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Test and Validation Studies of Mathematical Software Libraries

Test and Validation Studies of Mathematical Software Libraries. A summary of my work as a technical student at CERN. LCG AA Meeting, 22. September 2004. Special Functions. Comparison of numerical results Performance GSL-NAG-C and GSL-TMath Bessel functions

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Test and Validation Studies of Mathematical Software Libraries

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  1. Test and Validation Studies of Mathematical Software Libraries A summary of my work as a technical student at CERN LCG AA Meeting, 22. September 2004

  2. Special Functions • Comparison of numerical results • Performance • GSL-NAG-C and GSL-TMath • Bessel functions • Gamma, Logarithm of Gamma, Error function and Complementary Error function

  3. Results • GSL performed very well compared to NAG-C • Difference usually less than the estimated error • Bigger differences between GSL and TMath • Little difference in time for GSL • NAG-C and TMath are faster

  4. Timing of Special Functions • 100 000 function calls for Bessel I0, I1, J0 and J1 • 50 000 function calls for the rest

  5. Distributions • Comparison of numerical results • Performance in the evaluation • Generation of random numbers according to distribution • Comparison and Kolmogorov-Smirnov test • Normal distribution, Landau distribution, Gamma distribution, Poisson distribution and Chi-Square distribution

  6. Timing Results for some Distributions • 1 000 000 function calls

  7. Random Numbers According to Distribution

  8. Random Numbers • Two tests • Frequency test • Point test • Main generators from GSL • gsl_rng_mt19937 • gsl_rng_cmrg • gsl_rng_mrg • gsl_rng_taus • gsl_rng_taus2 • gsl_rng_gfsr4 • gsl_rng_ranlux389 • gsl_rng_ranlux • gsl_rng_ranlxd2

  9. Frequency Test Fill space in d dimensions with points formed from a sequence of random numbers. Look in a small volume and the frequency as the number of bins which maximize |Nodd-Neven|. gsl_rng_minstd

  10. Frequency Test • With this frequency, look other places in the space and compute Nodd. • Nodd should be normal distributed

  11. Results • 10 results for Nodd • Kolmogorov-Smirnov test • Taus, 8 dim and Ranlux389, 6 dim • New test for poor results • All passed

  12. Point Test • Arrange a sequence of random numbers into multidimensional points • Define distance between two points as • Find all points Pi that are closer to P1 than the mean-n*standard deviation (n=3,4,5) • Calculate the distance between Pi+1 and P2

  13. Point test • For the distance should be normal distributed. • Use Kolmogorov-Smirnov test • All generators pass

  14. Numerical Integration • Wrapper for existing gsl algorithms • Tested on a few number of integrals • Compare numerical results with analytical results • No difference larger than 10-7 (input tolerance), but need further testing

  15. Integrals

  16. Quadrature routines QAG – adaptive integration QAGUI – adaptive integration from zero to infinity QAGS – adaptive integration with singularities QNG – non-adaptive Gauss-Kronrod integration NB! Different integrals are used, marked with (*) on last slide Performance in Numerical Integration

  17. Linear Algebra • E. Myklebust summer student 2003 • A Comparative study of Numerical Linear Algebra Libraries • Particle Track Reconstruction, Kalman filter update equations • Multiplication, addition, inversion and transpose • Originally 2x2, 2x5, 5x2 and 5x5 • Extended to bigger matrices • 2x5, 4x10, 10x25 and 20x50 • CLHEP, uBLAS, LAPACK, GSL and ROOT • Used timer from SEAL base

  18. Results (Linux, P4 1.8 MHz ) • High RMS for GSL and LAPACK in 2x5 • Error with ROOT for 20x50

  19. Conclusions • GSL performs reasonably good • Both tests of randomness were passed by all the main generators from GSL • More testing is needed for the numerical integration • All test programs are in the SEAL cvs repository • A test suite can be easily created and automatically run for new SEAL releases • A written report of my work will be put on the webpage when finished

  20. Results

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