1. Linear Algebra Libraries: BLAS, LAPACK, ScaLAPACK, PLASMA, MAGMA
CPS5401 Fall 2015
November 19 Class

2. 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

3. 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

4. BLAS Basic Linear Algebra Subprograms
De facto standard (all implementations use the same calling interface)
First published in 1979
BLAS 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

5. BLAS Functionality and Levels
This level contains vector operations of the form
as well as scalar dot products and vector norms, among other things.
Level 2
This 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.

6. 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

7. LAPACK Linear Algebra PACKage www.netlib.org/lapack/ 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)

8. 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
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

9. Intel MKL Stands for Math Kernel Library
Vectorized and threaded linear algebra, FFTs, and statistics functions
Uses standard BLAS and LAPACK APIs
MIT’s FFTW C interface
Direct sparse solver (not standardized)
Support for AVX-512 Advanced Vector Extensions

10. ACML AMD Core Math Library
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

11. ScaLAPACK Scalable Linear Algebra PACKage www.netlib.org/scalapack/
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

12. Current Efforts
Parallel Linear Algebra Software for Multicore Architectures (PLASMA)
Matrix Algebra on GPU and Multicore Architectures (MAGMA)
OpenBLAS

13. MKL on Stampede Libraries in $TACC_MKL_DIR
(This environment variable should be defined
if you have the Intel compiler module loaded).
Examples in $TACC_MKL_DIR/examples
See documentation at

14. Scalapack
Introduction to MPI Chapter 10