Simulation of Simultaneous Equation Using Crammers Rule
CHAPTER ONE: INTRODUCTION
1.1 BACKGROUND OF THE STUDY
In linear algebra, Cramer's rule is
an explicit formula for the solution of a system of
linear equations with
as many equations as unknowns, valid whenever the system has a unique solution.
It expresses the solution in terms of the determinants of the (square) coefficient and of matrices obtained from it by replacing one
column by the column vector of right-hand-sides of the equations. It is named
after Gabriel Cramer (1704–1752), who published the
rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in
1748 (and possibly knew of it as early as 1729).
Cramer's rule implemented in a naïve
way is computationally inefficient for systems of more than two or three
equations.[7] In the case of n equations
in n unknowns, it requires
computation of n + 1 determinants,
while Gaussian elimination produces the result with the
same computational
complexity as
the computation of a single determinant. Cramer's rule can also be numerically unstable even for 2×2 systems.
However, it has recently been shown that Cramer's rule can be
implemented in O(n3) time, which is comparable to more
common methods of solving systems of linear equations, such as Gaussian elimination (consistently requiring 2.5
times as many arithmetic operations for all matrix sizes), while exhibiting
comparable numeric stability in most cases.
In
mathematics, Gaussian elimination,
also known as row reduction,
is an algorithm for solving systems of linear
equations. It
consists of a sequence of operations performed on the corresponding matrix of coefficients. This method
can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855) although some
special cases of the method albeit presented without proof were known to Chinese
mathematicians as
early as circa 179 CE.
To perform
row reduction on a matrix, one uses a sequence of elementary row
operations to
modify the matrix until the lower left-hand corner of the matrix is filled with
zeros, as much as possible. There are three types of elementary row operations:
·
Swapping
two rows,
·
Multiplying
a row by a nonzero number,
·
Adding
a multiple of one row to another row.
Using these operations, a matrix can
always be transformed into an upper triangular matrix, and in fact one that is in row echelon form. Once all of the leading coefficients (the leftmost nonzero
entry in each row) are 1, and every column containing a leading coefficient has
zeros elsewhere, the matrix is said to be in reduced row echelon
form. This final form is unique; in
other words, it is independent of the sequence of row operations used. For
example, in the following sequence of row operations (where two elementary
operations on different rows are done at the first and third steps), the third
and fourth matrices are the ones in row echelon form, and the final matrix is
the unique reduced row echelon form.
Using row operations to convert a matrix
into reduced row echelon form is sometimes called Gauss–Jordan elimination. In this case, the term Gaussian
elimination refers to the process until it has reached its upper
triangular, or (unreduced) row echelon form. For computational reasons, when
solving systems of linear equations, it is sometimes preferable to stop row
operations before the matrix is completely reduced.
The process of row reduction makes use
of elementary row
operations, and can
be divided into two parts. The first part (sometimes called forward
elimination) reduces a given system to row echelon form, from which
one can tell whether there are no solutions, a unique solution, or infinitely
many solutions. The second part (sometimes called back substitution) continues to use row operations
until the solution is found; in other words, it puts the matrix into reduced row
echelon form.
Another point of view, which turns
out to be very useful to analyze the algorithm, is that row reduction produces
a matrix decomposition of the original matrix. The
elementary row operations may be viewed as the multiplication on the left of
the original matrix by elementary matrices. Alternatively, a sequence of elementary operations that
reduces a single row may be viewed as multiplication by a Frobenius
matrix. Then the first part of the algorithm computes an 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.
1.2 STATEMENT OF THE PROBLEM
In computation
of linear equations many student find the difficult to calculate linear
equations using crammers rule correctly and accurately. Therefore computations
of linear equations using crammers rule in scientific calculator become impossible,
the ties several kinds of problems in solving tedious linear equations with
scientific calculator is not do able. Having been facing a lot of problems on solving linear equations with
crammer’s rule method will become very easy with the help of this newly develop
system to carry out the computations accurately and correctly.
1.3 AIMS AND OBJECTIVES OF THE STUDY
The aim of this
project is to implement the simulation of crammer's rule for linear equations
(2x2 & 3x3).
The objectives of the study include:
i.
To
provide graphical user interface (GUI) that will be very friendly to the user’s
to easily understand how to use the application.
ii.
To
examine the current procedures of calculating linear equations and allow the
user to provide values for manipulation within the system
iii.
To
improve the speed of calculating linear equations using crammer’s rule in such
a way to reduce the complexity of calculating linear equations.
iv.
Finally,
to reduce the problems immensely and provides a release working environment.
1.4 SCOPE AND LIMITATIONS OF THE
STUDY
The project will
only focus on the calculation of linear equations using crammer’s rule (2x2
& 2x3) and therefore the system lacks the ability to calculate any other
combinations rather than this two.
1.5 SIGNIFICANCE OF THE STUDY
This project work
will function to help and assist the student for calculation of linear
equations using crammer’s rule method easily. Therefore the student can use
this system to ensure the accuracy of their manual calculations of linear
equations using crammer’s rule in the aspect of their study.
1.6 DEFINITION OF TERMS
Crammer’s Rule: is an explicit
formula for the solution of a system of linear equations with as many equations
as unknowns, valid whenever the system has a unique solution.
Linear Equations: An equation between two variables
that gives a straight line when plotted on a graph.
Simulation: A simulation is the
imitation of the operation of a real-world process or system over time. Simulations
require the use of models; the model represents the key characteristics or
behaviors of the selected system or process, whereas the simulation represents
the evolution of the model over time
Rows: In the context of a
relational database, a row also called a tuple represents a single, implicitly
structured data item in a table. In simple terms, a database table can be
thought of as consisting of rows and columns.
Columns: In a relational
database, a column is a set of data values of a particular type, one value for
each row of the database. A column may contain text values, numbers, or even
pointers to files in the operating system.
Calculation: A calculation is a
deliberate process that transforms one or more inputs into one or more results.
Determinant: the determinant is
a scalar value that is a function of the entries of a square matrix. It allows
characterizing some properties of the matrix and the linear map represented by
the matrix
Constant: In mathematics, the
word constant can have multiple meanings. As an adjective, it refers to
non-variance; as a noun, it has two different meanings: A fixed and
well-defined number or other non-varying mathematical object.
Row Operation: Row operations are calculations
we can do using the rows of a matrix in order to solve a system of
equations, or later, simply row reduce the matrix for other purposes.
#Note: This project is complete chapter 1 - 5 and the Software. Hence the software is developed using vb.net.
Contact Project Developer
Whatapp: (+234) 07067066709
Email: lawansalisucs@gmail.com or drlawanonline@gmail.com
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