# Gennadi MalaschonokNational University of Kyiv-Mohyla Academy · Department of Computer Science

Gennadi Malaschonok

PhD

## About

39

Publications

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304

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## Publications

Publications (39)

In this paper, we describe general characteristics of the MathPartner computer algebra system (CAS) and Mathpar programming language thereof. MathPartner can be used for scientific and engineering calculations, as well as in high schools and universities. It allows one to carry out both simple calculations (acting as a scientific calculator) and co...

We introduce new features in the MathPartner service that have recently become available to users. We highlight the functions for calculating both arithmetic-geometric mean and geometric-harmonic mean. They allow calculating complete elliptic integrals of the first kind. They are useful for solving many physics problems, for example, one can calcul...

LU-factorization of matrices is one of the fundamental algorithms of linear algebra. The widespread use of supercomputers with distributed memory requires a review of traditional algorithms, which were based on the common memory of a computer. Matrix block recursive algorithms are a class of algorithms that provide coarse-grained parallelization. T...

We give an overview of the theoretical results for matrix block-recursive algorithms in commutative domains and present the results of experiments that we conducted with new parallel programs based on these algorithms on a supercomputer MVS-10P at the Joint Supercomputer Center of the Russian Academy of Science. To demonstrate a scalability of thes...

A modified Gauss's algorithm for solving a system of linear equations in an integral ring is proposed, as well as an appropriate algorithm for calculating the elements of the adjoint matrix.

Solution methods for linear equation systems in a commutative ring are discussed. Four methods are compared, in the setting of several different rings: Dodgson's method [1], Bareiss's method [2] and two methods of the author - method by forward and back-up procedures [3] and a one-pass method [4]. We show that for the number of coefficient operatio...

Two known computation methods and one new computation method for matrix determinant over an integral domain are discussed. For each of the methods we evaluate the computation times for different rings and show that the new method is the best.

New solution method for the systems of linear equations in commutative integral domains is proposed. Its complexity is the same that the complexity of the matrix multiplication.

In this paper, we describe general characteristics of the MathPartner computer algebra system (CAS) and Mathpar programming language thereof. MathPartner can be used for scientific and engineering calculations, as well as in high schools and universities. It allows one to carry out both simple calculations (acting as a scientific calculator) and co...

Deterministic recursive algorithms for the computation of matrix triangular decompositions with permutations like LU and Bruhat decomposition are presented for the case of commutative domains. This decomposition can be considered as a generalization of LU and Bruhat decompositions because they both may easily be obtained from this triangular decomp...

Deterministic recursive algorithms for the computation of generalized Bruhat decomposition of the matrix in commutative domain are presented. This method has the same complexity as the algorithm of matrix multiplication.

The deterministic recursive pivot-free algorithms for computing the generalized Bruhat decomposition of the matrix in the
field and for the computation of the inverse matrix are presented. This method has the same complexity as algorithm of matrix
multiplication, and it is suitable for the parallel computer systems.

We investigate multiplication algorithms for dense and sparse polynomials and polynomial matrices over different numerical
domains and obtain expressions for the complexity of multiplication of polynomials and polynomial matrices understood as the
expectation of the number of arithmetic operations. These expressions for a set of parameters of pract...

This paper is a review of results on computational methods of linear algebra over commutative domains. Methods for the following problems are examined: solution of systems of linear equations, computation of determinants, computation of adjoint and inverse matrices, computation of the characteristic polynomial of a matrix.

The best method for computing the adjoint matrix of an order n matrix in an arbitrary commutative ring requires O(n
β + 1/3 log n log log n) operations, provided that the complexity of the algorithm for multiplying two matrices is γn
β
+ o(n
β
). For a commutative domain – and under the same assumptions – the complexity of the best method is 6γn
β...

Given an m×n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see [1]).
Fig. 1. The row spaces and the nullspaces of A and AT ; a1 through an and h1 through hm are abbreviations of the alignerframe and hangerframe vectors respectively (see [2]).
The Fundamental Theorem of Linear Algebra tells us that N(A) is the orth...

Let A be an m×n matrix with m≥n. Then one form of the singular-value decomposition of A is A=UTΣV,where U and V are orthogonal and Σ is square diagonal. That is, UUT=Irank(A), VVT=Irank(A), U is rank(A)×m, V is rank(A)×n and is a rank(A)×rank(A) diagonal matrix. In addition σ1≥σ2≥⋯≥σrank(A)>0. The σi’s are called the singular values of A and their...

In this work, we study the efficiency of the Karat- suba (1) and Strassen (2) algorithms for the multiplica- tion of sparse polynomials and matrices. Specifically, we study how the efficiency of the algorithms depends on the density coefficient ρ , which is the ratio of the number of nonzero coefficients to the total number of matrix elements or co...

A new algorithm for solving systems of linear equations Ax = b in an Euclidean domain is suggested. In the case of the ring \(\mathbb{Z}\) of integers, the complexity of this algorithm is O
∼(n
3mlog2||A||), where \(A \in \mathbb{Z}^{n \times m} (m >n)\) is a matrix of rank n and \(\left\| A \right\| = \mathop {\max }\limits_{i,j} \left| {A_{i,j} }...

We present two methods of computing the characteristic polynomial f of an endomorphism of a free module over a commutative domain. The first method is based on the transition to a basis in which the matrix of the endomorphism f is tridiagonal. For the initial matrix, any disposition of zero entries is allowed. The second method is based on the tran...

Two new sequential methods are given for computing the characteristic polynomial of an endomorphism of a free finite rank-n module over a domain, that require O(n3) ring operations with exact divisions.

Two new methods to solve linear systems of Diophantine equations are proposed - modular (CRT) and p-adic (Hensel). Each of them allows to obtain solutions of a system with the size n x m with the complexity O(nßm). For quasi-square systems, the p-adic method allows to obtain solution with the complexity O(n
3), and the modular method with complexit...

Effective matrix methods for solving standard linear algebra problems in a commutative domains are discussed. Two of them are new. There are a methods for computing adjoined matrices and solving system of linear equations in a commutative domains.

Despite the fact that the importance of Sylvester's determinant identity has been recognized in the past, we were able to find only one proof of it in English (Bareiss, 1968), with reference to some others. (Recall that Sylvester (1857) stated this theorem without proof.) Having used this identity, recently, in the validity proof of our new, improv...

We present an improved variant of the matrix-triangularization subresultant prs method [1] for the computation of a greatest
common divisor of two polynomialsA andB (of degreesm andn, respectively) along with their polynomial remainder sequence. It is improved in the sense that we obtain complete theoretical
results, independent of Van Vleck’s theo...

A new method of computing determinants in commutative rings is developed. This method is compared with two known methods. For each method estimates of efficiency in various commutative rings are given.

We present an improved variant of the matrix-triangularization subresultant prs method [2] for the computation of a greatest common divisor of two poly-nomials A and B (of degrees m and n, respectively) along with their polynomial remainder sequence. It is improved in the sense that we obtain complete theoretical results, independent of Van Vleck's...

Solution methods for linear equation systems in a commutative ring are discussed. Four methods are compared, in the setting of several different rings: Dodgson’s method [1], Bareiss’s method [2] and two methods of the author — method by forward and back-up procedures [3] and a one-pass method [4].
We show that for the number of coefficient operati...

The main features of current version of parallel computer algebra system developed in Tambov university are presented.

We present two deterministic recursive pivot-free algorithms for matrix inversion. These algorithms improve on previous methods that required invertible on-diagonal blocks, or required row-or column-based pivoting. These algorithms have the same complexity as matrix multiplication, and they are recursive algorithms that do not require pivoting. The...