scholarly journals Communication-Optimal Parallel 2.5D Matrix Multiplication and LU Factorization Algorithms

2011 ◽  
pp. 90-109 ◽  
Author(s):  
Edgar Solomonik ◽  
James Demmel
Keyword(s):  
Matrix Multiplication ◽  
Lu Factorization ◽  
Factorization Algorithms

DATA REPLICATION IN DENSE MATRIX FACTORIZATION

1993 ◽  
Vol 03 (04) ◽  
pp. 419-430 ◽  
Author(s):  
J. MALARD ◽  
C.C. PAIGE
Keyword(s):  
Matrix Factorization ◽  
Message Passing ◽  
Data Replication ◽  
Lu Factorization ◽  
Dense Matrix ◽  
Software Libraries ◽  
Numerical Software ◽  
Factorization Algorithms ◽  
Performance Gains

Gossiping is proposed as the preferred communication primitive for replicating pivot data in dense matrix factorization on message passing multicomputer. Performance gains are demonstrated on a hypercube for LU factorization algorithms based on gossiping as opposed to broadcasting. This finding has consequences for the design of numerical software libraries.

scholarly journals TRADING REPLICATION FOR COMMUNICATION IN PARALLEL DISTRIBUTED-MEMORY DENSE SOLVERS

2002 ◽  
Vol 12 (01) ◽  
pp. 79-94 ◽  
Author(s):  
DROR IRONY ◽  
SIVAN TOLEDO
Keyword(s):  
Linear Systems ◽  
Message Passing ◽  
Distributed Memory ◽  
Matrix Multiplication ◽  
Lu Factorization ◽  
Linear Solvers ◽  
Factorization Algorithm ◽  
Temporary Storage ◽  
Right Hand ◽  
Cluster Of Workstations

We present new communication-efficient parallel dense linear solvers: a solver for triangular linear systems with multiple right-hand sides and an LU factorization algorithm. These solvers are highly parallel and they perform a factor of 0.4P1/6 less communication than existing algorithms, where P is number of processors. The new solvers reduce communication at the expense of using more temporary storage. Previously, algorithms that reduce communication by using more memory were only known for matrix multiplication. Our algorithms are recursive, elegant, and relatively simple to implement. We have implemented them using MPI, a message-passing libray, and tested them on a cluster of workstations.

Stable and Supported Semantics in Continuous Vector Spaces

2020 ◽  
Author(s):  
Yaniv Aspis ◽  
Krysia Broda ◽  
Alessandra Russo ◽  
Jorge Lobo
Keyword(s):  
Logic Program ◽  
Matrix Multiplication ◽  
Vector Spaces ◽  
Continuous Space ◽  
Continuous Vector ◽  
Novel Approach ◽  
Gradient Based ◽  
Normal Logic ◽  
Parameter Values ◽  
Normal Logic Program

We introduce a novel approach for the computation of stable and supported models of normal logic programs in continuous vector spaces by a gradient-based search method. Specifically, the application of the immediate consequence operator of a program reduct can be computed in a vector space. To do this, Herbrand interpretations of a propositional program are embedded as 0-1 vectors in $\mathbb{R}^N$ and program reducts are represented as matrices in $\mathbb{R}^{N \times N}$. Using these representations we prove that the underlying semantics of a normal logic program is captured through matrix multiplication and a differentiable operation. As supported and stable models of a normal logic program can now be seen as fixed points in a continuous space, non-monotonic deduction can be performed using an optimisation process such as Newton's method. We report the results of several experiments using synthetically generated programs that demonstrate the feasibility of the approach and highlight how different parameter values can affect the behaviour of the system.

Modular Matrix Multiplication on a Linear Array.

1983 ◽  
Author(s):  
I. V. Ramakrishnan ◽  
P. J. Varman
Keyword(s):  
Linear Array ◽  
Matrix Multiplication
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