Utilizing Quadruple-Precision Floating Point Arithmetic Operation for the Krylov Subspace Methods. SIAM Conference on Applied 1
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1 Utilizing Quadruple-Precision Floating Point Arithmetic Operation for the Krylov Subspace Methods SIAM Conference on Applied 1
2 Outline Krylov Subspace methods Results of Accurate Computing Tools for Accurate Computing Cost of Quadruple Floating Point Arithmetic Conclusion SIAM Conference on Applied 2
3 Krylov Subspace methods Series of residual vectors r 0, r 1, r 2,, r k, Finding a basis: they should be orthogonal Converge at most N iterations if its computation is accurate SIAM Conference on Applied 3
4 To test What happens to Krylov Subspace Methods in Accurate Computing We Compare Convergence History in different Mantissa s bit for changing Computing Accuracy Omni Fortran Compiler is used for this purpose SIAM Conference on Applied 4
5 Test problem SIAM Conference on Applied 5
6 Condition Number norm norm smallest i i i i largest i i SIAM Conference on Applied 6
7 BiCG Gamma = 1.3 SIAM Conference on Applied 7
8 BiCG Gamma = 1.3 SIAM Conference on Applied 8
9 Alpha in BiCG Gamma = 1.3 SIAM Conference on Applied 9
10 Beta in BiCG Gamma = 1.3 SIAM Conference on Applied 10
11 BiCG Gamma = 1.7 SIAM Conference on Applied 11
12 BiCG Gamma = 2.1 SIAM Conference on Applied 12
13 BiCG Gamma = 2.5 SIAM Conference on Applied 13
14 BiCG Gamma = 2.5 SIAM Conference on Applied 14
15 Alpha in BiCG Gamma = 2.5 SIAM Conference on Applied 15
16 Beta in BiCG Gamma = 2.5 SIAM Conference on Applied 16
17 CGS Gamma = 1.3 SIAM Conference on Applied 17
18 BiCGSTAB Gamma = 1.3 SIAM Conference on Applied 18
19 BiCGSTAB Gamma = 2.5 SIAM Conference on Applied 19
20 GPBiCG Gamma = 2.5 SIAM Conference on Applied 20
21 Observations Fast and smooth convergence are gained from More accurate computations. Required Mantissa is based on the problems: BiCG 53 bit for Gamma = bit for bit for bit for 2.5 Required Mantissa depends on Algorithms: BiCG 200 bit and 190 iterations CGS 300 bit and 160 x BiCGSTAB 1500 bit and 210 x GPBiCG 300 bit and 310 ( Gamma = 2.5 ) SIAM Conference on Applied 21
22 Tools for Accurate Computing Multiple Precision Package (Gnu MP) Symbolic Computing (Computer Algebra) Interval Arithmetic Quadruple Floating-Point Operations SIAM Conference on Applied 22
23 Sun Enterprise 3000 n=10000 MATVEC Double with ILU 4.69*10-3 ( 17.5% ) Quad ( 24.7% ) ratio 31.7 MATVECT 4.89*10-3 ( 18.2% ) ( 26.4% ) 32.5 DAXPY DDOT DNORM2 PSOLVE PSOLVET 7.47*10-3 ( 27.8% ) 4.87*10-3 ( 18.1% ) 4.92*10-3 ( 18.3% ) ( 48.1% ) Total 2.68* SIAM Conference on Applied 23
24 HITACHI MP5800 n=10000 MATVEC Double with ILU 0.135*10-2 ( 16.9% ) Quad *10-2 ( 27.3% ) ratio 3.4 MATVECT DAXPY DDOT DNORM2 PSOLVE PSOLVET 0.134*10-2 ( 16.8% ) 0.244*10-2 ( 30.6% ) 0.146*10-2 ( 18.3% ) 0.135*10-2 ( 16.9% ) 0.472*10-2 ( 27.6% ) 0.719*10-2 ( 42.0% ) Total 0.796* * SIAM Conference on Applied 24
25 Fujitsu VPP800 n=10000 MATVEC MATVECT DAXPY DDOT DNORM2 PSOLVE PSOLVET Double with ILU 3.92*10-5 ( 1.06% ) 3.93*10-5 ( 1.07% ) 1.21*10-3 ( 32.9% ) 1.35*10-3 ( 36.7% ) 1.03*10-3 ( 28.0% ) Quad. 4.52*10-3 ( 11.9% ) 4.52*10-3 ( 11.9% ) 2.86*10-2 ( 75.6% ) ratio Total 3.67* * SIAM Conference on Applied 25
26 HITACHI SR8000 n=10000 MATVEC Double with ILU 0.101*10-3 ( 3.5% ) Quad *10-3 ( 11.3% ) ratio 6.0 MATVECT DAXPY DDOT DNORM2 PSOLVE PSOLVET 0.996*10-4 ( 3.4% ) 0.781*10-3 ( 27.2% ) 0.738*10-3 ( 25.7% ) 0.115*10-2 ( 40.9% ) 0.601*10-3 ( 11.1% ) 0.419*10-2 ( 77.5% ) Total 0.287* * SIAM Conference on Applied 26
27 Double with ILU vs Quadruple Sun ( WS ) Double with ILU 0.268*10-1 Quad ratio 22.4 VPP500 ( Vector ) VPP300 * ( Vector ) VPP800 ( Vector ) SR8000 ( SMP ) MP5800 ( Mainframe ) 0.341* * * * * * * SIAM Conference on Applied 27
28 Conclusion (tentative) 1. Fast and Smooth Convergence are gained from Accurate Computing. 2. Quadruple arithmetic is economically powerful tool, also easy and simple to use. 3. The best Algorithm varies depending on Computational Environment. 4. The simple Bi-CG is good for more Accurate Computing Environment. SIAM Conference on Applied 28
29 Future works 1. Analysis: alpha, beta and other vectors. 2. Real problems: 3. Refinement of Implementation: 4. Find a good tool: easy and effective SIAM Conference on Applied 29
30 Advertisement High performance should be used not only for Speeding but also Quality of Computation. To test effectiveness of Krylov Subspace Methods, try to change Computing Accuracy. Try to use Quadruple Arithmetic in High- Performance Computers and legacy machines. SIAM Conference on Applied 30
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Z Electronic Transactions on Numerical Analysis. olume 16, pp. 191, 003. Copyright 003,. ISSN 10689613. etnamcs.kent.edu A BLOCK ERSION O BICGSTAB OR LINEAR SYSTEMS WITH MULTIPLE RIGHTHAND SIDES A. EL
Iterative methods for Linear System of Equations. Joint Advanced Student School (JASS-2009)
Iterative methods for Linear System of Equations Joint Advanced Student School (JASS-2009) Course #2: Numerical Simulation - from Models to Software Introduction In numerical simulation, Partial Differential
Key words. conjugate gradients, normwise backward error, incremental norm estimation.
Proceedings of ALGORITMY 2016 pp. 323 332 ON ERROR ESTIMATION IN THE CONJUGATE GRADIENT METHOD: NORMWISE BACKWARD ERROR PETR TICHÝ Abstract. Using an idea of Duff and Vömel [BIT, 42 (2002), pp. 300 322