Basic Linear Algebra Subprograms for Fortran Usage

Abstract-This paper examines the mapping of algorithms encountered when solving dense linear systems and linear leastsquares problems to a custom Linear Algebra Processor. Specifically, the focus is on Cholesky, LU (with partial pivoting), and QR factorizations. As part of the study, we expose the benefits of redesigning floating point units and their surrounding datapaths to support these complicated operations. We show how adding moderate complexity to the architecture greatly alleviates complexities in the algorithm. We study design trade-offs and the effectiveness of architectural modifications to demonstrate that we can improve power and performance efficiency to a level that can otherwise only be expected of full-custom ASIC designs.A feasibility study shows that our extensions to the MAC units can double the speed of required vector-norm operations while reducing energy by 60%. Similarly, up to 20% speedup with 15% savings in energy can be achieved for LU factorization. We show how such efficiency is maintained even in the complex inner kernels of these operations.

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“…However, it still needs to perform division. More details about how this is computed in software are discussed in [15], [17].…”

Section: In the First Iteration We Partitionmentioning

confidence: 99%

Floating Point Architecture Extensions for Optimized Matrix Factorization

Pedram

Gerstlauer

Geijn

2013

2013 IEEE 21st Symposium on Computer Arithmetic

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“…• ATLAS [26] and Sparsity [18] optimize the dense and sparse BLAS [19], respectively. ATLAS relies on a one-time expensive installation, after which its use at runtime engenders no overhead.…”

Section: Related Workmentioning

confidence: 99%

Self-Adapting Numerical Software and Automatic Tuning of Heuristics

Dongarra¹,

Eijkhout²

2003

Lecture Notes in Computer Science

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Self-Adapting Numerical Software (SANS) systems aim to bridge the knowledge gap that exists between the expertise of domain scientists, and the know-how that is needed to fulfill efficiently their computational demands. This know-how extends to algorith choice, computational grid utilization, and use of properly optimized kernels. A SANS system is a piece of meta software that mediates between the application program and the computational platform so that application scientists -with disparate levels of knowledge of algorithmic and programmatic complexities of the underlying numerical software -can easily realize numerical solvers and efficiently solve their problem.The main component of a SANS system is an Intelligent Agent that automates method selection based on data, algorithm and system attributes. The IA uses heuristics to make its decisions. In this paper we explain how the heuristics of the IA can be tuned over time by redundant testing and using the nature of many applications. IntroductionIn numerous technologically important areas, such as aerodynamics (vehicle design), electrodynamics (semiconductor device design), magnetohydrodynamics (fusion energy device design), and porous media (petroleum recovery), production runs on expensive, high-end systems last for hours or days, and a major portion of the execution time is usually spent inside of numerical routines, such as for the solution of large-scale nonlinear and linear systems that derive from discretized systems of nonlinear partial differential equations. These numerical parts of the code can contain a large number of tuning parameters, the choice of which greatly influences the efficiency of the total code, or can even make the difference between obtaining a solution and obtaining none.Such numerical concerns, however, are artifactual from the perspective of the scientific and engineering users, who are usually more concerned with modeling and discretization issues. The classic response to numerics was to encode the requisite mathematical, algorithmic and programming expertise into libraries that could be easily reused by a broad spectrum of domain scientists. However, in high-performance computing this solution is no longer sufficient. There is typically more than one algorithm for the stated purpose, and since several levels of algorithms are needed in a large-scale application; the different algorithms can have interlocking parameter settings, and the availability of parallel computing platforms influences algorithmic decisions. Since the difference in performance between an optimal choice of algorithm and hardware, and a less than optimal one, can span orders of magnitude, it is unfortunate that selecting the right solution strategy requires specialized knowledge of both numerical analysis and of computing platform characteristics. Our SANS system aims to assist the application user to navigate this maze of computational possibilities. *

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“…For computers that require greater vector lengths to achieve maximum performance it is possible to write (4) as a set of "vector = vector + scalar*vector" (SAXPY [31]) operations of length n 2 to N~ or even n 2 N to N 3 [2]. It is also possible to perform the first half-transformation (4a, 6) efficiently on a parallel architecture, by generating and processing subsets of integrals (such as [//i/|Aa], V A > o and fixed /j > v] on each processor.…”

Section: Computational Considerationsmentioning

confidence: 99%

Integral processing in beyond‐Hartree‐Fock calculations

Taylor

1987

Int J of Quantum Chemistry

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The increasing rate at which improvements in processing capacity outstrip improvements in input/output performance of large computers has led to recent attempts to bypass generation of a disk-based integral file. The "direct" SCF method of Almlof and co-workers represents a very successful implementation of this approach. -The present work is concerned with the extension of this general approach to CI and MCSCF calculations. After a discussion of the particular types of MO integrals for which -at least for most current generation machines -disk-based storage seems unavoidable, it is shown how all the necessary integrals can be obtained as matrix elements of Coulomb and exchange operators that can be calculated using a direct approach. Computational implementations of such a scheme are discussed.

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Basic Linear Algebra Subprograms for Fortran Usage

Cited by 1,277 publications

References 6 publications

Floating Point Architecture Extensions for Optimized Matrix Factorization

Floating Point Architecture Extensions for Optimized Matrix Factorization

Self-Adapting Numerical Software and Automatic Tuning of Heuristics

Integral processing in beyond‐Hartree‐Fock calculations

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