Gaussian elimination with partial pivoting and load balancing on a multiprocessor

Solving a system of nonlinear equations is a common operation in many practical applications such as analyzing physics experiments or running simulations of analog electronic circuits. Applications become more and more complex, both in terms of variables and number of involved equations, severely limiting the applicability of the sequential algorithms. As both the processing power and the available bandwidth in modern network increase, the distributing solution becomes more and more appealing. This paper presents a parallel algorithm for solving systems nonlinear of equations based on the Newton-Raphson method. The core of this algorithm is the Gaussian reduction. Our implementation attempts to minimize the overall amount of data to be transferred during both the Gauss pivoting operation and each Newton-Raphson iteration.

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“…The pivot element is required whenever any A[k, k] is null or very small. Typical strategies are pivot columns, rows or total pivot [9], [10].…”

Section: End For 9: End Formentioning

confidence: 99%

A Distributed Approach for Solving Systems of Nonlinear Equations

Mocanu

Țăpuș

2011

2011 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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“…As in the first step, a new (temporary) value for A [1,2] is calculated and stored in-place, on the basis of e 2 and A [1,2]. The fourth step is very similar to Step 2.…”

Section: Data Dependenciesmentioning

confidence: 99%

“…Parallel solutions to Singular Value Decomposition in general, and Householder bidiagonalization in particular, have been studied extensively in the literature [2], [3], [4], [5], [6]. In contrast to such earlier efforts, our main focus is on the integration of a parallel solution to Householder bidiagonalization behind a user transparent parallel programming interface.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Parallel Householder Bidiagonalization

Liu

Seinstra²

2009

Lecture Notes in Computer Science

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Abstract.With the increasing use of large image and video archives and high-resolution multimedia data streams in many of today's research and application areas, there is a growing need for multimedia-oriented high-performance computing. As a consequence, a need for algorithms, methodologies, and tools that can serve as support in the (automatic) parallelization of multimedia applications is rapidly emerging.This paper discusses the parallelization of Householder bidiagonalization, a matrix factorization method which is an integral part of full Singular Value Decomposition (SVD) -an important algorithm for many multimedia problems. Householder bidiagonalization is hard to parallelize efficiently because the total number of matrix elements taking part in the calculations reduces during runtime. To overcome the growing negative performance impact of load imbalances and overprovisioning of compute resources, we apply adaptive runtime techniques of periodic matrix remapping and process reduction for improved performance. Results show that our adaptive parallel execution approach provides a significant improvement in efficiency, even when applying a set of compute resources which is (initially) very large.

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“…The communication cost is therefore given by n (4) TiC(n,p) 2d + A 2d(n k + l) k--1 k--1 2nd + (n 2 + n) d.…”

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confidence: 99%

“…According to the timing results to be discussed in 7, our idea of using otherwise idle processors to perform redundant computation appears to be effective in keeping the communication algorithm simple, efficient, and versatile for use in a generalized version of Algorithm I and other algorithms. 4. Performance analysis of Algorithm I.…”

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confidence: 99%

QR Factorization of a Dense Matrix on a Hypercube Multiprocessor

Chu¹,

George²

1990

SIAM J. Sci. and Stat. Comput.

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In this article a new algorithm for computing the QR factorization of a rectangular matrix on a hypercube multiprocessor is described. The hypercube network is configured as a two-dimensional subcube-grid in the proposed scheme. A global communication scheme that uses redundant computation to maintain data proximity is employed, and the mapping strategy is such that for a fixed number of processors the processor idle time is small and either constant or grows linearly with the dimension of the matrix. A complexity analysis shows what the aspect ratio of the configured grid should be in terms of the shape of the matrix and the relative speeds of communication and computation. Numerical experiments performed on an Intel Hypercube multiprocessor support the theoretical results.

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Gaussian elimination with partial pivoting and load balancing on a multiprocessor

Cited by 39 publications

References 2 publications

A Distributed Approach for Solving Systems of Nonlinear Equations

A Distributed Approach for Solving Systems of Nonlinear Equations

Adaptive Parallel Householder Bidiagonalization

QR Factorization of a Dense Matrix on a Hypercube Multiprocessor

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