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

6. Timing of Special Functions
function calls for Bessel I0, I1, J0 and J1
function calls for the rest

7. 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

9. Timing Results for some Distributions
function calls

10. Random Numbers According to Distribution

11. Random Numbers Two tests Main generators from GSL 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

12. 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

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

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

15. 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

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

17. 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

18. Integrals

19. Performance in Numerical Integration
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

20. 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

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

22. 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

23. Results UBLAS 1 UBLAS 2 UBLAS 3 LAPACK CLHEP GSL ROOT 2x5
UBLAS 1
UBLAS 2
UBLAS 3
LAPACK
CLHEP
GSL
ROOT
2x5
(1.59)
(2.00)
(1.89)
(1.44) )
(1.72)
(1.00)
4x10
(6.69)
(7.02)
(2.46)
(3.06)
(3.24)
(2.12)
10x25
(89.0)
(17.2)
(30.6)
(22.0)
(15.2)
20x50
(815)
(68.4)
(166)
(78.5)