scholarly journals Parallelization of the Gaussian Elimination Algorithm on Systolic Arrays

1996 ◽  
Vol 33 (1) ◽  
pp. 69-75 ◽  
Author(s):  
J-C. Bermond ◽  
C. Peyrat ◽  
I. Sakho ◽  
M. Tchuenté
Keyword(s):  
Gaussian Elimination ◽  
Systolic Arrays ◽  
Elimination Algorithm

scholarly journals Gaussian Elimination versus Greedy Methods for the Synthesis of Linear Reversible Circuits

2021 ◽  
Vol 2 (3) ◽  
pp. 1-26
Author(s):  
Timothée Goubault De Brugière ◽  
Marc Baboulin ◽  
Benoît Valiron ◽  
Simon Martiel ◽  
Cyril Allouche
Keyword(s):  
Quantum Computing ◽  
State Of The Art ◽  
Gaussian Elimination ◽  
Random Operators ◽  
Lu Factorization ◽  
Large Problem ◽  
Computational Costs ◽  
Elimination Algorithm ◽  
Reversible Circuits ◽  
Greedy Methods

Linear reversible circuits represent a subclass of reversible circuits with many applications in quantum computing. These circuits can be efficiently simulated by classical computers and their size is polynomially bounded by the number of qubits, making them a good candidate to deploy efficient methods to reduce computational costs. We propose a new algorithm for synthesizing any linear reversible operator by using an optimized version of the Gaussian elimination algorithm coupled with a tuned LU factorization. We also improve the scalability of purely greedy methods. Overall, on random operators, our algorithms improve the state-of-the-art methods for specific ranges of problem sizes: The custom Gaussian elimination algorithm provides the best results for large problem sizes (n > 150), while the purely greedy methods provide quasi optimal results when n < 30. On a benchmark of reversible functions, we manage to significantly reduce the CNOT count and the depth of the circuit while keeping other metrics of importance (T-count, T-depth) as low as possible.

A SOLVER OF LINEAR SYSTEMS OF EQUATIONS (REBSM) FOR LARGE-SCALE ENGINEERING PROBLEMS

2012 ◽  
Vol 09 (01) ◽  
pp. 1240011 ◽  
Author(s):  
XIAO-WEI GAO ◽  
LINGJIE LI
Keyword(s):  
Large Scale ◽  
Solution Method ◽  
Gaussian Elimination ◽  
Systems Of Equations ◽  
The Finite Element Method ◽  
Elimination Algorithm ◽  
Engineering Problems ◽  
Sparse Systems ◽  
Linear Systems Of Equations ◽  
Element Method

In this paper, a novel linear equation solution method is proposed based on a row elimination back-substitution method (REBSM). The elimination and back-substitution procedures are carried out on individual row levels. The advantage of the proposed method is that it is much faster and requires less storage than the Gaussian elimination algorithm and, therefore, is capable of solving larger systems of equations. The method is particularly efficient for solving band diagonal sparse systems with symmetric or nonsymmetric coefficient matrices, and can be easily applied to popular numerical methods, such as the finite element method and the boundary element method. Detailed Fortran codes and examples are given to demonstrate the robustness and efficiency of the proposed method.

Benchmarking a Sparse Elimination Routine on the Cyber 205 and the Cray 1-S

1983 ◽  
Vol 23 (05) ◽  
pp. 743-745 ◽  
Author(s):  
P.T. Woo ◽  
John M. Levesque
Keyword(s):  
Gaussian Elimination ◽  
Assembly Language ◽  
Triangular Matrix ◽  
Floating Point ◽  
Full Potential ◽  
Vector Computer ◽  
Solution Vector ◽  
Elimination Algorithm ◽  
Lower Triangular ◽  
Vector Computers

Abstract An existing general sparse-elimination algorithm was benchmarked on the Cyber 205 and the Cray 1-S. Computation rates on the Cyber 205 reached 16. 7 million floating point operations per second (MFLOPS) and 9. 8 MFLOPS on the Cray 1-S. It is concluded that the existing algorithm does not exploit the full potential of the vector computer. Several schemes are under investigation to improve the performance of the algorithm. Introduction General-purpose sparse-elimination techniques have been used successfully in solving systems of linear equations in reservoir simulation. These techniques allow Gaussian elimination to be performed efficiently with any given matrix ordering. The implementation of these techniques on a scalar computer such as the IBM 370/158 and on an array processor such as the Floating Point Systems AP120B are described in Refs. 1 and 2. respectively. The purpose of this paper is to report the results of a benchmark test to determine the performance of a known sparse-elimination algorithm on the vector computers Cyber 205 and Cray 1-S. The ground rules for the benchmark were as follows.The sparse-elimination algorithm was to remain unchanged.The use of vectorizable FORTRAN or vector syntax FORTRAN was allowed.The use of assembly-language coding was not allowed unless the assembly-language code was part of a systems or mathematics library callable by FORTRAN.The alternate diagonal grid ordering was to be used. Implementation on the Vector Computer The algorithm NNF in the Yale sparse matrix packages was used for benchmarking. The matrix equation to be solved is A is factored into the lower triangular matrix, L, and the upper triangular matrix, U. The elements of A, L, and U are stored in a compressed storage mode as described by Gustavson. The solution vector x is obtained by forward substitution by using L, and backward substitution by using U. The Yale algorithm uses the row operations approach, which consists of two steps:multiply one equation by a nonzero number, andadd (or subtract) the equation from above to (or from) another equation. Step 2 reduces the nonzeros in the lower triangular portion of A to zeros as factorization proceeds. portion of A to zeros as factorization proceeds. SPEJ P. 743

scholarly journals Are There Elimination Algorithms for the Permanent?

1990 ◽  
Vol 19 (306) ◽  
Author(s):  
Carl Sturtivant
Keyword(s):  
Gaussian Elimination ◽  
Multivariate Polynomials ◽  
Elimination Algorithm ◽  
Algebraic Algorithms ◽  
No Elimination ◽  
Special Case

We define the class of elimination algorithms. These are algebraic algorithms for evaluating multivariate polynomials, and include as a special case, Gaussian elimination for evaluating the determinant. We show how to find the linear symmetries of a polynomial, defined appropriately, and use these methods to find the linear symmetries of the permanent and determinant. We show that in contrast to Gaussian elimination for the determinant, there is no elimination algorithm for the permanent.

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