Freshly launched Scilab window

Let us start with some very trivial examples. Type x=5 followed by Enter key, followed by y=6 followed by Enter key, followed by x*y. On the Scilab window, the commands would appear as follows. You can also see that the result 30, is displayed by Scilab.

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Fig 2.5 Scilab window displaying results of x*y

There are a few things that may be noted. The syntax of Scilab commands is very similar to that of C , C++, Java etc. However, there are no variable declarations, no print statements etc. x*y produces output, by just suppressing the semicolon after it. Why Scilab is “ quick and dirty” is very evident from this example.

Let us now run through a set of selected examples, with brief comments on each. The reader is encouraged to work on these examples in his/her personal computer with a bit of experimentation too. Here is a minor modification of the first example.

>p=3.7;>q=6.7;>ptimesq=p*q;>2*ptimesqWe can see that Scilab permits the use of identifiers as in C programming language. Identifiers are variables where values of mathematical operations are stored. They can be later referred to in a program as can be seen by the statement 2*ptimesq . The result 49.58 is displayed by Scilab.

Scilab is case-sensitive. See this example.>a=5;>a>A

The reference to a is correctly identified, whereas A produces !--error 4 undefined variable : A

Mathematical operators which are very frequently used and accepted by Scilab are

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+ for addition- for subtraction* for multiplication/ for division^ for power operation

See some illustrations>2.5+4.75 ans = 7.25

-->3.38+5.49 ans = 8.87

-->5.0-3.75 ans = 1.25 >>8*5.3 ans = 42.4

-->89.34*43.67 ans = 3901.4778

-->24/3 ans = 8.

-->5/2.5 ans = 2.

-->5.0/2.5 ans = 2.

-->5.021/2.5 ans = 2.0084

-->8/5 ans = 1.6

-->8.0/6.0 ans = 1.3333333

-->8.0/5.0 ans = 1.6

-->3^10 ans = 59049.

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Parenthesis can be liberally used to ensure clarity of what is expected.>2+3/5 ans = 2.6

-->(2+3)/5 ans = 1.

If you have been typing out the commands we discussed so far, your Scilab window is possibly cluttered. Use the clear command to clear the window . You may note that the clear command only changes the appearance of the window. All variables and their values are intact in Scilab memory. You may test this by typing ptimesq and verifying that Scilab returns the original value.

You may also note that commands can not only be corrected using backspace key while typing , but after the Enter key is pressed, using the arrow keys we can go to the previous commands , correct and reuse them. This is convienient in correcting long and cumbersome commands which have trivial typing mistakes. Of course these are the classic features of command driven softwares.

On a lighter side, the most important command that one needs to know with any software package is how to wind up and come out ! . In Scilab typing exit would take you out of the current session with all variables and data in memory lost ( There are ways to save them, if you wish).

As already mentioned, Scilab is a free software alternative to Matlab. The termMatlab is an acronym for Matrix Laboratory. Matlab and therefore Scilab are basically matrix processing softwares. A matrix , in layman’s language is tabular data many times arranged in rows and columns. Let us see how Scilab operates simple row matrix, also known as vector.

Consider the marks scored by 5 students in an examination. These marks when written in a sequence separated by commas is known as a vector. Scilab can define a vector mark as follows

-->m= [35, 45, 65, 79, 23] m = 35. 45. 65. 79. 23.

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Notice the square brackets in the beginning and at the end of the sequence. True to its quick and dirty fame, Scilab wouldn’t mind if you skip the commas.

-->m= [35 45 65 79 23] m = 35. 45. 65. 79. 23Can you recall what commotion C and Java compilers would have made on such an occasion!

Suppose we want to calculate the average of these marks. no variables, no algorithms, no loops! Just one command mean(m)

>m= [35, 45, 65,79,23] m = 35. 45. 65. 79. 23.

-->mean(m) ans = 49.4

Most of the popular statistical calculations can be done using similar commands. If we want to add the marks, we perform

-->sum(m) ans = 247. We can also find out the deviation of the marks from the mean value (standard deviation) using

-->stdev(m) ans = 22.600885

If you are interested in finding out the maximum and minimum marks max and min commands helps->max(m) ans = 79.

-->min(m) ans = 23.

The simplicity of these codes might be of obvious interest to the social science research community.

Till now, we have been discussing a sequence of numbers arranged in a row called as vector. We have also discussed that a matrix is tabular data , many times arranged in rows and columns. Let us see the representation of a matrix,

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-->b=[5,10,15; 20,25,30;35,40,45]OR>b=[ 5 10 15 20 25 30 35 40 45]The above matrix has 3 rows and 3 columns and therefore is a 3*3 matrix.

If you want to interchange the rows and columns of the above matrix (mathematically known as transpose) , we use the ’ operator as follows:

b’ans = 5. 20. 35. 10. 25. 40. 15. 30. 45. You can see the rows have now become columns and columns have become rows.There are some special matrices that are frequently used in mathematical computations.Scilab has commands to handle these matrices.

>zeros(3,1) ans =

0. 0. 0. You can see that a 3*1 matrix of all elements zeros are generated.

-->ones(2,2) ans =

1. 1. 1. 1.

A 2*2 matrix of all elements ones is generated.. This is also known as an identity matrix.

-->eye(2,2) ans =

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1. 0. 0. 1. A 2*2 matrix with diagonal elements as ones is generated. Similarly random numbers can be generated in a matrix as follows:

-->rand(2,2) ans = 0.2113249 0.0002211 0.7560439 0.3303271A 2*2 random matrix is generated by Scilab.

Pictorial representation of data is a very common requirement in all fields where data arises from business to engineering and technology. Scilab has a host of commands to satisfy these requirements. Suppose the marks of 5 students are defined as a vector m=[33, 77, 45, 61,64] . We can plot this using the following command

plot(m)

Fig 2.6 Plot of m

Note that the given values are taken as y values and x values are assumed as 1, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0 Similarly, if the votes polled by different candidates in an election are represented in a vector as n=[125000,35678,45678,56789,6000] ,We can plot a pie chart of the same using:

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pie(n)

Fig 2.7 A pie chart showing the votes polled by different candidates

In the case of scientific calculations two dimensional and three dimensional plots are very common. Consider the following table of x and y values

x y1 341.5 272 682.5 47

These can be defined as two vectors in Scilabx=[1, 1.5, 2, 2.5]y=[34, 27 ,68, 47]The plot of y against x is obtained using the commandplot(x,y)

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Fig 2.8 Plot of (x,y)We have so far been discussing examples where Scilab manipulated numbers. Let us now see how Scilab manages a different kind of data, digital images. Of course, a digital image ia basically a matrix of numbers. In fact every digital object, text, images, sound, video or softwares ultimately boil down to numbers, to be specific zeros and ones. To try out image processing, you need to have your favorite image on the same folder from where Scilab has been launched, in our example 1.jpg. The image can be in any of the following formats .jpg, .bmp, .gif, .pnpSee how easy it is to bring the image on the screen. x=imread(“1.jpg”)imshow(x)

The command imshow reads an image into a Scilab variable x (which is essentially a matrix )and imshow displays this image using the matrix x. We will conclude this chapter rather abruptly, as the aim of giving a quick taste of Scilab has been fairly done. We will now proceed to a methodical study of Scilab.

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Fig 2.9 Displaying an image using imshow()

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-->m= [35 45 65 79 23] m = 35. 45. 65. 79. 23.

The same data can also be arranged a column vector. Here the sequence of numbers are separated by semicolon

-->m= [ 35; 45; 65; 79; 23] m =35. 45. 65. 79. 23. Scilab has some short hand notations if the vector is am arithmetic progression of numbers. To define [1, 2, 3, 4, 5, 6, 7, 8 ,9 10], we can command as follows:

A=[1:1:10]A=1, 2, 3, 4, 5, 6, 7, 8 ,9 10

Here the starting number, increment value and upper limit are specified separated by colon. Suppose we want to plot all the even numbers from 2 to 20 we proceed as follows:

B=[2:2:20]B= 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. The increment (and the starting and ending values too) can also be a fractions. The ending value need not be exactly calculated, the initialization would use the value closest to the ending value. See example below

C=[0;0.3;1]C=0.0, 0.3, 0.6, 0.9 If we are interested in creating a row vector or column vector of all elements zeros, it can be done using a special zeros function. Let’s see an example of a zero row vector of 5 elements

-->D= zeros (1,5) D = 0. 0. 0. 0. 0.Alternatively a zero column vector of 2 elements can be created as

-->zeros (2, 1) ans = 0. 0.We will later see that this function is defined generally on matrices and vectors are matrices with number of columns or rows as one.

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3.3 Mathematical operations on vectors

The various mathematical operations possible on vectors are listed as follows:

Addition Subtraction Multiplication Division. Element wise multiplication Element wise division Transpose

Now consider two vector sequences a and b to demonstrate the above operations:a = [10,20,30] b =[ 1, 2, 3]

3.3.1 Addition Addition of two vectors a and b is performed using the + operator. Both vectors must be of the same size. It is not possible to add a row vector and a column vector.-->a=[10,20,30]-->b=[5,10,15] -->a+b ans = 15. 30. 45. 3.3.2 Subtraction Subtraction uses the – operator. Here also both vectors have to be of the same size.-->a=[10,20,30]-->b=[5,10,15] -->a-b ans = 5. 10. 15.

3.3.3 Multiplication When multiplication is performed, one vector either a or b has to be a row vector and the other a column vector. In this example a is a row vector and b is a column vector.

-->a=[10,20,30]-->b=[5;10;15] -->a*bans = 700.

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3.3.4 Division Division is usually performed with a and b as row vectors. -->a=[10,20,30]; -->b=[5, 10,15]; -->a/b ans = 2.

3.3.5 Element wise multiplication Element wise multiplication on two sequences can be done using the .*operator. Both vectors must necessarily of the same size.

-->a=[10,20,30]-->b=[5, 10,15] -->a.*b ans = 50. 200. 450.

3.3.6 Element wise division Element wise division is done on two sequences using ./operator. Both vectors must necessarily of the same size.-->a=[10,20,30]->b=[5, 10,15]-->a./b ans = 2. 2. 2. 3.3.7 Transpose A row vector can be transformed into a column vector and vice versa using ’operator.Which is mathematically called as transpose? >a=[10,20,30]-->a' ans =

10. 20. 30.

-->b=[5;10;15]

-->b' ans = 5. 10. 15. 3.4 Relational operations on vectors

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Scilab uses six relational operators for mathematic comparison of vectors. The operations result in a vector of the same size, with the answer T when the relation is true and F when it is false.

3.4.1 less than relational operator <>x=[ 2 3 7 8], y=[2 4 5 1]-->z=x<y z = F T F F In the above operation we can see that the second number 3 is less than 4. The mathematic comparison 3<4 yields True. In all the other cases the result is False.

3.4.2 less than or equal relational operator <=

x=[ 2 3 7 8], y=[2 4 5 1];-->z=x<=y z = T T F F Here the first number in vector x , 2 is equal to the first number 2 in vector y. The comparison results in True as the result. Similarly the second number in vector x , 3 is less than the second number 4 in vector y. Here also the comparison results in True as the result. The third and the fourth numbers in vector x is less than their counter parts in y. Hence results in answer False.

3.4.3 greater than relational operator >>x=[ 2 3 7 8], y=[2 4 5 1] -->z=x>y z = F F T T Here the third number 7 and the fourth number 8 in vector x are greater than their counterparts in vector y and hence their mathematical comparison yields True value.

3.4.4 greater than or equal to relational operator >=x=[ 2 3 7 8], y=[2 4 5 1]-->z=x>=y z = T F T T

Here all the numbers in vector x except the second number is greater than the corresponding numbers in vector y. All the comparisons results in True value except for the second where the result is False.

3.4.5 equal to relational operator ==>x=[ 2 3 7 8], y=[2 4 5 1] -->z= x==y z = T F F F Here the first number in vector x is equal to the first number in vector y .The comparison results in True value.

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3.4.6 not equal relational operator ~ = x=[ 2 3 7 8], y=[2 4 5 1] -->z=x~=y z = F T T T

All the numbers in vector x are not equal to the corresponding numbers in vector y except for the first number 2 in vector x which is same as the first number 2 in vector y. All comparisons result in True value except for the first where the result is False.

3.5 Logical operations on vectors.

3.5.1 logical AND operator &The operations result in a vector of the same size, with the answer T when both the expressions are true and F when it is false...> x=[ 2 3 7 8], y=[2 4 5 1] -->z=(x>y)&(x>2) z = F F T T

Here we can see the condition x is greater than y and also that x is greater than 2 holds good for the third and fourth elements of vector x

3.5.2 logical OR operator | The operation result in answer T if either of the expressions are truex=[ 0 3 7 8], y=[0 4 5 1] -->z=x|y z = F T T T Here all the elements of vector x and vector y are non zero except for the first element where x|y yields the answer F

3.5.3 logical Complement ~ This operation results in answer T if the expression is false and vice versax=[ 0 3 7 8], y=[0 4 5 1]-->z=~(x|y) z = T F F F

From the operation we can see that the result is the complement of x|y.

3.6 Built- in logical functions isempty() find() isreal() isglobal()

3.6.1 isempty()This function returns a True value for an empty matrix-->x=[0 3 7 8]

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isempty(x)ans = F

3.6.2 find()This function finds the index of the non zero elements of a matrix-->x=[0 3 7 8]-->i=find(x>5) i =3. 4.3.6.3 isreal()This function returns a True value for all the real elements of a matrix -->x=[0 3 7 8]-->isreal(x) ans = T

3.6.4 isglobal() This function returns a True value if the variable declared is global-->x=[0 3 7 8]-->isglobal(x) ans = F

3.7 ELEMENTARY MATHEMATICAL FUNCTIONS

In this section , we will take a close look at a number of elementary mathematical functions and related commands that are applicable to vectors.

3.7.1 clean()This function rounds all small values down to zero. Consider a matrix A and the computation of A*A-1 in Scilab:

A=[ 1. 3. 2. - 1. 2. 1. 4. 2. 1.] ;

-->B=A*inv(A) B = 1. - 3.053D-16 - 8.327D-17 - 2.776D-17 1. - 2.776D-17 1.110D-16 - 3.331D-16 1.

Due to approximations2, some elements are shown as small nonzero numbers. The function clean will round these numbers to zero, cleaning the result to produce identity matrix., as expected

2 Though integers can be represented exactly by their binary equivalents in digital computers, real numbers need not be. This limitation of the binary representation (indeed, of any number system) manifests itself very visibly in some cases as seen here.

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-->B=clean(B)B = 1. 0. 0. 0. 1. 0. 0. 0. 1.

3.7.2 ceil()This function rounds a floating point number to the next higher integer, irrespective of the magnitude of the decimal point. ceil(1.x) is always 2 irrespective of the value of x.

>ceil([1.2 1.5 1.9 -1.2]) ans = 2. 2. 2. - 1. 3.7.3 floor()This function rounds down a floating point number, to the next lowest integer.

-->floor([1.2 1.5 1.9 -1.2]) ans = 1. 1. 1. - 2.

3.7.4 round()This function rounds to the nearest integer, the usual way.

-->round([1.2 1.5 1.9 -1.2]) ans = 1. 2. 2. - 1.

3.7.5 fix()This function rounds a floating point number towards zero>fix([1.2 1.5 1.9 -1.2]) ans = 1. 1. 1. - 1.

The floor and fix functions rounds down to the nearest integer number. Both gives the same result as with function int which returns the integer part of a number. -->int([1.2 1.5 1.9 -1.2]) ans = 1. 1. 1. - 1. 3.7.6 sign()

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This function sign returns 1,-1 or 0 for a real number depending on whether the number is positive, negative or zero

-->sign([1.2 1.5 1.9 -1.2]) ans = 1. 1. 1. - 1.

3.8 Mathematical functions on scalarsMost of the functions discussed in section 3.7 are applicable on scalars (ciel, floor …hema to add). There are a small set of functions applicable to scalars alone. We introduce them below.

3.8.1 modulo ()This function is used to calculate the remainder of division of two numbers.-->modulo(5,2) ans = 1.

3.8.2 Rat ()This function is used to find the numerator and denominator of therational approximation of a floating point number.->[n,d]=rat(0.353535) d = 28529. n = 10086. The result can be verified by calculating n/d >n/d ans = 0.353535

Yet another example is 0.33333333333333-->[n,d]=rat(0.333333333333) d = 3. n = 1.

Here we have to provide a relatively large number of decimals to force the result. A small number of decimal points will not produce the required result. The function rat can be called using a second argument which represents error or tolerance allowed in the approximation.

-->[n,d]=rat(0.33333333,1e-4) d = 3. n = 1.

Here the tolerance of 1/10000 produces the approximation 1/3

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3.8.3 sqrt() This function returns the square root of a real or complex number. Let us see an example of a real numbersqrt(2345.67) ans = 48.432117

The remaining functions in this chapter are exponential functions.

3.8.4 exp() The exponential function exp returns the value of ℮x for a real number where e constitutes the basis of natural logarithm. This value is given as a constant %e in Scilab

-->%e %e = 2.7182818 Let us see an example of finding the exponent of a real number.>exp(2.5) ans = 12.182494

-->exp(-3.6) ans = 0.0273237

3.8.5 log() Natural logarithm is referred as log(). Let us see the natural logarithm of a real number-->log(2.35) ans = 0.8544153 iθFor a complex number represented as z=re , ln(z) =ln(r)+iθ

-->log(2+3*%i) ans =1.2824747 + 0.9827937i

Besides the natural logarithm(logarithms of base e) , Scilab also provides functions log10() and log2() that calculate logarithms of base 10 and base 2 respectively.

3.8.6 log10() Here the base of logarithm is taken as10. If x=10y then y=log10(x).Here is an example to illustrate this-->log10(1000) ans = 3.

3.8.7 log2()

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Here 2 is the base of logarithm. If x=2y then y=log2(x). Let us see an example

-->log2(8) ans = 3. 3. 9 Complex numbers

A complex number z can be written as z=x+iy where x and y are real numbers and i is √ -1 , an imaginary number. widely used in Mathematics. Scilab supports complex numbers. We can say that x is the real part of z and y is the imaginary part of z. A complex number z=2.3+5.5 i can be written in Scilab as follows:

z=2.3+5.5*%iz = 2.3 + 5.5i

We can separate out the real and imaginary parts with the real ( ) and imag( ) functions.x=real(z)x= 2.3y=imag(z).y=5.5

real(0.6-0.8*%i)ans = 0.6

-->imag(0.6-0.8*%i)ans = - 0.8

The sign function when applied to a complex numbers, returns z/|z|. The resulting complex number will have magnitude of unity.

-->sign(3-4*%i) ans = 0.6 - 0.8i

The abs() function returns the magnitude of the complex number-->abs(0.6-0.8*i) ans = 1.

The functions clean, ceil, floor , int, round and fix can be applied to complex numbers also

z=2.3+5.5*%i z = 2.3 + 5.5i -->ceil(z)

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ans = 3. + 6.i-->floor(z) ans = 2. + 5.i

-->int(z) ans = 2. + 5.i

-->round(z) ans = 2. + 6.i

-->fix(z) ans = 2. + 5.i

The square root of a negative real number(x<0) is calculated as √-x=i√x-->sqrt(-2345.67) ans = 48.432117i

Logarithms also can be calculated for complex numbers. For a complex number represented as z=reiθ , ln(z) =ln(r)+iθ

-->log(2+3*%i) ans =1.2824747 + 0.9827937i 3.10 Trigonometric functionsThe basic trigonometric functions sine(sin), cosine(cos), and tangent(tan) are supported in Scilab.. The trigonometric functions are defined in terms of the angles and sides of a right triangle.

Hypotenuse Opposite side

Adjascent side

Sin(α)=opposite side/Hypotenuse; cos(α)=adjascent side/ Hypotenuse and tan(α)=opposite side/Adjascent side. Scilab assumes the angles to be in degrees. We shall for example calculate these functions for a value of α=45

->sin(45) ans = 0.8509035 -->cos(45)

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ans =.5253220 -->tan(45) ans = 1.6197752

sec, cosec and cot ----???

3.11 Inverse trigonometric functions If sin(y) =x then y=sin-1 (x) =asin(x), If cos(y) =x then y=cos-1 (x) =acos(x), If tan(y) =x, then y=tan-1 (x) =atan(x). We will find these values for x=0.5

>asin(0.5) ans =0.5235988 -->acos(0.5) ans =1.0471976 -->atan(0.5) ans =0.4636476

Chapter 4MATRICES

4.1 IntroductionWe have already mentioned that matrices are central to both Matlab and Scilab. In this chapter we will attempt to cover a wide variety of Scilab features on matrix handling. A matrix, in layman’s language is tabular data arranged in rows and columns. Let us see an example of a matrix ┌ ┐

│ 11 12 13 │ A = │ 21 22 23 │

│ 31 32 33 │└ ┘

This can be defined in Scilab in a single line, with rows separated by semi-colons or in three lines, with a line break after each row, as shown below.

-->A= [ 11 12 13 ; 21 22 23 ; 31 32 33]A= [ 11 12 13 21 22 23 31 32 33]

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In chapter 3 we have seen that a vector is a special case of a matrix arranged in a single row or single column. For instance, B= [ 23, 34, 56, 78, 90] represents a vector. Matrix having a single element is treated as a scalar. A=39 and A=[39] are treated the same way by Scilab.

Any element of the matrix can be referred to by specifying its row and column indices. Note that the indices start from 1 and not 0, as in many modern programming languages. If we want to access the second element of the first row, we may referred to it as ‘A(1,2)’.

Column 1 2 3 Row ↓ ┌ ┐

│ 11 12 13 │ 1 A(1,2) A = │ 21 22 23 │ 2

│ 31 32 33 │ 3└ ┘

The first number in parenthesis refers to the row to which the element belongs and the second number indicates the column containing the element.

-->A(1,2) ans =12.

Similarly we can display the third element of the second row using-->A(2,3) ans =23

To overwrite any defined value, we can simply reassign----A(2,3)=0A = 11. 12. 13. 21. 22. 0. 31. 32. 33.

We will now look at the following categories of matrix commands Matrix Arithmetic and Relational operators Basic matrix processing Mathematical functions of matrices

4.2 Arithmetic operators for MatricesIn matrix arithmetic we have basically three kinds of commands to be aware of

Basic arithmetic Element wise arithmetic and Left and right division Relational and Logical operators

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4.2.1 Basic arithmeticAddition, subtraction and multiplication are done on matrices in Scilab in exactly the same way as they are traditionally defined in mathematics. These operations work only if the size of the matrices are compatible. In case of addition and subtraction, the matrices have to be of exactly same dimensions. In case of multiplication, an (m × k) matrix is compatible with only an (k×n) matrix, ie, the second matrix must have same number rows as the number of columns of the first matrix. The following examples illustrate addition subtraction and multiplication.

--> a= [1 2; 3 4] -->b=[2 3; 4 5]-->c=a+b c = 3. 5. 7. 9. -->d=b-a d = 1. 1. 1. 1. -->e=a*b e = 10. 13. 22. 29.Division of matrices is not straightforward and involves a concept of matrix inverses, which we will discuss in another section. However, here it may be noted that division operator has the same effect as multiplication by inverse. By using the operator / and \, we can choose which matrix to invert. a\b is equivalent to inv(a)*b and a/b is equivalent to a*inv(b). We show below the equivalence using the inv( ) function which we will look at closely in later sections.

-->a=[1 2; 3 4]-->b=[2 3; 4 5]-->a\b ans = 0. - 1. 1. 2.-->inv(a)*b ans = 0. - 1. 1. 2. -->a/b ans = 1.5 - 0.5 0.5 0.5

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-->a*inv(b)ans= 1.5 - 0.5 0.5 0.5 4.2.2 Element wise arithmeticEven though element wise operations on matrices may not always be mathematically meaningful, in practice many uses of such operations may occur. Scilab differentiates elementwise matrix operations with a dot preceding the operator. It may be noted that the traditional addition and subtraction are element wise operations.

See the examples below which demonstrate element wise multiplication and divison. Be sure to compare the results with those of traditional multiplication and division in section 4.2.1. Also note that the matrices only need to be of the same size to be compatible, unlike in the case of traditional multiplication where they need to have dimensions (m×k) and (k×n).

-->a=[1 2; 3 4] -->b=[2 3; 4 5]---> a.*b ans = 2. 6. 12. 20. Element wise division has left and right options, indicated by / and \ operators. See examples below. Both matrices a and b must be of the same size. In left division, the elements of b are divided by the elements of a. In right division, the elements of a are divided by the elements of b-->a=[1 2; 3 4] -->b=[2 3; 4 5]-->a.\b ans = 2. 1.5 1.3333333 1.25-->a./b ans = 0.5 0.6666667 0.75 0.8

4.3 Basic matrix processing

In this section we will focus on matrices with various data types and then the basic matrix operations- transpose, trace, determinant, eigen value etc

4.3.1 Matrices with various data types

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Matrices can hold a variety of data types. The elements all need to be of the same type except in the case of Lists. The following are the major types of data that can be held by matrices such as constants, characters, strings, boolean, polynomials. Lists can hold a variety of data and also matrices themselves. Constant matrices are already familiar to you. They can hold numbers as in the following example

┌ ┐ │ 11 12 13 │A= │ 21 22 23 │ │ 31 32 33 │ └ ┘The numbers can have decimal parts also. In addition, variables (with values pre-defined) can also appear as elements of matrices. For example,

-->x=22

-->A= [ 11 12 13--> 21 x x+1--> 31 32 33] A = 11. 12. 13. 21. 22. 23. 31. 32. 33. Character data defined in single quotes such as ‘a’ , ‘+’, ‘o’ etc can be elements of matrices.-->B= [ 'm' 'a' 't'--> 'h' 'a' 't'--> 'r' 'a' 't' ] B =! m a t !! ! ! h a t !! !! r a t !

It may be noted that character matrices are displayed with special brackets built with ‘ ! ’ mark. Character strings can also be elements in matrices as follows. -->C=[ 'cat' 'rat'--> 'mat' 'hat' ]

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C =! cat rat !! !!mat hat ! A very special kind of matrix that Scilab supports is a polynomial matrix. The use of this other than in mathematics, shall be dealt with in Chapter 12 on Control Toolbox. For defining a polynomial matrix D given by ┌ ┐ │ │D = │ 1+x 2 -3+x2 │ │ 1 +x x-5 │ └ ┘we must make x a polynomial variable. This is done as follows:

-->x=poly(0, 'x');-->D=[ x^2+1 x^2-3--> x+1 x-5] D = 2 2 1 + x - 3 + x

1 + x - 5 + x We can do mathematical operations like addition, subtraction, multiplication and inverse operations on polynomial matrices.--->x=poly(0, 'x');-->D1=[ x^2+1 x^2-3--> x+1 x-5 ] D1 = 2 2 1 + x - 3 + x

1 + x - 5 + x

-->D2= [ x^3+2 x^3-6--> x^2+2 x^2-5 ] D2 =

3 3 2 + x - 6 + x

2 2 2 + x - 5 + x

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-->D1+D2 ans =

2 3 2 3 3 + x + x - 9 + x + x

2 2 3 + x + x - 10 + x + x -->D1-D2 ans =

2 3 2 3 - 1 + x - x 3 + x - x

2 2 - 1 + x - x x - x

-->D1*D2 ans =

2 3 4 5 2 3 4 5 - 4 + x + x + x + x 9 - 14x + x + x + x

2 3 4 2 3 4 - 8 + 4x - 5x + 2x + x 19 - 11x - 5x + 2x + x

-->inv(D1) ans =

2 - 5 + x 3 - x ----------- ----------- 2 2 - 2 + 4x - 6x - 2 + 4x - 6x

2 - 1 - x 1 + x ----------- ----------- 2 2 - 2 + 4x - 6x - 2 + 4x - 6x Boolean values of True and False are defined as constants %t and %f In Scilab. They can be used as elements in matrices.

-->E=[ %t %f

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--> %t %f ] E = T F T F Such matrices are usually generated when we do element wise relational operations on matrices.-->[1 2]==[1 3] ans = T FSo far we have seen matrices of different data types, but in all cases every matrix had a single data type. List is a special facility used to group a set of heterogeneous objects into a single object. It is similar to struct in C programming language. Let us consider creating a list containing two data types, say a string and a number. Let us assume they are the name and marks of a student. Let us denote the list as student. We can create such a list with data ‘Gayathri’ for name and 95 as marks by using the following command.

F=tlist(‘student’, ‘name’, ‘marks’), “Gayathri”, 95) ↑ List name ↑ List creation command↑ Variablename

-->F=tlist(['student', 'name', 'marks'], "Gayathri', 95) F =

F(1)!student name marks !

F(2) Gayathri

F(3) 95.

We can extract an entry from the list by specifying the item, as follows:

>F('name') ans = Gayathri

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-->F('marks') ans = 95. 4.3.2 Basic Matrix Operations

In this section we shall see how to handle a matrix, including navigationg through it in special ways and implementing common operations such as transpose , inverse etc.

One of the basic information about a matrix is its dimensions. For two dimensional matrices, it is of the form [m×n], where m is the number of rows and n is the number of columns. We can use size() function in scilab to determine the dimensions of a matrix, as demonstrated below

A= [ 11 12 13--> 21 22 23 --> 31 32 33]

-->size(A) ans =

3. 3.

The answer shows that there are three rows and three columns for the matrix A, or A is a [3*3] matrix.

Square matrices (matrices with equal number of rows and columns) have their diagonal elements defined as indicated below

┌ ┐ │ \ 11 \ │A= │ \ 22 \ │ │ \ 33 \ │ └ \ ┘

If a ij is the matrix element then in general all elements with i=j define the diagonal elements. In Scilab we use the function diag() to find the diagonal elements.-->diag(A) ans = 11. 22. 33.

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We can find out the trace of a matrix, once the diagonal elements are found out. Trace is the sum of all the diagonal elements trace(A)ans=66 The lower triangular matrix of A which is all elements in the diagonal and below it, can be found with tril( ) function.-->tril(A)ans = 11. 0. 0. 21. 22. 0. 31. 32. 33.

Similarly, the upper triangular matrix which is a square matrix whose elements below the diagonal elements are zeros is obtained using the triu( ) function.-->triu(a) ans = 11. 12. 13. 0. 22. 23. 0. 32. 33. Another basic operation is transpose of a matrix which is obtained by inter changing rows and columns. of a given matrix. The figure below illustrates a matrix A and its transpose AT

┌ ┐ │ 11 12 13 │A= │ 21 22 23 │ │ 31 32 33 │ └ ┘

→ ┌ ┐ ↓ │ 11 21 31 │AT= 12 22 32 │ │ 13 23 33 │ └ ┘In Scilab we use ’ operator to find the transpose

--->A' ans = 11. 21. 31. 12. 22 32. 13. 23. 33.

Scilab uses the : operator to specify range of rows and columns

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Suppose we want to select rows rows 2 and 3 alone and all columns ie; 1-3, we proceed as follows:-->A(2:3, 1:3) ans = 21. 22. 23. 31. 32. 33. If you want to navigate through the entire rows, but only columns 1 and 2 type colon followed by the columns( ie; 1:2) to be displayed-->A(: , 1:2) ans = 11. 12. 21. 22. . 31. 32. See an example of specifying entire columns and rows 1 and 2--->A (1:3, :) ans =

11. 12. 13. 21. 22. 23. The entire matrix can be reshaped into a vector column wise>b=a(:) b = 11. 12. 13. 21. 22. 23. 31. 32. 33. The ‘:’ facility makes it possible to extract a sub matrix of A, by specifying the required rows and columns of it. ┌ ┐

│ 11 12 13 │ A = │ 21 22 23 │ C

│ 31 32 33 │└ ┘

-->C=A(2:3, 2:3) C =

22. 23. 32. 33

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We can delete a row or column of a matrix by setting it to a null vector [ ]. A null vector is a vector having no elements in it . Here is an example of deleting the second row.-->A(:, 2)=[ ] A = 11 12 13. 31. 32 33.

You can append a row or column to an existing matrix if the row or column has the same length as the existing matrix. See this example of appending a row to the matrix A

-->P=[41 42 43]-->A=[A ; P] B = 11 12 13 21 22 23 31 32 33 41 42 43.

Like wise, we can see how a column can be appended

-->Q= [14 24 34] -->B=[A Q] B =

11 12 13 14 21 22 23 24 31 32 33 34We can add elements to a null vector, which is a vector having no elements in it ..-->e=[ ]--->e=[e; 1 2 3]e = 1. 2. 3.A comparatively involved concept related to matrices is their eigen value and eigen vectors. For every square matrix A, the equation A-λI =0 is known as the characteristic equation. λ is a vector whose values are known as eigen values and columns of the matrix with λ ‘s as diagonals are known as eigen vectors. An insightful explanation of eigen values and vectors is beyond the scope of this book. We will only briefly note that every matrix can be considered as a transformation operator as in the example below. A

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point with coordinate x,y can be transformed into x’, y’ by the transformation

[x’,y’]=A[x y]

If A=I, then [x’y’]=[x,y]. If A=[-1,0; 0,1] then [x’y’]=[-x,y], resulting in a reflection of the point with respect to y axis. A general matrix A=[a, b; c, d] will result in x’=ax+by and y’=cx+dy. It is well known that such transformations have certain axes which are special, as points on these axes are transformed as points along the axes only. These axes represent the eigen vectors of the transformation and their scaling factors represent eigen values. The eigen vectors and eigen values have wide applications in applied mathematics, computer graphics, soft computing and control systems. We will limit our discussion to a simple example showing evaluation of eigen values and vectors. Consider A=[1 2 ; 3 4] Its eigen values can be calculated as below based on the equation A-λI =0

[1 2; 3 4] –[ λ1 0; 0 λ2]= **************

The above computation can be achieved with spec( ) function in Scilab.

-->A=[ 1 2; 3 4];-->spec(A) ans = 5.3722813 - 0.3722813 The eigen vectors corresponding to the eigen values can be computed by the same command by specifying variables for storing both.

-->[x,y]=spec(A) y = 5.3722813 0 0 - 0.3722813 x = 0.4159736 - 0.8245648 0.9093767 0.5657675

We find that eigen values appear as diagonal elements in the variable y. The eigen vectors appear as columns in the matrix x.

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