Simulation of Simultaneous Equation Using Crammers Rule

 

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