Samuel D. Relton

The Design and Performance of Batched BLAS on Modern High-Performance Computing Systems

A current trend in high-performance computing is to decompose a large linear algebra problem into batches containing thousands of smaller problems, that can be solved independently, before collating the results. To standardize the interface to these routines, the community is developing an extension to the BLAS standard (the batched BLAS), enabling users to perform thousands of small BLAS operations in parallel whilst making efficient use of their hardware. We discuss the benefits and drawbacks of the current batched BLAS proposals and perform a number of experiments, focusing on a general matrix-matrix multiplication (GEMM), to explore their affect on the performance. In particular we analyze the effect of novel data layouts which, for example, interleave the matrices in memory to aid vectorization and prefetching of data. Utilizing these modifications our code outperforms both MKL1 CuBLAS2 by up to 6 times on the self-hosted Intel KNL (codenamed Knights Landing) and Kepler GPU architectures, for large numbers of double precision GEMM operations using matrices of size 2 × 2 to 20 × 20.

Estimating the Condition Number of the Frechet Derivative of a Matrix Function

The Frechet derivative

Optimized Batched Linear Algebra for Modern Architectures

L_f

Estimating the Largest Elements of a Matrix

of a matrix function

A BLOCK KRYLOV METHOD TO COMPUTE THE ACTION OF THE FRECHET DERIVATIVE OF A MATRIX FUNCTION ON A VECTOR WITH APPLICATIONS TO CONDITION NUMBER ESTIMATION

f : \mathbb{C}^{n\times n} \to \mathbb{C}^{n\times n}

Ranking the importance of nuclear reactions for activation and transmutation events

is used in a variety of applications and several algorithms are available for computing it. We define a condition number for the Frechet derivative and derive upper and lower bounds for it that differ by at most a factor

New Algorithms for Computing the Matrix Sine and Cosine Separately or Simultaneously

2

A Proposed API for Batched Basic Linear Algebra Subprograms

. For a wide class of functions we derive an algorithm for estimating the 1-norm condition number that requires

Higher order fréchet derivatives of matrix functions and the level-2 condition number

O(n^3)

A Comparison of Potential Interfaces for Batched BLAS Computations

flops given