LINPACK

LINPACK

Author	Jack Don Gullah, gym Bunty, クリーヴ モラー (English version), Gilbert Stewart
Programming language	FORTRAN
Classification	Library
Official site	www.netlib.org/linpack/
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LINPACK (Linpack) is software library to perform a mathematical operation of the linear algebra on a computer.

Table of contents

Summary

Like MINPACK, EISPACK, I was developed as FORTRAN library in American Argonne Natl. Lab. It is really Jack Don Gullah, gym Bunty, クリーヴ モラー (English version), Gilbert Stewart to have developed. It was designed for super computers from the 1970s through the early 1980s [¹] and was replaced [²], afterwards more refined library LAPACK.

LINPACK performs a vector operation and line operation using BLAS (Basic Linear Algebra Subprograms, basic linear algebra subprogram group) library.

It is a beginning that the LINPACK benchmark to mention later was shown as some User Manual of LINPACK.

Benchmark

LINPACK benchmarks

Author	Jack Don Gullah, gym Bunty, クリーヴ モラー (English version), Gilbert Stewart
First edition	1979 (1979)
Official site	www.netlib.org/benchmark/performance.ps
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The LINPACK benchmark evaluates the floating point arithmetic performance of the system by a benchmark program based on LINPACK. Jack Don Gullah devised it, and measures speed to remove linear equation system [³] Ax = b of general n X n in science, engineering. The latest edition of this benchmark is used for a ranking for a performance price of the world high-speed computer in TOP500 [⁴].

It is intended to get an approximate value of the performance that a computer solves a problem of the real world. However, it is impossible to let every performance of the computing system be representative with one measurements and is one simplification to the last. Still LINPACK benchmark provides good revision for the peak performance level that a maker provides. Peak performance is the most high efficiency that the computer can achieve theoretically, and it is calculated at clock frequency, number of the fetch cycle, 1 cycle of one second by the feasible operation number of times. Real performance is always lower than peak performance [⁵]. The performance of the computer is the complicated problem that various factors were connected with each other mutually. It is the practice number of times per one second of the 64 bits floating point arithmetic (it usually multiplies it by with the addition) to measure in LINPACK benchmark (FLOPS). However, it is more likely to lower than the most high efficiency when the performance when I carried out true application sets LINPACK benchmark appropriately and measured [⁶].

History

It is a beginning that the LINPACK benchmark was shown as an appendix of the User Manual of LINPACK in 1979 [⁷]. The time to take when I solved a problem using LINPACK package was designed for the purpose of giving you the clue which a user estimated. Therefore I prepared for 23 kinds of issue of line of size 100 at first. The size was chosen in consideration of memory capacity at the time. -I generate 10,000 floating point number within 1 from 1 at random, and several dense people in charge form a line. And I use the LU resolution by the partial pivot choice. I relaxed a limit and, with the size of the line as 300 and 1000, did it, and optimization by the hardware architecture that came to implement a line and the operation of the vector was made use of afterwards [⁸]. In the case of a problem of the size of order 1000, I call the benchmark LINPACK 1000 and can revise algorithm to solve a problem [⁵]. The benchmark which solved a problem of any size came up in 1991 [⁹]. I came to be provided, and TOP500 list came to be shown to the extreme performance of the super computer on a near value in this two years later.

Concrete benchmark

LINPACK 100

An original benchmark announced in User Manual [¹⁰] of LINPACK in 1979 is approximately near. I solve a problem in method of elimination of the gauss by the partial pivot choice and perform the floating point arithmetic of the 2/3ⁿ3 + 2n² time. n is a degree of line A defining the problem with 100. Because size is small, and flexibility of the software was lacking, I cannot draw limit performance with many latest computers. However, I am helpful to some extent to estimate the performance of the calculation-centered user's program [⁵].

LINPACK 1000

Because I increased the size of the problem and assumed the degree of the line 1000 and I could change algorithm and did it, I became able to draw the performance that the limit of the computer was near. A point that the remaining limit does not lower relative precision and the operation number of times are points to be 2/3ⁿ3 + 2n² time (n = 1000) [⁵].

HPLinpack

The conventional benchmark is not suitable for the performance measurement of the parallel computer [¹¹]. Therefore new benchmark Linpack' s Highly Parallel Computing benchmark for parallel computers or HPLinpack benchmark was devised. A value to be able to draw the most high efficiency with the machine can make size n big in HPLinpack. The operation number of times is 2/3ⁿ3 + 2n² time, but the algorithm is selectable. Because the algorithm of Strassen reduces the operation number of times, I am not usable [¹²]. It is necessary for the next ceremony to come to be managed as for the precision.

${\displaystyle {\lVert Ax-b\rVert \over \lVert A\rVert \lVert x\rVert n\epsilon }\leq O(1)}$

Here ${\displaystyle \epsilon}$ Precision, n of the はその machine the size [¹³] of the problem,${\displaystyle \lVert \cdot \rVert}$ は line norm,${\displaystyle O(1)}$ It is は O- scale.

About each system, I report the next numerical value [⁵].

• R_max: Performance (GFLOPS) about the problem that is biggest with the machine
• N_max: Size of the biggest problem with the machine
• Size of the problem when it becomes half performance of N_1/2: Rmax
• R_peak: Theoretical peak performance (GFLOPS) of the machine

Using these results, a list of TOP500 is updated 2 times a year [⁴].

Implementation

I comment on the basic rule of the benchmark in the foregoing paragraph. The implementation of the program based on those rules varies, and there are implementation [¹⁴] by FORTRAN, implementation [¹⁵] by the C language, implementation [¹⁶] by the JAVA.

HPL

HPL implemented HPLinpack by C language, and it is originally a thing written as guidelines, but is used widely in TOP500. In addition, it does not have the problem to use technology and the package except HPL. HPL generates linear equation system of the n next and clears it up by the LU resolution by the partial pivot choice. With the need [¹⁷] that installs MPI and BLAS or VSIPL to use it.

If say roughly, there is the following characteristic in this algorithm [¹⁸]; [¹⁹].

• I perform periodic data distribution by a two-dimensional block.
• The LU resolution using kind of the right-looking method with the look-ahead of various depth
• The recursive panel resolution
• The panel broadcast has six kinds of methods.
• Swap broadcast algorithm to reduce a bandwidth
• Retreat substitution with the look-ahead of depth 1

Criticism

As for the factor that LINPACK benchmark succeeded, I include that the database which accumulated a fact that a comparison is easy, previous measurements exists [²¹] to generate a scalability [²⁰] of HPLinpack, one numerical value. However, and I provided the performance level that only the programmer who gave optimization to LINPACK benchmark steadily from the release direct next could achieve and criticism that there was not a meaning was poured on the optimization only with the machine [²²]. In addition, as for the problem solving, there is the criticism not to be the linear equation thing that pro-it, it represents the technology calculation whole of the dense coefficient line [²³]. Only peak performance and the number of of the CPU the CPU's are emphasized, and the Jack Don Gullah who promoted LINPACK benchmark states that stress to a bandwidth and the network is not enough [²⁴]. Tom ダニング of the Institute for U.S. national super computer application stated, "most all the members who were one of the interesting phenomena, and knew it laughed at it, and they understood the limit, but a kind of mind share produces it because we crossed it for several years, and we came using the value" about LINPACK benchmark [²⁵]. , because "it is important that I clarify the performance of the system from various aspects," according to the Don Gullah, "the sponsor of TOP500 explores a method to expand the range of the benchmark positively" [²⁰]. As a thing having high possibility about the expansion of the benchmark of TOP500 with the HPC challenge benchmark [²⁶]. In addition, the unit called Traversed Edges Per Second (English version) (TEPS) which was a performance level of the Graph500 benchmark to depend on communication performance for more came to be used because the FLOPS value to evaluate in LINPACK had low contribution ratio of the communication performance.

The matter of time that the measurement requires

According to Jack Don Gullah, it is said that time comes to suffer from the measurement when I am going to get a good performance level in HPLinpack. In a meeting held in 2010, he performed expectation that it came to take two and a half days in the measurement within "several years" [²⁷].

Footnote

1. ^ Matlis, Jan (May 30, 2005). "Sidebar: The Linpack Benchmark". ComputerWorld. http://www.computerworld.com/s/article/102050/Sidebar_The_Linpack_Benchmark 
2. ^ Markoff, John (September 22, 1991). "Technology; Measuring How Fast Computers Really Are". New York Times. http://select.nytimes.com/gst/abstract.html?res=F20616FF39550C718EDDA00894D9494D81 
3. But a coefficient line dense ^. Generally, pro-a local low sparse line of the reference described by a connection list, the large-scale problem solved by law or a finite element method for a difference hardly receives the benefit of the cache memory. Therefore it is not necessarily a thing indicating the performance of the true application (in other words, performance is decided by memory band width.), and what think as one of the indexes will be proper.
4. ^ ^{a b} "The Linpack Benchmark, TOP500 Supercomputing Sites." January 11, 2012 reading.
5. ^ ^{a b c d e} Dongarra, Jack J.; Luszczek, Piotr; Petitet, Antoine (2003), "The LINPACK Benchmark: past, present and future," it is Concurrency and Computation: Practice and Experience (John Wiley & Sons, Ltd.): 803–820, http://www.netlib.org/utk/people/JackDongarra/PAPERS/hplpaper.pdf 
6. ^ Jack Dongarra interview by Sander Olson, http://nextbigfuture.com/2010/06/jack-dongarra-interview-by-sander-olson.html 
7. ^ Dongarra, J.J.; Moler, C.B.; Bunch, J.R.; Stewart, G.W. (1979), LINPACK: users' guide, SIAM, http://books.google.ch/books?id=AmSm1n3Vw0cC&lpg=PR5&ots=EDFdqJhr8x&dq=info%3Ahttp%3A%2F%2Fs3da3171290b34600.scholar.google.com%2F0&lr&pg=SL2-PA1#v=onepage&q&f=false 
8. ^ Dongarra, Jack (1988), "The LINPACK benchmark: An explanation" Supercomputing (Springer Berlin/Heidelberg): 456–474, http://www.netlib.org/utk/people/JackDongarra/PAPERS/The-LINPACK-Benchmark-An-Explanation.pdf 
9. ^ High Performance Linpack Benchmark, http://icl.cs.utk.edu/graphics/posters/files/SC2010-HPL.pdf January 10, 2012 reading. 
10. ^ LINPACK users' manual
11. ^ Bailey, D.H.; Barszcz, E.; Barton, J.T.; Browning, D.S.; Carter, R.L.; Dagum, L.; Fatoohi, R.A.; Frederickson, P.O. et al. (1991), The NAS parallel benchmarks summary and preliminary results, "Supercomputing '91 is Supercomputing: Proceedings of the 1991 ACM/IEEE Conference" 158–165, http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5348941&isnumber=5348846 
12. ^ "LINPACK FAQ - Can I use Strassen' s Method when doing the matrix multiples in the HPL benchmark or for the Top500 run?." January 12, 2012 reading.
13. ^ "LINPACK FAQ - To what accuracy must be the solution conform?." January 12, 2012 reading.
14. ^ "Linpack benchmark program in Fortran". January 12, 2012 reading.
15. ^ "Linpack benchmark program in C". January 12, 2012 reading.
16. ^ "Linpack benchmark program in JAVA". January 12, 2012 reading.
17. ^ "HPL - A Portable Implementation of the High-Performance Linpack Benchmark for Distributed-Memory Computers." January 12, 2012 reading.
18. ^ "HPL algorithm." January 12, 2012 reading.
19. ^ "HPL overview." January 12, 2012 reading.
20. ^ ^{a b} Meuer, Martin (May 24, 2002). "An interview with supercomputer legend Jack Dongarra". January 13, 2012 reading.
21. ^ Haigh, Thomas (2004), An interview with Jack J. Dongarra, http://history.siam.org/pdfs2/Dongarra_%20returned_SIAM_copy.pdf, "LINPACK is a benchmark that people often cite because there' s such a historical data base of information there, because it' s fairly easy to run, it' s fairly easy to understand, and it captures in some sense the best and worst of programming." 
22. ^ Hammond, Steven (1995), Beyond Machoflops: Getting MPPs Into the Production Environment, http://nldr.library.ucar.edu/repository/collections/TECH-NOTE-000-000-000-227 
23. ^ Gahvari, Hormozd; Hoemmen, Mark; Demmel, James; Yelick, Katherine (2006), "Benchmarking Sparse Matrix-Vector Multiply in Five Minutes," it is SPEC Benchmark Workshop, http://bebop.cs.berkeley.edu/pubs/gahvari2007-spmvbench-spec.pdf 
24. ^ Dongarra, Jack J. (2007), "The HPC Challenge Benchmark: A Candidate for Replacing Linpack in the Top500?," it is SPEC Benchmark Workshop, http://www.spec.org/workshops/2007/austin/slides/Keynote_Jack_Dongarra.pdf 
25. ^ Christopher Mims (November 8, 2010). "Why China's New Supercomputer Is Only Technically the World's Fastest". September 22, 2011 reading.
26. ^ Luszczek, Piotr; Dongarra, Jack J.; Koester, David; Rabenseifner, Rolf; Lucas, Bob; Kepner, Jeremy; Mccalpin, John; Bailey, David et al. (2005), Introduction to the HPC Challenge Benchmark Suite, http://icl.cs.utk.edu/projectsfiles/hpcc/pubs/hpcc-challenge-benchmark05.pdf 
27. ^ Dongarra, Jack J. (2010), "LINPACK Benchmark with Time Limits on Multicore & GPU Based Accelerators," it is http://www.netlib.org/utk/people/JackDongarra/SLIDES/isc-talk-06102.pdf 

Outside link

• LINPACK
• BLAS
• TOP500 Supercomputing Sites
• Linpack Benchmark - - JAVA Version a web-based LINPACK benchmark
• The HPL benchmark used in the TOP500

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