Mathematical Software Libraries Research Articles

Scalability Issues Affecting the Design of a Dense Linear Algebra Library

This paper discusses the scalability of Cholesky, LU, and QR factorization routines on MIMD distributed memory concurrent computers. These routines form part of the ScaLAPACK mathematical software library that extends the widely used LAPACK library to run efficiently on scalable concurrent computers. To ensure good scalability and performance, the ScaLAPACK routines are based on block-partitioned algorithms that reduce the frequency of data movement between different levels of the memory hierarchy, and particularly between processors. The block cyclic data distribution, that is used in all three factorization algorithms, is described. An outline of the sequential and parallel block-partitioned algorithms is given. Approximate models of algorithms′ performance are presented to indicate which factors in the design of the algorithm have an impact upon scalability. These models are compared with timings results on a 128-node Intel iPSC/860 hypercube. It is shown that the routines are highly scalable on this machine for problems that occupy more than about 25% of the memory on each processor, and that the measured timings are consistent with the performance model. The contribution of this paper goes beyond reporting our experience: our implementations are available in the public domain.

Linear algebra software on IBM and CRAY computers

Mathematical software libraries represent collections of subprograms for the solution of frequently occurring numerical problems. The acceptance of mathematical software libraries strongly depends on the quality of their components, because the library routines are often used as building blocks in more complex computer codes. This approach can simplify code development and introduce know-how to the user's code. Results on the performance and accuracy of library subroutines from IMSL and NAG for matrix multiplication, solution of linear equations, and eigenvalue problems on a CRAY X-MP and an IBM 3081 are reported.

Linear algebra software on a vector computer

Mathematical software libraries (IMSL, NAG) supply subprograms for the solution of a broad class of frequently occurring numerical problems. Therefore, the library routines are normally used as building blocks in more complex computer codes. Users taking advantage of this approach can simplify code development and introduce expertise to their codes. The acceptance of mathematical software libraries is strongly related to the quality of their components. Results on the performance and accuracy of library subroutines for matrix multiplication, the solution of linear equations and eigenvalue problems on a CRAY X-MP are reported.

Some experiments in multitasking on an ELXSI system 6400

In the next six years, Sandia National Laboratories plans to acquire several of the most powerful supercomputers available at that particular time. We anticipate that these machines will have multitasking capabilities. To ensure that the Department of Energy programmatic goals will be met, it is essential that these machines be used efficiently. For this purpose, the Applied Mathematics Division and the Parallel Processing Division at Sandia have underway research programs to investigate the interaction between algorithms, the software environment, and advanced computer architecture for numerical-analysis applications in both mathematical-software libraries and large production codes. This report presents some preliminary results of running a code for solving two-point boundary-value problems in a multitasking environment on the ELXSI 6400.

Mathematical software libraries for vector computers

The principles of vector computation are reviewed including a definition of “depth of parallelism”. The latter concept is related to algorithm performance on various computer architectures (in particular the CRAY-1 and the FPS AP120B). The challenge of providing efficient mathematical software libraries on vector computers and the subsequent trade-offs introduced by vector architecture are discussed. Boeng's multi-level library approach to meet vector computer user's needs is explained. Some timing comparisons for software on the CRAY-1 are also given.

Notes from the second department of energy library Workshop

The U.S. Atomic Energy Commission (AEC) and its successors, first the U.S. Energy Research and Development Administration and now the U.S. Department of Energy (DOE) and Nuclear Regulatory Commission, has been and continues to be one of the nation's major purchasers and users of large scale scientific computers. Historically, each of the more than dozen computer centers at different laboratories evolved independently of the others so that each was self-contained. In particular, each computer center developed mathematical software libraries according to its own needs. In 1975, representatives for the mathematical software libraries at the various AEC computer centers met, with Argonne National Laboratory as the host, to hold the first Workshop on the Operational Aspects of Mathematical Software Libraries. Among the purposes of the first Workshop were: (1) to meet colleagues doing similar work at other AEC computer centers, (2) to share experiences in the management of mathematical software libraries, and (3) to discuss ideas and issues in the operation of libraries. The first Workshop was sufficiently rewarding that the participants felt it appropriate to hold a second Workshop in three years, an interval that would encompass sufficient progress in library operation that new experiences and ideas could be discussed.

Extending reliability (Extended Abstract>

Over the past decade, mathematical software libraries have matured from small, usually locally-assembled, collections of subroutines to large, commercially-provided libraries which are approaching the status of standards [Aird; Du Croz; Fox]**. Despite the high quality of such libraries and the obvious economic advantages of using routines whose development cost has been shared with many other users, applications programmers, when asked: “Why don't you use routine XYZ from IMSL, or from NAC, or from PORT?” frequently reply that library routines are too general, that they need a routine which takes advantage of special features of their problem, and that since they could not use a library routine without modifying it, they might as well write their own routine from scratch. In many, if not most, instances, the latter assertion could be easily refuted by a simple competition on selected test problems. However, the need for a routine adapted, or tailored, to a particular problem is more difficult to dismiss. It usually arises from considerations of efficiency, which may range from the perceived inefficiency of the presence of unused options in a routine to the practical impossibility of using a routine whose data representation is utterly incompatible with that needed in the rest of the applications program.

Evaluation of functions on microcomputers: square root

This paper treats the evaluation of one of the elementary functions on short wordlength computers. The setting is a binary fixed point short wordlength (8–16 bits) machine where the intent is to suggest improvements in ROM- or microcode-based software which include the square root function as part of a more general mathematical software library or for special computation in real-time applications. This paper focuses on the evaluation of square roots and features a careful treatment of Newton's method with linear initialization. Comparisons with other popular algorithms are made based on storage requirements, speed, and accuracy, with some indication of the effect that special hardware features have on the performance of these routines.