Linear Algebra Libraries: BLAS, LAPACK, ScaLAPACK, PLASMA, MAGMA Shirley Moore CPS5401 Fall 2013 svmoore.pbworks.com November 12,

Learning Objectives After completing this lesson, you should be able to – List and describe advantages of using linear algebra libraries – List types of computations performed by linear algebra libraries – Describe functionality of the BLAS – Locate and use documentation on linear algebra libraries for your platform – Insert calls to linear algebra library routines into your program and compile and run the resulting program – Describe current research on numerical linear algebra for multicore and heterogeneous architectures 2

Numerical Linear Algebra Algorithms for performing matrix operations on computers Widely used in scientific, engineering, and financial applications Fundamental algorithms – Basic matrix and vector operations – LU decomposition – QR decomposition – Singular value decomposition – Eigenvalues 3

BLAS Basic Linear Algebra Subprograms De facto standard (all implementations use the same calling interface) First published in BLA Quick Reference Guide: Tuned versions implemented by vendors (Intel MKL, AMD ACML, Cray LibSci, IBM ESSL) Routines to perform basic operations such as vector and matrix multiplication 4

BLAS Functionality and Levels Level 1 This level contains vector operations of the form as well as scalar dot products and vector norms, among other things. Level 2 Th is level contains matrix-vector operations of the form as well as solving for with being triangular, among other things. Level 3 This level contains matrix-matrix operations of the form as well as solving for triangular matrices, among other things. This level contains the widely used General Matrix Multiply (GEMM) operation. 5

General Matrix Multiply (GEMM) where TRANSA and TRANSB determine if the matrices A and B are to be transposed M is the number of rows in matrix C and, depending on TRANSA, the number of rows in the original matrix A or its transpose. N is the number of columns in matrix C and, depending on TRANSB, the number of columns in the matrix B or its transpose. K is the number of columns in matrix A (or its transpose) and rows in matrix B (or its transpose). LDA, LDB and LDC specify the size of the first dimension of the matrices, as laid out in memory; meaning the memory distance between the start of each row/column, depending on the memory structure. Precision (x) – S for single, D for double, C for complex single, Z for complex double 6

LAPACK Linear Algebra PACKage De facto standard Successor to the linear equations and linear least-squares routines of LINPACK and the eigenvalue routines of EISPACK Routines for solving systems of linear equations, linear least squares, eigenvalue problems, and singular value decomposition Routines to implement the associated matrix factorizations such as LU, QR, Cholesky and Schur decomposition Handles real and complex matrices in both single and double precision Depends on the BLAS to effectively exploit caches on modern cache-based architectures Tuned versions implemented in vendor libraries (e.g., AMD ACML, Intel MKL, Cray LibSci, IBM ESSL) 7

LAPACK Naming Scheme A LAPACK subroutine name is in the form pmmaaa, where: – p is a one-letter code denoting the type of numerical constants used. S, D stand for real floating point arithmetic respectively in single and double precision, while C and Z stand for complex arithmetic with respectively single and double precision. – mm is a two-letter code denoting the kind of matrix expected by the algorithm. The actual data are stored in a different format depending on the specific kind; e.g., when the code DI is given, the subroutine expects a vector of length n containing the elements on the diagonal, while when the code GE is given, the subroutine expects an n×n array containing the entries of the matrix. – aaa is a one- to three-letter code describing the actual algorithm implemented in the subroutine, e.g. SV denotes a subroutine to solve linear system, while R denotes a rank-1 update. For example, the subroutine to solve a linear system with a general (non-structured) matrix using real double-precision arithmetic is called DGESV. For details, see the LAPACK User’s Guide at 8

ACML AMD Core Math Library core-math-library-acml/ core-math-library-acml/ ACML consists of the following main components: – A full implementation of Level 1, 2 and 3 Basic Linear Algebra Subprograms (BLAS), with optimizations for AMD Opteron processors. – A full suite of Linear Algebra (LAPACK) routines. – A comprehensive suite of Fast Fourier transform (FFTs) in single-, double-, single-complex and double-complex data types. – Fast scalar, vector, and array math transcendental library routines – Random Number Generators in both single- and double- precision /shared/acml on Griffin 9

ScaLAPACK Scalable Linear Algebra PACKage Library of high-performance linear algebra routines for parallel distributed memory machines Solves dense and banded linear systems, least squares problems, eigenvalue problems, and singular value problems Key ideas – block cyclic data distribution for dense matrices and a block data distribution for banded matrices, parameterizable at runtime – block-partitioned algorithms to ensure high levels of data reuse Efficient low-level communication implemented by BLACS (Basic Linear Algebra Communication Subprograms) Will run on any machine with BLAS, LAPACK, and BLACS 10

Current Efforts Parallel Linear Algebra Software for Multicore Architectures (PLASMA) – – icl.cs.utk.edu/plasma/ icl.cs.utk.edu/plasma/ Matrix Algebra on GPU and Multicore Architectures (MAGMA) – icl.cs.utk.edu/magma/ icl.cs.utk.edu/magma/ OpenBLAS –