Families of algorithms related to the inversion of a Symmetric Positive Definite matrix

Bientinesi, Paolo; Gunter, B. C.; Geijn, Robert A.

doi:10.1145/1377603.1377606

Cited by 54 publications

(42 citation statements)

References 27 publications

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“…However, libflame differs from LAPACK in two impor-tant ways. First, as mentioned, it provides families of algorithms for each operation so that the best can be chosen for a given circumstance [3]. Second, it provides a framework for building complete custom linear algebra codes.…”

Section: What Is Differentmentioning

confidence: 99%

“…In our publications and performance graphs, we do our best to dispel the myth that user-and programmer-friendly linear algebra codes cannot yield high performance. Our FLAME implementations of operations such as Cholesky factorization and triangular matrix inversion often outperform the corresponding implementations currently available in LAPACK [3]. In Section 3 and numerous papers, we give numerous examples of how performance is facilitated by the FLAME methodology.…”

Section: What Is Differentmentioning

confidence: 99%

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Introducing: The Libflame Library for Dense Matrix Computations

et al. 2019

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As part of the FLAME project, we have been dilligently developing new methodologies for analyzing, designing, and implementing linear algebra libraries. While we did not know it when we started, these techniques appear to solve many of the programmability problems that now face us with the advent of multicore and many-core architectures. These efforts have culminated in a new library, libflame, which strives to replace similar libraries that date back to the late 20th century. With this paper, we introduce the scientific computing community to this library. Why a New LibraryHow do we convince people that in programming simplicity and clarity -in short: what mathematicians call "elegance" -are not a dispensable luxury, but a crucial matter that decides between success and failure?Edsger W. DijkstraDuring the past decade, the FLAME project, a collaborative effort between The University of Texas at Austin and Universidad Jaime I de Castellon, has developed a unique methodology, notation, tools, and set of application programming interfaces (APIs) for deriving and representing linear algebra libraries. In an effort to better promote the techniques characteristic to the FLAME project, we have implemented a functional library that demonstrates findings and insights from the last decade of research. We call this library libflame. The primary purpose of libflame is to provide the scientific and numerical computing communities with a modern, high-performance dense linear algebra library that is extensible, easy to use, and available under an open source license. Its developers have published two books, numerous papers, and even more working notes over the last decade documenting the challenges and motivations that led to the APIs and implementations present within the libflame library [11]. Seasoned users within scientific and numerical computing circles will quickly recognize the general set of functionality targeted by libflame. In short, in libflame we wish to provide not only a framework for developing dense linear algebra solutions, but also a ready-made library that is, by almost any metric, easier to use and offers competitive (and in many cases superior) real-world performance when compared to the more traditional Basic Linear Algebra Subprograms (BLAS) [8,9,16] and Linear Algebra PACKage (LAPACK) libraries [1].The purpose of this article is to briefly introduce both the philosophy that underlies the library and the library itself. Evidence of how easily it can be retargeted to what are often considered "hostile" environments is presented in form of performance results on different architectures.

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Section: What Is Differentmentioning

confidence: 99%

Section: What Is Differentmentioning

confidence: 99%

Introducing: The Libflame Library for Dense Matrix Computations

et al. 2019

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“…For other operations, targeting sequential, multithreaded, and distributed memory architectures, we have consistently shown the benefit of having a choice of algorithms. Several illuminating examples, including derivations and performance comparisons, can be found in [GGHvdG01, BGM + 05, QOvdG03,vdGQO08,BGdG]. …”

Section: Performancementioning

confidence: 99%

Deriving dense linear algebra libraries

Bientinesi

Gunnels²,

Myers

et al. 2013

Form. Asp. Comput.

Self Cite

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Abstract. Starting in the late 1960s computer scientists including Dijkstra and Hoare advocated goaloriented programming and the formal derivation of algorithms. The chief impediment to realizing this for loop-based programs was that a priori determination of loop-invariants, a prerequisite for developing loops, was a task too complex for any but the simplest of operations. Around 2000, these techniques were for the first time successfully applied to the domain of high-performance dense linear algebra libraries. This has led to a multitude of papers, mostly published in the ACM Transactions for Mathematical Software, a system for the mechanical derivation of algorithms, and a high-performance linear algebra library, libflame, that includes more than a thousand variants of algorithms for more than a hundred linear algebra operations. To our knowledge, this success story has unfolded with limited awareness on the part the formal methods community. This paper reports on ten years of experience and is meant to raise that awareness.

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“…Moreover, as part of the FLAME project we have long advocated that it is important to have multiple algorithmic variants at our disposal so that the best algorithm can be chosen for each situation [4]. The FLAME methodology advocates systematic derivation of these variants [3,11].…”

Section: Accelerating the Cublasmentioning

confidence: 99%

Level-3 BLAS on a GPU: Picking the low hanging fruit

Igual

Quintana-Ortí

Geijn

2012

AIP Conference Proceedings

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The arrival of hardware accelerators has created a new gold rush to be the first to deliver their promise of high performance for numerical applications. Since they are relatively hard to program, with limited language and compiler support, it is generally accepted that one needs to roll up one's sleeves and tough it out, not unlike the early days of distributed memory parallel computing (or any other period after the introduction of a drastically different architecture). In this paper we remind the community that while this is a noble endeavor, there is a lot of low hanging fruit that can be harvested easily. Picking this low hanging fruit benefits the scientific computing community immediately and prototypes the approach that the further optimizations may wish to follow. We demonstrate this by focusing on a widely used set of operations, the level-3 BLAS, targeting the NVIDIA family of GPUs.Insanity: doing the same thing over and over again and expecting different results.

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Families of algorithms related to the inversion of a Symmetric Positive Definite matrix

Cited by 54 publications

References 27 publications

Introducing: The Libflame Library for Dense Matrix Computations

Introducing: The Libflame Library for Dense Matrix Computations

Deriving dense linear algebra libraries

Level-3 BLAS on a GPU: Picking the low hanging fruit

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