Sparse matrices in computer algebra when using distributed memory: theory and applications: [preprint] (original) (raw)

Recursive Matrix Algorithms in Commutative Domain for Cluster with Distributed Memory

2018 Ivannikov Memorial Workshop (IVMEM), 2018

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

2356-5608 Efficient Recursive Implementations for Linear Algebra Operations

2014

Divide and Conquer' (D&C) is a famous paradigm for designing efficient algorithms and improving the effectiveness of computer memory hierarchies. Indeed, D&C-based matrix algorithms operate on submatrices or blocks, so that data loaded into the faster memory levels are reused. In this paper, we design recursive D&C algorithms for solving four basic linear algebra problems, namely matrix multiplication (MM), triangular matrix system solving, LU factorization, dense matrix system solving. Our solution is based on the use of matrix block decomposition and Strassen MM algorithm in the top decomposition level and BLAS routines invocation in the bottom decomposition level. The theoretical complexity of our algorithms is O(ૠ). In an experimental part, we compared our implementations with the equivalent kernels in the BLAS library. This latter study achieved on different machines permits to evaluate the practical interest of our contribution.

Effective Matrix Methods in Commutative Domains

2017

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.

FACTORIZATION, MULTIPLICATION AND VIEWS ON PARALLEL FOR SPARE MATRICES

The complexity of multiplying two matrices with the natural algorithm that of order O() has been improved to generate Stassen's algorithm to improve the complexity to be of order O(()) ≈ O(.) using recursive procedure for divide-and –conquer algorithm[1]. Multiplication of matrices is essential when solving a large system of equation using iterative techniques or direct factorization method. For iterative technique the conjugate-gradient method is viewed [3]. For direct factorization of matrices, the Gram-Schmidt's with complexity (2 m) and Householder's algorithms with complexity (2 m −) , the MATLAB PROGRAM are given. When m and n are large it can be seen that the flops for the three algorithm Stassen , Schmidt and Householder are not differ very much while the iterative techniques specially conjugate-gradient method may be executed in a less time than the others as in [4]. If one looks deeper in these algorithms , all apply the operations to multiply two matrices. So if We reduce the number of flops for multiplication ,the efficiency for times can be achieved. To achieved such goal either to write PAREALLEL algorithms for sparse matrices or investigate LIBARAY ROUTINES IN DIFFERENT LANGUAGES But the problem with parallel computing to get more efficient programs , all must be run on SUPPER or PARALLEL computing machine which is not available to us to run problems with intensive data ,never the less beginners to parallel computing can grab a little lot of information. Here , We first , give both the Schmidt's and Householder's algorithms with Matlab Program for them. In these algorithms , all apply the operations to multiply two matrices. Second ,factorization and linear system of equations are considered with some analysis considering Cholesky factors the matrix A=LL T where L is a lower triangular matrix and the system can be solved with the forward and backward algorithm while the inverse for an upper triangular matrix is given in Fig.(3)for the file INV_LOWER.m Third, sparse matrices representation as Compressed Sparse Column format CSC and Compressed Sparse Row format CSR will be given with an algorithm to multiply two matrices as in Fig.(6.). Fourth, parallelism and multination of matrices algorithm such as FOX'S algorithm and CUDA program are given in Fig.(10) &Fig.(11), respectively. Multiple approaches to demonstrate Parallelism on matrices and solve a system of equation are given in section IV part C. Finally computational remarks are given .

Triangular -basis decompositions and derandomization of linear algebra algorithms over

Journal of Symbolic Computation, 2012

Deterministic algorithms are given for some computational problems that take as input a nonsingular polynomial matrix A over K[x], K an abstract field, including solving a linear system involving A and computing a row reduced form of A. The fastest known algorithms for linear system solving based on the technique of high-order lifting by Storjohann (2003), and for row reduction based on fast minimal approximant basis computation algorithm by Giorgi, Jeannerod and Villard (2003), use randomization to find either a linear or small degree polynomial that is relatively prime to det A. We derandomize these algorithms by first computing a factorization of A = U H, with x not dividing det U and x − 1 not dividing det H. A partial linearization technique, that is applicable also to other problems, is developed to transform a system involving H, which may have some columns of large degrees, to an equivalent system that has degrees reduced to that of the average column degree.

Block recursive computation of generalized inverses

The electronic journal of linear algebra ELA

A fully block recursive method for computing outer generalized inverses of a given square matrix is introduced. The method is applicable even in the case when some of the main diagonal minors of A are singular or A is singular. The cmputational complexity of the method is not harder than the matrix multiplication, under the assumption that the Strassen matrix inversion algorithm is used. A partially recursive algorithm for computing various classes of generalized inverses is also developed. This method can be efficiently used for the acceleration of the known methods for computing generalized inverses.

A recursive algorithm for the inversion of matrices with circulant blocks

Applied Mathematics and Computation, 2007

We investigate the recursive inversion of matrices with circulant blocks. Matrices of this type appear in several applications of Computational Electromagnetics and in the numerical solution of integral equations with the boundary-element method. The inversion is based on the diagonalization of each circulant block by means of the discrete Fourier transform and the application of a recursive algorithm for the inversion of the matrix with diagonal blocks, determined by the eigenvalues of each block. The efficiency of the recursive inversion is exhibited by determining its computational complexity. An implementation of the algorithm in MATLAB is given and numerical results are presented to demonstrate the efficiency in terms of CPU time of our approach.

A new algorithm for the factorization and inversion of recursively generated matrices

In this paper we present a recursive algorithm for factorization and inversion of matrices generated by adding dyads (elementary or rank-one matrices) as it happens in recursive array signal processing. The algorithm is valid for any recursively generated rectangular matrix and it has two parts: one valid for rank-deficient and the other for full rank matrices. The second part is used to obtain a generalization of the Sherman-Morrison algorithm for the recursive inversion of the covariance matrix. From the proposed algorithm we derive two others: one to compute the inverse (pseudo-inverse) of any matrix and the other to invert simultaneously two matrices. Zusammenfassung In diesem Beitrag prbentieren wir einen rekursiven Algorithmus fiir die Faktorisierung und Inversion von Matrizen, die durch die Addition von Dyaden (Elementar-Matrizen oder Matrizen mit dem Rang eins) generiert werden, wie dies bei der rekursiven Array-Signalverarbeitung geschieht. Der Algorithmus ist fiir jede rekursiv generierte Rechteck-Matrix giiltig und besteht aus zwei Teilen: der eine gilt fiir Matrizen mit nicht vollem Rang und der andere fiir Matrizen mit vollem Rang. Der zweite Teil wird benutzt, urn eine Verallgemeinerung des Sherman-Morrison Algorithmus fiir die rekursive inversion der Kovarianz-Matrix zu erhalten. Aus dem vorgeschlagenen Algorithmus leiten wir zwei weitere ab: einer berechnet die Inverse (Pseudo-Inverse) jeder beliebigen Matrix und der andere invertiert gleichzeitig zwei Matrizen. On prBsente dans cet article un algorithme rCcursif pour la factorisation et l'inversion de matrices gCntrCes par l'addition de diades (matrices Clkmentaires ou de rang unitaire), comme dans le traitement de signal matriciel rCcursif. L'algorithme convient pour toute matrice rectangulaire g&ntrCe rt.cursivement et contient deux parties: une pour les matrices qui ne sont pas de plein rang et l'autre pour les matrices de plein rang. La seconde partie est consacr6e g l'obtention d'une g&nCralisation de l'algorithme de Sherman-Morrison pour l'inversion r6cursive de la matrice de covariance. Deux autres algorithmes sont d&iv&s B partir du premier: l'un pour calculer l'inverse (ou le pseudo-inverse) de n'importe quelle matrice, et l'autre pour inverser simultantment deux matrices.

A New Algorithm for the Computation of Canonical Forms of Matrices over Fields

Journal of Symbolic Computation, 1997

A new algorithm is presented for the computation of canonical forms of matrices over fields. These are the Primary Rational, Rational, and Jordan canonical forms. The algorithm works by obtaining a decomposition of the vector space acted on by the given matrix into primary cyclic spaces (spaces whose minimal polynomials with respect to the matrix are powers of irreducible polynomials). An efficient implementation of the algorithm is incorporated in the Magma Computer Algebra System.

Triangular Decomposition of Matrices in a Domain

Lecture Notes in Computer Science, 2015

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 be easily obtained from this triangular decomposition. Algorithms have the same complexity as the algorithm of matrix multiplication.