ISSN NO. 2456-3129
International Journal of Engineering, Pure and Applied Sciences,
Vol. 1, No. 3, Dec-2016
Mathematical Computational for System of Linear
Equation
1,
1
Sofia Monika
Department of Mathematics, CGC Landran, Mohali, Punjab, India1,2
Email: [email protected] ,[email protected],
Abstract - In this paper straight conditions are talked about in detail alongside end strategy. This paper contains
network presentation, and the immediate techniques for straight conditions. The objective of this exploration
was to dissect distinctive end systems of straight conditions and measure the execution of Guassian end and
Guass Jordan strategy, with a specific end goal to locate their relative significance and favorable position in the
field of typical and numeric calculation. The motivation behind this examination is to reconsider an early on
idea of direct conditions, grid theorey and types of Guassian elimination through which the execution of Guass
Jordan and Guassian end can be measured.
Keywords: Matrix, Direct, Indirect, Backward Stage, Forward
1. INTRODUCTION
Arrangement of condition is a set or gathering of
conditions comprehended together. Gathering of
straight conditions is named as arrangement of direct
conditions. They are regularly in view of same
arrangement of factors. Different techniques have
been developed to comprehend the straight conditions
however there is no best strategy yet proposed for
understanding arrangement of direct equations[1].
Different strategies are proposed by various
mathematicians in light of the speed and exactness.
However speed is an essential component for settling
direct conditions where volume of calculation is so
vast. Direct condition techniques are separated into
two classifications. Immediate and Aberrant. Every
classification involves a few end strategies utilized for
tackling conditions. this paper manages Guassian
elimination technique, an immediate strategy for
illuminating arrangement of straight conditions.
A starting part of Guass Jordan end is likewise
completed keeping in mind the end goal to break
down the execution of both techniques. Roundabout
techniques are fundamentally iterative strategies and
these strategies have preference it might be said that
they require less duplication ventures for extensive
calculations. Iterative strategies can be executed in
littler projects and are sufficiently quick. With
investigation of arrangement of straight condition one
must be acquainted with framework hypothesis that
how diverse operations are performed on a coveted
network to ascertain the outcome. Exemplary books
on the Historical backdrop of Science, and in addition
late reviews regarding this matter, put the birthplaces
of Guassian End in an assortment of antiquated
writings from better places and times: China, Greece,
Rome, India, medieval Arabic nations, and European
Renaissance. Nonetheless, it is not correct to state that
these antiquated writings depict what we see today as
the strategy for Guassian elimination, since these
writings basically display some particular issues that
are explained in a way that is acknowledged as
Guassian End, however they do exclude any
unequivocal articulation of the arrangement of
standards that constitute the technique for GE. The
textbook disposal time frame compares to the
improvement of GE basically as it is introduced in
current secondary school reading material. This
period began with Isaac Newton who addressed on
Variable based math's it showed up in Renaissance
writings. Isaac Newton set up first the tenets of
Gaussian end as they are still displayed in current
secondary school reading material. Carl Friedrich
Gauss created productive strategies for unraveling
typical conditions, i.e., the extraordinary kind of
direct conditions that may emerge in arrangements of
minimum square issues, through Gaussian elimination
of straight conditions by means of hand calculations.
The standard strategy for fathoming straight
conditions. As it is an omnipresent calculation and
assumes a principal part in logical calculation.
Guassian end is a device for getting the arrangement
of conditions, to register the determinant, for
concluding rank of coefficient grid. However,
Guassian elimination depends more on framework
investigation and calculation. It accentuation on
square turning, strategies for cycle and a way to
enhance the registered arrangement quality. It
includes two phases forward and in reverse stage.
Forward stage: Questions are wiped out in this phase
by control of conditions and constitute an echelon
shape.
In reverse stage: it is connected with back
substitution prepare on the diminished upper
triangular strategy bringing about an answer of
condition.
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ISSN NO. 2456-3129
2456
International Journal of Engineering, Pure and Applied Sciences,
Sciences
Vol. 1, No. 3, Dec-2016
2. STEPS TO UNDERSTAND GUASSIAN
DISPOSAL
for j = i+1 to n Guassian Elimination is orderly use of
basic column operations in arrangement of conditions
[2]. It changes over the directt arrangement of
conditions to upper triangular frame, from which
arrangement of condition is resolved. Guassian
disposal is condensed in the previously mentioned
strides III.
i). Augmented network must be composed for the
arrangement of straight conditions.
ii). Transform A to upper triangular frame utilizing
line operations on {A/b}. Slanting components
may not be zero.
iii). Use back substitution for finding the
arrangement of issue.
Consider
nsider the arrangement of direct conditions with
including n factors [3].
a11x1+ a12x2+ … + a1nxn= a1,n+1
a21x1+ a22x2+ … + a2nxn= a2,n+1
a31x1+ a32x2+ … + a3nxn= a3,n+1
………………
an1x1+ an3x2+ … + annxn= an,n+1
Where aij and ai,j+1 are constants, xi’s are variables.
The system becomes equal to:
AX=B
Step 3 :
New elements are written aij indicate that this is the
third time that a number has been stored in the matrix
at location ( i , j ) .
Final result after the row operation may result in
above form:
Sequential Algorithm – Gauss Elimination Method[4]
Input : Given Matrix a[1 : n, 1: n+1]
Output : x[1 : n]
Step 1:
Store the coefficients in an augmented matrix. The
superscript on aij means that this is the first time that a
number is stored in location (i, j).
Step 2:
If necessary, shift rows so that a≠110,
≠110, then eliminate
x1 in row2 through n .The new elements are written
aij to indicate that this is the second time that a
number has been stored in the matrix at location ( i , j
).
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
for k = 1 to n -1
for i = k+1 to n
u = aik/akk
for j = k to n+1
aij = aij – u * akj
next j
next i
next k
xn = a n,n+1/ann
for i = n to 1 step -1
sum = 0
sum = sum + aij * xj
next j
xi = ( ai,n+1 - sum )/aii
next i
end
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ISSN NO. 2456-3129
2456
International Journal of Engineering, Pure and Applied Sciences,
Sciences
Vol. 1, No. 3, Dec-2016
3. GUASSIAN ELIMINATION
Through Partial Pivoting In actual computational
practice, it is necessary to permute the rows of the
matrix A (equivalently, the equations of the
system Ax = b) for obtaining a reliable algorithm.
Th e permutations are performed on line as GE
proceeds and several permutations (or pivoting).
Partial pivoting involves the following steps:
Transform the matrix A into lower and upper
triangular matrix for the further computations. Let
lower triangular matrix be L and upper triangular
matrix be U.
Step 1: Select the equation having the larger 1st
coefficient in system of equation and place it at the 1st
entity of matrix
Step 2 : Now perform the elementary row operations
and convert the matrix into upper triangular form by
using the pivot equation. The resultant matrix after
operations may result in the form:
Step 3 : Make the equation equal to the number of
variables and determine the solution of equation.
d) LU Factorization Guassian Elimination
LU factorization is the most important mathematical
concept used in Guassian Elimination method. It plays
a key role in the implementation of GE in modern
computers, and, finally, it is essential to facilitate the
rounding error analysis of the algorithm. LU
factorization method is performed in three steps [5]:
i). A=LU, compute the LU factorization.
ii). For y solve the lower triangular matrix as
Ly=b by using forward substitution method
(i.e, start by computing the first unknown
u
as
y1=b1 from the first equation, after that
compute the second unknown using the value
of previous variable and so on.
iii). Compute x for the upper triangular matrix
using the relation Ux=y by using backward
substitution method.
Consider the following matrix
atrix of the form Ax=b:
From these upper and lower triangular matrix perform
the computations for both equations listed in step ii.
Calculate the values of unknown.
4. GUASSIAN ELIMINATION AS
COMPUTATIONAL PARADIGM
Sparse Guassian Elimination was studied in the early
70’s. For the vertex elimination on the undirected
graphs a graph model was proposed. The structural
property of the vertices has been a major research in
the last decades. Also work on optimal elimination
tree was carried out, which proved of no importance
in sparsity preserving elimination trees, looking
towards optimal elimination
mination trees could result in non
linear fill. There has been minimum use of tree related
graphs elimination outside the sparse Guassian
Elimination. Cholesky factor is described in terms of
different set of vertices: sets of predecessor and
successor, chain
ain elimination and elimination sets. The
model of Guassian Elimination gives a precise
description of interaction between master, sub
problems which are hidden in formulation of dynamic
programming. In case of solving blocked linear
equations with PDS matrices,
rices, proposed model of
computation is a straightforward extension of
Guassian Elimination (point wise). Transformation
associated with elimination of vertex is simply block
elimination using sub-matrix
matrix of block diagonal as
block pivot. One application regarding
re
computational
model is in the context of solution of asymmetric
blocked structural system of linear system of
equations which demonstrates an indirect use in
process of solution, rules of assignments of columns
to block for block elimination process.
proces These rules
provide a new concept of pivoting. Consider zero sub
columns in original data and non zero in partially
reduced matrix. Computed sub column remains in the
column space of some sub column in the original data.
Substituting one of these sub columns
co
to the
considered sub column is appealing. It is
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ISSN NO. 2456-3129
International Journal of Engineering, Pure and Applied Sciences,
Vol. 1, No. 3, Dec-2016
complementary to sparse preserving elimination.
Cholesky Factorization method is motivated by
solution of so called normal equations that come from
linearalized KKT system in the context of Newton
method. While considering interior point for solving
large scale block problems, from a numerical point of
view to solve linear system of equations is of great
consideration. Smallest height elimination trees tend
to have maximum number of leaves. Block Cholesky
includes block LU factorizations, the coefficients of
submatrix that correspond to the leaves is the original
data. Incomplete factorization consists of factorization
from leaves up to the level where data is to be
transformed several times by preceding block
eliminations.
[5]
[6]
[7]
J. F. Grcar, “How ordinary elimination became
Gaussian
elimination”.
Historia
Math.,
38(2):163_218, 2011.
Etienne Loute, “guassian elimination as
computational paradigm”, August 2003.
Nae Kaun Tsao, Ditroit Michigan, “On the
Equivalence of Gaussian Eliminatio and GaussJordan
Reduction
in
Solving
Linear
Equations”,February 1989.
5. CONCLUSION
There are different direct and indirect methods which
are used to compute the linear system of equations.
Guassian Elimination is a type of direct method used
to calculate the unknown variables. Many scientific
and engineering domains of computation may take the
form of linear equations. The equations in this field
may contain large number of variables and hence it is
important to solve these equations in an efficient
manner. This paper comprises of Guassian
Elimination method an efficient method to solve these
equations. Although the comparison on the basis of
execution time is carried out along with the Guass
Jordan method and it has been concluded that
Guassian Elimination is faster than the other
elimination methods and it is used
in various scientific fields where large number of
computations are performed by elimination of
variables. Our future directions are to use and develop
the simple and efficient method for non linear system
of equations.
REFERENCES
[1]
[2]
[3]
[4]
Noreen Jamil,“ direct and indirect solvers for
linear system equations”, june 2012.
Froilán M. Dopico, “Alan Turing and the origins
of modern Gaussian elimination”, Instituto de
Ciencias Matemáticas CSIC-UAM-UC3MUCM, December 14, 2012.
Yadanar Mon, Lai lai win kyi, “performance
comparison of guassian elimination and
guassJordan”, International Journal of Computer
& Communication Engineering Research
(IJCCER), Information Technology Department,
Volume 2, 02-march-2014.
Luke Smith and Joan Powell: “An Alternative
Method
to
Gauss-Jordan
Elimination:
Minimizing
Fraction
Arithmetic”,
The
Mathematics Educator, 2011
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