Scilab/C4/Linear-equations-Iterative-Methods/English-timed

Time	Narration
00:01	Dear Friends,
00:02	Welcome to the Spoken Tutorial on “Solving System of Linear Equations using Iterative Methods”
00:10	At the end of this tutorial, you will learn how to:
00:14	Solve system of linear equations using iterative methods
00:18	Develop Scilab code to solve linear equations
00:22	To record this tutorial, I am using
00:25	Ubuntu 12.04 as the operating system
00:28	and Scilab 5.3.3 version
00:33	Before practising this tutorial, a learner should have basic knowledge of
00:38	Scilab
00:39	and Solving Linear Equations
00:42	For Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website.
00:50	The first iterative method we will be studying is Jacobi method.
00:56	Given a system of linear equations, with n equations and n unknowns
01:02	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
01:24	We assume values for each x of i
01:27	Then we substitute the values in the equations obtained in the previous step.
01:34	We continue the iteration until the solution converges.
01:39	Let us solve this example using Jacobi Method
01:44	Let us look at the code for Jacobi Method.
01:48	We use format method to specify the format of the displayed answers on the Scilab console.
01:56	Here e denotes the answer should be in scientific notation.
02:01	And twenty specifies the number of digits to be displayed.
02:06	Then we use input function to get the values for
02:10	the matrices coefficient matrix,
02:12	right hand side matrix,
02:14	initial values matrix,
02:17	maximum number of iteration and
02:19	convergence tolerance
02:22	Then we use size function to check if A matrix is a square matrix.
02:29	If it isn't, we use error function to display an error.
02:34	We then check if matrix A is diagonally dominant.
02:40	The first half calculates the sum of each row of the matrix.
02:45	Then it checks if twice the product of the diagonal element is greater than the sum of the elements of that row.
02:54	If it isn't, an error is displayed using error function.
03:01	Then we define the function Jacobi Iteration with input arguments
03:07	A, b , x zero,
03:09	maximum iteration and tolerance level.
03:14	Here x zero is the initial values matrix.
03:19	We check if the size of A matrix and initial values matrix are compatible with each other.
03:28	We calculate the value for x k p one and then check if the relative error is lesser than tolerance level.
03:38	If it is lesser than tolerance level, we break the iteration and the solution is returned.
03:45	Finally we end the function.
03:48	Let us save and execute the function.
03:51	Switch to Scilab console.
03:54	Let us enter the values at each prompt.
03:57	The coefficient matrix A is open square bracket two space one semi colon five space seven close square bracket
04:08	Press Enter.
04:10	Then we type open square bracket eleven semicolon thirteen close square bracket
04:17	Press Enter.
04:20	The initial values matrix is open square bracket one semi colon one close square bracket
04:28	Press Enter.
04:30	The maximum number of iterations is twenty five.
04:34	Press Enter.
04:36	Let the convergence tolerance level be zero point zero zero zero zero one
04:44	Press Enter.
04:46	We call the function by typing
04:48	Jacobi Iteration open paranthesis A comma b comma x zero comma M a x I t e r comma t o l close paranthesis
05:04	Press Enter.
05:06	The values for x one and x two are shown on the console.
05:11	The number of iterations are also shown.
05:14	Let us now study Gauss Seidel method.
05:19	'Given a system of linear equations, with n equations and n unknowns
05:26	We rewrite the equations for each unknown
05:29	by subtracting the other variables and their coefficients from the corresponding right hand side element.
05:37	Then we divide this by the coefficient a i i of the unknown variable' for that variable.
05:45	This is done for every given equation.
05:49	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
06:03	In Gauss Seidel method, we over write the value of x of i k with x of i k plus one
06:12	Let us solve this example using Gauss Seidel Method
06:17	Let us look at the code for Gauss Seidel Method
06:21	The first line specifies the format of the displayed answer on the console using format function.
06:29	Then we use input function to get the values of
06:32	coefficient matrix,
06:34	right hand side matrix,
06:36	initial values of the variables matrix,
06:38	maximum number of iterations and
06:40	tolerance level
06:43	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
06:58	We check if matrix A is square and the sizes of initial vector and matrix A are compatible using size and length function.
07:10	Then we start the iterations.
07:13	We equate the initial values vector x zero to x k.
07:19	We create a matrix of zeros with the same size of x k and call it x k p one.
07:28	We solve for each equation to get the value of the unknown variable for that equation using x k p one.
07:38	At each iteration, the value of x k p one gets updated.
07:44	Also, we check if relative error is lesser than specified tolerance level.
07:50	If it is, we break the iteration.
07:54	Then equate x k p one to the variable solution.
07:59	Finally, we end the function.
08:02	Let us save and execute the function.
08:06	Switch to Scilab console
08:09	For the first prompt, we type matrix A.
08:12	Type open square bracket two space one semi colon five space seven close square bracket
08:21	Press Enter
08:22	For the next prompt,
08:24	type open square bracket eleven semi colon thirteen close square bracket
08:31	Press Enter.
08:33	We provide the values of initial value vector by typing
08:38	open square bracket one semicolon one close square bracket
08:43	Press Enter.
08:45	Then we specify the maximum number of iterations to be twenty five
08:50	Press Enter.
08:52	Let us define 'tolerance level to be zero point zero zero zero zero one
08:58	Press Enter
09:01	Finally we call the function by typing
09:04	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
09:24	Press Enter.
09:26	The values of x one and x two are displayed.
09:30	The number of iterations to solve the same problem are lesser than Jacobi method.
09:37	Solve this problem on your own using Jacobi and Gauss Seidel methods
09:43	In this tutorial, we have learnt to:
09:47	Develop Scilab code for solving system of linear equations
09:52	Find the value of the unknown variables of a system of linear equations
09:58	Watch the video available at the following link
10:01	It summarises the Spoken Tutorial project
10:04	If you do not have good bandwidth, you can download and watch it
10:09	The spoken tutorial project Team
10:11	Conducts workshops using spoken tutorials
10:15	Gives certificates to those who pass an online test
10:18	For more details, please write to contact@spoken-tutorial.org
10:25	Spoken Tutorial Project is a part of the Talk to a Teacher project
10:30	It is supported by the National Mission on Eduction through ICT, MHRD, Government of India.
10:37	More information on this mission is available at http://spoken-tutorial.org/NMEICT-Intro
10:49	This is Ashwini Patil signing off.
10:51	Thank you for joining.

Contributors and Content Editors

Gaurav, PoojaMoolya, Pratik kamble, Sandhya.np14