MathLib Requirements proposal (LHC++ meeting 9-10 June 1997)

MathLib requirements proposal

This page is setup to give the HEP community a chance to give feedback on the list of math. functions presented at the LHC++ meeting of 9-10 June 1997 at CERN by Rob Veenhof. for the MathLib requirements. The list of math. functions is heavily biased by his personnel use of CERNLIB now, although Rob Veenhof did get information on usage from colleagues. The list of math. functions, comments and conclusions are Rob Veenhof's personnel

As was agreed during the LHC++ meeting of 9-10 June 1997, a list of most dessirable math. functions will be presented to NAG who will incorporated them into their libraries. For questions relating to the comments and conclusions please contact: Rob Veenhof. For all other questions regarding this page please contact: Antoni S. Baranski.

Types of math. functions:

Integration + Differential eq.

Special Functions

Zero Finding

Interpolation

Linear Algebra, vectors

Random number Generators

Various

Comparison with commercial math. librariers:

Integration + Differential eq. comparison
Special Functions comparison
Zero Finding comparison
Interpolation comparison
Linear Algebra, vector comparison
Random number Generators comparison

Summary

Integration + Differential eq.

Routine	Routines with documented usage:
GAUSS	Gauss integration 1 variable.
RKNYS	Runge-Kutta-Nystroem 2nd. order.
DGMLT	n-Dimensional integration.
SIMPS	Simpson integration.
	Probably used:
RADAPT	Adaptive Gauss integration.
RGS56P
CGAUSS	Complex Gauss integration.
RIWIAD	M-C integration
RRKSTP	1st. Order differential equation.
RDEQBS
RDEQMR
ELDANY	Elliptic differential eq.

For a comparison with possible replacement libraries click here.

Special Functions

Routine	Routines with documented usage:
BESJ0	Bessel function order 0, 1.
CCLBES	Bessel function complex arg.
RBZEJY	Zeroes of Bessel I, Y functions.
	Cylindrical configurations, usually integer order, sometimes complex.
BESI0	Modified Bessel functions
BSIR3	Modified Bessel functions non-integer order.
	Order 5/3: Synchroton radiation power. Integer order: Space charge in tubes.
CELFUN	Jacobian elliptic functions (complex).
RTHETA	Theta functions.
CELINT	"Inverse" of CELFUN (complex).
	2D Electrostatics, doubly periodic configurations.
ERF	Error function.
FREQ	Frequency function.
	Error calculation, probability intervals.
GAMMA	Gamma function.
GAMMF
ALGAMA
GGAMMA
BINOMI	Binomial coefficients.
RASLGF	Legendre polynomials.
	Multipole expansions, atomic physics.
	Probably used: *

Note: There are functions which Rob Veenhof has not been able to identify as used, but they are likeliy to be used. They are also highly valuable for their numerical accuracy.
For a comparison with possible replacement libraries click here.

Zero Finding

Routine	Routines with documented usage or likely to be used:
RTEQ3	Tartagia cubic equation.
RTEQ4	Ferrari quartic equation.
RZERO	Zero of 1 variable.
NERFZ	Zero count of complex function.

For a comparison with possible replacement libraries click here.

Interpolation

Routine	Routines with documented usage:
DIVDIF	I-D interpolation (polynomial).
POLINT
FINT	Linear n-dimensional interpolation.
RCHECF	Chebyshev series.
RCHSUM
SPLINE	(various routines).

For a comparison with possible replacement libraries click here.

Linear Algebra, vectors

Routine	Routines with documented usage:
RINV etc.	Solving linear equations.
VECMAN	Vector manipulations.
SORTZV	Various sorting routines
FLPSOR
SORTRQ

For a comparison with possible replacement libraries click here.

Random number Generators

Routine	Routines with documented usage:
RANF	Uniform Random numbers.
RNDM	Uniform Random numbers.
RANLUX,RN48	Uniform Random numbers.
NORRAN	Gaussian Random numbers.
RNORML	Gaussian Random numbers.
	Basic generators, called many times within a single job (+/- 10^9 and more).
RNGAMA	Gamma Rangom numbers.
	Simulate polya, confidence etc.
POISSN	Poisson Random numbers.
RNPSSN	Poisson Random numbers.
	Number of things happening.
RANLAN	Landav Random numbers.
VVILOV	Vavilov Random numbers.
	Energy loss fluctuations.
HISRAN	Histogram Random numbers.
FUNRAN	Function Random numbers.
RN3DIM	Spherical Random numbers.

For a comparison with possible replacement libraries click here.

Various

Bit manipulation

Heavily used when dealing with testbeam and experimental data (too langauge dependent for this talk).

Histograming

Likely to be replaced by ROOT, etc.

Function minimisation, fitting

Heavily used, need for a well tested and reliable system (Seperate presentation).

Fourier transformations

Better covered by NAG and ESSL. LAPLACE + inverse would be desirable (is in NAG-Fortran [C06LAF, C06LBF, C06LCF]).

Integration + Differential eq. comparison

Comparison with NAG :

Integration looks compleet.
Specific methods for 2nd. order (1D) and Elliptic (2D) equations not found.

Comparison with DiffPack :

Caters well for 2D Differential equations, using F.E.M., nothing found about 1D equations using Runge-Kutta-Nystroem.

Comparison with ESSL :

Good collection of quadrature routines (1D and 2D).

Comparison with BLITZ :

Conclusion:

Quadrature: well covered with both analytic and MC methods, mostly by NAG but also by ESSL .
Differential equations: Finite elements: well represented by DiffPack .
2nd. Order methods for 1D differential equations not found. Also missing: Complex quadrature and (trivial) simpson rule.

Special Functions comparison

Comparison with NAG :

Contains y, J0,1 , Y0,1 , I0,1 , K0,1 , Fresnel
Has in addition: ber, bei, her, hei and airy.
But misses many routines:

Complex Bessel.
Bessel function
Modified Bessel of non-integer order.
Zeroes of Bessel.
Elliptic Functions (Has some Elliptic integrals).
Theta functions.
Legendre functions.
most of the undocumented routines.

Comparison with BLITZ, ESSL and DiffPack :

No reference to special functions.

Conclusion:

Many items with documented usage are missing in alternative librariers.
Experience shows that contact with a mathematician is helpful for a physicist solving, a problem in which a special function occurs.

Zero Finding comparison

Comparison with NAG :

Has Zero Finding for 1 variable.
Misses 3rd. + 4th. order explict formula.
Misses "Principle of Argument" type treatment of complex functions.

Comparison with BLITZ, ESSL, DiffPack :

No mention of these.

Conclusion:

Some missing functionality.
Cubic, Quartic and complex Zero counting should be trivial to introduce in a library that has complex integration.

Interpolation comparison

Comparison with NAG :

Has 1D and 2D interpolation, and splines
Misses n-Dim. and Chebyshev.

Comparison with ESSL :

Has 1D, 2D and Splines.
Very much faster than CERNLIB (x100!).

Comparison with BLITZ and DiffPack :

Conclusion:

An area where CERNLIB is easy to beat.
Why doesn'y any of the replacement libraries have Chebyshev ? (Is found in ACM, is also found in NAG-Fortran librarie,C06DBF).

Linear Algebra, vectors comparison

Comparison with NAG, ESSL, BLITZ and Lapack :

All, plus more.
ESSL is easily x20 faster for n >= 500.
BLITZ wants to offer high performance in this.

Conclusion:

No worry.

Random number Generators

Comparison with NAG :

Has Uniform.
Gaussian (also multivariate).
Exp(trivial).
Gamma.
Function.
Binomial.
Poisson.
Misses Landav & Vavilov.

Comparison with ESSL :

Has (series of) uniform + Gaussian random numbers including long sequence

Comparison with BLITZ, Lapack and DiffPack :

Conclusion:

Need to verify sequence length of alternatives.
Landav & Vavilov missing, their inverse should be added to the special functions.

Summary

NAG and ESSL offer in general a wide choice of numerical routines, Lapack and DiffPack too in their respective domains.
NAG and ESSL are according to Rob Veenhof well documented and relaible.
Areas of concern:

Special functions.
Complex number specifics.
2nd. Order PDE.
Random number Generators.

Packages for Minimisation and for Histograming to be reviewed separately.

Replacement librariers

A list of possible replacement librariers:
NAG
ESSL
BLITZ
Lapack
DiffPack

Last changed: June 11, 1997.
This page was setup on behalve of Rob Veenhof by Antoni S. Baranski.