New Sequential and Parallel Division Free Methods for Determinant of Matrices

A determinant plays an important role in many applications of linear algebra. Finding determinants using non division free methods will encounter problems if entries of matrices are represented in rational or polynomial expressions, and also when floating point errors arise. To overcome this problem, division free methods are used instead. The two commonly used division free methods for finding determinant are cross multiplication and cofactor expansion. However, cross multiplication which uses the Sarrus Rule only works for matrices of order less or equal to three, whereas cofactor expansion requires lengthy and tedious computation when dealing with large matrices. This research, therefore, attempts to develop new sequential and parallel methods for finding determinants of matrices. The research also aims to generalise the Sarrus Rule for any order of square matrices based on permutations which are derived using starter sets. Two strategies were introduced to generate distinct starter sets namely the circular and the exchanging of two elements operations. Some theoretical works and mathematical properties for generating permutation and determining determinants were also constructed to support the research. Numerical results indicated that the new proposed methods performed better than the existing methods in term of computation times. The computation times in the newly developed sequential methods were dominated by generating starter sets. Therefore, two parallel strategies were developed to parallelise this algorithm so as to reduce the computation times. Numerical results showed that the parallel methods were able to compute determinants faster than the sequential counterparts, particularly when the tasks were equally allocated. In conclusion, the newly developed methods can be used as viable alternatives for finding determinants of matrices

The division free parallel algorithm for finding determinant

A cross multiplication method for determinant was generalized for any size of square matrices using a new permutation strategy.The permutation is generated based on starter sets.However, via permutation, the time execution of sequential algorithm became longer.Thus, in order to reduce the computation time, a parallel strategy was developed which is suited for master and slave paradigm of the high performance computer.A parallel algorithm is integrated with message passing interface.The numerical results showed that the parallel methods computed the determinants faster than the sequential counterparts particularly when the tasks were equally allocated

SEALiP: A simple and efficient algorithm for listing permutation via starter set method

Algorithm for listing permutations for n elements is an arduous task.This paper attempts to introduce a novel method for generating permutations.The fundamental concept for this method is to seek a starter set to begin with as an initial set to generate all distinct permutations. In order to demonstrate the algorithm, we are keen to list the permutations with the special references for cases of three and four objects.Based on this algorithm, a new method for listing permutations is developed and analyzed.This new permutation method will be compared with the existing lexicographic method.The results revealed that this new method is more efficient in terms of computation time

Persembahan kumpulan-2 yang berperingkat 32

Research on the presentation of 2-groups of order 2 n, (n ≤ 6) was founded by Hall et al.[4] and was continued by Sag et al [7].This paper focuses on the presentation of 2-groups of order 25 where there are 51 groups comprising of 7 Abelian groups and 44 non-Abelian groups.We are more concerned in finding the structures for each of the every 44 non-Abelian groups to show that these groups are not isomorphic to each other using GAP (Groups, Algorithms, and Programming) software

New recursive circular algorithm for listing all permutations

Linear array of permutations is hard to be factorised. However, by using a starter set, the process of listing the permutations becomes easy. Once the starter sets are obtained, the circular and reverse of circular operations are easily employed to produce distinct permutations from each starter set. However, a problem arises when the equivalence starter sets generate similar permutations and, therefore, willneed to be discarded. In this paper, a new recursive strategy is proposed to generate starter sets that will not incur equivalence by circular operation. Computational advantages are presented that compare the results obtained by the new algorithm with those obtained using two other existing methods. The result indicates that the new algorithm is faster than the other two in time execution

On the existence of a cyclic near-Rrsolvable (6n+4)-cycle system of 2K12n+9

In this article, we prove the existence of a simple cyclic near-resolvable - cycle system of for by the method of constructing its starter. Then, some new properties and results related to this construction are formulated

A new method for starter sets generation by fixing any element

a new permutation technique based on distinct starter sets was introduced by employing circular and reversing operations.The crucial task is to generate the distinct starter sets by eliminating the equivalence starter sets.Meanwhile new strategies for starter sets generation without generating its equivalence starter sets were developed and more efficient in terms of computation time compared to old method.However all these algorithms have limitations in terms of fixing element to construct the first set (starter set) to begin with.It would be interesting to derive new strategy by fixing an element in any position. A new method is developed for starter sets generation namely STARSET1 based on circular where any element can be selected randomly to be fixed. The result showed that no redundancy of starter sets is occurring and no equivalence starter sets are obtained

Integrated strategy for generating permutation

An integrated strategy for generating permutation is presented in this paper. This strategy involves exchanging two consecutive elements to generate the starter sets and then applying circular and reversing operations to list all permutations. Some theoretical works are also presented

Some modifications of Sarrus’s rule method via permutation for finding determinant of 4 by 4 square matrix

Sarrus rule is well known method for finding determinant of square matrix.This method is also known as a cross multiplication method.However this method is not applicable for n > 3.With this motivation, we attempt to extend this method by employing some modifications using permutation for the case of 4 by 4 square matri

The construction of distinct circuits of length six for complete graph K6

The emergent applications of complete graph in diverse domains have invited numerous works in this subject matter.Studies related to the decomposition of complete graph such as one-factor, several ɳ-gons and Cartesian product have been solved.Yet, the decomposition of complete graph into distinct circuits still has not been done.Thus, this paper aims to investigate the structure of complete graph, and in particular, the decomposing of complete graph of length six.The decomposition algorithm will be presented to enumerate distinct circuits of length six. Along this process, the adjacency matrices will be used to clarify distinct structures of circuits in a complete graph of length six