Maths Term Papr

8/3/2019 Maths Term Papr

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LOVELY

PROFESSIONAL

UNIVERTY (TRANSFORMING EDUCATION, TRANSFORMING INDIA)

SUBMITTED TO:

SUBMITTED BY:

MAM GURPREET KAUR

SHISHIR KHEROD Department of engineering

RB4005

Mathematics

A16

MTH:101REG.NO. 11005897


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TERM PAPER ON GAUSSIAN

ELIMINATION METHOD AND

GAUSS-JORDAN METHOD


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Acknowledgement

Gratitude cannot be seen or expressed. It can only

be felt in heart and is beyond description. Often

words are inadequate to serve as a model of

expression of ones feeling, specially the sense ofindebtedness and gratitude to all those who help

us in our duty.

It is of immense pleasure and profound

privilege to express my gratitude and

indebtedness along with sincere thanks to mamgurpreet kaur, Faculty of Lovely University for

providing me the opportunity to work for a project

on GAUSSIAN ELIMINATIONMETHOD AND GAUSS-

JORDAN METHOD

In particular I would like to mention the efforts of

mam gurpreet kaur, Lecturer of LPU Jalandhar,

without whose encouragement the project could

not have been started. He helped me on the


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project as an advisor and offered his help when

needed in every aspect of project.

I am beholden to my family and friends for theirblessings and encouragement.

Always Obediently

SHISHIR KHEROD

REG.NO-11005897

CONTENTS

HISTORY

GAUSSIAN ELIMINATION METHOD

ALGORITHM OVERVIEW

GAUSS-JORDAN ELIMINATION METHOD

APPLICATION TO FINDING INVERSES

REFRENCES


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HISTORY:

The first reference to the book by this title is dated to 179 CE, but parts of it were

written as early as approximately 150 BCE. It was commented on byLiu Huiin

the 3rd century.

However, the method was invented in Europe independently by Carl Friedrich

Gauss when developing the method of least squares in his 1809 publication

Theory of Motion of Heavenly Bodies.


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Gaussian elimination:

In linear algebra, Gaussian elimination is an algorithm for solving systems of

linear equations, finding the rankof a matrix, and calculating the inverse ofan

invertible square matrix. Gaussian elimination is named after German

mathematician and scientist Carl Friedrich Gauss.

Elementary row operations are used to reduce a matrix to row echelon form.

GaussJordan elimination, an extension of this algorithm, reduces the matrix

further to reduced row echelon form. Gaussian elimination alone is sufficient formany applications.

The process of Gaussian elimination has two parts. The first part (Forward

Elimination) reduces a given system to eithertriangularor echelon form, or results

in a degenerate equation with no solution, indicating the system has no solution.

This is accomplished through the use of elementary row operations. The second

step usesback substitution to find the solution of the system above.

Stated equivalently for matrices, the first part reduces a matrix to row echelon form

using elementary row operations while the second reduces it to reduced rowechelon form, orrow canonical form.
http://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/System_of_linear_equationshttp://en.wikipedia.org/wiki/System_of_linear_equationshttp://en.wikipedia.org/wiki/Rank_(linear_algebra)http://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Elementary_row_operationshttp://en.wikipedia.org/wiki/Triangular_formhttp://en.wikipedia.org/wiki/Triangular_matrix#Forward_and_back_substitutionhttp://en.wikipedia.org/wiki/Elementary_row_operationshttp://en.wikipedia.org/wiki/Row_canonical_formhttp://en.wikipedia.org/wiki/System_of_linear_equationshttp://en.wikipedia.org/wiki/System_of_linear_equationshttp://en.wikipedia.org/wiki/Rank_(linear_algebra)http://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Elementary_row_operationshttp://en.wikipedia.org/wiki/Triangular_formhttp://en.wikipedia.org/wiki/Triangular_matrix#Forward_and_back_substitutionhttp://en.wikipedia.org/wiki/Elementary_row_operationshttp://en.wikipedia.org/wiki/Row_canonical_formhttp://en.wikipedia.org/wiki/Algorithm


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Algorithm overview:

Another point of view, which turns out to be very useful to analyze the algorithm,

is that Gaussian elimination computes decomposition.The three elementary row

operations used in the Gaussian elimination (multiplying rows,switching rows,

and adding multiples of rows to other rows) amount to multiplying the original

matrix with invertible matrices from the left. The first part of the algorithm

computes LU decomposition, while the second part writes the original matrix as

the product of a uniquely determined invertible matrix and a uniquely determined

reduced row-echelon matrix

Example:

Suppose the goal is to find and describe the solution(s), if any, of the following

system of linear equations:

The algorithm is as follows: eliminate x from all equations below L1, and theneliminate y from all equations below L2. This will put the system into triangular

form. Then, using back-substitution, each unknown can be solved for.

In the example, x is eliminated from L2 by adding to L2. x is then eliminated

from L3 by adding L1 to L3. Formally:
http://en.wikipedia.org/wiki/Triangular_formhttp://en.wikipedia.org/wiki/Triangular_formhttp://en.wikipedia.org/wiki/Triangular_formhttp://en.wikipedia.org/wiki/Triangular_form


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The result is:

Now y is eliminated from L3 by adding 4L2 to L3:

The result is:

This result is a system of linear equations in triangular form, and so the first part of

the algorithm is complete.

The second part, back-substitution, consists of solving for the unknowns in reverse

order. It can thus be seen that

Then, z can be substituted into L2, which can then be solved to obtain

Next, z and y can be substituted into L1, which can be solved to obtain

The system is solved.


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Some systems cannot be reduced to triangular form, yet still have at least one valid

solution: for example, if y had not occurred in L2 and L3 after the first step above,

the algorithm would have been unable to reduce the system to triangular form.

However, it would still have reduced the system to echelon form. In this case, the

system does not have a unique solution, as it contains at least one free variable.

The solution set can then be expressed parametrically (that is, in terms of the free

variables, so that if values for the free variables are chosen, a solution will be

generated).

GaussJordan elimination:

In linear algebra, GaussJordan elimination is a version ofGaussian elimination

that puts zeros both above and below eachpivot element as it goes from the top

row of the given matrix to the bottom. Every matrix has a reduced row echelon

form, and both algorithms are guaranteed to produce it.

It is named afterCarl Friedrich Gauss and Wilhelm Jordan, because it is a

modification of Gaussian elimination as described by Jordan in 1887. However, themethod also appears in an article by Clasen published in the same year. Jordan and

Clasen probably discovered GaussJordan elimination independently.

In computer science, GaussJordan elimination as an algorithm has a time

complexity of O (n3) for an n by n matrix.
http://en.wikipedia.org/wiki/Echelon_formhttp://en.wikipedia.org/wiki/Free_variablehttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Gaussian_eliminationhttp://en.wikipedia.org/wiki/Pivot_elementhttp://en.wikipedia.org/wiki/Reduced_row_echelon_formhttp://en.wikipedia.org/wiki/Reduced_row_echelon_formhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Wilhelm_Jordanhttp://en.wikipedia.org/wiki/Echelon_formhttp://en.wikipedia.org/wiki/Free_variablehttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Gaussian_eliminationhttp://en.wikipedia.org/wiki/Pivot_elementhttp://en.wikipedia.org/wiki/Reduced_row_echelon_formhttp://en.wikipedia.org/wiki/Reduced_row_echelon_formhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Wilhelm_Jordan


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Application to finding inverses

If GaussJordan elimination is applied on a square matrix, it can be used to

calculate the matrix's inverse. This can be done by augmenting the square matrixwith the identity matrix of the same dimensions, and through the following matrix

operations:

If the original square matrix, A, is given by the following expression:

Then, after augmenting by the identity, the following is obtained:

By performing elementary row operations on the [AI] matrix until it reaches

reduced row echelon form, the following is the final result:
http://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Augmented_matrixhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Elementary_row_operationshttp://en.wikipedia.org/wiki/Reduced_row_echelon_formhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Augmented_matrixhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Elementary_row_operationshttp://en.wikipedia.org/wiki/Reduced_row_echelon_form


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The matrix augmentation can now be undone, which gives the following:

A matrix is non-singular(meaning that it has an inverse matrix)if and only ifthe

identity matrix can be obtained using only elementary row operations.

References

1. Althoen, Steven C.; McLaughlin, Renate (1987), "GaussJordan reduction: a

brief history".

Lipschutz, Seymour, and Lipson, Mark. "Schaum's Outlines: Linear

Algebra".

Strang, Gilbert (2003), Introduction to Linear Algebra (3rd ed.), Wellesley,

Massachusetts.
http://en.wikipedia.org/wiki/Non-singular_matrixhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Non-singular_matrixhttp://en.wikipedia.org/wiki/If_and_only_if

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