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- Implemented support for the user manually overriding the automatic
  subconfiguration selection that happens at runtime. This override
  can be requested by setting the BLIS_ARCH_TYPE environment variable.
  The variable must be set to the arch_t id (as enumerated in
  bli_type_defs.h) corresponding to the desired subconfiguration. If a
  value outside this enumerated range is given, BLIS will abort with an
  error message. If the value is in the valid range but corresponds to a
  subconfiguration that was not activated at configure-time/compile-time,
  BLIS will abort with a (different) error message. Thanks to decandia50
  for suggesting this feature via issue #451.
- Defined a new function bli_gks_lookup_id to return the address of an
  internal data structure within the gks. If this address is NULL, then
  it indicates that the subconfig corresponding to the arch_t id passed
  into the function was not compiled into BLIS. This function is used
  in the second of the two abort scenarios described above.
- Defined the enumerated error code BLIS_UNINITIALIZED_GKS_CNTX, which
  is returned for the latter of the two abort scenarios mentioned above,
  along with a corresponding error message and a function to perform
  the error check.
- Added cpp macro branching to bli_env.c to support compilation of the
  auto-detect.x executable during configure-time. This cpp branch is
  similar to the cpp code already found in bli_arch.c and bli_cpuid.c.
- Cleaned up the auto_detect() function to facilitate easier maintenance
  going forward. Also added a convenient debug switch that outputs the
  compilation command for the auto-detect.x executable and exits.
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README.md

The BLIS cat is sleeping.

Build Status Build Status

Contents

Introduction

BLIS is a portable software framework for instantiating high-performance BLAS-like dense linear algebra libraries. The framework was designed to isolate essential kernels of computation that, when optimized, immediately enable optimized implementations of most of its commonly used and computationally intensive operations. BLIS is written in ISO C99 and available under a new/modified/3-clause BSD license. While BLIS exports a new BLAS-like API, it also includes a BLAS compatibility layer which gives application developers access to BLIS implementations via traditional BLAS routine calls. An object-based API unique to BLIS is also available.

For a thorough presentation of our framework, please read our ACM Transactions on Mathematical Software (TOMS) journal article, "BLIS: A Framework for Rapidly Instantiating BLAS Functionality". For those who just want an executive summary, please see the Key Features section below.

In a follow-up article (also in ACM TOMS), "The BLIS Framework: Experiments in Portability", we investigate using BLIS to instantiate level-3 BLAS implementations on a variety of general-purpose, low-power, and multicore architectures.

An IPDPS'14 conference paper titled "Anatomy of High-Performance Many-Threaded Matrix Multiplication" systematically explores the opportunities for parallelism within the five loops that BLIS exposes in its matrix multiplication algorithm.

For other papers related to BLIS, please see the Citations section below.

It is our belief that BLIS offers substantial benefits in productivity when compared to conventional approaches to developing BLAS libraries, as well as a much-needed refinement of the BLAS interface, and thus constitutes a major advance in dense linear algebra computation. While BLIS remains a work-in-progress, we are excited to continue its development and further cultivate its use within the community.

The BLIS framework is primarily developed and maintained by individuals in the Science of High-Performance Computing (SHPC) group in the Institute for Computational Engineering and Sciences at The University of Texas at Austin. Please visit the SHPC website for more information about our research group, such as a list of people and collaborators, funding sources, publications, and other educational projects (such as MOOCs).

Education and Learning

Want to understand what's under the hood? Many of the same concepts and principles employed when developing BLIS are introduced and taught in a basic pedagogical setting as part of LAFF-On Programming for High Performance (LAFF-On-PfHP), one of several massive open online courses (MOOCs) in the Linear Algebra: Foundations to Frontiers series, all of which are available for free via the edX platform.

What's New

  • BLIS awarded SIAM Activity Group on Supercomputing Best Paper Prize for 2020! We are thrilled to announce that the paper that we internally refer to as the second BLIS paper,

    "The BLIS Framework: Experiments in Portability." Field G. Van Zee, Tyler Smith, Bryan Marker, Tze Meng Low, Robert A. van de Geijn, Francisco Igual, Mikhail Smelyanskiy, Xianyi Zhang, Michael Kistler, Vernon Austel, John A. Gunnels, Lee Killough. ACM Transactions on Mathematical Software (TOMS), 42(2):12:1--12:19, 2016.

    was selected for the SIAM Activity Group on Supercomputing Best Paper Prize for 2020. The prize is awarded once every two years to a paper judged to be the most outstanding paper in the field of parallel scientific and engineering computing, and has only been awarded once before (in 2016) since its inception in 2015 (the committee did not award the prize in 2018). The prize was awarded at the 2020 SIAM Conference on Parallel Processing for Scientific Computing in Seattle. Robert was present at the conference to give a talk on BLIS and accept the prize alongside other coauthors. The selection committee sought to recognize the paper, "which validates BLIS, a framework relying on the notion of microkernels that enables both productivity and high performance." Their statement continues, "The framework will continue having an important influence on the design and the instantiation of dense linear algebra libraries."

  • Multithreaded small/skinny matrix support for dgemm now available! Thanks to contributions made possible by our partnership with AMD, we have dramatically accelerated gemm for double-precision real matrix problems where one or two dimensions is exceedingly small. A natural byproduct of this optimization is that the traditional case of small m = n = k (i.e. square matrices) is also accelerated, even though it was not targeted specifically. And though only dgemm was optimized for now, support for other datatypes and/or other operations may be implemented in the future. We've also added new graphs to the PerformanceSmall document to showcase multithreaded performance when one or more matrix dimensions are small.

  • Performance comparisons now available! We recently measured the performance of various level-3 operations on a variety of hardware architectures, as implemented within BLIS and other BLAS libraries for all four of the standard floating-point datatypes. The results speak for themselves! Check out our extensive performance graphs and background info in our new Performance document.

  • BLIS is now in Debian Unstable! Thanks to Debian developer-maintainers M. Zhou and Nico Schlömer for sponsoring our package in Debian. Their participation, contributions, and advocacy were key to getting BLIS into the second-most popular Linux distribution (behind Ubuntu, which Debian packages feed into). The Debian tracker page may be found here.

  • BLIS now supports mixed-datatype gemm! The gemm operation may now be executed on operands of mixed domains and/or mixed precisions. Any combination of storage datatype for A, B, and C is now supported, along with a separate computation precision that can differ from the storage precision of A and B. And even the 1m method now supports mixed-precision computation. For more details, please see our ACM TOMS journal article submission (current draft).

  • BLIS now implements the 1m method. Let's face it: writing complex assembly gemm microkernels for a new architecture is never a priority--and now, it almost never needs to be. The 1m method leverages existing real domain gemm microkernels to implement all complex domain level-3 operations. For more details, please see our ACM TOMS journal article submission (current draft).

What People Are Saying About BLIS

"I noticed a substantial increase in multithreaded performance on my own machine, which was extremely satisfying." ... "[I was] happy it worked so well!" (Justin Shea)

"This is an awesome library." ... "I want to thank you and the blis team for your efforts." (@Lephar)

"Any time somebody outside Intel beats MKL by a nontrivial amount, I report it to the MKL team. It is fantastic for any open-source project to get within 10% of MKL... [T]his is why Intel funds BLIS development." (@jeffhammond)

"So BLIS is now a part of Elk." ... "We have found that zgemm applied to a 15000x15000 matrix with multi-threaded BLIS on a 32-core Ryzen 2990WX processor is about twice as fast as MKL" ... "I'm starting to like this a lot." (@jdk2016)

"I [found] BLIS because I was looking for BLAS operations on C-ordered arrays for NumPy. BLIS has that, but even better is the fact that it's developed in the open using a more modern language than Fortran." (@nschloe)

"The specific reason to have BLIS included [in Linux distributions] is the KNL and SKX [AVX-512] BLAS support, which OpenBLAS doesn't have." (@loveshack)

"All tests pass without errors on OpenBSD. Thanks!" (@ararslan)

"Thank you very much for your great help!... Looking forward to benchmarking." (@mrader1248)

"Thanks for the beautiful work." (@mmrmo)

"[M]y software currently uses BLIS for its BLAS interface..." (@ShadenSmith)

"[T]hanks so much for your work on this! Excited to test." ... "[On AMD Excavator], BLIS is competitive to / slightly faster than OpenBLAS for dgemms in my tests." (@iotamudelta)

"BLIS provided the only viable option on KNL, whose ecosystem is at present dominated by blackbox toolchains. Thanks again. Keep on this great work." (@heroxbd)

"I want to definitely try this out..." (@ViralBShah)

Key Features

BLIS offers several advantages over traditional BLAS libraries:

  • Portability that doesn't impede high performance. Portability was a top priority of ours when creating BLIS. With virtually no additional effort on the part of the developer, BLIS is configurable as a fully-functional reference implementation. But more importantly, the framework identifies and isolates a key set of computational kernels which, when optimized, immediately and automatically optimize performance across virtually all level-2 and level-3 BLIS operations. In this way, the framework acts as a productivity multiplier. And since the optimized (non-portable) code is compartmentalized within these few kernels, instantiating a high-performance BLIS library on a new architecture is a relatively straightforward endeavor.

  • Generalized matrix storage. The BLIS framework exports interfaces that allow one to specify both the row stride and column stride of a matrix. This allows one to compute with matrices stored in column-major order, row-major order, or by general stride. (This latter storage format is important for those seeking to implement tensor contractions on multidimensional arrays.) Furthermore, since BLIS tracks stride information for each matrix, operands of different storage formats can be used within the same operation invocation. By contrast, BLAS requires column-major storage. And while the CBLAS interface supports row-major storage, it does not allow mixing storage formats.

  • Rich support for the complex domain. BLIS operations are developed and expressed in their most general form, which is typically in the complex domain. These formulations then simplify elegantly down to the real domain, with conjugations becoming no-ops. Unlike the BLAS, all input operands in BLIS that allow transposition and conjugate-transposition also support conjugation (without transposition), which obviates the need for thread-unsafe workarounds. Also, where applicable, both complex symmetric and complex Hermitian forms are supported. (BLAS omits some complex symmetric operations, such as symv, syr, and syr2.) Another great example of BLIS serving as a portability lever is its implementation of the 1m method for complex matrix multiplication, a novel mechanism of providing high-performance complex level-3 operations using only real domain microkernels. This new innovation guarantees automatic level-3 support in the complex domain even when the kernel developers entirely forgo writing complex kernels.

  • Advanced multithreading support. BLIS allows multiple levels of symmetric multithreading for nearly all level-3 operations. (Currently, users may choose to obtain parallelism via either OpenMP or POSIX threads). This means that matrices may be partitioned in multiple dimensions simultaneously to attain scalable, high-performance parallelism on multicore and many-core architectures. The key to this innovation is a thread-specific control tree infrastructure which encodes information about the logical thread topology and allows threads to query and communicate data amongst one another. BLIS also employs so-called "quadratic partitioning" when computing dimension sub-ranges for each thread, so that arbitrary diagonal offsets of structured matrices with unreferenced regions are taken into account to achieve proper load balance. More recently, BLIS introduced a runtime abstraction to specify parallelism on a per-call basis, which is useful for applications that want to handle most of the parallelism.

  • Ease of use. The BLIS framework, and the library of routines it generates, are easy to use for end users, experts, and vendors alike. An optional BLAS compatibility layer provides application developers with backwards compatibility to existing BLAS-dependent codes. Or, one may adjust or write their application to take advantage of new BLIS functionality (such as generalized storage formats or additional complex operations) by calling one of BLIS's native APIs directly. BLIS's typed API will feel familiar to many veterans of BLAS since these interfaces use BLAS-like calling sequences. And many will find BLIS's object-based APIs a delight to use when customizing or writing their own BLIS operations. (Objects are relatively lightweight structs and passed by address, which helps tame function calling overhead.)

  • Multilayered API, exposed kernels, and sandboxes. The BLIS framework exposes its implementations in various layers, allowing expert developers to access exactly the functionality desired. This layered interface includes that of the lowest-level kernels, for those who wish to bypass the bulk of the framework. Optimizations can occur at various levels, in part thanks to exposed packing and unpacking facilities, which by default are highly parameterized and flexible. And more recently, BLIS introduced sandboxes--a way to provide alternative implementations of gemm that do not use any more of the BLIS infrastructure than is desired. Sandboxes provide a convenient and straightforward way of modifying the gemm implementation without disrupting any other level-3 operation or any other part of the framework. This works especially well when the developer wants to experiment with new optimizations or try a different algorithm.

  • Functionality that grows with the community's needs. As its name suggests, the BLIS framework is not a single library or static API, but rather a nearly-complete template for instantiating high-performance BLAS-like libraries. Furthermore, the framework is extensible, allowing developers to leverage existing components to support new operations as they are identified. If such operations require new kernels for optimal efficiency, the framework and its APIs will be adjusted and extended accordingly.

  • Code re-use. Auto-generation approaches to achieving the aforementioned goals tend to quickly lead to code bloat due to the multiple dimensions of variation supported: operation (i.e. gemm, herk, trmm, etc.); parameter case (i.e. side, [conjugate-]transposition, upper/lower storage, unit/non-unit diagonal); datatype (i.e. single-/double-precision real/complex); matrix storage (i.e. row-major, column-major, generalized); and algorithm (i.e. partitioning path and kernel shape). These "brute force" approaches often consider and optimize each operation or case combination in isolation, which is less than ideal when the goal is to provide entire libraries. BLIS was designed to be a complete framework for implementing basic linear algebra operations, but supporting this vast amount of functionality in a manageable way required a holistic design that employed careful abstractions, layering, and recycling of generic (highly parameterized) codes, subject to the constraint that high performance remain attainable.

  • A foundation for mixed domain and/or mixed precision operations. BLIS was designed with the hope of one day allowing computation on real and complex operands within the same operation. Similarly, we wanted to allow mixing operands' numerical domains, floating-point precisions, or both domain and precision, and to optionally compute in a precision different than one or both operands' storage precisions. This feature has been implemented for the general matrix multiplication (gemm) operation, providing 128 different possible type combinations, which, when combined with existing transposition, conjugation, and storage parameters, enables 55,296 different gemm use cases. For more details, please see the documentation on mixed datatype support and/or our ACM TOMS journal paper on mixed-domain/mixed-precision gemm (linked below).

How to Download BLIS

There are a few ways to download BLIS. We list the most common four ways below. We highly recommend using either Option 1 or 2. Otherwise, we recommend Option 3 (over Option 4) so your compiler can perform optimizations specific to your hardware.

  1. Download a source repository with git clone. Generally speaking, we prefer using git clone to clone a git repository. Having a repository allows the user to periodically pull in the latest changes and quickly rebuild BLIS whenever they wish. Also, implicit in cloning a repository is that the repository defaults to using the master branch, which contains the latest "stable" commits since the most recent release. (This is in contrast to Option 3 in which the user is opting for code that may be slightly out of date.)

    In order to clone a git repository of BLIS, please obtain a repository URL by clicking on the green button above the file/directory listing near the top of this page (as rendered by GitHub). Generally speaking, it will amount to executing the following command in your terminal shell:

    git clone https://github.com/flame/blis.git
    
  2. Download a source repository via a zip file. If you are uncomfortable with using git but would still like the latest stable commits, we recommend that you download BLIS as a zip file.

    In order to download a zip file of the BLIS source distribution, please click on the green button above the file listing near the top of this page. This should reveal a link for downloading the zip file.

  3. Download a source release via a tarball/zip file. Alternatively, if you would like to stick to the code that is included in official releases, you may download either a tarball or zip file of any of BLIS's previous tagged releases. We consider this option to be less than ideal for most people since it will likely mean you miss out on the latest bugfix or feature commits (in contrast to Options 1 or 2), and you also will not be able to update your code with a simple git pull command (in contrast to Option 1).

  4. Download a binary package specific to your OS. While we don't recommend this as the first choice for most users, we provide links to community members who generously maintain BLIS packages for various Linux distributions such as Debian Unstable and EPEL/Fedora. Please see the External Packages section below for more information.

Getting Started

NOTE: This section assumes you've either cloned a BLIS source code repository via git, downloaded the latest source code via a zip file, or downloaded the source code for a tagged version release---Options 1, 2, or 3, respectively, as discussed in the previous section.

If you just want to build a sequential (not parallelized) version of BLIS in a hurry and come back and explore other topics later, you can configure and build BLIS as follows:

$ ./configure auto
$ make [-j]

You can then verify your build by running BLAS- and BLIS-specific test drivers via make check:

$ make check [-j]

And if you would like to install BLIS to the directory specified to configure via the --prefix option, run the install target:

$ make install

Please read the output of ./configure --help for a full list of configure-time options. If/when you have time, we strongly encourage you to read the detailed walkthrough of the build system found in our Build System guide.

Documentation

We provide extensive documentation on the BLIS build system, APIs, test infrastructure, and other important topics. All documentation is formatted in markdown and included in the BLIS source distribution (usually in the docs directory). Slightly longer descriptions of each document may be found via in the project's wiki section.

Documents for everyone:

  • Build System. This document covers the basics of configuring and building BLIS libraries, as well as related topics.

  • Testsuite. This document describes how to run BLIS's highly parameterized and configurable test suite, as well as the included BLAS test drivers.

  • BLIS Typed API Reference. Here we document the so-called "typed" (or BLAS-like) API. This is the API that many users who are already familiar with the BLAS will likely want to use. You can find lots of example code for the typed API in the examples/tapi directory included in the BLIS source distribution.

  • BLIS Object API Reference. Here we document the object API. This is API abstracts away properties of vectors and matrices within obj_t structs that can be queried with accessor functions. Many developers and experts prefer this API over the typed API. You can find lots of example code for the object API in the examples/oapi directory included in the BLIS source distribution.

  • Hardware Support. This document maintains a table of supported microarchitectures.

  • Multithreading. This document describes how to use the multithreading features of BLIS.

  • Mixed-Datatypes. This document provides an overview of BLIS's mixed-datatype functionality and provides a brief example of how to take advantage of this new code.

  • Performance. This document reports empirically measured performance of a representative set of level-3 operations on a variety of hardware architectures, as implemented within BLIS and other BLAS libraries for all four of the standard floating-point datatypes.

  • PerformanceSmall. This document reports empirically measured performance of gemm on select hardware architectures within BLIS and other BLAS libraries when performing matrix problems where one or two dimensions is exceedingly small.

  • Release Notes. This document tracks a summary of changes included with each new version of BLIS, along with contributor credits for key features.

  • Frequently Asked Questions. If you have general questions about BLIS, please read this FAQ. If you can't find the answer to your question, please feel free to join the blis-devel mailing list and post a question. We also have a blis-discuss mailing list that anyone can post to (even without joining).

Documents for github contributors:

  • Contributing bug reports, feature requests, PRs, etc. Interested in contributing to BLIS? Please read this document before getting started. It provides a general overview of how best to report bugs, propose new features, and offer code patches.

  • Coding Conventions. If you are interested or planning on contributing code to BLIS, please read this document so that you can format your code in accordance with BLIS's standards.

Documents for BLIS developers:

  • Kernels Guide. If you would like to learn more about the types of kernels that BLIS exposes, their semantics, the operations that each kernel accelerates, and various implementation issues, please read this guide.

  • Configuration Guide. If you would like to learn how to add new sub-configurations or configuration families, or are simply interested in learning how BLIS organizes its configurations and kernel sets, please read this thorough walkthrough of the configuration system.

  • Sandbox Guide. If you are interested in learning about using sandboxes in BLIS--that is, providing alternative implementations of the gemm operation--please read this document.

External Packages

Generally speaking, we highly recommend building from source whenever possible using the latest git clone. (Tarballs of each tagged release are also available, but we consider them to be less ideal since they are not as easy to upgrade as git clones.)

That said, some users may prefer binary and/or source packages through their Linux distribution. Thanks to generous involvement/contributions from our community members, the following BLIS packages are now available:

  • Debian. M. Zhou has volunteered to sponsor and maintain BLIS packages within the Debian Linux distribution. The Debian package tracker can be found here. (Also, thanks to Nico Schlömer for previously volunteering his time to set up a standalone PPA.)

  • Gentoo. M. Zhou also maintains the BLIS package entry for Gentoo, a Linux distribution known for its source-based portage package manager and distribution system.

  • EPEL/Fedora. There are official BLIS packages in Fedora and EPEL (for RHEL7+ and compatible distributions) with versions for 64-bit integers, OpenMP, and pthreads, and shims which can be dynamically linked instead of reference BLAS. (NOTE: For architectures other than intel64, amd64, and maybe arm64, the performance of packaged BLIS will be low because it uses unoptimized generic kernels; for those architectures, OpenBLAS may be a better solution.) Dave Love provides additional packages for EPEL6 in a Fedora Copr, and possibly versions more recent than the official repo for other EPEL/Fedora releases. The source packages may build on other rpm-based distributions.

  • OpenSuSE. The copr referred to above has rpms for some OpenSuSE releases; the source rpms may build for others.

  • GNU Guix. Guix has BLIS packages, provides builds only for the generic target and some specific x86_64 micro-architectures.

  • Conda. conda channel conda-forge has Linux, OSX and Windows binary packages for x86_64.

Discussion

You can keep in touch with developers and other users of the project by joining one of the following mailing lists:

  • blis-devel: Please join and post to this mailing list if you are a BLIS developer, or if you are trying to use BLIS beyond simply linking to it as a BLAS library. Note: Most of the interesting discussions happen here; don't be afraid to join! If you would like to submit a bug report, or discuss a possible bug, please consider opening a new issue on github.

  • blis-discuss: Please join and post to this mailing list if you have general questions or feedback regarding BLIS. Application developers (end users) may wish to post here, unless they have bug reports, in which case they should open a new issue on github.

Contributing

For information on how to contribute to our project, including preferred coding conventions, please refer to the CONTRIBUTING file at the top-level of the BLIS source distribution.

Citations

For those of you looking for the appropriate article to cite regarding BLIS, we recommend citing our first ACM TOMS journal paper (unofficial backup link):

@article{BLIS1,
   author      = {Field G. {V}an~{Z}ee and Robert A. {v}an~{d}e~{G}eijn},
   title       = {{BLIS}: A Framework for Rapidly Instantiating {BLAS} Functionality},
   journal     = {ACM Transactions on Mathematical Software},
   volume      = {41},
   number      = {3},
   pages       = {14:1--14:33},
   month       = {June},
   year        = {2015},
   issue_date  = {June 2015},
   url         = {http://doi.acm.org/10.1145/2764454},
}

You may also cite the second ACM TOMS journal paper (unofficial backup link):

@article{BLIS2,
   author      = {Field G. {V}an~{Z}ee and Tyler Smith and Francisco D. Igual and
                  Mikhail Smelyanskiy and Xianyi Zhang and Michael Kistler and Vernon Austel and
                  John Gunnels and Tze Meng Low and Bryan Marker and Lee Killough and
                  Robert A. {v}an~{d}e~{G}eijn},
   title       = {The {BLIS} Framework: Experiments in Portability},
   journal     = {ACM Transactions on Mathematical Software},
   volume      = {42},
   number      = {2},
   pages       = {12:1--12:19},
   month       = {June},
   year        = {2016},
   issue_date  = {June 2016},
   url         = {http://doi.acm.org/10.1145/2755561},
}

We also have a third paper, submitted to IPDPS 2014, on achieving multithreaded parallelism in BLIS (unofficial backup link):

@inproceedings{BLIS3,
   author      = {Tyler M. Smith and Robert A. {v}an~{d}e~{G}eijn and Mikhail Smelyanskiy and
                  Jeff R. Hammond and Field G. {V}an~{Z}ee},
   title       = {Anatomy of High-Performance Many-Threaded Matrix Multiplication},
   booktitle   = {28th IEEE International Parallel \& Distributed Processing Symposium
                  (IPDPS 2014)},
   year        = {2014},
   url         = {https://doi.org/10.1109/IPDPS.2014.110},
}

A fourth paper, submitted to ACM TOMS, also exists, which proposes an analytical model for determining blocksize parameters in BLIS (unofficial backup link):

@article{BLIS4,
   author      = {Tze Meng Low and Francisco D. Igual and Tyler M. Smith and
                  Enrique S. Quintana-Ort\'{\i}},
   title       = {Analytical Modeling Is Enough for High-Performance {BLIS}},
   journal     = {ACM Transactions on Mathematical Software},
   volume      = {43},
   number      = {2},
   pages       = {12:1--12:18},
   month       = {August},
   year        = {2016},
   issue_date  = {August 2016},
   url         = {http://doi.acm.org/10.1145/2925987},
}

A fifth paper, submitted to ACM TOMS, begins the study of so-called induced methods for complex matrix multiplication (unofficial backup link:

@article{BLIS5,
   author      = {Field G. {V}an~{Z}ee and Tyler Smith},
   title       = {Implementing High-performance Complex Matrix Multiplication via the 3m and 4m Methods},
   journal     = {ACM Transactions on Mathematical Software},
   volume      = {44},
   number      = {1},
   pages       = {7:1--7:36},
   month       = {July},
   year        = {2017},
   issue_date  = {July 2017},
   url         = {http://doi.acm.org/10.1145/3086466},
}

A sixth paper, submitted to ACM TOMS, revisits the topic of the previous article and derives a superior induced method (unofficial backup link:

@article{BLIS6,
   author      = {Field G. {V}an~{Z}ee},
   title       = {Implementing High-Performance Complex Matrix Multiplication via the 1m Method},
   journal     = {SIAM Journal on Scientific Computing},
   volume      = {42},
   number      = {5},
   pages       = {C221--C244},
   month       = {September}
   year        = {2020},
   issue_date  = {September 2020},
   url         = {https://doi.org/10.1137/19M1282040}
}

A seventh paper, submitted to ACM TOMS, explores the implementation of gemm for mixed-domain and/or mixed-precision operands (unofficial backup link:

@article{BLIS7,
   author      = {Field G. {V}an~{Z}ee and Devangi N. Parikh and Robert A. van~de~{G}eijn},
   title       = {Supporting Mixed-domain Mixed-precision Matrix Multiplication
within the BLIS Framework},
   journal     = {ACM Transactions on Mathematical Software},
   note        = {submitted}
}

Funding

This project and its associated research were partially sponsored by grants from Microsoft, Intel, Texas Instruments, AMD, HPE, Oracle, Huawei, and Facebook, as well as grants from the National Science Foundation (Awards CCF-0917167, ACI-1148125/1340293, CCF-1320112, and ACI-1550493).

Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

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