Scilab/C4/Linear-equations-Gaussian-Methods/Gujarati

Time	Narration
00:01	Dear Friends,
00:02	Welcome to the spoken tutorial on Solving System of Linear Equations using Gauss Elimination and Gauss-Jordan Methods.
00:12.	At the end of this tutorial, you will learn how to:
00:15	Solve system of linear equations using Scilab
00:20	Develop Scilab code to solve linear equations.
00:25	To record this tutorial, I am using
00:27	Ubuntu 12.04 as the operating system
00:31	with Scilab 5.3.3 version.
00:36	To practice this tutorial, a learner should have basic knowledge of Scilab
00:40	and should know how to solve Linear Equations.
00:45	To learn Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website.
00:52	A system of linear equations is a
00:55	finite collection of linear equations of the same set of variables.
01:00	Let us study Gauss elimination method.
01:04	Given a system of equations
01:06	A x equal to b
01:08	with m equations and
01:10	n unknowns.
01:12	We write the coefficients of the variables a one to a n
01:16	along with the constants b one to b n of the system of equations
01:22	in one matrix called the augmented matrix.
01:27	How do we convert the augmented matrix to an upper triangular form matrix?
01:33	We do so by performing row wise manipulation of the matrix.
01:40	Let us solve this system of equations using Gaussian elimination method.
01:45	Before we solve the system, let us go through the code for Gaussian elimination method.
01:52	The first line of the code is format e comma twenty.
01:58	This defines how many digits should be displayed in the answer.
02:04	The letter 'e' within single quotes denotes that the answer should be displayed in scientific notation.
02:12	The number twenty is the number of digits that should be displayed.
02:17	The command funcprot is used to let Scilab know what to do when variables are redefined.
02:26	The argument zero specifies that Scilab need not do anything if the variables are redefined.
02:33	Other arguments are used to issue warnings or errors if the variables are redefined.
02:40	Next we use the input function.
02:43	It will display a message to the user and get the values of A and b matrices.
02:51	The message should be placed within double quotes.
02:55	The matrices that the user enters, will be stored in the variables A and b.
03:02	Here A is the coefficient matrix and b is the right-hand-side matrix or the constants matrix.
03:11	Then we define the function naive gaussian elimination.
03:15	And we state that A and b are the arguments of the function naive gaussian elimination.
03:22	We store the output in variable x.
03:27	Then we find the size of matrices A and b using the size command.
03:34	Since they are two dimensional matrices, we use n and n one to store the size of matrix A.
03:42	Similarly we can use m one and p for matrix b.
03:48	Then we have to determine if the matrices are compatible with each other and
03:53	if A is a square matrix.
03:57	If n and n one are not equal then we display a message that Matrix A must be square.
04:05	If n and m one are not equal, we display a message
04:10	incompatible dimension of A and b.
04:15	If the matrices are compatible, we place matrices A and b in one matrix, C.
04:23	This matrix C is called augmented matrix.
04:28	The next block of code performs forward elimination.
04:32	This code converts the augmented matrix to upper triangular matrix form.
04:39	Finally, we perform back substitution.
04:42	Once the upper triangular matrix is obtained, we take the last row and find the value of the variable in that row.
04:52	Then once one variable is solved, we take this variable to solve the other variables.
04:59	Thus the system of linear equations is solved.
05:03	Let us save and execute the file.
05:06	Switch to Scilab console to solve the example.
05:10	On the console, we have a prompt to enter the value of the coefficient matrix.
05:17	So we enter the values of matrix A.
05:20	Type: square bracket three point four one space one point two three space minus one point zero nine semi colon
05:33	two point seven one space two point one four space one point two nine semi colon
05:41	one point eight nine space minus one point nine one space minus one point eight nine close square bracket.
05:53	Press Enter.
05:54	The next prompt is for matrix b.
05:57	So we type
05:58	open square bracket four point seven two semi colon three point one semi colon two point nine one close square bracket.
06:10	Press Enter.
06:13	Then we call the function by typing
06:16	naive gaussian elimination open parenthesis A comma b close parenthesis
06:24	Press Enter.
06:26	The solution to the system of linear equations is shown on Scilab console.
06:32	Next we shall study the Gauss-Jordan method.
06:36	In Gauss–Jordan Method,
06:38	the first step is to form the augmented matrix.
06:42	To do this, place the coefficient matrix A and the right hand side matrix b together in one matrix.
06:50	Then we perform row operations to convert matrix A to diagonal form.
06:56	In diagonal form, only the elements a i i are non-zero. Rest of the elements are zero.
07:05	Then we divide the diagonal element and corresponding element of right hand side element, by the diagonal element.
07:14	We do this to get diagonal element equal to one.
07:19	The resulting value of the elements of each row of the right hand side matrix gives the value of each variable.
07:27	Let us solve this example using Gauss-Jordan Method.
07:33	Let us look at the code first.
07:36	The first line of the code uses format function to specify the format of the displayed answers.
07:44	The parameter e specifies the answer should be in scientific notation.
07:49	Twenty (20) denotes that only twenty digits should be displayed.
07:55	Then we get the A and b matrix using the input function.
08:00	We define the function Gauss Jordan Elimination with input arguments A and b and output argument x.
08:11	We get the size of matrix A and store it in m and n.
08:17	Similarly, we get the size of matrix b and store it in r and s.
08:23	If the sizes of A and b are not compatible, we display an error on the console using error function.
08:33	Then we perform row operations to get diagonal form of the matrix.
08:38	Here pivot refers to the first non-zero element of a column.
08:45	Then we create a matrix of zeros called x with m rows and s columns.
08:52	Once we have the diagonal form,
08:54	we divide the right hand side part of augmented matrix by the corresponding diagonal element to get the value of each variable.
09:04	We store the value of each variable in x.
09:08	Then we return the value of x.
09:11	Finally, we end the function.
09:13	Now let us save and execute the function.
09:18	The prompt requires us to enter the value of matrix A.
09:22	So we type
09:23	open square bracket zero point seven comma one seven two five semi colon
09:31	zero point four three five two comma minus five point four three three close square bracket.
09:41	Press Enter.
09:43	The next prompt is for vector b.
09:45	So we type open square bracket one seven three nine semi colon
09:51	three point two seven one close square bracket.
09:55	Press Enter.
09:58	Then we call the function by typing
10:01	Gauss Jordan Elimination open parenthesis A comma b close parenthesis
10:08	Press Enter.
10:10	The values of x one and x two are shown on the console.
10:15	Let us summarize this tutorial.
10:18	In this tutorial, we have learnt to:
10:21	Develop Scilab code for solving system of linear equations.
10:25	Find the value of the unknown variables of a system of linear equations.
10:32	Watch the video available at the link shown below.
10:35	It summarizes the Spoken Tutorial project.
10:38	If you do not have good bandwidth, you can download and watch it.
10:43	The spoken tutorial project Team:
10:45	Conducts workshops using spoken tutorials.
10:48	Gives certificates to those who pass an online test.
10:52	For more details, please write to conatct@spoken-tutorial.org.
10:59	Spoken Tutorial Project is a part of the Talk to a Teacher project.
11:03	It is supported by the National Mission on Eduction through ICT, MHRD, Government of India.
11:10	More information on this mission is available at http://spoken-tutorial.org/NMEICT-Intro.
11:21	This is Ashwini Patil, signing off.
11:23	Thank you for joining.

Contributors and Content Editors

Jyotisolanki, PoojaMoolya