
Moawwad ElmikkawyMansoura University · Department of Mathematics
Moawwad Elmikkawy
Full Professor
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71
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Introduction
El-Mikkawy holds the following qualifications: B.Sc., 1974 and M.Sc., 1979 degrees in Mathematics from Mansoura Univ., EGYPT; Ph.D. degree in Mathematics & Computer Science, 1986 from Teesside Univ., U.K . After years of specializing in research fields of Numerical Analysis & Scientific Computing and Computational Number Theory, he published more than 60 papers. Some of his papers are selected as hottest articles, and some others are used as bases of software codes like RKN1210 code in Matlab.
Additional affiliations
June 1974 - January 2015
Publications
Publications (71)
This article aims to present novel identities for elementary and complete symmetric polynomials and explore their applications, particularly to generalized Vandermonde and special tri-diagonal matrices. It also extends existing results on Jacobi polynomials $ P_n^{(\alpha, \beta)}(x) $ and introduces an explicit formula based on the zeros of $ P_{n...
This research addressed the need to enhance template-matching performance in e-learning and automated assessments within Egypt’s evolving educational landscape, marked by the importance of e-learning during the COVID-19 pandemic. Despite the widespread adoption of e-learning, robust template-matching feedback mechanisms should still be developed fo...
The main research object of this paper is to present a systematic computational procedure for computing the inverse of a generalized Vandermonde matrix. Short and rigorous proofs for the formulas of the determinant and the inverse of a generalized Vandermonde matrix are proposed. The computational cost of this method is O(n2). The proposed method c...
In the current paper the authors linked two methods in order to evaluate general n-th order tridiagonal determinants. A breakdown free numerical algorithm is developed for computing the inverse of any nxn general nonsingular tridiagonal matrix without imposing any constrains. The algorithm is suited for implementation using any computer language su...
The principal minors of a tridiagonal matrix satisfy two-term and three-term recurrences [1, 2]. Based on these facts, the current article presents a new efficient and reliable hybrid numerical algorithm for evaluating general n-th order tridiagonal determinants in linear time. The hybrid numerical algorithm avoid all symbolic computations. The alg...
The main objective of the current article is to present a fast and reliable
algorithm for evaluating n-th order k-tridiagonal determinants with Toeplitz structure. Additionally, a modified algorithm for evaluating the general n-th order k-tridiagonal determinants is proposed. Numerical
tests and illustrative examples are also given.
Abstract
In the current paper, the authors present a symbolic algorithm for solving doubly bordered k-tridiagonal
linear system having n equations and n unknowns. The proposed algorithm is derived by using partition together with UL factorization. The cost of the algorithm is O(n). The algorithm is implemented using the computer algebra system, MAP...
The main objective of the current article is to present a fast and reliable algorithm for evaluating n-th order k-tridiagonal determinants with Toeplitz structure. Additionally, a modified algorithm for evaluating the general n-th order k-tridiagonal determinants is proposed. Numerical tests and illustrative examples are also given.
A New Symbolic Algorithm for Solving General Opposite-Bordered Tridiagonal
Linear Systems
In the current article we propose a new efficient, reliable and breakdown-free algorithm for solving
general opposite-bordered tridiagonal linear systems. An explicit formula for computing the
determinant of an opposite-bordered tridiagonal matrix is investigated. Some illustrative examples
are given.
The present article is mainly devoted for solving bordered k-tridiagonal linear systems of equations. Two efficient and reliable symbolic algorithms for solving such systems are constructed. The computational cost of the algorithms is obtained. Some illustrative examples are given.
This paper presents new numeric and symbolic algorithms for solving doubly bordered tridiagonal
linear system. The proposed algorithms are derived using partition together with UL factorization.
Inversion algorithm for doubly bordered tridiagonal matrix is also considered based on the
Sherman-Morrison-Woodbury formula. The algorithms are implemente...
In the present article we give a new breakdown-free recursive algorithm for inverting general k-tridiagonal matrices without imposing any simplifying assumptions. The implementation of the algorithm in Computer Algebra Systems (CAS) such as Maple, Mathematica and Macsyma is straightforward. Two illustrative examples are given.
Abstract. In this paper we present some efficient computational methods to compute the
power sum. Some new identities are given.
In the current paper we focus on the study of three special matrices and two symmetric polynomials. As a consequence, a recurrence relation satisfied by the entries of the n × n inverse matrix, Q n of the n × n symmetric Pascal matrix, P n is obtained. Moreover, a new proof for El-Mikkawy conjecture [14] is investigated. Finally, some identities ar...
In this paper we present a novel algorithm, that will never fail, for inverting a general nonsingular k-tridiagonal matrix. The computational cost of the algorithm is given. Some illustrative examples are introduced.
The current article focus on the ordinary Bernoulli, Euler and Genocchi numbers and polynomials. It introduces a new approach to obtain identities involving these special polynomials and numbers via generating functions. As an application of the new approach, an easy proof for the main result in [6] is given. Relationships between the Genocchi and...
By means of complete symmetric polynomials this paper gives a new proof for the Vandermonde determinant formula. Another alternative proof for this formula is obtained via the collocation matrices. It also gives a generalized relationship between the Vandermonde, the Pascal and the Stirling matrices. A new approach to obtain the explicit inverse of...
In this paper, we consider an inverse problem with the k-tridiagonal Toeplitz matrices. A theoretical result is obtained that under certain assumptions the explicit inverse of a k-tridiagonal Toeplitz matrix can be derived immediately. Two numerical examples are given to demonstrate the validity of our results.
The current paper is mainly devoted for solving centrosymmetric linear systems of equations. Formulae for the determinants of tridiagonal centrosymmetric matrices are obtained explicitly. Two efficient computational algorithms are established for solving general centrosymmetric linear systems. Based on these algorithms, a MAPLE procedure is written...
We introduce k-tridiagonal Toeplitz matrices and give algorithms for determinants and permanents of these matrices. We give illustrative examples for these algorithms. In the last section, we give examples of matrices whose determinants are powers of generalized Fibonacci numbers. These algorithms can be implemented in all computer algebra systems.
Minimizing service delivery and travel time during rush hours downtown is strategic target for several organizations, especially the emergency organizations. This paper presents an On-line and Real-time Dynamic Route System (ORDRS) which benefits from the advantages and integration between information system and communications technology. It utiliz...
In this note, we give comments on a very recent paper by J. S. Respondek [1]. In [1], the author claims that an algorithm in [2] contains a severe error. We show that the algorithm in [2] can be implemented properly without causing any errors by using vectors (one-dimensional arrays) rather than using 2-dimensional arrays. To enable users and progr...
The current paper is mainly devoted to construct a generalized symbolic Thomas algorithm that will never fail. Two new efficient and reliable computational algorithms are given. The algorithms are suited for implementation using computer algebra systems (CAS) such as Mathematica, Macsyma and Maple. Some illustrative examples are given.
In the present paper, we give a fast algorithm for block diagonalization of k-tridiagonal matrices. The block diagonalization provides us with some useful results: e.g., another derivation of a very recent result on generalized k-Fibonacci numbers in [M.E.A. El-Mikkawy, T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010)...
A database intrusion detection system (DIDS) is used to detect potential violations in database security. DIDS follows other traditional database security mechanisms and network security mechanisms such as firewall and network intrusion detection. Therefore, it faces the intrusion from internal users or the intrusion that can be passed through othe...
In this paper, by using parallel computing along with recursion, we describe a reliable symbolic computational algorithm for inverting cyclic pentadiagonal matrices. The algorithm is implemented in MAPLE. Two other symbolic algorithms are developed and the computational costs for all algorithms are given. An example is presented for the sake of ill...
In the present paper, we give a new family of k-Fibonacci numbers and establish some properties of the relation to the ordinary Fibonacci numbers. Furthermore, we describe the recurrence relations and the generating functions of the new family for k=2 and k=3, and presents a few identity formulas for the family and the ordinary Fibonacci numbers.
This paper presents some applications using several properties of three important symmetric polynomials: elementary symmetric polynomials, complete symmetric polynomials and the power sum symmetric polynomials. The applications includes a simple proof of El-Mikkawy conjecture in [M.E.A. El-Mikkawy, Appl. Math. Comput. 146 (2003) 759–769] and a very...
This short note describes new properties of the elementary symmetric polynomials, and reveals that the properties give an answer to the conjecture raised by El-Mikkawy in [M.E.A. El-Mikkawy, On a connection between the Pascal, Vandermonde and Stirling matrices—II, Appl. Math. Comput. 146 (2003) 759–769].
In the current article, the authors present a new recursive symbolic computational algorithm, that will never break down, for inverting general periodic pentadiagonal and anti-pentadiagonal matrices. It is a natural generalization of the work presented in [M.E.A. El-Mikkawy, E.D. Rahmo, A new recursive algorithm for inverting general tridiagonal an...
In the current article, the authors present a new recursive symbolic computational algorithm for inverting general tridiagonal and anti-tridiagonal matrices. An illustrative example is given.
In the current article we present a fast and reliable algorithm for evaluating nth order pentadiagonal determinants in linear time. It is a natural generalization of the DETGTRI algorithm [M. El-Mikkawy, A fast algorithm for evaluating nth order tri-diagonal determinants, J. Comput. Appl. Math. 166 (2004) 581-584]. The algorithm is suited for imple...
In this short note, we present a fast and reliable algorithm for evaluating special nth-order pentadiagonal Toeplitz determinants in linear time. The algorithm is suited for implementation using Computer Algebra Systems (CAS) such as MACSYMA and MAPLE. Two illustrative examples are given.
By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we obtain very simple connections between the Stirling numbers of both kinds and other generalized harmonic numbers. Further, we suggest that Riordan arrays associated with such generalized harmonic numbers allow us to find new generatin...
In this paper, we obtain important combinatorial identities of generalized harmonic numbers using symmetric polynomials. We also obtain the matrix representation for the generalized harmonic numbers whose inverse matrix can be computed recursively.
In the current work, the authors present a symbolic algorithm for finding the inverse of any general nonsingular tridiagonal matrix. The algorithm is mainly based on the work presented in [Y. Huang, W.F. McColl, Analytic inversion of general tridiagonal matrices, J. Phys. A 30 (1997) 7919–7933] and [M.E.A. El-Mikkawy, A fast algorithm for evaluatin...
In the current article we present a modified version of the DETGTRI algorithm in (1). Based on the modified algorithm , an efficient and reliable Maple procedure is written to compute any nth order tri-diagonal determinant. As an application, we show that all orthogonal polynomials can be computed efficiently using this procedure.
In this paper we establish some new identities involving Stirling numbers of both kinds. These identities are obtained via Riodan arrays with a variable x. Some well-known identities are obtained as special cases of the new identities for the specific number x.
In this paper old and new combinatorial and hypergeometric identities are obtained via the Legendre polynomials as one of the most commonly used orthogonal polynomials. The results are tested using MAPLE Computer Algebra System (CAS).
In this article we present how to reduce the evaluation of the special infinite series Sn=∑∞k=1kn(k+1)!, n=1,2,3,… which are very time consuming in Computer Algebra Systems (CAS). An algorithm is given for this purpose. Based on the new algorithm, a fast and reliable MAPLE procedure is designed and S1,S2,…,S50 are given as sample output of this pro...
In this article we present a new efficient computational algorithm for solving periodic tri-diagonal linear systems. The implementation of the algorithm using Computer Algebra Systems (CAS) such as MAPLE, MACSYMA, MATHEMATICA and MATLAB is straightforward. An example is given in order to illustrate the algorithm.
In this paper, we give a matrix representation of the hypergeometric functions of the type F-2(1) (a, b; c; x). As a result, we obtain a connection between the hypergeometric functions, the Legendre polynomials and the Delannoy numbers. Moreover, it is shown that each entry of P-n(x, y)P-n(x, y)(T) can be represented by the hypergeometric functions...
The cost of all existing algorithms for evaluating the nth order determinants [cf. R. L. Burden and J. D. Faires, Numerical analysis, 7th Edition, Brooks & Cole Publishing, Pacific Grove, CA (2001)] is at most O(n 3 ). In the current article we present a new efficient computational algorithm for evaluating the nth order tri-diagonal determinants wi...
In the current paper a new efficient computational algorithm to find the inverse of a general tridiagonal matrix is presented. The algorithm is suited for implementation using computer algebra systems such as MAPLE, MATHEMATICA, MATLAB and MACSYMA. Symbolic and numeric examples are given.
In this article a new algorithm for generating the elements of the n+1 by n+1 Stirling matrix of the second kind is presented. Based on this algorithm, a computational MAPLE procedure is written and a sample output of this procedure is also given for some values of n.
In this paper the author gives an explicit closed form expression for the n×n inverse matrix (VG(k)(n))−1 of the generalized n×n Vandermonde matrix VG(k)(n) by using the elementary symmetric functions. Symbolic and numerical results are presented.
This article is an extension of the work presented in [On a connection between the Pascal, Vandermonde and Stirling matrices-I, Applied Mathematics and Computation, 2003, to appear]. It gives a full matrix T(t) for which T(1) is exactly the lower triangular stochastic matrix T given in [On a connection between the Pascal, Vandermonde and Stirling m...
In the current paper a new 4-parameter Runge–Kutta–Nystrom (RKN) family of the non-FSAL type is constructed. The strategy used for the construction is based on the criteria listed by Dormand et al. [IMA J. Numer. Anal. 7 (1987) 235]. By an appropriate choice of the free parameters we obtained an optimized non-FSAL RKN 6(4) 6 embedded algorithm. The...
The n x n generalized Pascal matrix P(t) whose elements are related to the hypergeometric function F-2(1) (a,b; c; x) is presented and the Cholesky decomposition of P(t) is obtained. As a result, it is shown that F-2(1) (-a, -b; 1; x) = Sigma(k=0)(min(a,b)) ((a)(k))((b)(k))x(k) is the solution of the Gauss's hypergeometric differential equation, x(...
The well-known Dormand–Prince embedded RK 5(4) 7FM algorithm [J. Comput. Appl. Math. 6 (1980) 19] is of the FSAL type and uses seven stages per step. This algorithm has been recommended by Shampine [Math. Comput. 46 (1986) 135] as a candidate for an efficient production RK code. In fact the new MATLAB function ode45 is based on this algorithm. Late...
In this article the author gives a new computational approach for constructing Vandermonde interpolating polynomials by using special associated matrices. An illustrative example is given.
In this article the author shows that under certain conditions a three-term recurrence for a tridiagonal matrix becomes a two-term recurrence. Using this new recurrence, the possibility of the LU factorization of any tridiagonal matrix is now easy to investigate. The positive definiteness of any real symmetric tridiagonal matrix is now easy to chec...
In this paper the author gives an explicit closed form expression for the $n\times n$ inverse matrix $(V_{G}^{(k)})^{-1}(n)$ of the $n\times n$ Vandermonde matrix $V_{G}^{(k)}(n)$ by using the elementary symmetric functions. Symbolic and numerical results are presented.
This paper gives a connection between symmetric polynomials, generalized Stirling numbers and the Newton general divided difference interpolation polynomial. The generalized Stirling numbers of the first and second kind denoted sn,k[a] and Sn,k[a] respectively are given symbolically for the case n=5, as an illustrative example, by using MAPLE progr...
In this paper the author introduces a new approach to obtain the Cotes numbers for both the open and closed Newton–Cotes quadrature formulae. He also introduces a new definition for the degree of exactness for a quadrature formula. Some numerical results are presented.
In the current paper we study the Pascal matrix of order n. An algorithm is developed to find its inverse in an explicit form. Another algorithm for solving any linear system with coefficient matrix of this type is also developed. An illustrative example for solving a linear system of the Pascal type is given for the case n=6. The implementation of...
In this article the author shows that under certain conditions a three-term recurrence for a tri-diagonal matrix becomes a two-term recurrence. Using this new recurrence, the possibility of the LU factorization of any tri-diagonal matrix is now easy to investigate. The positive definiteness of any real symmetric tri-diagonal matrix of any positive...
In the current paper, the truncation error (TE) for both the closed and open Newton-Cotes quadrature formulae for numerical integration is investigated and calculated for n = 1, 2, @, 12 in both cases.
The author [J. Inst. Math. Comput. Sci. 3, No. 3, 293-297 (1990; (*) ibid. 4, No. 2, 205-210 (1991)] developed two algorithms for solving special linear systems of Vandermonde and tridiagonal type, respectively. In the current paper a third algorithm for solving pentadiagonal systems is given. The algorithm is useful for being used in digital compu...
A better approach than the approach in [1] for the derivation of the Runge-Kutta-Nystrom (RKN) embedded pair of orders 6 and 4 is given. Using the new approach a family of RKN 6(4) pairs possessing the same number of free parameters as in [1] is shown to be available. Some remarks are also given.
This paper is mainly a part of the author’s Ph. D. thesis [Embedded Runge-Kutta-Nyström methods, Dept. Computer Sci., Tesside Polytechnic, Middlesbrough, Cleveland U.K. (1986)] with some modifications. It describes an algorithm for solving linear systems with Vandermonde coefficient matrix. A FORTRAN 77 program based on this algorithm is also given...
Two typographical errors were contained in Table 1 on page 425. Use of the values for a73 and b7′ will invalidate the RKN8(6)9FM formula pair. The correct values for these parameters are
The authors are grateful to Steve Stalos of Laurel, MD, USA, for reporting these errors.
Criteria to be satisfied by efficient embedded Runge-Kutta-Nystrom formulae are presented, and new families are derived. Test
results indicate their improved efficiency relative to other RKN formulae in current use.