High performance computation: numerical music or numerical noise?

Transcription

1 High performance computation: numerical music or numerical noise? N.S. Scott*, C. Denis, F. Jèzèquel, J.-M. Chesneaux *School of Electronics, Electrical Engineering and Computer Science The Queen's University of Belfast, Belfast, BT7 1NN, UK Laboratoire d'informatique de Paris 6, Universitè Pierre et Marie Curie - Paris 6, 4 place Jussieu,75252 Paris Cedex 05, France 1

2 Scientific computation: numerical music or numerical noise? I have little doubt that about 80% of all the results printed from the computer are in error to a much greater extent than the user would believe.. Leslie Fox,

3 Scientific computation is inherently flawed. continuum ionization threshold e - + H(1s) H - e - + H(nl) nl r 1 Internal region 2-D R-matrix Approach External region ground state 1s outer region e - r 2 Mathematical model e - Real world inner region Computational model Computer implementation 3

4 Precariousness of relying on extended precision f(x,y) =333.75y 6 +x 2 (11x 2 y 2 -y 6-121y 4-2)+5.5y 8 +x/(2y) Method Fortran:single precision Fortran:double precision Fortran:quad precision VP Interval Arithmetic f(77617,33096) * [ , ] S.M. Rump, Reliability in Computing. The role of Interval Methods in Scientific Computing, Academic Press,

5 Numerical health check recommended Ideally, we seek a numerical screening tool that will: report gradual and catastrophic loss of precision; report the accuracy of intermediate and final results; be of acceptable efficiency; and be non invasive to the source code. Accuracy vs Precision has eight decimal digit precision, irrespective of what it represents. 22/7 - accurate to eight decimal digits, π - accurate to three decimal digits only. 5

6 The CESTAC methodology Where no overflow occurs, the exact result, r, of any non exact floating-point arithmetic operation is bounded by two consecutive floating-point values R - and R +. The basic idea of the method is to perform each arithmetic operation N times, randomly rounding each time, with a probability of 0.5, to R - or R +. R 1 =x*y R 2 =x*y R 3 =x*y _ R C = 1 N N i= 1 Ri ( = f R, σ ({ R R i })) 6

7 The CANDA Library PROGRAM f!use cadna double precision :: y,x,res!type(double_st) :: y,x,res!call cadna_init(-1) x=77617d0; y=33096d0 res=333.75*y*y*y*y*y*y+x*x*(11*x*x*y*y-y*y*y*y*y*y- & 121*y*y*y*y-2.0)+5.5*y*y*y*y*y*y*y*y+x/(2*y) print *, res! print *, str(res)! call cand_end() END PROGRAM f 7

8 CADNA f(x,y) =333.75y 6 +x 2 (11x 2 y 2 -y 6-121y 4-2)+5.5y 8 +x/(2y) Method Fortran:single precision Fortran:double precision Fortran:quad precision VP Interval Arithmetic CADNA: single precision CADNA: double precision f(77617,33096) * Control of Accuracy and Debugging for Numerical Applications (CADNA) 8

9 CANDA analysis of the Slater integrals To compute the Slater integrals 2DRMP uses, rs, a legacy subroutine that has been used R-matrix codes for over 30 years. The computation of I λ with λ in {0, 2, 4, 6, 8} in double precision using 1025 equally spaced integration points. λ I λ E E E E E+002 9

10 CANDA analysis of the Slater integrals 10

11 CANDA analysis of the Slater integrals In the 1970s, for reasons of storage economy and computational efficiency was replaced by 11

12 CANDA analysis of the Slater integrals An alternative is to compute directly but in the direction of decreasing y Accuracy vs Precision Even points poor algorithm, well computed Odd points better algorithm, poorly computed 12

13 CANDA analysis of the Slater integrals λ I λ E E E E E+002 I λ improved E E E E E-001 How do we know that the results are accurate? 13

14 Dynamical control of step size using CADNA For Newton-Cotes type approximations it can be shown that in a series of successive iterations, if I n -I n+1 then the significant digits in common to I n and I n+1 are also common to the exact result, I, up to one bit. This can be used to generate benchmark results. λ I λ using 2 17 integration points E E E E E

15 Concluding remarks report gradual and catastrophic loss of precision? report the accuracy of intermediate and final results? be of acceptable efficiency? be non invasive to the source code? 15

16 Moral Scientific codes used in high performance and grid environments need regular health checks to determine if they are fit for purpose. This is particularly important when legacy routines are used in new situations. CADNA is not a panacea - but it provides one way of achieving this. It is especially beneficial for screening legacy software where no documentation is available and where the author has long since departed. Scott NS, Jèzèquel F, Denis C and Chesneaux, J-M, Numerical health check for scientific codes: the CADNA approach, Compt. Phys. Commun (2007) 16