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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
M. Hatlo / CERN
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
M. Hatlo / CERN
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
M. Hatlo / CERN
M. Hatlo / CERN
M. Hatlo / CERN
Timing of Special Functions
• 100 000 function calls for Bessel I0, I1, J0 and
J1
• 50 000 function calls for the rest
M. Hatlo / CERN
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
M. Hatlo / CERN
M. Hatlo / CERN
Timing Results for some Distributions
• 1 000 000 function calls
M. Hatlo / CERN
Random Numbers According to Distribution
M. Hatlo / CERN
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
M. Hatlo / CERN
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
M. Hatlo / CERN
Frequency Test
• With this frequency, look other places in the
space and compute Nodd.
• Nodd should be normal distributed
M. Hatlo / CERN
Results
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•
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10 results for Nodd
Kolmogorov-Smirnov test
Taus, 8 dim and Ranlux389, 6 dim
New test for poor results
All passed
M. Hatlo / CERN
Point Test
• Arrange a sequence of random numbers into
(1)
(d )
multidimensional points Pi  ( pi ,..., pi )
• Define distance between two points as
d
Pi  Pj   dist k ,

k 1
dist k  min pi( k )  p (jk ) ,1  pi( k )  p (jk )
1 k  d

• 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
M. Hatlo / CERN
Point test
• For d  3 the distance should be normal
distributed.
• Use Kolmogorov-Smirnov test
• All generators
pass
M. Hatlo / CERN
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
M. Hatlo / CERN
Integrals
1
 1
1 


0  log x 1  x dx  C  0.5772157...
log x
(*) 
dx  4.0
x
0
1
(*)
2
x
t
e

2
/2
dt  erf (
x
1
x
)
2
dx


0 1  x  x 2 27
1

x p 1

(*) 
dx 
1 x
sin( p )
0

dx
0 (1  x) x  

dx


0 a 2 sin 2 x  b 2 cos 2 x ab
1
p
 x log
0
1
1
dx 
x
( p  1) 2


xe  x dx 
0

x
 e dx 
2
0


2

2
2
0 e x1 dx  6
x

x3
4
0 e x1 dx  15

sin x

dx

0 x
2
(*) Lorenzianp eak  Background
M. Hatlo / CERN
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
M. Hatlo / CERN
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
M. Hatlo / CERN
Results (Linux, P4 1.8 MHz )
• High RMS for GSL and LAPACK in 2x5
• Error with ROOT for 20x50
M. Hatlo / CERN
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
M. Hatlo / CERN
Results
UBLAS 1
UBLAS 2
UBLAS 3
LAPACK
CLHEP
GSL
ROOT
2x5
0.0000291446
(1.59)
0.000036635
(2.00)
0.0000345755
(1.89)
0.0000262477
(1.44)
0.0000254471
1.39)
0.0000315174
(1.72)
0.000018277
(1.00)
4x10
0.000122262
(6.69)
0.000128379
(7.02)
0.00004496
(2.46)
0.0000558439
(3.06)
0.0000591721
(3.24)
0.0000387867
(2.12)
10x25
0.00162649
(89.0)
0.000313892
(17.2)
0.000559981
(30.6)
0.000401781
(22.0)
0.000277101
(15.2)
20x50
0.0148939
(815)
0.00125005
(68.4)
0.00302708
(166)
0.0014349
(78.5)
M. Hatlo / CERN