1. MathMore Lorenzo Moneta, Andràs Zsenei ROOT Meeting 19/8/2005

2. 19/8/2005ROOT meeting, Andràs Zsenei2 MathMore components  Special functions (extension of MathCore)  Statistical functions (extension of MathCore)  Derivation  Integration  Interpolation  Root finding  1 dimensional minimization

3. 19/8/2005ROOT meeting, Andràs Zsenei3 The Current Implementation  The relevant GSL code extracted into mathmore/src/gsl-xxx and compiled automatically (mathmore/Module.mk)  Easily maintainable and updateable compared to direct copy of the algorithms into ROOT classes

4. 19/8/2005ROOT meeting, Andràs Zsenei4 Special Functions  Those functions from the N1687 Technical Report on Standard Library Extensions that use GSL implementa- tion ( http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1687.pdf ) http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1687.pdf  For example:  Bessel Functions  Elliptic Integrals  Legendre Polynomials, etc

5. 19/8/2005ROOT meeting, Andràs Zsenei5 Special Functions cont’d  Free functions following the C++ standard’s naming convention (N1687)  Trivial use: root [0] gSystem->Load("libMathMore.so"); root [1] ROOT::Math::cyl_bessel_i(1.2, 2.4) (double)2.05567401212170076e+00

6. 19/8/2005ROOT meeting, Andràs Zsenei6 Statistical Functions  Cumulative Distribution Functions (CDF) of those distributions from MathCore which were missing:  Chi-square, T-distribution, etc  The inverses of the CDF-s  Free functions callable in a trivial way  Naming convention in N1668, but might change (following up…)  http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1069.pdf http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1069.pdf

7. 19/8/2005ROOT meeting, Andràs Zsenei7 Function Interface  Minimal interface for functions used by all the nu- merical algorithms: IGenFunction, ParamFunction, Polynomial (see previous presentations)  It would be nice but not necessary if TF1 would implement this interface  class WrappedFunction which wraps any C++ callable object (C free functions, functors, etc...)  Reviewed by CMS — several of the recommendations implemented  Still a few questions:  Gradients: interface or feature testing  NVI (Non-Virtual Interface)

8. 19/8/2005ROOT meeting, Andràs Zsenei8 Derivation — an example of the overall design  Class Derivator which defines the user interface (constructors, member functions) and has an instance of GSLDerivator  Class GSLDerivator which calls the appropri- ate GSL functions  Both GSLDerivator.h and GSLDerivator.cxx are in mathmore/src so that implementa- tion details do not pollute ROOT header files

9. 19/8/2005ROOT meeting, Andràs Zsenei9 Derivation — an example of the overall design, cont’d  Usage with function inheriting from IGenFunction: ROOT::Math::Polynomial *f1 = new ROOT::Math::Polynomial(2); … ROOT::Math::Derivator *der = new ROOT::Math::Derivator(*f1); double step = 1E-8; double x0 = 2; double result = der->EvalCentral(x0, step); double error = der->Error();

10. 19/8/2005ROOT meeting, Andràs Zsenei10 Derivation — an example of the overall design, cont’d  Usage with a function pointer: double myfunc ( double x, void * ) { return std::pow( x, 1.5); } ROOT::Math::Derivator *der = new ROOT::Math::Derivator(myfunc); double step = 1E-8; double x0 = 2; double result = der->EvalCentral(x0, step);

11. 19/8/2005ROOT meeting, Andràs Zsenei11 Derivation — an example of the overall design, cont’d  Usage with modified TF1 (or any other callable implementing operator() ): TF1 *fun1 = new TF1("fun1","x*x+3*x",0,10); ROOT::Math::Derivator *der = new ROOT::Math::Derivator(ROOT::Math::WrappedFunction (*fun1)); double step = 1E-8; double x0 = 2; double result = der->EvalCentral( x0, step);

12. 19/8/2005ROOT meeting, Andràs Zsenei12 Derivation — an example of the overall design, cont’d  Straightforward to add the functionality into TF1’s Derivative() so that it is trans-parent to the old user (disadvantage: cre-ates a Derivator object for each call)  If performance is a major issue there is always the possibility to create the object oriented way as shown previously + use function pointers

13. 19/8/2005ROOT meeting, Andràs Zsenei13 Integration  Non-adaptive, adaptive and adaptive singular (i.e. taking into account singularities) integration  Different Gauss-Konrod rules can be selected  Possibility to use infinite and semi- infinite ranges

14. 19/8/2005ROOT meeting, Andràs Zsenei14 Interpolation  Linear, polynomial, Akima and Akima periodic interpola- tions

15. 19/8/2005ROOT meeting, Andràs Zsenei15 Root Finding  Root finding of one dimensional functions  Bracketing algorithms: bisection, false position, Brent-Dekker  Polishing algorithms (derivatives): Newton, secant, Steffenson

16. 19/8/2005ROOT meeting, Andràs Zsenei16 Minimization in 1 D  1 dimensional minimization for cases where Minuit or Fumili would be too much overhead  Golden section algorithm  Brent algorithm

17. 19/8/2005ROOT meeting, Andràs Zsenei17 In progress — Implementation type  How can we select the various implementations (GSL, original ROOT, Cernlib, etc..)?  An additional constructor for class Derivator which takes implementation type so that Derivator constructs an instance of worker class according to implementation chosen (i.e. GSL)  Have also a default implementation (for example used by TF1) which would allow to move Derivator.h into MathCore  Introduce a class MathSelector which manages centrally all the default implementations for the numerical algorithms; PluginManager would be convenient but not standalone