QR factorization of dense matrices is a ubiquitous tool in high performance computing (HPC). From solving linear systems and least squares problems to eigenvalue problems, and singular value decompositions, the impact of a high performance QR ...

The efficient utilization of mixed-precision numerical linear algebra algorithms can offer attractive acceleration to scientific computing applications. Especially with the hardware integration of low-precision special-function units designed for machine ...

This article describes a standard API for a set of Batched Basic Linear Algebra Subprograms (Batched BLAS or BBLAS). The focus is on many independent BLAS operations on small matrices that are grouped together and processed by a single routine, called a ...

With the acquisition and widespread use of more resources that rely on accelerator/wide vector–based computing, there has been a strong demand for science and engineering applications to take advantage of these latest assets. This, however, has been ...

Efficient processing of Irregular Matrices on Single Instruction, Multiple Data (SIMD)-type architectures is a persistent challenge. Resolving it requires innovations in the development of data formats, computational techniques, and implementations that ...

The SLATE (Software for Linear Algebra Targeting Exascale) library is being developed to provide fundamental dense linear algebra capabilities for current and upcoming distributed high-performance systems, both accelerated CPU-GPU based and CPU based. ...

We parallelize the LU factorization of a hierarchical low-rank matrix ( H -matrix) on a distributed-memory computer. This is much more difficult than the H -matrix-vector multiplication due to the dataflow of the factorization, and it is much harder than the ...

This work presents two implementations of linear solvers for distributed-memory machines with GPU accelerators—one based on the Cholesky factorization and one based on the LU factorization with partial pivoting. The routines are developed as part ...

This work presents an implementation of a linear least squares solver for distributed-memory machines with GPU accelerators, developed as part of the Software for Linear Algebra Targeting Exascale (SLATE) package. From the algorithmic standpoint, the ...

As parallel computers approach exascale, power efficiency in high-performance computing (HPC) systems is of increasing concern. Exploiting both the hardware features and algorithms is an effective solution to achieve power efficiency, and to ...

• Accelerates all three phases of the singular value decomposition using a GPU.
• ...

The increasing gap between memory bandwidth and computation speed motivates the choice of algorithms to take full advantage of today’s high performance computers. For dense matrices, the classic algorithm for the singular value ...

We formulate an implementation of a Jacobi iterative solver for sparse linear systems that iterates the distinct components of the solution with different precision in terms of mantissa length. Starting with very low accuracy, and using an inexpensive ...

The mixed-precision Cholesky QR (CholQR) can orthogonalize the columns of a dense matrix with the minimum communication cost. Moreover, its orthogonality error depends only linearly to the condition number of the input matrix. However, when the desired ...

A low-rank approximation of a dense matrix plays an important role in many applications. To compute such an approximation, a common approach uses the QR factorization with column pivoting (QRCP). Though the reliability and efficiency of QRCP have been ...

As modern hardware keeps evolving, an increasingly effective approach to develop energy efficient and high-performance solvers is to design them to work on many small size independent problems. Many applications already need this functionality, ...

As hardware evolves, an increasingly effective approach to develop energy efficient, high-performance solvers, is to design them to work on many small and independent problems. Indeed, many applications already need this functionality, especially for ...

This paper presents the design and implementation of several fundamental dense linear algebra (DLA) algorithms in OpenCL. In particular, these are linear system solvers and eigenvalue problem solvers. Further, we give an overview of the clMAGMA library, ...

We describe and analyze a novel symmetric triangular factorization algorithm. The algorithm is essentially a block version of Aasen's triangular tridiagonalization. It factors a dense symmetric matrix $A$ as the product $A=PLTL^{T}P^{T},$ where $P$ is a ...

The enormous gap between the high-performance capabilities of GPUs and the slow interconnect between them has made the development of numerical software that is scalable across multiple GPUs extremely challenging. We describe a successful methodology on ...

Four routines called DPOTF3i, i = a,b,c,d, are presented. DPOTF3i are a novel type of level-3 BLAS for use by BPF (Blocked Packed Format) Cholesky factorization and LAPACK routine DPOTRF. Performance of routines DPOTF3i are still increasing when the ...