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

'''Title of script''': Solving System of Linear Equations using Iterative Methods

'''Author: Shamika'''

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



{|border=1
! <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 Iterative 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 '''iterative methods'''
* 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;"| Before practising this tutorial, a learner should have basic knowledge of 
*'''Scilab '''
*and '''Solving Linear Equations'''


For '''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- Jacobi Method
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * The first '''iterative method''' we will be studying is '''Jacobi method.'''
* Given a '''system of linear equations, with n equations and n unknowns'''
* We rewrite the equations such that '''x of i k plus one is equal to b i minus summation of a i j x j k from j equal to one to n divided by a i i''' where '''i''' is from '''one to n'''
* We assume values for each''' x of i'''
* Then we substitute the values in the equations obtained in the previous step.
* We continue the iteration until the solution converges.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 6- 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 '''Jacobi 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 JacobiIteration_final.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 for '''Jacobi Method'''.

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

'''format('e',20)'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|We use '''format''' method to specify the format of the displayed answers on the '''Scilab console'''.


Here '''e '''denotes the answer should be in '''scientific notation'''.

And '''twenty''' specifies the number of digits to 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 of nxn: ")'''

'''b=input("Enter the right-hand side matrix nx1: ")'''

'''x0=input("Initial approximation nx1: ")'''

'''MaxIter=input("Maximum no. of iterations: ")'''

'''tol=input("Enter the convergence tolerance :")//stop if norm <nowiki>change in x < tol</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|Then we use '''input''' function to get the values for 
*'''the matrices coefficient matrix''', 
*'''right hand side matrix''', 
*'''initial values matrix''', 
*'''maximum number of iteration''' and 
*'''convergence tolerance'''

|-
| 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;"|Then we use''' size '''function to check if '''A matrix''' is a '''square matrix.'''


If it isn't, we use '''error function''' to display an error.

|-
| 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'''

'''sum=0;'''

'''for j=1:1:m'''

'''sum=sum+abs(A(i,j));'''

'''end'''

'''<nowiki>if 2*abs(A(i,i))<sum then</nowiki>'''

'''error("the matrix is not diagonally dominant")'''

'''end'''

'''end'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|
* We then check if '''matrix A''' is '''diagonally dominant'''.
* The first half calculates the sum of each row of the '''matrix'''.
* Then it checks if twice the product of the '''diagonal element''' is greater than the sum of the elements of that row.
* If it isn't, an error is displayed using '''error function'''.

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

'''<nowiki>function [ solution ] = JacobiIteration( A, b, x0, MaxIter, tol )</nowiki>'''
| 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 '''Jacobi Iteration''' with input arguments 
'''A, b , x zero,'''

''' maximum iteration and tolerance level'''.


Here '''x zero''' is the '''initial values matrix'''.

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

'''if ( length(x0) ~= n )'''

'''error( "Sizes of input matrix and input vector do not match" )'''

'''end'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|We check if the size of '''A matrix''' and '''initial values matrix''' are compatible with each other.

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

'''xk = x0'''

'''for k = 1 : 1 : MaxIter'''

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

'''xkp1( i ) = (b(i) - A(i, 1:i-1)*xk(1:i-1) - A(i, i+1:n)*xk(i+1:n)) / A(i,i) //Computes the jacobian updates'''

'''end'''


'''RelativeError = norm( xkp1 - xk, 'inf' ) / norm( xkp1, 'inf' )'''

'''printf( "iter = %d, Relative Error = %g\n", k, RelativeError )'''

'''xk = xkp1'''


'''<nowiki>if ( RelativeError < tol )</nowiki>'''

'''break'''

'''end'''

'''end'''


'''solution = xkp1'''


'''endfunction'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|
* We calculate the value for''' x k p one''' and then check if the '''relative error''' is lesser than '''tolerance level.'''
* If it is lesser than '''tolerance level''', we break the iteration and the solution is returned.
* 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;"|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;"| Switch to 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'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Enter these values on Scilab console for each prompt

Enter the coeffiecient matrix of nxn: '''<nowiki>[2 1;5 7]</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|Let us enter the values at each prompt.

The '''coefficient matrix A''' is '''open square bracket two space one semi colon five space seven 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

Enter the right-hand side matrix nx1: '''<nowiki>[11; 13]</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|
Then we type '''open square bracket eleven semicolon thirteen'''

Press '''Enter'''.

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

Initial approximation nx1: '''<nowiki>[1;1]</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The '''initial values matrix''' is '''open square bracket one semi colon 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

Maximum no. of iterations: '''25'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|The '''maximum number of iterations''' is twenty five.

Press '''Enter'''.

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

Enter the convergence tolerance :'''0.00001'''



| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|Let the '''convergence tolerance level''' be''' zero point zero zero zero zero one'''

Press '''Enter'''.

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

'''JacobiIteration( A, b, x0, MaxIter, tol )'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|We call the function by typing 
'''Jacobi Iteration open paranthesis A comma b comma x zero comma M a x I t e r comma t o l close paranthesis'''

Press '''Enter'''.


The values for '''x one''' and '''x two''' are shown on the '''console'''. 

The number of iterations are also shown.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 7- Gauss Seidel 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 now study '''Gauss Seidel method'''. 
* Given a '''system of linear equations''', with '''n equations''' and '''n unknowns'''
* We rewrite the equations for each unknown 
**by subtracting the other variables and their coefficients 
**from the corresponding right hand side element.
* Then we divide this by the '''coefficient a i i'' of the '''unknown variable''' for that variable.


This is done for every given equation.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 8- Gauss Seidel
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|In '''Jacobi method''', for the computation of '''x of i k plus one''', every element of '''x of i k''' is used except '''x of i k plus one'''


In '''Gauss Seidel method''', we over write the value of '''x of i k''' with '''x of i k plus one'''

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

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Switch to Scilab editor and open GaussSeidel.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 for '''Gauss Seidel Method'''

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

'''format('e',20)'''
| 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 specifies the '''format''' of the displayed answer on the '''console''' using '''format function'''.

|-
| 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 of nxn: ")'''

'''b=input("Enter the right-hand side matrix nx1: ")'''

'''x0=input("Initial approximation nx1: ")'''

'''MaxIter=input("Maximum no. of iterations: ")'''

'''tol=input("Enter the convergence <nowiki>tolerance :")//stop if norm change in x < tol</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|Then we use '''input function''' to get the values of 
*'''coefficient matrix''', 
*'''right hand side matrix''', 
*'''initial values of the variables matrix''', 
*'''maximum number of iterations''' and 
*'''tolerance level'''

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

'''<nowiki>function [ solution ] = GaussSeidel( A, b, x0, MaxIter, tol )</nowiki>'''
| 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 '''Gauss Seidel '''with''' input arguments A comma b comma x zero comma max iterations '''and '''tolerance level''' and output argument solution


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

'''// Check that the matrix is square'''

'''<nowiki>[m, n] = size( A )</nowiki>'''

'''if ( m ~= n )'''

'''error( "The input matrix is not square" )'''

'''end'''


'''// Check that the initial vector is of the same size'''

'''if ( length(x0) ~= n )'''

'''error( "Sizes of input matrix and input vector do not match" )'''

'''end'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|We check if '''matrix A is square''' and the sizes of '''initial vector and matrix A''' are compatible using''' size and length function'''.

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

'''xk = x0'''

'''xkp1 = zeros( xk )'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|Then we start the iterations. 

* We equate the '''initial values vector x zero to x k.'''
* We create a '''matrix of zeros''' with the same size of '''x k''' and call it '''x k p one.'''

|-
| 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 : MaxIter


'''// Computes the Gauss-Seidel update'''

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

'''xkp1( i ) = (b(i) - A(i, 1:i-1)*xkp1(1:i-1) - A(i,i+1:n)*xk(i+1:n)) / A(i,i)'''

'''// x^{k+1}'''

'''end'''

'''// Applies the relaxation'''

'''RelativeError = norm( xkp1 - xk, 'inf' ) / norm( xkp1, 'inf' )'''

'''printf( "iter = %d, Relative Error = %g\n", k, RelativeError )'''

'''xk = xkp1'''

'''<nowiki>if ( RelativeError < tol )</nowiki>'''

'''break'''

'''end'''

'''end'''


'''solution = xkp1'''


'''endfunction'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|We solve for each equation to get the value of the '''unknown variable''' for that equation using '''x k p one'''.
* At each iteration, the value of '''x k p one''' gets updated.
* Also, we check if '''relative error''' is lesser than specified '''tolerance level'''.
* If it is, we break the iteration.
* Then equate '''x k p one''' to the variable '''solution'''.
* 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;"|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;"| Switch to 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'''

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

For first prompt

'''<nowiki>[2 1;5 7]</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|For the first prompt, we type '''matrix A.'''

Type '''open square bracket two space one semi colon five space seven 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;"| Second prompt

'''<nowiki>[11; 13]</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|For the next prompt, 

type
'''left square bracket eleven semi colon thirteen right square bracket'''

Press '''Enter'''.

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

'''<nowiki>[1;1]</nowiki>'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|We provide the values of '''initial value vector '''by typing 

'''open square bracket one semicolon 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;"| For fourth prompt

'''25'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|Then we specify the''' maximum number of iterations''' to be twenty five


Press '''Enter'''.

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

'''0.00001'''
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|Let us define '''tolerance level''' to be zero point zero zero zero zero one


Press '''Enter'''.

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

'''GaussSeidel( A, b, x0, MaxIter, tol )'''



| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|Finally we call the function by typing

''' G a u s s S e i d e l open paranthesis A comma b comma x zero comma M a x I t e r comma t o l close paranthesis'''


Press '''Enter'''.


The values of '''x one''' and '''x two '''are displayed.


The number of iterations to solve the same problem are lesser than '''Jacobi method'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 13- Solve
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"|Solve this problem on your own using '''Jacobi''' and '''Gauss Seidel methods'''

|-
| 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;"| 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 15'''

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

* Watch the video available at [http://spoken-tutorial.org/What_is_a_Spoken_Tutorial http://spoken-][http://spoken-tutorial.org/What_is_a_Spoken_Tutorial 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;"| * About the Spoken Tutorial Project

* 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 16'''

'''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

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* 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 17'''

'''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'''


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


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