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1 International Journal of Computer & Communication Engineering Research (IJCCER) Volume 2 - Issue 2 March 2014 Performance Comparison of Gauss and Gauss-Jordan Yadanar Mon 1, Lai Lai Win Kyi 2 1,2 Information Technology Department Mandalay Technological University,Mandalay, Myanmar 1 yadanarmon2012@gmail.com, 2 laelae83@gmail.com Abstract This paper examines the comparisons of execution time between Gauss and Gauss Jordan Methods for solving system of linear equations. It tends to calculate unknown variables in linear system. It was noted for the solved problems that both s gave the same answers. Many different types of linear equations have been solved with the help of these two s using computer program developed in Java Programming language. Numbers of operations involved in the solutions of linear simultaneous equations have also been calculated. This paper also explicitly reveals that these s could be applied to different systems of linear equations arising in fields of study like Physics, Business, Economics, Chemistry, etc. Keywords: Gauss, Gauss Jordan, Java Programming Language, Linear system. I. INTRODUCTION Solving systems of linear equation is probably one of the most visible applications of Linear Algebra. Systems of linear equations appear in almost every branch of Science and Engineering. Many scientific and engineering problems can take the form of a system of linear equations. Also many practical problems in economics, engineering, biology, electronics, communication, etc can be reduced to solving a system of linear equations. These equations may contain thousands of variables, so it is important to solve them as efficiently as possible. In linear algebra, Gaussian elimination is an algorithm for solving system of linear equations, finding the rank of a matrix and calculating the inverse of an invertible square matrix. Gaussian elimination is considered as the workhorse of computational science for the solution of a system of linear equations. Carl Friedrich Gauss, a great 19th century mathematician suggested this elimination as a part of his proof of a particular theorem. Computational scientists use this proof as a direct computational. Gaussian elimination is a systematic application of elementary row operations to a system of linear equations in order to convert the system to upper triangular form. Once the coefficient matrix is in upper triangular form, we use back substitution to find a solution. Gaussian elimination places zeros below each pivot in the matrix starting with the top row and working downwards. Matrices containing zeros below each pivot are said to be in row echelon form. The process of Gaussian elimination has two parts. The first part (forward elimination) reduces a given system to either triangular or echelon form. The second step uses back substitution to find the solution of system of linear equation. The Gauss Jordan is a modification of the Gaussian elimination. It is named after Carl Friedrich Gauss and Wilhelm Jordan because it is a variation of Gaussian elimination as Jordan described in 1887 while Gaussian elimination places zeros below each pivot in the matrix starting with the top row and working downwards, Gauss Jordan elimination goes a step further by placing zeroes above and below each pivot. Every matrix has a reduced row echelon form and Gauss Jordan elimination is guaranteed to find it. II. RELATED WORKS T.J. Dekker and W. Hoffmann presented and analyzed the rehabilitation of the Gauss-Jordan algorithm. In this paper a Gauss-Jordan algorithm with column interchanges. They showed that, in contrast with Gaussian elimination, the Gauss-Jordan algorithm has essentially differing properties when using column interchanges instead of row interchanges for improving the numerical stability. For solutions obtained by Gauss-Jordan with column interchanges, a more satisfactory bound for the residual norm can be given. The analysis gives theoretical evidence that the algorithm yields numerical solutions as good as those obtained by Gaussian elimination and that, in most practical situations, the residuals are equally small. This is confirmed by numerical experiments. The results showed that the Gauss -Jordan algorithm is rather efficient on a vector computer and that the processing time is competitive with Gaussian elimination for order up to 25 [1]. Luke Smith discussed an alternative to Gauss - Jordan elimination by minimizing fraction arithmetic. He said that many teachers presented a Gauss-Jordan elimination approach to row reducing matrices that can involve painfully tedious operations with fractions (the traditional ) when solving systems of equations by using matrices. In this paper, he presented an alternative to row reduce matrices that does not introduce additional fractions until the very last steps. Here, the researcher spent more time focusing on designing an appropriate system of equations for a given problem instead of interpreting the results of their calculations. It is found that the alternative has few arithmetic mistakes as compared to the traditional. And this may increase the accessibility of matrix material with weaknesses in fractions. Finally, the has the potential to increase the speed and accuracy of computations alike by the substitution of integer computations for rational number computations [2]. Adenegan and his colleagues studied Gaus s and Gauss- Jordan elimination s for solving system of linear equation. Here, it was noted very remarkably that both Gauss elimination and Gaussian-Jordan elimination e-issn: p-issn: X Page 67

2 gave the same answers for each worked example and both with pivoting and partial pivoting equally gave the same answers. The study has concluded that when performing calculation by hand, Gauss-Jordan is more preferable to Gaussian elimination version because it avoids the need for back substitution [3]. R. B. Srivastava and Vinod Kumar examined the comparison of numerical efficiencies of Gaussian and Gauss-Jordan s for the solutions of linear simultaneous equations. In this paper, twenty linear simultaneous equations have been solved with the help of these two s using computer program developed in C++ programming language. They conclude that the efficiency of Gaussian elimination depends on the number of calculation involved in the solution of linear simultaneous equation. As the number of calculations increases; the efficiency of Gaussian elimination decreases and vice-versa [4]. K. RAJALAKSHMI, proposed the parallel algorithm for solving large system of simultaneous linear equations. In this paper, the systems of linear equations are solved by using Gauss and Gauss-Jordan sequential algorithms and parallel algorithms by calculation the execution times. Parallel algorithm has good speedups and less time complexity than sequential algorithm for both of the algorithms. Similarly, Gauss is better than Gauss Jordan [5]. According to all above researches, this paper tends to evaluate the performance comparison between Gauss and Gauss Jordan sequential algorithm for solving system of linear equations. In the future, it has a plan to develop these s in parallel. III. BACKGROUND THEORY This section describes the background theory of the Gauss elimination and Gauss-Jordan elimination s for solving system of linear equations. A system of linear equations (or linear system) is a collection of linear equation involving the same set of variables. A solution of a linear system is an assignment of values to the variable x 1, x 2, x 3,, x n equivalently, such that {x i } n i = 1, each of the equation is satisfied. The set of all possible solutions is called the solution set. A linear system solution may behave in any one of three possible ways: 1. The system has infinitely many solutions 2. The system has a single unique solution also known as an undetermined system. A system with the same number of equation and unknowns has a single unique solution. A system with more equations than unknowns has no solution. There are many s of solving linear systems. Among them, Gauss and Gauss Jordan elimination s shall be considered [2].So, they are discussed in details in the following sections. A. Gauss elimination Gaussian elimination is a systematic application of elementary row operations to a system of linear equations in order to convert the system to upper triangular form. Once the coefficient matrix is in upper triangular form, we use back substitution to find a solution. The general procedure for Gaussian elimination can be summarized in the following steps: 1. Write the augmented matrix for the system of linear equation. 2. Use elementary operation on {A/b} to transform A into upper triangular form. If a zero is located on the diagonal, switch the rows until a non zero is in that place. If you are unable to do so, stop; the system has either infinite or no solution. 3. Use back substitution to find the solution of the problem. Consider the n linear equations in n unknowns a 11 x 1 + a 12 x a 1n x n = a 1,n+1 a 21 x 1 + a 22 x a 2n x n = a 2,n+1 a 31 x 1 + a 32 x a 3n x n = a 3,n+1 a n1 x 1 + a n3 x a nn x n = a n,n+1 Where a ij and a i,j+1 are known constant and x i s are unknowns. The system is equivalent to AX = B Step 1: Store the coefficients in an augmented matrix. The superscript on a ij means that this is the first time that a number is stored in location (i, j). 3. The system has no solution For three variables, each linear equation determines a plane in three dimensional spaces and the solution set is the Step 2 : If necessary, switch rows so that a 11 0, then intersection of these planes. Thus, the solution set may be a eliminate x 1 in row2 through n. In this process m i1 is the line, a single point or the empty set. For variables, each linear multiple of row1 that is subtracted from row i. equation determines a hyper plane in n-dimensional space. for i = 2 to n The solution set is the intersection of these hyper-planes, mi1 = a i1 / a 11 which may be a flat of any dimension. The solution set for ai1 = 0 two equations in three variables is usually a line. for j = 2 to n+1 In general, the behavior of a linear system is determined by a ij = a ij m i1 * a 1j the relationship between the number of equations and the number of unknowns. Usually, a system with fewer equations than unknowns has infinitely many solutions such system is e-issn: p-issn: X Page 68

3 The new elements are written a ij to indicate that this is the second time that a number has been stored in the matrix at location ( i, j ). The result after step 2 is Step 3: If necessary, switch the second row with some row below it so that a 22 0, then eliminate x 2 in row 3 through n. In this process u i2 is the multiple of row 2 that is subtracted from row i. for i = 3 to n u i2 = a i2 / a 22 a i2 = 0 for j = 3 to n+1 a ij = a ij u i2 * a 2j The new elements are written a ij indicate that this is the third time that a number has been stored in the matrix at location ( i, j ). The after step 3 is Step k+1: This is the general step. If necessary, switch row k with some row beneath it so that a kk!= 0; then eliminate x k in rows k+1 through n. Here u ik is the multiple of row k that is subtracted from row i. for i = k +1 to n u ik = a ik / a kk a ik = 0 for j = k + 1 to n+1 a ij = a ij u ik * a kj The final result after x n-1 has been eliminated from row n is Perform the back substitution, get the values of x n, x n-1, x n- 2,, x 1. Sequential Algorithm Gauss Method Input : Given Matrix a[1 : n, 1: n+1] Output : x[1 : n] 1. for k = 1 to n-1 2. for i = k+1 to n 3. u = a ik /a kk 4. for j = k to n+1 5. a ij = a ij u * a kj 6. next j 7. next i 8. next k 9. x n = a n,n+1 /a nn 10. for i = n to 1 step sum = for j = i+1 to n 13. sum = sum + a ij * x j 14. next j 15. x i = ( a i,n+1 - sum )/a ii 16. next i 17. end B. Gauss Jordan elimination Gauss Jordan elimination is a modification of Gaussian elimination. Again we are transforming the coefficient matrix into another matrix that is much easier to solve and the system represented by the new augmented matrix has the same solution set as the original system of linear equations. In Gauss Jordan elimination, the goal is transform the coefficient matrix into a diagonal matrix and the zeros are introduced into the matrix one column at a time. We work to eliminate the elements both above and below the diagonal elements of a given column in one passes through the matrix. The general procedure for Gauss Jordan elimination can be summarized in the following steps: 1. Write the augmented matrix for a system of linear equation 2. Use elementary row operation on the augmented matrix [A/b] to the transform A into diagonal form (pivoting). If a zero is located on the diagonal, switch the rows until a non-zero is in that place. If you are unable to do so, stop; the system has either infinite or no solution. 3. By dividing the diagonal elements and a right-hand side s element in each row by the diagonal elements in the row, make each diagonal elements equal to one. Given a system of equation 4 x 4 matrix of the form: The upper triangularization process is now complete x n =a n,n+1 / a n for i = n to 1 step -1 sum = 0 for j = i+1 to n sum = sum + a ij * x j x i = ( a i,n+1 - sum )/a ii Step 1: Write the above as augmented matrix, we have e-issn: p-issn: X Page 69

4 Step 2: Pivoting the matrix, that is, Interchanging rows, if necessary, to obtain an augmented matrix in which the first entry in the first row is nonzero (e.g. Ri Rj). Add one row to another row, or multiply one row first and then adding it to another (e.g. krj + Ri Ri). Multiplying a row by any constant greater than zero (e.g. kri Ri). (NOTE: means replaces, and means interchange ) Continue until the final matrix is in row-reduced form, i.e. B. Implementation This system tends to calculate unknown variables in linear equations with the help of Gauss and Gauss Jordan algorithms. It is implemented by using Java programming language. When the system is started, the user will see the home window form as illustrated in figure 2. In this form, the user can choose the desire algorithm and number of unknown variables. On the other way, the user can type the desire unknown equations in the text boxes in this form. Sequential Algorithm Gauss Jordan Method Input : Given Matrix a[1 : n, 1 : n+1] Output : x[1 : n] 1. for i = 1 to n 2. for j = 1 to n+1 3. if ( i j ) then 4. u = a ik /a kk 5. for k = 1 to n+1 6. a jk = a jk u * a ik 7. end if 8. next k 9. next j 10. next i 11. for i = 1 to n 12. xi = a i,n+1 /a ii 13. end IV. DES IGN AND IMPLEMENTATION OF THE PROPOS ED S YSTEM Figure 2: Implementation of the Gauss and Gauss-Jordan Methods Under the Gauss calculation, the user must choose Gauss option and define the number of unknown variables. Figure 3 mention the sample of calculation result for five variables. And it can also display the calculation time. A. Design of the proposed system The design of the proposed system is illustrated in figure 1. In this system, the algorithm has first selected to solve the problem. Then send the input problem or the system of linear equations for solving by each algorithm. Here the problem is calculated both of Gauss Jordan and Gauss algorithm. It is noted that these two algorithms give the same results for the same problem. But they give the different execution time for the solution. So, the system will give the comparison of execution time for the problem solving these algorithms. Figure3: The result for five unknown variables using Gauss Method Under the Gauss Jordan calculation, the user must choose Gauss Jordan option and define the number of unknown variables. Figure 4 mention the sample of calculation result for five variables. And it can also display the calculation time. Figure 1: The Proposed System Design e-issn: p-issn: X Page 70

5 Milliseconds Yadanar Mon, et al International Journal of Computer and Communication Engineering Research [Volume 2, Issue 2 March 2014] Comparison of Execution time Number of Variables Gauss Jordan Gauss Figure 4: The results for five variables using Gauss and Gauss-Jordan Methods Finally, in figure 5, the system shows the comparison of execution time or calculation time of these two algorithms. Figure 6: The Comparison of Execution times IV. CONCLUS IONS Many scientific and engineering problems can take the form of a system of linear equations. These equations may contain thousands of variables, so it is important to solve them as efficiently as possible. In this paper, the unknown variables in linear system are carried out by using Gauss Method and Gauss-Jordan Method. And the experiment is demonstrated with the help of java programming language. The results are carried out by solving 2, 3, 4, 5, 6 and 7 unknown variables. According to these experimental results, Gauss Method is faster than the Gauss Jordan. Therefore Gauss Method is more efficient than the Gauss Jordan. Figure 5: The chart for the comparison of two algorithms The comparison of execution time between the Gauss and Gauss Jordan is mentioned in Table 1 and figure 5.These results are carried out on the calculation of unknown variables 2, 3, 4, 5, 6 and 7. TABLE I: The Comparison of Execution Time between Gauss and Gauss-Jordan Method Numbers of Variables Execution Time for Gauss (Milliseconds) Execution Time for Gauss-Jordan (Milliseconds) According to these results, the Gauss is faster than Gauss Jordan. ACKNOWLEDGMENT First of all, the author is highly grateful to Dr. Myint Thein, the Pro-Rector of the Mandalay Technological University for his permission for completion of this paper. The author wants to express her gratitude to Dr. Lai Lai Win Kyi for her help and advice regarding of this topic and excellent guidance, valuable suggestions and advices. The author is deeply thankful to all teachers fro m Department of Information Technology at Mandalay Technological University for their overall supporting during the writing of this paper. REFERENCES [1] T.J. Dekker and W. Hoffmann: Rehabilitation of the Gauss- Jordan algorithm, Department of Computer Systems, University of Amsterdam, Kruislann 409, 1098 SJ Amsterdam, The Netherlands, 1989 [2] Luke Smith and Joan Powell: An Alternative Method to Gauss-Jordan : Minimizing Fraction Arithmetic, The Mathematics Educator, 2011 [3] Adenegan, Kehinde Emmanuel1, Aluko, Tope Moses: Gauss and Gauss-Jordan elimination s for solving system of linear equations: comparisons and applications, Journal of Science and Science Education, Ondo, 19th November, 2012 [4] R. B. Srivastava and Vinod Kumar, Comparison of Numerical Efficiencies of Gaussian and Gauss-Jordan s for the Solutions of linear Simultaneous Equations, Department of Mathematics M. L. K. P. G. College, Balrampur, U. P., India, November 2012 [5] K. RAJALAKSHMI, Parallel Algorithm for Solving Large System of Simultaneous Linear Equations, IJCSNS International Journal of Computer Science and Network Security, VOL.9 No.7, July e-issn: p-issn: X Page 71