View source for Scilab/C4/Linear-equations-Gaussian-Methods/English

'''Title of script''': Solving System of Linear Equations using Gauss Elimination and Gauss-Jordan Methods

'''Author: Shamika'''

'''Keywords: System of linear equations, Gaussian Methods'''


{| style="border-spacing:0;"
! <center>Visual Cue</center>
! <center>Narration</center>

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 1
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Dear Friends,

Welcome to the Spoken Tutorial on “'''Solving System of Linear Equations using Gauss Elimination and Gauss-Jordan Methods'''”

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 2 -Learning Objective Slide
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| At the end of this tutorial, you will learn how to: 

* Solve system of linear equations using '''Scilab'''
* Develop '''Scilab''' code to solve linear equations 



|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 3-System Requirement slide
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| To record this tutorial, I am using 

*'''Ubuntu 12.04''' as the operating system 
*and '''Scilab 5.3.3''' version 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 4- Prerequisites slide
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| To practise this tutorial, a learner 
*should have basic knowledge of '''Scilab '''
*and should know how to solve''' Linear Equations.'''


To learn '''Scilab''', please refer to the relevant tutorials available on the '''Spoken Tutorial '''website. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 5- System of Linear Equations
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| A system of '''linear equations''' is a 
* Finite collection of
* '''linear equations'''
* of the same set of '''variables'''



|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 7- Gaussian Elimination Method 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us study '''Gauss elimination method'''

Given a system of equations

* '''a x equal to b'''
* with '''m '''equations and''' '''
* '''n '''unknowns



|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 8- Gaussian Elimination Method
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * We write the coefficients of the '''variables''' '''a one''' to''' a n''' 
* along with the '''constants b one''' to''' b n''' of the system of equations 
* in one '''matrix''' called the '''augmented matrix'''



|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 9- Gaussian Elimination Method



| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| How do we convert the '''augmented matrix '''to an '''upper triangular form''' '''matrix?'''


We do so''' '''by performing row wise manipulation of the '''matrix'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 10- Example
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us solve this system of equations using '''Gaussian elimination method'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Switch to Scilab and open naivegaussianelimination.sci
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Before we solve the system, let us go through the code for '''Gaussian elimination method'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight '''format e comma twenty'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The first line of the code is '''format e comma twenty.'''


This''' '''defines how many digits should be displayed in the answer. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Point to '''e '''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The letter '''e''' within single quotes denotes that the answer should be displayed in '''scientific notation '''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Point to '''twenty'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The number '''twenty''' is the number of digits that should be displayed. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight '''funcprot'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The command '''funcprot''' is used to let '''Scilab '''know what to do when variables are redefined.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Point to '''zero'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The argument '''zero''' specifies that '''Scilab''' need not do anything if the variables are redefined.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| No visual clue. This is extra information that is being offered to the listener. 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Other arguments are used to issue warnings or errors if the variables are redefined.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight '''input'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Next we use the '''input''' function.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight '''input'''


Poin to '''“ “'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| It will display a message to the user and get the values of '''A''' and '''B''' matrices. 


The message should be placed within '''double quotes'''. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The matrices that the user enters, will be stored in the variables '''A''' and '''B'''. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight '''A'''

Highlight '''B'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Here '''A''' is the '''coefficient matrix''' and '''B''' is the right-hand-side matrix or the '''constants''' '''matrix'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight '''naivegaussianelimination'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we define the function '''naive gaussian elimination.'''




|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight '''A'''

Highlight '''B'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| And we state that '''A''' and '''B''' are the arguments of the function '''naive gaussian elimination'''. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight '''x'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We store the output in variable '''x'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight '''size'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we find the size of matrices '''A''' and '''B''' using the '''size''' command.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight''' n and n one'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Since they are two dimensional matrices, we use '''n''' and '''n one''' to store the size of matrix '''A.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight '''m one '''and '''p'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Similarly we can use '''m one '''and '''p''' for matrix '''b'''. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''if n ~= n1 '''

'''error('gaussianelimination - Matrix A must be square');'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we have to determine if the matrices are compatible with each other and if '''A''' is a '''square matrix'''. 


If '''n''' and '''n one''' are not equal , then we display a message that '''Matrix A''' must be '''square'''. 




|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''elseif n ~= m1 '''

'''error('gaussianelimination - incompatible dimension of A & b');'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| If '''n''' and '''m''' one are not equal, we display a message incompatible dimension of '''A''' and '''b.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''<nowiki>C=[A b];</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| If the matrices are compatible, we place matrices '''A''' and '''b''' in one matrix '''C'''. 


This matrix '''C''' is called '''augmented matrix'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''n=size(A,1); '''

'''for k=1:n-1 '''

'''for i=k+1:n '''

'''factor=A(i,k)/A(k,k); '''

'''for j=k+1:n '''

'''A(i,j)=A(i,j)-factor*A(k,j); '''

'''end '''

'''b(i)=b(i)-factor*b(k); '''

'''end '''

'''end'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The next block of code performs '''forward elimination'''. 

This code converts the '''augmented matrix''' to '''upper triangular matrix''' form. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''x(n)=b(n)/A(n,n); '''

'''for i=n-1:-1:1 '''

'''sum=0; '''

'''for j=i+1:n '''

'''sum=sum+A(i,j)*x(j); '''

'''end '''

'''x(i)=(b(i)-sum)/A(i,i); '''

'''end'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Finally we perform '''back substitution'''. 
* Once the '''upper triangular matrix''' is obtained, we take the last row and find the value of the variable in that row. 
* Then once one variable is solved, we take this variable to solve the other variables. 
* Thus the system of '''linear equations''' is solved.



|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Click on Execute and select Save and Execute
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us save and execute the file. 



|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Show Scilab Console
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Switch to '''Scilab''' console to solve the example. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Show Scilab Console
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| On the console, we have a prompt to enter the value of the '''coefficient matrix'''. 


So we enter the values of '''matrix A'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Type'''

'''<nowiki>[3.41 1.23 -1.09;2.71 2.14 1.29;1.89 -1.91 -1.89]</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Type '''square bracket three point four one space one point two three space minus one point zero nine '''

'''semi colon '''

'''two point seven one space two point one four space one point two nine semi colon '''

'''one point eight nine space minus one point nine one space minus one point eight nine close square bracket.'''

'''Press enter'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Type'''

'''<nowiki>[4.72;3.1;2.92]</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The next prompt is for '''matrix b'''.


So we type 

'''open square bracket four point seven two semi colon three point one semi colon two point nine one close square bracket.'''

'''Press enter'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Type'''

'''naivegaussianelimination(A,b)'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we call the function by typing 

'''naive gaussian elimination open paranthesis A comma b close paranthesis '''

'''Press enter'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Show answer on Scilab console
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The solution to the system of linear equations is shown on Scilab console. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Next we shall study the '''Gauss- Jordan''' '''method'''. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 11 – Gauss- Jordan Method
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| In '''Gauss – Jordan Method'''

* The first step is to form the '''augmented matrix. '''
* To do this place the coefficient '''matrix A''' and the right hand side '''matrix b''' together in one '''matrix'''.
* Then we perform '''row operations t'''o convert '''matrix A '''to diagonal form. 
* In diagonal form, only the elements '''a i i '''are non-zero. Rest of the elements are zero.



|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 12- Gauss-Jordan Method
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we divide the '''diagonal element and corresponding element of right hand side element, by the diagonal element'''.


We do this to get '''diagonal element''' equal to one.


The resulting value of the elements of each row of the right hand side matrix gives the value of each variable.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 13- Example



| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us solve this example using '''Gauss- Jordan Method'''. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Switch to Scilab console and open GaussJordan Elimination.sci
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us look at the code first. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''format'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The first line of the code uses '''format function''' to specify the format of the displayed answers. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

''''e''''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The parameter '''e''' specifies the answer should be in '''scientific notation'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight '''20'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| '''Twenty''' denotes that only '''twenty digits''' should be displayed.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''A=input("Enter the coeffiecient matrix : ") '''

'''b=input("Enter the right-hand side matrix : ")'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we get the '''A''' and '''b matrix''' using the '''input function'''. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''<nowiki>function [x] = GaussJordanElimination( A, b )</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We define the function '''Gauss Jordan Elimination''' with input arguments '''A '''and''' b''' and output argument''' x.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''<nowiki>[m, n] = size( A )</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We get the size of '''matrix A''' and store it in '''m''' and '''n'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''<nowiki>[r, s] = size( b )</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Similarly we get the size of '''matrix b '''and store it in '''r''' and '''s'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''<nowiki>if ( m <> r ) then </nowiki>'''

'''error("Error: matrix A and vector b are incompatible sizes") '''

'''end'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| If the sizes of '''A '''and '''b''' are not compatible, we display an error on the console using '''error function'''. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Highlight '''

'''for k = 1 : 1 : m '''

'''<nowiki>indices = [ 1 : 1 : k-1, </nowiki>k+1 : 1 : m ] '''


'''// For all rows below and above the pivot, subtract a multiple '''

'''// of the pivoting row to get a zero '''

'''for i = indices '''

'''multiplier = C(i, k) / C(k,k) '''

'''for j = k+1 : n '''

'''C(i, j) = C(i, j) - multiplier * C(k, j) '''

'''end '''

'''end '''

'''end'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we perform '''row operations''' to get diagonal form of the '''matrix'''.


Here '''pivot''' refers to the first non-zero element of a column. 

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''x = zeros( m, s )'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we create a '''matrix''' of zeros called '''x''' with '''m''' rows and '''s''' columns.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''for i = 1 : 1 : m '''

'''for j = 1 : 1 : s '''

'''x(i, j) = C(i, m+j) / C(i, i) '''

'''end '''

'''end'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Once we have the diagonal form, 

* we divide the right hand side part of '''augmented matrix''' 
* by the corresponding '''diagonal element '''
* to get the value of each variable.



|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''x(i, j) = C(i, m+j) / C(i, i) '''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We store the value of each variable in '''x.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''return x'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we return the value of '''x.''' 




|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight 

'''endfunction'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Finally we end the function.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Click on Execute and select Save and Execute
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Now let us save and execute the function.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type the following

'''<nowiki>[0.7, 1725;0.4352,-5.433]</nowiki>'''



| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The prompt requires us to enter the value of '''matrix A'''. 


So we type

'''open square bracket zero point seven comma one seven two five '''

'''semi colon '''

'''zero point four three five two comma minus five point four three three close square bracket.'''

'''Press enter'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type the following

'''<nowiki>[1739;3.271]</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The next prompt is for '''vector b'''. 


So we type 

'''open squre bracket one seven three nine '''

'''semi colon '''

'''three point two seven one close square bracket'''

'''Press enter'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type the following

'''GaussJordanElimination(A,b)'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we call the function by typing 

'''Gauss Jordan Elimination open paranthesis A comma b close paranthesis'''

'''Press enter'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Show answers on Scilab console
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The values of '''x one''' and '''x two''' are shown on the console.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 14- Summary
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us summarize this tutorial.

In this tutorial, we have learnt to: 

* Develop Scilab code for solving system of '''linear equations''' 
* Find the value of the unknown variables of a system of '''linear '''

'''equations '''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Show Slide 16'''

'''Title: About the Spoken Tutorial Project''' 

* Watch the video available at [http://spoken-tutorial.org/What_is_a_Spoken_Tutorial http://spoken-tutorial.org/What_is_a_Spoken_Tutorial] 

* It summarises the Spoken Tutorial project 

* If you do not have good bandwidth, you can download and watch it 


| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Watch the video available at the following link 
* It summarises the Spoken Tutorial project 
* If you do not have good bandwidth, you can download and watch it 
* 



|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Show Slide 17'''

'''Title: Spoken Tutorial Workshops''' 

The Spoken Tutorial Project Team 

* Conducts workshops using spoken tutorials 

* Gives certificates for those who pass an online test 

* For more details, please write to contact@spoken-tutorial.org 


| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The Spoken Tutorial Project Team 

* Conducts workshops using spoken tutorials 

* Gives certificates for those who pass an online test 

* For more details, please write to contact at spoken hyphen tutorial dot org 



|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Show Slide 18'''

'''Title: Acknowledgement''' 

* Spoken Tutorial Project is a part of the Talk to a Teacher project 

* It is supported by the National Mission on Education through ICT, MHRD, Government of India 

* More information on this Mission is available at 

* [http://spoken-tutorial.org/NMEICT-Intro http://spoken-tutorial.org/NMEICT-][http://spoken-tutorial.org/NMEICT-Intro Intro] 


| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Spoken Tutorial Project is a part of the Talk to a Teacher project 
* It is supported by the National Mission on Education through ICT, MHRD, Government of India 
* More information on this Mission is available at 
* spoken hyphen tutorial dot org slash NMEICT hyphen Intro <br/> 




|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| 
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| This is mandatory. This is Ashwini Patil signing off. Thanks for joining.

|}