1 Performance of Various Computers Using Standard Linear Equ

This report compares the performance of different computer systems in solving dense systems of linear equations. The comparison involves approximately a hundred computers, ranging from a Cray Y-MP to scientific workstations such as the Apollo and Sun to IB

|||||||| CS - 89 - 85||||||||

Performance of Various Computers Using Standard Linear Equations SoftwareJack J. Dongarra*Computer Science Department University of Tennessee Knoxville, TN 37996-1301 and Mathematical Sciences Section Oak Ridge National Laboratory Oak Ridge, TN 37831 CS - 89 - 85 January 18, 2001

* Electronic mail address: dongarra@cs.utk.edu. An up-to-date version of this report can be found at http:///benchmark/performance.ps This work was supported in part by the Applied Mathematical Sciences subprogram of the O ce of Energy Research, U.S. Department of Energy, under Contract DE-AC05-96OR22464, and in part by the Science Alliance a state supported program at the University of Tennessee. 1

This report compares the performance of different computer systems in solving dense systems of linear equations. The comparison involves approximately a hundred computers, ranging from a Cray Y-MP to scientific workstations such as the Apollo and Sun to IB

1

Performance of Various Computers Using Standard Linear Equations SoftwareComputer Science Department University of Tennessee Knoxville, TN 37996-1301 and Mathematical Sciences Section Oak Ridge National Laboratory Oak Ridge, TN 37831 January 18, 2001This report compares the performance of di erent computer systems in solving dense systems of linear equations. The comparison involves approximately a hundred computers, ranging from a Cray Y-MP to scienti c workstations such as the Apollo and Sun to IBM PCs.Jack J. Dongarra

Abstract

1 Introduction and ObjectivesThe timing information presented here should in no way be used to judge the overall performance of a computer system. The results re ect only one problem area: solving dense systems of equations. This report provides performance information on a wide assortment of computers ranging from the home-used PC up to the most powerful supercomputers. The information has been collected over a period of time and will undergo change as new machines are added and as hardware and software systems improve. The programs used to generate this data can easily be obtained over the Internet. While we make every attempt to verify the results obtained from users and vendors, errors are bound to exist and should be brought to our attention. We encourage users to obtain the programs and run the routines on their machines, reporting any discrepancies with the numbers listed here. The rst table reports three numbers for each machine listed (in some cases the numbers are missing because of lack of data). All performance numbers re ect arithmetic performed in full precision (usually 64-bit), unless noted. On some machines full precision may be single precision, such as the Cray, or double precision, such as the IBM. The rst number is for the LINPACK 1] benchmark program for a matrix of order 100 in a Fortran environment. The second number is for solving a system of equations of order 1000, with no restriction on the method or its implementation. The third number is the theoretical peak performance of the machine. LINPACK programs can be characterized as having a high percentage of oating-point arithmetic operations. The routines involved in this timing study, SGEFA and SGESL, use columnori

ented algorithms. That is, the programs usually reference array elements sequentially down a

This work was supported in part by the Applied Mathematical Sciences subprogram of the O ce of Energy Research, U.S. Department of Energy, under Contract DE-AC05-84OR21400, and in part by the Science Alliance a state supported program at the University of Tennessee. An up-to-date version of this report can be found at http:///benchmark/performance.ps

This report compares the performance of different computer systems in solving dense systems of linear equations. The comparison involves approximately a hundred computers, ranging from a Cray Y-MP to scientific workstations such as the Apollo and Sun to IB

January 18, 2001

2

column, not across a row. Column orientation is important in increasing e ciency because of the way Fortran stores arrays. Most oating-point operations in LINPACK take place in a set of subprograms, the Basic Linear Algebra Subprograms (BLAS) 3], which are called repeatedly throughout the calculation. These BLAS, referred to now as Level 1 BLAS, reference one-dimensional arrays, rather than two-dimensional arrays. In the rst case, the problem size is relatively small (order 100), and no changes were made to the LINPACK software. Moreover, no attempt was made to use special hardware features or to exploit vector capabilities or multiple processors. (The compilers on some machines may, of course, generate optimized code that itself accesses special features.) Thus, many high-performance machines may not have reached their asymptotic execution rates. In the second case, the problem size is larger (matrix of order 1000), and modifying or replacing the algorithm and software was permitted to achieve as high an execution rate as possible. Thus, the hardware had more opportunity for reaching near-asymptotic rates. An important constraint, however, was that all optimized programs maintain the same relative accuracy as standard techniques, such as Gaussian elimination used in LINPACK. Furthermore, the driver program (supplied with the LINPACK benchmark) had to be run to ensure that the same problem is solved. The driver program sets up the matrix, calls the routines to solve the problem, veri es that the answers are correct, and computes the total number of operations to solve the problem (independent of the method) as 2n3=3+ 2n2, where n= 1000. The last column is …… 此处隐藏：38668字，全部文档内容请下载后查看。喜欢就下载吧 ……