# Introduction¶

## Overview¶

Elemental is a library for distributed-memory dense linear algebra that draws heavily from the PLAPACK approach of building a graph of matrix distributions with a simple interface for redistributions (much of the syntax of the library is also inspired from FLAME). Elemental is also similar in functionality to ScaLAPACK, which is the very widely used effort towards extending LAPACK onto distributed-memory architectures. Unlike PLAPACK and ScaLAPACK, Elemental performs all computations using element-wise, rather than block, matrix distributions (please see the first journal publication on Elemental, Elemental: A new framework for distributed memory dense matrix computations, for a detailed discussion of this design choice). Some of the unique features of Elemental include distributed implementations of:

• Bunch-Kaufman and Bunch-Parlett for accurate symmetric factorization
• LU and Cholesky with full pivoting
• Column-pivoted QR and interpolative/skeleton decompositions
• Quadratically Weighted Dynamic Halley iteration for the polar decomposition
• Spectral Divide and Conquer Schur decomposition and Hermitian EVD
• Multi-shift Lanczos-based inverse iteration for computing pseudospectra
• Many algorithms for Singular-Value soft-Thresholding (SVT)
• Tall-skinny QR decompositions
• Hermitian matrix functions
• Sign-based Lyapunov/Ricatti/Sylvester solvers

For the sake of objectivity, here are a few reasons why one might want to use ScaLAPACK or PLAPACK instead:

• Elemental currently supports a divide-and-conquer scheme for parallel Schur decompositions, but not a QR-based algorithm. The Aggressive Early Deflation implementation of the Hessenberg QR algorithm in ScaLAPACK should be significantly more accurate and faster for small to medium sized matrices, but the divide-and-conquer scheme should likely be preferred for very large-scale Schur decompositions using several thousand cores. Ideally Elemental will eventually contain implementations of both algorithms, with an appropriate switching mechanism.
• Some applications exploit the block distribution format used by ScaLAPACK and PLAPACK in order to increase the efficiency of matrix construction. Though it is clearly possible to redistribute the matrix into an element-wise distribution format after construction, this might add an unnecessary level of complexity.
• Elemental is primarily intended to be used from C++11, though interfaces to other languages, such as C, Fortran 90, and Python, are in various stages of development. ScaLAPACK and PLAPACK routines are currently significantly more straightforward to call from C and Fortran.

Note

At this point, the vast majority of Elemental’s source code is in header files, so do not be surprised by the sparsity of the src/ folder; please also look in include/. There were essentially two reasons for moving as much of Elemental as possible into header files:

1. In practice, most executables only require a small subset of the library, so avoiding the overhead of compiling the entire library beforehand can be significant. On the other hand, if one naively builds many such executables with overlapping functionality, then the mainly-header approach becomes slower.
2. Though Elemental does not yet fully support computation over arbitrary fields, the vast majority of its pieces do. Moving templated implementations into header files is a necessary step in the process and also allowed for certain templating techniques to exploited in order to simplify the class hierarchy.

## Dependencies¶

• Functioning C++11 and ANSI C compilers.
• A working MPI2 implementation.
• BLAS and LAPACK (ideally version 3.3 or greater) implementations. If a sufficiently up-to-date LAPACK implementation is not provided, then a working F90 compiler is required in order to build Elemental’s eigensolvers (the tridiagonal eigensolver, PMRRR, requires recent LAPACK routines).
• CMake (version 2.8.8 or later).

If a sufficiently up-to-date C++11 compiler is used (e.g., recent versions of g++ or clang++), Elemental should be straightforward to build on Unix-like platforms. Building on Microsoft Windows platforms should also be possible with minor effort.