# Gaussian elimination method and gauss jordan method computer science essay

He has had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians. Gauss was a child prodigy.

Carre To cite this article: Carre A comparison of gaussian and gauss- jordan elimination in regular algebra, International Journal of Computer Mathematics, Submit your article to this journal Article views: Received February A comparison is presented in regular algebra of the Gaussian and Gauss-Jordon elimination techniques for solving sparse systems of simultaneous equations.

This result generalises an earlier one due to Brayton, Gustavson and Willoughby in which it is shown that the product form of the inverse A-' of a matrix A is never more sparse than the elimination form of the inverse.

Our result applies both in linear algebra and, more generally, to path-finding problems. Gaussian elimination, Gauss-Jordan elimination, regular algebra, linear algebra, path-finding, sparsity.

Recently it has been observed that many path-finding 31 1 algorithms arc in fact variants of such elimination techniques Carre, ; Backhouse and Carre, [3].

Examplcs are Floyd's shortest path algorithm Floyd, [4] and Warshall's transitive closure algorithm Warshall, [5]both of which arc variants of Gauss-Jordan elimination. Most recently, interest in sparse matrix techniques has been aroused in a wide cross- section of computer scientists since the rcalisation that "new" algorithms developed to solve global data flow analysis problems can be regarded as applications of Gaussian elimination Tarjan, [6]or the equally well- known Gauss-Seidel iterative technique Tarjan, [7].

The rramework for this unification is thc algebra of regular languages. A concrete example is finding shortest paths through a graph. Here by interpreting u - t b as the rniniinum of a and h, a. Note, however, that the numerical analysts' formulation of Gauss-Jordan elimination cannot be applied because thcre is no meaningful interpretation of u '.

## Solution Preview:

In this paper wc employ regular algebra to give a novel presentation of Brayton et d ' s comparison of Gaussian and Gauss-Jordan elimination Brayton, Gustavson and Willoughby, Our comparison adds insight to their result as well as being relevant to many path-finding problems.

The paper contains five sections. Section 2 reviews the properties of a regular algebra we require and Section 3 summarises the two algorithms.

The fclrmal comparison is presented in Section 4 whilst Section 5 discusses the meaning and implications of the comparison. The following properties will be used without mention in the sequel. From AlO, we have Note that addition is assumed to be idempotent property A6.

In the algebra R. The null matrix, denoted cD, is that matrix all of whose elements are 4. The ith row and jth column of a matrix A will be denoted by a, and aoj,respectively. Certain elementary matrices, which differ from the null matrix in only one column or one row, are the primary tool in both algorithms.

Specifically, and To justify the algorithms for obtaining the PFS and EFS, and to allow us to compare these forms, we shall in each case first give a concise derivation of it, originally presented in Backhouse and Carre [3]. Our notation here follows closely that in Carrt [9], where the "path algebra" used is an example of a regular algebra.

We express A'k-" in the partitioned form where the diagonal submatrices Alk;", Ark,-'' and AT;' are square, of order k - 1, 1 and n - k respectively. Note that in A''', the first row and R. The fact that the first k columns of A k are null suggests a simple and compact method of forming and storing the Q k'-factors: It is also demonstrated there that for triangular matrices it is particularly easy to obtain a PFS: Furthermore, it is easily proved by induction on k that in each matrix A'k', all entries on and below the principal diagonal, in the first k columns, are null.

Thus A'"' is strictly upper triangular, and if we denote this matrix by U, Eq. Construction of the matrix M " ,by repeated use of 8again conveniently gives the factors of the EFS, this time in the form shown in Figure 2. To make elimination methods feasible in such circumstances, it is important to exploit sparsity, by storing and manipulating only non-null matrix elements at each stage of the computation.

The effectiveness of this technique depends on the extent to which sparsity is "preserved" in constructing the EFS or PFS, and therefore we are interested in the relative sparsity of these two forms of the star.

In numerical linear algebra, it is well-known that the elimination form of the inverse E F I t w h i c h corresponds to our EFS-has no more non-null entries, and often considerably less, than the product form of the inverse P F I t w h i c h corresponds to our PFS.

This was rigorously established by Brayton et al. Our purpose in this section is to present a relationship between the EFS and PFS, analogous to that which exists between the EFI and PFI; this algebraic relationship enables us immediately to compare the sparsities of the two forms of the star.

To distinguish between the M'k -matrices produced in the Gauss and Gauss-Jordan methods, we shall henceforth denote these by MG k'and MJ'k' respectively; similar notations will be used for the Q k and R k matrices.

Figure 3 summarises the argument which follows.Jul 18,  · In this video explaining example and by using Gauss elimination method. In this method find the x, y and z values. Please watch: "Simple . topic: gaussian elimination method and gauss-jordan method acknowledement:i would like to be thankful to my teacher who gave me such a nice topic by which i able to learn these things and ofcourse tell you about these things.

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## Solutions:

The Solution of Large System of Linear Equations by using several Methods and its applications. Gaussian elimination method, Gauss-Jordan method, Iterative improvement method,, applications 24T. 1. Introduction as would be the case for computer-generated solutions, a pivot element that is small compared to the entries.

Yes, you can apply row operations to convert every $m\times n$ matrix to reduced echelon form. Furthermore, there is one and only one reduced echelon form for each matrix.

The operations predate Gauss and Jordan whose names are associated to them. I would like to have a macro for gaussian elimination, but I don't like the notation of the gauss package.

I've found a great macro here, but I would like to have several steps in one urbanagricultureinitiative.com I just remove \\ it doesn't look good, the spaces between matrices and row operations are not equal and the line doesn't wrap: Also, I would like to decrease spaces between matrices and row operations.

Faculty of Mathematics and Computer Science University of Amsterdam Implementation of Gauss-Huard's the implementation of a variant of Gaussian elimination, Gauss-Huard elimination, on the IBM SP1 is discussed. As programming environment, PVMe is is as stable as Gauss-Jordan elimination [Dekker93].

Gaussian elimination - Wikipedia