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

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

Author: Shamika

Keywords: System of linear equations, Gaussian Methods


Visual Cue
Narration
Slide 1 Dear Friends,

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

Slide 2 -Learning Objective Slide 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
Slide 3-System Requirement slide To record this tutorial, I am using
  • Ubuntu 12.04 as the operating system
  • and Scilab 5.3.3 version
Slide 4- Prerequisites slide 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.


Slide 5- System of Linear Equations A system of linear equations is a
  • Finite collection of
  • linear equations
  • of the same set of variables
Slide 7- Gaussian Elimination Method Let us study Gauss elimination method

Given a system of equations

  • A x equal to b
  • with m equations and
  • n unknowns
Slide 8- Gaussian Elimination Method
  • We write the coefficients of the variables a one to a n
  • along with the constants b one to b m of the system of equations
  • in one matrix called the augmented matrix
Slide 9- Gaussian Elimination Method


How do we convert the augmented matrix to an upper triangular form matrix?


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

Slide 10- Example Let us solve this system of equations using Gaussian elimination method
Switch to Scilab and open naivegaussianelimination.sci Before we solve the system, let us go through the code for Gaussian elimination method.
Highlight format e comma twenty The first line of the code is format e comma twenty.


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

Point to e The letter 'e' within single quotes denotes that the answer should be displayed in scientific notation
Point to twenty The number twenty is the number of digits that should be displayed.
Highlight funcprot The command funcprot is used to let Scilab know what to do when variables are redefined.
Point to zero The argument zero specifies that Scilab need not do anything if the variables are redefined.
No visual clue. This is extra information that is being offered to the listener. Other arguments are used to issue warnings or errors if the variables are redefined.
Highlight input Next we use the input function.
Highlight input


Point to “ “

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.

The matrices that the user enters, will be stored in the variables A and b.
Highlight A

Highlight B

Here A is the coefficient matrix and b is the right-hand-side matrix or the constants matrix.
Highlight naivegaussianelimination Then we define the function naive gaussian elimination.
Highlight A

Highlight b

And we state that A and b are the arguments of the function naive gaussian elimination.
Highlight x We store the output in variable x.
Highlight size Then we find the size of matrices A and b using the size command.
Highlight n and n one Since they are two dimensional matrices, we use n and n one to store the size of matrix A.
Highlight m one and p Similarly we can use m one and p for matrix b.
Highlight

if n ~= n1

error('gaussianelimination - Matrix A must be square');

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 Matrix A must be square.

Highlight

elseif n ~= m1

error('gaussianelimination - incompatible dimension of A & b');

If n and m one are not equal, we display a message

incompatible dimension of A and b.

Highlight

C=[A b];

If the matrices are compatible, we place matrices A and b in one matrix, C.


This matrix C is called augmented matrix.

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

The next block of code performs forward elimination.

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







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

  • 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.


Click on Execute and select Save and Execute Let us save and execute the file.
Show Scilab Console Switch to Scilab console to solve the example.
Show Scilab Console On the console, we have a prompt to enter the value of the coefficient matrix.


So we enter the values of matrix A.

Type

[3.41 1.23 -1.09;2.71 2.14 1.29;1.89 -1.91 -1.89]

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

Type

[4.72;3.1;2.92]

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

Type

naivegaussianelimination(A,b)

Then we call the function by typing

naive gaussian elimination open paranthesis A comma b close paranthesis


Press Enter

Show answer on Scilab console The solution to the system of linear equations is shown on Scilab console.
Next we shall study the Gauss- Jordan method.
Slide 11 – Gauss- Jordan Method 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 to convert matrix A to diagonal form.
  • In diagonal form, only the elements a i i are non-zero. Rest of the elements are zero.
Slide 12- Gauss-Jordan Method 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.

Slide 13- Example


Let us solve this example using Gauss- Jordan Method.
Switch to Scilab console and open GaussJordan Elimination.sci Let us look at the code first.
Highlight

format

The first line of the code uses format function to specify the format of the displayed answers.
Highlight

'e'

The parameter e specifies the answer should be in scientific notation.
Highlight 20 Twenty (20) denotes that only twenty digits should be displayed.
Highlight

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

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

Then we get the A and b matrix using the input function.
Highlight

function [x] = GaussJordanElimination( A, b )

We define the function Gauss Jordan Elimination with input arguments A and b and output argument x.
Highlight

[m, n] = size( A )

We get the size of matrix A and store it in m and n
Highlight

[r, s] = size( b )

Similarly, we get the size of matrix b and store it in r and s
Highlight

if ( m <> r ) then

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

end

If the sizes of A and b are not compatible, we display an error on the console using error function.
Highlight

for k = 1 : 1 : m

indices = [ 1 : 1 : k-1, 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

Then we perform row operations to get diagonal form of the matrix.


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

Highlight

x = zeros( m, s )

Then we create a matrix of zeros called x with m rows and s columns.
Highlight

for i = 1 : 1 : m

for j = 1 : 1 : s

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

end

end

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.


Highlight

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

We store the value of each variable in x.
Highlight

return x

Then we return the value of x.


Highlight

endfunction

Finally, we end the function.
Click on Execute and select Save and Execute Now let us save and execute the function.
Type the following

[0.7, 1725;0.4352,-5.433]


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

Type the following

[1739;3.271]

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

Type the following

GaussJordanElimination(A,b)

Then we call the function by typing

Gauss Jordan Elimination open paranthesis A comma b close paranthesis

Press Enter

Show answers on Scilab console The values of x one and x two are shown on the console.
Slide 14- Summary 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
Show Slide 16

Title: About the Spoken Tutorial Project

  • It summarises the Spoken Tutorial project
  • If you do not have good bandwidth, you can download and watch it


* 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


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


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


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


* 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



This is mandatory. This is Ashwini Patil signing off. Thanks for joining.

Contributors and Content Editors

Lavitha Pereira, Nancyvarkey, Pratik kamble

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