Set Of Test Matrices Research Articles

Sparse Matrix-Vector Multiplication on GPGPUs

The multiplication of a sparse matrix by a dense vector (SpMV) is a centerpiece of scientific computing applications: it is the essential kernel for the solution of sparse linear systems and sparse eigenvalue problems by iterative methods. The efficient implementation of the sparse matrix-vector multiplication is therefore crucial and has been the subject of an immense amount of research, with interest renewed with every major new trend in high-performance computing architectures. The introduction of General-Purpose Graphics Processing Units (GPGPUs) is no exception, and many articles have been devoted to this problem.With this article, we provide a review of the techniques for implementing the SpMV kernel on GPGPUs that have appeared in the literature of the last few years. We discuss the issues and tradeoffs that have been encountered by the various researchers, and a list of solutions, organized in categories according to common features. We also provide a performance comparison across different GPGPU models and on a set of test matrices coming from various application domains.

Incomplete LU Preconditioner Based on Max-Plus Approximation of LU Factorization

We present a new method for the a priori approximation of the orders of magnitude of the entries in the LU factors of a complex or real matrix $A$. This approximation is used to determine the positions of the largest entries in the LU factors of $A$, and these positions are used as the sparsity pattern for an incomplete LU factorization preconditioner. Our method uses max-plus algebra and is based solely on the moduli of the entries of $A$. We also present techniques for predicting which permutation matrices will be chosen by Gaussian elimination with partial pivoting. We exploit the strong connection between the field of Puiseux series and the max-plus semiring to prove properties of the max-plus LU factors. Experiments with a set of test matrices from the University of Florida Sparse Matrix Collection show that our max-plus LU preconditioners outperform traditional level of fill methods and have similar performance to those preconditioners computed with more expensive threshold-based methods.

The BiConjugate gradient method on GPUs

In a wide variety of applications from different scientific and engineering fields, the solution of complex and/or nonsymmetric linear systems of equations is required. To solve this kind of linear systems the BiConjugate Gradient method (BCG) is especially relevant. Nevertheless, BCG has a enormous computational cost. GPU computing is useful for accelerating this kind of algorithms but it is necessary to develop suitable implementations to optimally exploit the GPU architecture. In this paper, we show how BCG can be effectively accelerated when all operations are computed on a GPU. So, BCG has been implemented with two alternative routines of the Sparse Matrix Vector product (SpMV): the CUSPARSE library and the ELLR-T routine. Although our interest is focused on complex matrices, our implementation has been evaluated on a GPU for two sets of test matrices: complex and real, in single and double precision data. Experimental results show that BCG based on ELLR-T routine achieves the best performance, particularly for the set of complex test matrices. Consequently, this method can be useful as a tool to efficiently solve large linear system of equations (complex and/or nonsymmetric) involved in a broad range of applications.

Smt: a Matlab toolbox for structured matrices

We introduce the smt toolbox for Matlab. It implements optimized storage and fast arithmetics for circulant and Toeplitz matrices, and is intended to be transparent to the user and easily extensible. It also provides a set of test matrices, computation of circulant preconditioners, and two fast algorithms for Toeplitz linear systems.

Parameter-Estimation Algorithms for Characterizing a Class of Isotropic and Anisotropic Viscoplastic Material Models

A number of robust, and computationally efficient, algorithms arepresented for the development of an overall strategy to estimate thematerial parameters characterizing a class of complex viscoplasticmaterial models (i.e., rate dependent plastic flow, nonlinear kinematichardening, thermal/static recovery, isotropic and anisotropic, etc.). Theentire procedure is automated through the integrated software COMPARE(an acronym for COnstitutive Material PARameter Estimator) to enable thedetermination of an `optimum' set of material parameters by minimizingthe errors between the experimental test data and the predictedresponse. The key ingredients of COMPARE are: (i) primal analysisutilizing the unconditionally-stable, fully implicit, integration schemefor the models' underlying flow and evolutionary rate equations; (ii) sensitivity analysis utilizing a direct-differentiation approach (i.e.,explicit, `exact' expressions are derived); (iii) a gradient-basedoptimization technique of an error/cost function; and (iv) graphicaluser interface. The estimation of the material parameters is cast as aminimum-error, weighted-multi-objective, nonlinear optimization problemwith constraints. Comparison between the sensitivities obtained by theproposed direct scheme and those produced by conventional finitedifference techniques is presented to assess accuracy. Detailedderivations are given, together with the results generated by applyingthe developed algorithms to a comprehensive set of test matrices. Theseinclude constant strain-rate tension, creep, and relaxation tests, forboth isotropic as well as anisotropic behaviors.

Partitioning Rectangular and Structurally Unsymmetric Sparse Matrices for Parallel Processing

A common operation in scientific computing is the multiplication of a sparse, rectangular, or structurally unsymmetric matrix and a vector. In many applications the matrix-transpose-vector product is also required. This paper addresses the efficient parallelization of these operations. We show that the problem can be expressed in terms of partitioning bipartite graphs. We then introduce several algorithms for this partitioning problem and compare their performance on a set of test matrices.

Comparison of Two Algorithms for Solving Large Linear Systems

Assume that the coefficient matrix A in the system $Ax = b$ is large and sparse. Consider the following two algorithms: DS (where the system is solved by a direct use of Gaussian elimination) and IR (where the use of Gaussian elimination is combined with the use of a large drop tolerance and followed by iterative refinement). Assume that some sparse technique is implemented with both DS and IR. The performance of two codes, the NAG subroutines (which are based on DS) and the RECKU subroutines (which are based on IR) are compared on a wide set of test matrices. The comparison shows that the second algorithm, IR, performs better in general. The computing time and/or the storage needed may be reduced considerably when the IR algorithm is used. Moreover, this algorithm normally provides a reliable estimate of the accuracy of the computed solution. When the problems are time and storage consuming, IR is much better (it gives a reduction in the computing time of up to 10 times and a reduction in the storage of up to 2–3 times). It is shown that IR is very efficient when linear least-squares problems are solved by the use of augmented matrices.

Additional remarks on algorithm 52: a set of test matrices

Additional remarks on algorithm 52: a set of test matrices

Remarks on and certification of algorithm 52: a set of test matrices

Remarks on and certification of algorithm 52: a set of test matrices

Certification of Algorithm 52: A set of test matrices

Certification of Algorithm 52: A set of test matrices

Remark on algorithm 52: a set of test matrices

Remark on algorithm 52: a set of test matrices

Certification of algorithm 52: a set of test matrices

Certification of algorithm 52: a set of test matrices

Algorithm 52: A set of test matrices

Algorithm 52: A set of test matrices

A set of test matrices

A set of test matrices