Gennadi Malaschonok - Academia.edu (original) (raw)
Papers by Gennadi Malaschonok
Наукові записки НаУКМА, Dec 10, 2021
Ця робота пропонує реалізацію паралельного алгоритму SVD для тридіагональної матриці на відеокарт... more Ця робота пропонує реалізацію паралельного алгоритму SVD для тридіагональної матриці на відеокарті з використанням архітектури Nvidia CUDA для роботи з великими матрицями. Для цього було досліджено роботу послідовного алгоритму, розроблено модель паралельного алгоритму на Java, який враховує особливості роботи відеокарти, і реалізовано та протестовано алгоритми для відеокарти з використанням різних типів пам'яті відеокарти, які можна використовувати у програмах на Java та С/C++.
Social Science Research Network, 2023
У цій статті досліджено блочно-рекурсивний паралельний алгоритм розкладання Холецького для суперк... more У цій статті досліджено блочно-рекурсивний паралельний алгоритм розкладання Холецького для суперкомп'ютера з розподіленою пам'яттю. Для зручності масштабування, кількість ядерце деяка ступінь двійки. Розглянуто стійкість алгоритму до накопичення помилок обчислень і ефективність масштабування статичного блочно-рекурсивного алгоритму. Показано, що для щільних матриць, починаючи з розміру 64, використання подвійної точності (стандартна арифметика з плаваючою точкою і 64-розрядним машинним словом), немає змоги отримувати похибку, меншу ніж 1, навіть у простих випадках. Зі збільшенням розмірів коефіцієнтів похибка тільки зростає, тому використання такої арифметики для матриць ще більшого розміру втрачає сенс. Щоб розв'язати цю проблему, для матриць великого розміру ми використовували арифметику BigDecimal. Вона дає змогу програмно задавати точність, яка вказується як кількість десяткових знаків після коми. Спочатку ми визначаємо необхідну кількість знаків для BigDecimal так, щоб похибка обчислень цієї матриці не перевищувала одиницю, а потім робимо експерименти з різною кількістю ядер для такої матриці. Ми даємо рекомендації для щільних матриць, елементи яких задавались випадково з рівномірним розподілом. Для таких матриць, зважаючи на їхній розмір і кількість десяткових знаків у їхніх елементів, ми рекомендуємо вибір точності для машинної арифметики і кількість ядер для обчислень. Ключові слова: паралельне програмування, алгоритм Холецького, блочно-рекурсивний алгоритм, факторизація матриць, обчислення на кластері з розподіленою пам'яттю, накопичення похибки.
arXiv (Cornell University), Nov 26, 2017
Two known computation methods and one new computation method for matrix determinant over an integ... more 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.
Springer eBooks, Sep 1, 2007
This paper is a review of results on computational methods of linear algebra over commutative dom... more 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.
arXiv (Cornell University), Apr 25, 2022
![Research paper thumbnail of A sympy/sage Module for Computing Polynomial Remainder Sequences: [preprint]](https://attachments.academia-assets.com/107937401/thumbnails/1.jpg)
](Given the polynomials f, g ∈ Z[x], we are interested in the following four polynomial remainder s... more Given the polynomials f, g ∈ Z[x], we are interested in the following four polynomial remainder sequences (prs's): (a) Euclidean prs, (b) Modified Euclidean prs, (c) Subresultant prs, and (d) Modified Subresultant prs.
Serdica Journal of Computing, Nov 10, 2014
In 1900 E. B. Van Vleck proposed a very efficient method to compute the Sturm sequence of a polyn... more In 1900 E. B. Van Vleck proposed a very efficient method to compute the Sturm sequence of a polynomial p (x) ∈ [x] by triangularizing one of Sylvester's matrices 1 of p (x) and its derivative p ′ (x). That method works fine only for the case of complete sequences provided no pivots take place. In 1917, A. J. Pell 2 and R. L. Gordon pointed out this "weakness" in Van Vleck's theorem, rectified it but did not extend his method, so that it also works in the cases of: (a) complete Sturm sequences with pivot, and (b) incomplete Sturm sequences. Despite its importance, the Pell-Gordon Theorem for polynomials in É[x] has been totally forgotten and, to our knowledge, it is referenced by us for the first time in the literature. In this paper we go over Van Vleck's theorem and method, modify slightly the formula of the Pell-Gordon Theorem and present a general triangularization method, called the VanVleck-Pell-Gordon method, that correctly computes in [x] polynomial Sturm sequences, both complete and incomplete.
We present an improved (faster) variant of the matrix-triangularization subresultant prs method f... more We present an improved (faster) variant of the matrix-triangularization subresultant prs method for the computation of a greatest common divisor of two polynomials A and B (of degrees dA and dB, respectively) along with their polynomial remainder sequence [1]. The computing time of our fast method is 0(n2+slog ∥C∥2), for standard arithmetic and 0(((n1+s+n 3 log ∥C∥)(log n+ log ∥C∥)2) for the Chinese remainder method, where n = d A + d B, ∥C∥ is the maximal coefficient of the two polynomials and the best known s < 2.356. By comparison, the computing time of the old version is 0(n 5 log ∥C∥2 ).
![Research paper thumbnail of Sparse matrices in computer algebra when using distributed memory: theory and applications: [preprint]](https://attachments.academia-assets.com/107937399/thumbnails/1.jpg)
](put attansion on the several difficult challenges. The task of managing calculations on a cluster... more put attansion on the several difficult challenges. The task of managing calculations on a cluster with distributed memory for algorithms with sparse matrices is today one of the most difficult challenges. Here we must also add problems with the type of the basic algebra: matrices can be over fields or over commutative rings. For sparse matrices, it is not true that all computations over polynomials or integers can be reduced to computations in finite fields. Such reduction may be not effective for sparse matrices. We consider the class of block-recursive matrix algorithms. The most famous of them are standard and Strassen's block matrix multiplication, Schur and Strassen's block-matrix inversion [2]. Class of block-recursive matrix algorithms Block-recursive algorithms were not so important as long as the calculations were performed on computers with shared memory. The generalization of Strassen's matrix inversion algorithm [2] with additional permutations of rows and columns by J. Bunch and J. Hopkroft [3] is not a block-recursive algorithm. Only in the nineties it became clear that block-recursive matrix algorithms are required to operate with sparse super large matrices on a supercomputer with distributed memory. The block recursive algorithm for the solution of systems of linear equations and for adjoint matrix computation which is some generalisation of Schur inversion in commutative domains was discraibed in [7], [8] and [10]. See also at the book [9]. However, in all these algorithms, except matrix multiplication, a very strong restriction are imposed on the matrix. The leading minors, which are on the main diagonal, should not be zero. This restriction was removed later. The algorithm that computes the adjoint matrix, the echelon form, and the kernel of the matrix operator for the commutative domains was proposed in [11]. The block-recursive algorithm for the Bruhat decomposition and the LDU decomposition for the matrix over the field was obtained in [12], and these algorithms were generaized for the matrices over commutative domains in [14] and in [15].
![Research paper thumbnail of Subresultant Polynomial Remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x]](https://attachments.academia-assets.com/107937398/thumbnails/1.jpg)
](Serdica Journal of Computing, Nov 3, 2017
* The second author was partially supported by RFBR grant No 16-07-00420. 1 Also known as general... more * The second author was partially supported by RFBR grant No 16-07-00420. 1 Also known as generalized Sturmian prs. 2 Defined by equation (4) in Section 1. 3 Defined by equation (1) in Section 1. 4 To distinguish it from sylvester2(f, g, x), Sylvester's matrix of 1853 of dimensions (2 • n) × (2 • n) [15].
Наукові записки НаУКМА, Oct 16, 2018
The report is devoted to the concept of creating block-recursive matrix algorithms for computing ... more The report is devoted to the concept of creating block-recursive matrix algorithms for computing on a super-computer with distributed memory and dynamic decentralized control.
We give an overview of the theoretical results for matrix block-recursive algorithms in commutati... more 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 these programs we measure the running time of the program for a different number of processors and plot the graphs of efficiency factor. Also we present the main application areas in which such parallel algorithms are used. It is concluded that this class of algorithms allows to obtain efficient parallel programs on clusters with distributed memory. Index Terms-block-recursive matrix algorithms, commutative domain, factorization of matrices, matrix inversion, distributed memory
Reliable Computing, Dec 1, 1995
We present an impr{wed variant of the matrix-triangularization subresultant prs method [1] fi~r t... more We present an impr{wed variant of the matrix-triangularization subresultant prs method [1] fi~r the computation of a greatest comnum divi~w of two polynomials A and B (of degrees m and n, respectively) along with their polynomial remainder ~quence. It is impr~wed in the sense that we obtain complete theoretical results, independent {}f Van Vleck's theorem [13] (which is not always tnle [2, 6]), and, instead of transfornfing a matrix of order 2 .max(m, n) [1], we are now transforming a matrix of order m+ n. An example is al.,a~ induded to clarify the concepts. MaTp qHOe SbIq CAeH e cy6pe3yAt, TaHTHBIX IIOAHHOMIIaAbHbIX nocAeAOBaTeABHOCTefl OCTaTKOB B HHTeFpaABHblX 06AaCTXX A. F. AKP//rrAc, E. K. AKPHTAC, F. I/'l. MAAAmOHOK Flpe~lcraBJ'lerl yJlyqnleHH/:,ll~ Bapl.laHT MaTpl.lqHo-rpHaHlyJl.qpH3alIHOHHOlO cy6pe3yJlbTanTHOl'O MeToaa no-:lllHOMtlaJlbl-n~x ,u~c:xea<>marem, ti<m'refi <RTraTKOB ([I['IO) [1] a;xa ma,~nc.aemta Hal.16O/lbnlero ~gSntexo ae-:lnTedl~! 21ByX MHOrOttdleH(}B m n B (CTelleHelTI m ii n C{R)TBC~'I'(YrBeHHO) C O}IHOBpeMeHHI~M HaxoxK31eHHeM HX ~lOl']. YJlytlnleHne 3aKdiiottaeTcI, l B TOM, qTO IlOdlyqeHl~I 3aKOHt,IeHHIMe TeOpeTrtqecKne pe3yJlt, TaTIM, ae3aBr, lCrlMl:,le OX T~}pCMbl Baa B.aer.a [13] (KOTopaa tie Bcerlta cupaBeaJmma, CM [fi, 6]). KI.~Me XOn~, B.~ecxo npe~6pa2o~anu~ Maxpnm,1 nopsaKa 2-max(m, n) [1] renepb upeo6pa2yexc~ Maxpnua nop~taKa D2 + '/Z. I"Ipe, llCTaB.rleH qlIUleHtibllYl IlptiMep ~ldl~l IIdl211oCTpalll.n,I 3THX IIOJIo)KeHHI~I.
Journal of Pure and Applied Algebra, Feb 1, 2001
Two new sequential methods are given for computing the characteristic polynomial of an endomorphi... more Two new sequential methods are given for computing the characteristic polynomial of an endomorphism of a free ÿnite rank-n module over a domain, that require O(n 3) ring operations with exact divisions.
arXiv (Cornell University), Apr 25, 2022
In this paper, we describe the general characteristics of the MathPartner computer algebra system... more In this paper, we describe the general characteristics of the MathPartner computer algebra system (CAS) and its Mathpar programming language. The MathPartner can be used for scientific and engineering calculations, as well as in secondary schools and higher education institutions. It allows one to carry out both simple calculations (acting as a scientific calculator) and complex calculations with large-scale mathematical objects. The Mathpar is a procedural language that supports a large number of elementary and special functions, as well as matrix and polynomial operators. This service allows one to build function images and animate them. The MathPartner also makes it possible to solve some symbolic computation problems on supercomputers with distributed memory. We highlight the main differences of the MathPartner from other CASs and describe the Mathpar language along with the user service provided.
arXiv (Cornell University), Nov 8, 2020
LU-factorization of matrices is one of the fundamental algorithms of linear algebra. The widespre... more 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. The block recursive LU factorization algorithm was obtained in 2010. This algorithm is called LEU-factorization. It, like the traditional LU-algorithm, is designed for matrices over number fields. However, it does not solve the problem of numerical instability. We propose a generalization of the LEU algorithm to the case of a commutative domain and its field of quotients. This LDU factorization algorithm decomposes the matrix over the commutative domain into a product of three matrices, in which the matrices L and U belong to the commutative domain, and the elements of the weighted truncated permutation matrix D are the elements inverse to the product of some pair of minors. All elements are calculated without errors, so the problem of instability does not arise. The work was partially supported by the Academy of Sciences of Ukraine, project (2020) "Creation of an open data e-platform for the collective use centers".
Serdica Journal of Computing, Apr 18, 2016
In 1917 Pell 1 and Gordon used sylvester2, Sylvester's little known and hardly ever used matrix o... more In 1917 Pell 1 and Gordon used sylvester2, Sylvester's little known and hardly ever used matrix of 1853, to compute 2 the coefficients of a Sturmian remainder-obtained in applying in É[x], Sturm's algorithm on two polynomials f, g ∈ [x] of degree n-in terms of the determinants 3 of the corresponding submatrices of sylvester2. Thus, they solved a problem that had eluded both J. J. Sylvester, in 1853, and E. B. Van Vleck, in 1900. 4 In this paper we extend the work by Pell and Gordon and show how to compute 2 the coefficients of an Euclidean remainder-obtained in finding in É[x], the greatest common divisor of f, g ∈ [x] of degree n-in terms of the determinants 5 of the corresponding submatrices of sylvester1, Sylvester's widely known and used matrix of 1840.
Nonlinear Analysis-Modelling and Control, May 18, 2006
Given an m×n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see... more Given an m×n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see [1]). is unique. Given the importance of these subspaces, computing bases for them is the gist of Linear Algebra. In "Classical" Linear Algebra, bases for these subspaces are computed using Gaussian Elimination; they are orthonormalized with the help of the Gram-Schmidt method. Continuing our previous work [3] and following Uhl's excellent approach [2] we use SVD analysis to compute orthonormal bases for the four subspaces associated with A, and give a 3D explanation. We then state and prove what we call the "SVD-Fundamental Theorem" of Linear Algebra, and apply it in solving systems of linear equations.
Наукові записки НаУКМА, Dec 10, 2021
Ця робота пропонує реалізацію паралельного алгоритму SVD для тридіагональної матриці на відеокарт... more Ця робота пропонує реалізацію паралельного алгоритму SVD для тридіагональної матриці на відеокарті з використанням архітектури Nvidia CUDA для роботи з великими матрицями. Для цього було досліджено роботу послідовного алгоритму, розроблено модель паралельного алгоритму на Java, який враховує особливості роботи відеокарти, і реалізовано та протестовано алгоритми для відеокарти з використанням різних типів пам'яті відеокарти, які можна використовувати у програмах на Java та С/C++.
Social Science Research Network, 2023
У цій статті досліджено блочно-рекурсивний паралельний алгоритм розкладання Холецького для суперк... more У цій статті досліджено блочно-рекурсивний паралельний алгоритм розкладання Холецького для суперкомп'ютера з розподіленою пам'яттю. Для зручності масштабування, кількість ядерце деяка ступінь двійки. Розглянуто стійкість алгоритму до накопичення помилок обчислень і ефективність масштабування статичного блочно-рекурсивного алгоритму. Показано, що для щільних матриць, починаючи з розміру 64, використання подвійної точності (стандартна арифметика з плаваючою точкою і 64-розрядним машинним словом), немає змоги отримувати похибку, меншу ніж 1, навіть у простих випадках. Зі збільшенням розмірів коефіцієнтів похибка тільки зростає, тому використання такої арифметики для матриць ще більшого розміру втрачає сенс. Щоб розв'язати цю проблему, для матриць великого розміру ми використовували арифметику BigDecimal. Вона дає змогу програмно задавати точність, яка вказується як кількість десяткових знаків після коми. Спочатку ми визначаємо необхідну кількість знаків для BigDecimal так, щоб похибка обчислень цієї матриці не перевищувала одиницю, а потім робимо експерименти з різною кількістю ядер для такої матриці. Ми даємо рекомендації для щільних матриць, елементи яких задавались випадково з рівномірним розподілом. Для таких матриць, зважаючи на їхній розмір і кількість десяткових знаків у їхніх елементів, ми рекомендуємо вибір точності для машинної арифметики і кількість ядер для обчислень. Ключові слова: паралельне програмування, алгоритм Холецького, блочно-рекурсивний алгоритм, факторизація матриць, обчислення на кластері з розподіленою пам'яттю, накопичення похибки.
arXiv (Cornell University), Nov 26, 2017
Two known computation methods and one new computation method for matrix determinant over an integ... more 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.
Springer eBooks, Sep 1, 2007
This paper is a review of results on computational methods of linear algebra over commutative dom... more 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.
arXiv (Cornell University), Apr 25, 2022
Given the polynomials f, g ∈ Z[x], we are interested in the following four polynomial remainder s... more Given the polynomials f, g ∈ Z[x], we are interested in the following four polynomial remainder sequences (prs's): (a) Euclidean prs, (b) Modified Euclidean prs, (c) Subresultant prs, and (d) Modified Subresultant prs.
Serdica Journal of Computing, Nov 10, 2014
In 1900 E. B. Van Vleck proposed a very efficient method to compute the Sturm sequence of a polyn... more In 1900 E. B. Van Vleck proposed a very efficient method to compute the Sturm sequence of a polynomial p (x) ∈ [x] by triangularizing one of Sylvester's matrices 1 of p (x) and its derivative p ′ (x). That method works fine only for the case of complete sequences provided no pivots take place. In 1917, A. J. Pell 2 and R. L. Gordon pointed out this "weakness" in Van Vleck's theorem, rectified it but did not extend his method, so that it also works in the cases of: (a) complete Sturm sequences with pivot, and (b) incomplete Sturm sequences. Despite its importance, the Pell-Gordon Theorem for polynomials in É[x] has been totally forgotten and, to our knowledge, it is referenced by us for the first time in the literature. In this paper we go over Van Vleck's theorem and method, modify slightly the formula of the Pell-Gordon Theorem and present a general triangularization method, called the VanVleck-Pell-Gordon method, that correctly computes in [x] polynomial Sturm sequences, both complete and incomplete.
We present an improved (faster) variant of the matrix-triangularization subresultant prs method f... more We present an improved (faster) variant of the matrix-triangularization subresultant prs method for the computation of a greatest common divisor of two polynomials A and B (of degrees dA and dB, respectively) along with their polynomial remainder sequence [1]. The computing time of our fast method is 0(n2+slog ∥C∥2), for standard arithmetic and 0(((n1+s+n 3 log ∥C∥)(log n+ log ∥C∥)2) for the Chinese remainder method, where n = d A + d B, ∥C∥ is the maximal coefficient of the two polynomials and the best known s < 2.356. By comparison, the computing time of the old version is 0(n 5 log ∥C∥2 ).
put attansion on the several difficult challenges. The task of managing calculations on a cluster... more put attansion on the several difficult challenges. The task of managing calculations on a cluster with distributed memory for algorithms with sparse matrices is today one of the most difficult challenges. Here we must also add problems with the type of the basic algebra: matrices can be over fields or over commutative rings. For sparse matrices, it is not true that all computations over polynomials or integers can be reduced to computations in finite fields. Such reduction may be not effective for sparse matrices. We consider the class of block-recursive matrix algorithms. The most famous of them are standard and Strassen's block matrix multiplication, Schur and Strassen's block-matrix inversion [2]. Class of block-recursive matrix algorithms Block-recursive algorithms were not so important as long as the calculations were performed on computers with shared memory. The generalization of Strassen's matrix inversion algorithm [2] with additional permutations of rows and columns by J. Bunch and J. Hopkroft [3] is not a block-recursive algorithm. Only in the nineties it became clear that block-recursive matrix algorithms are required to operate with sparse super large matrices on a supercomputer with distributed memory. The block recursive algorithm for the solution of systems of linear equations and for adjoint matrix computation which is some generalisation of Schur inversion in commutative domains was discraibed in [7], [8] and [10]. See also at the book [9]. However, in all these algorithms, except matrix multiplication, a very strong restriction are imposed on the matrix. The leading minors, which are on the main diagonal, should not be zero. This restriction was removed later. The algorithm that computes the adjoint matrix, the echelon form, and the kernel of the matrix operator for the commutative domains was proposed in [11]. The block-recursive algorithm for the Bruhat decomposition and the LDU decomposition for the matrix over the field was obtained in [12], and these algorithms were generaized for the matrices over commutative domains in [14] and in [15].
Serdica Journal of Computing, Nov 3, 2017
* The second author was partially supported by RFBR grant No 16-07-00420. 1 Also known as general... more * The second author was partially supported by RFBR grant No 16-07-00420. 1 Also known as generalized Sturmian prs. 2 Defined by equation (4) in Section 1. 3 Defined by equation (1) in Section 1. 4 To distinguish it from sylvester2(f, g, x), Sylvester's matrix of 1853 of dimensions (2 • n) × (2 • n) [15].
Наукові записки НаУКМА, Oct 16, 2018
The report is devoted to the concept of creating block-recursive matrix algorithms for computing ... more The report is devoted to the concept of creating block-recursive matrix algorithms for computing on a super-computer with distributed memory and dynamic decentralized control.
We give an overview of the theoretical results for matrix block-recursive algorithms in commutati... more 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 these programs we measure the running time of the program for a different number of processors and plot the graphs of efficiency factor. Also we present the main application areas in which such parallel algorithms are used. It is concluded that this class of algorithms allows to obtain efficient parallel programs on clusters with distributed memory. Index Terms-block-recursive matrix algorithms, commutative domain, factorization of matrices, matrix inversion, distributed memory
Reliable Computing, Dec 1, 1995
We present an impr{wed variant of the matrix-triangularization subresultant prs method [1] fi~r t... more We present an impr{wed variant of the matrix-triangularization subresultant prs method [1] fi~r the computation of a greatest comnum divi~w of two polynomials A and B (of degrees m and n, respectively) along with their polynomial remainder ~quence. It is impr~wed in the sense that we obtain complete theoretical results, independent {}f Van Vleck's theorem [13] (which is not always tnle [2, 6]), and, instead of transfornfing a matrix of order 2 .max(m, n) [1], we are now transforming a matrix of order m+ n. An example is al.,a~ induded to clarify the concepts. MaTp qHOe SbIq CAeH e cy6pe3yAt, TaHTHBIX IIOAHHOMIIaAbHbIX nocAeAOBaTeABHOCTefl OCTaTKOB B HHTeFpaABHblX 06AaCTXX A. F. AKP//rrAc, E. K. AKPHTAC, F. I/'l. MAAAmOHOK Flpe~lcraBJ'lerl yJlyqnleHH/:,ll~ Bapl.laHT MaTpl.lqHo-rpHaHlyJl.qpH3alIHOHHOlO cy6pe3yJlbTanTHOl'O MeToaa no-:lllHOMtlaJlbl-n~x ,u~c:xea<>marem, ti<m'refi <RTraTKOB ([I['IO) [1] a;xa ma,~nc.aemta Hal.16O/lbnlero ~gSntexo ae-:lnTedl~! 21ByX MHOrOttdleH(}B m n B (CTelleHelTI m ii n C{R)TBC~'I'(YrBeHHO) C O}IHOBpeMeHHI~M HaxoxK31eHHeM HX ~lOl']. YJlytlnleHne 3aKdiiottaeTcI, l B TOM, qTO IlOdlyqeHl~I 3aKOHt,IeHHIMe TeOpeTrtqecKne pe3yJlt, TaTIM, ae3aBr, lCrlMl:,le OX T~}pCMbl Baa B.aer.a [13] (KOTopaa tie Bcerlta cupaBeaJmma, CM [fi, 6]). KI.~Me XOn~, B.~ecxo npe~6pa2o~anu~ Maxpnm,1 nopsaKa 2-max(m, n) [1] renepb upeo6pa2yexc~ Maxpnua nop~taKa D2 + '/Z. I"Ipe, llCTaB.rleH qlIUleHtibllYl IlptiMep ~ldl~l IIdl211oCTpalll.n,I 3THX IIOJIo)KeHHI~I.
Journal of Pure and Applied Algebra, Feb 1, 2001
Two new sequential methods are given for computing the characteristic polynomial of an endomorphi... more Two new sequential methods are given for computing the characteristic polynomial of an endomorphism of a free ÿnite rank-n module over a domain, that require O(n 3) ring operations with exact divisions.
arXiv (Cornell University), Apr 25, 2022
In this paper, we describe the general characteristics of the MathPartner computer algebra system... more In this paper, we describe the general characteristics of the MathPartner computer algebra system (CAS) and its Mathpar programming language. The MathPartner can be used for scientific and engineering calculations, as well as in secondary schools and higher education institutions. It allows one to carry out both simple calculations (acting as a scientific calculator) and complex calculations with large-scale mathematical objects. The Mathpar is a procedural language that supports a large number of elementary and special functions, as well as matrix and polynomial operators. This service allows one to build function images and animate them. The MathPartner also makes it possible to solve some symbolic computation problems on supercomputers with distributed memory. We highlight the main differences of the MathPartner from other CASs and describe the Mathpar language along with the user service provided.
arXiv (Cornell University), Nov 8, 2020
LU-factorization of matrices is one of the fundamental algorithms of linear algebra. The widespre... more 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. The block recursive LU factorization algorithm was obtained in 2010. This algorithm is called LEU-factorization. It, like the traditional LU-algorithm, is designed for matrices over number fields. However, it does not solve the problem of numerical instability. We propose a generalization of the LEU algorithm to the case of a commutative domain and its field of quotients. This LDU factorization algorithm decomposes the matrix over the commutative domain into a product of three matrices, in which the matrices L and U belong to the commutative domain, and the elements of the weighted truncated permutation matrix D are the elements inverse to the product of some pair of minors. All elements are calculated without errors, so the problem of instability does not arise. The work was partially supported by the Academy of Sciences of Ukraine, project (2020) "Creation of an open data e-platform for the collective use centers".
Serdica Journal of Computing, Apr 18, 2016
In 1917 Pell 1 and Gordon used sylvester2, Sylvester's little known and hardly ever used matrix o... more In 1917 Pell 1 and Gordon used sylvester2, Sylvester's little known and hardly ever used matrix of 1853, to compute 2 the coefficients of a Sturmian remainder-obtained in applying in É[x], Sturm's algorithm on two polynomials f, g ∈ [x] of degree n-in terms of the determinants 3 of the corresponding submatrices of sylvester2. Thus, they solved a problem that had eluded both J. J. Sylvester, in 1853, and E. B. Van Vleck, in 1900. 4 In this paper we extend the work by Pell and Gordon and show how to compute 2 the coefficients of an Euclidean remainder-obtained in finding in É[x], the greatest common divisor of f, g ∈ [x] of degree n-in terms of the determinants 5 of the corresponding submatrices of sylvester1, Sylvester's widely known and used matrix of 1840.
Nonlinear Analysis-Modelling and Control, May 18, 2006
Given an m×n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see... more Given an m×n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see [1]). is unique. Given the importance of these subspaces, computing bases for them is the gist of Linear Algebra. In "Classical" Linear Algebra, bases for these subspaces are computed using Gaussian Elimination; they are orthonormalized with the help of the Gram-Schmidt method. Continuing our previous work [3] and following Uhl's excellent approach [2] we use SVD analysis to compute orthonormal bases for the four subspaces associated with A, and give a 3D explanation. We then state and prove what we call the "SVD-Fundamental Theorem" of Linear Algebra, and apply it in solving systems of linear equations.