Parallel algorithms for certain matrix computations

1995

Abstract A parallel algorithm for the eigenvalue assignment problem (EAP) in single-input linear systems is presented. The algorithm is based on solution of the observer matrix equation, and it has a time complexity of 0 (n3/p), where n is the order of the system and p is the number of processors. The algorithm has been implemented on the hypercube parallel computer INTEL iPSC 860, with 8 processors, and expected speed up has been obtained.

Differential-Algebraic Equations: A Projector Based Analysis, 2013

Parallel Computing, 2006

ABSTRACT

Computers & Mathematics with Applications, 1989

Computers & Mathematics with Applications, 1990

We review some of the most important results in the area of fast parallel algorithms for the solution of linear systems and related problems, such as matrix inversion, computation of the determinant and of the adjoint matrix. We analyze both direct and iterative methods implemented in various models of parallel computation.

Journal of Computer Science, 2007

PRAM algorithms for Symmetric Gaussian elimination is presented. We showed actual testing operations that will be performed during Symmetric Gaussian elimination, which caused symbolic factorization to occur for sparse linear systems. The array pattern of processing elements (PE) in row major order for the specialized sparse matrix in formulated. We showed that the access function in 2 +jn+k contains topological properties. We also proved that cost of storage and cost of retrieval of a matrix are proportional to each other in polylogarithmic parallel time using P-RAM with a polynomial numbers of processor. We use symbolic factorization that produces a data structure, which is used to exploit the sparsity of the triangular factors. In these parallel algorithms number of multiplication/division in O(log 3 n), number of addition/subtraction in O(log 3 n) and the storage in O(log 2 n) may be achieved.

Parallel Computing, 2018

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Parallel Computing, 2001

This paper studies the parallelization of a recursive algorithm for triangular matrix inversion (TMI), using the``divide and conquer'' paradigm. For a (large scale) matrix of size n m2 k m; k P 1 and p 2 q 6 n=2 available processors, we ®rst construct an adequate 2-phases task segmentation and inducing a balanced layered task graph. Then, we design a greedy scheduling leading to a cost optimal parallel algorithm, i.e. whose eciency is equal to 1 for large n. The practical interest of the contribution is proven through an experimental study of two versions of the original algorithm on an IBM SP1 distributed memory multiprocessor. Ó 2001 Published by Elsevier Science B.V. 0167-8191/01/$ -see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 1 6 7 -8 1 9 1 ( 0 1 ) 0 0 1 1 1 -9 triangular systems of size n, a recursive algorithm, based on successive partitionings of the original matrix, was ®rst proposed by Heller in 1973 . Although, this algorithm globally reduces (in its original form) to a series of increasing size matrix multiplications, its complexity is the same as in the standard one, i.e. n 3 =3 On 2 .

Linear Algebra and its Applications, 1990

Here we offer a new randomized parallel algorithm that determines the Smith normal form of a matrix with entries being univariate polynomials with coefficients in an arbitrary field. The algorithm has two important advantages over our previous one: the multipliers relating the Smith form to the input matrix are computed, and the algorithm is probabilistic of Las Vegas type, i.e., always finds the correct answer. The Smith form algorithm is also a good sequential algorithm. Our algorithm reduces the problem of Smith form computation to two Hermite form computations. Thus the Smith form problem has complexity asymptotically that of the Hermite form problem. We also construct fast parallel algorithms for Jordan normal form and testing similarity of matrices. Both the similarity and non-similarity problems are in the complexity class RNC for the usual coefficient fields, i.e., they can be probabilistically decided in polylogarithmic time using polynomially many processors.

2005

In this chapter we discuss numerical software for linear algebra problems on parallel computers. We focus on some of the most common numerical operations: linear system solving and eigenvalue computations. Numerical operations such as linear system solving and eigenvalue calculations can be applied to two di erent kinds of matrix: dense and sparse. In dense systems, essentially every matrix element is nonzero. In sparse systems, a su ciently large number of matrix elements is zero that a specialized storage scheme is warranted; for an introduction to sparse storage, see [3]. Because the two classes are so di erent, usually di erent numerical softwares apply to them. We discuss ScaLAPACK and PLAPACK as the choices for dense linear system solving (see Section 13.1). For solving sparse linear systems, there exist two classes of algorithms: direct methods and iterative methods. We will discuss SuperLU as an example of a direct solver (see Section 13.2.1) and PETSc as an example of itera...