M. Hatlo / CERN 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

M. Hatlo / CERN 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

M. Hatlo / CERN 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

M. Hatlo / CERN 4

M. Hatlo / CERN 5

M. Hatlo / CERN 6

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 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

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Timing Results for some Distributions

• 1 000 000 function calls

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Random Numbers According to Distribution

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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

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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

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Frequency Test

• With this frequency, look other places in the space and compute Nodd.

• Nodd should be normal distributed

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Results

• 10 results for Nodd

• Kolmogorov-Smirnov test• Taus, 8 dim and Ranlux389, 6 dim• New test for poor results• All passed

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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

),...,( )()1( diii ppP

)()()()(

1

1

1,min

,

kj

ki

kj

ki

dkk

d

kkji

ppppdist

distPP

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Point test

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

pass

3d

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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 18

Integrals

1

02

02222

0

0

1

1

02

2/

1

0

)1(

11log

cossin

)1(

)sin(1(*)

271

)2

(2

1(*)

0.4log

(*)

2

pdxx

x

abxbxa

dx

xx

dx

pdxx

x

xx

dx

xerfdte

dxx

x

p

p

x

x

t

BackgroundeakLorenzianp

dxx

x

dxe

x

dxe

x

dxe

dxex

Cdxxx

x

x

x

x

(*)

2

sin

15

6

2

2

...5772157.01

1

log

1

0

0

4

1

3

0

2

1

0

0

1

0

2

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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

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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

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Results (Linux, P4 1.8 MHz )

• High RMS for GSL and LAPACK in 2x5• Error with ROOT for 20x50

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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 23

Results

  UBLAS 1 UBLAS 2 UBLAS 3 LAPACK CLHEP GSL ROOT

2x50.0000291446

(1.59)0.000036635

(2.00)0.000034575

5 (1.89)0.0000262477

(1.44)0.0000254471

1.39)0.0000315174

(1.72)0.000018277

(1.00)

4x100.000122262

(6.69)0.000128379

(7.02)  0.00004496

(2.46)0.0000558439

(3.06)0.0000591721

(3.24)0.0000387867

(2.12)

10x250.00162649

(89.0)    0.000313892

(17.2)0.000559981

(30.6)0.000401781

(22.0)0.000277101

(15.2)

20x500.0148939

(815)    0.00125005

(68.4)0.00302708

(166)0.0014349

(78.5)