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Enhancing ROOT with MathMore: Special and Statistical Functions Workshop Overview

This workshop presented a comprehensive enhancement of the ROOT framework through the MathMore library, focusing on special and statistical functions. Key highlights included the integration of GSL code into MathMore, allowing for better maintainability and robustness across platforms. The workshop covered advanced mathematical functions like error functions, Bessel functions, and integration techniques, along with comparative tests for accuracy against leading libraries like Mathematica. Attendees learned about enhanced interfaces for numerical algorithms and the potential for integration and interpolation methods, all designed to streamline computational tasks.

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Enhancing ROOT with MathMore: Special and Statistical Functions Workshop Overview

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  1. MathMore Lorenzo Moneta, Andràs Zsenei ROOT Workshop 30/9/2005

  2. MathMore components • Special functions • Statistical functions • Generic interfaces • Derivation • Integration • Interpolation • Root finding • Chebyshev polynomials ROOT2005 Workshop, Andràs Zsenei

  3. The Current Implementation • The relevant GSL code extracted into mathmore/src/gsl-xxx and compiled automatically • A GSL tar file is in CVS with the extracted code • Works on all supported platforms (thanks to Bertrand Bellenot for the Windows port) • Easily maintainable and updateable compared to direct copy of the algorithms into ROOT classes ROOT2005 Workshop, Andràs Zsenei

  4. Special Functions • Defined in the N1687 Technical Report on Standard Library Extensions • http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1687.pdf • Basic functions in mathcore: • Error functions, gamma functions • The less used and those that use GSL imple-mentation are in mathmore • Bessel functions, elliptic integrals, Legendre polynomials etc • Possibility to migrate functions between mathcore and mathmore transparently for the user ROOT2005 Workshop, Andràs Zsenei

  5. 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 ROOT2005 Workshop, Andràs Zsenei

  6. Mathematical Functions tests • New functions improve on the precision of TMath • Extensive tests of numerical accuracy, comparisons with other libraries (Mathematica, Nag) ROOT::Math::erfc and relative difference compared to Mathematica ( ≈ 10-15) TMath::Erfc and relative difference compared to Mathematica ( ≈ 10-7) ROOT2005 Workshop, Andràs Zsenei

  7. Statistical Functions • Common functions used in statistics with a coherent naming scheme : • Probability density functions (pdf) • Cumulative distributions (lower tail and upper tail) • Inverse of cumulative distributions • Examples: • chisquared_pdf • chisquared_prob, chisquared_quant • chisquared_prob_inv, chisquared_quant_inv • Naming convention proposed for the C++ standard in N1668, but might change (to be followed closely) • http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1069.pdf ROOT2005 Workshop, Andràs Zsenei

  8. Statistical Functions Tests  =1.0 10-16  =2.0 10-16  =3.0 10-16 Lognormal PDF (left) and its Relative Difference Compared to Mathematica (right) ROOT2005 Workshop, Andràs Zsenei

  9. Function Interfaces ROOT2005 Workshop, Andràs Zsenei

  10. Function Interface • Minimal interface for functions used by all the numerical algorithms: IGenFunction, ParamFunction, Polynomial (see previous presentations) • class WrappedFunction<T> which wraps any C++ callable object (C free functions, functors, etc...) • Reviewed by C++ experts — several of the recommendations implemented ROOT2005 Workshop, Andràs Zsenei

  11. Derivation • Adaptive central difference algorithm using a 5-point rule • Adaptive forward difference algorithm using a 4-point rule • Adaptive backward difference algorithm using a 4-point rule ROOT2005 Workshop, Andràs Zsenei

  12. Derivation — an example of the overall design • 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 x0 = 2; double result = der->EvalCentral(x0); double error = der->Error(); ROOT2005 Workshop, Andràs Zsenei

  13. 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 x0 = 2; double result = der->EvalCentral(x0); ROOT2005 Workshop, Andràs Zsenei

  14. 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 ROOT2005 Workshop, Andràs Zsenei

  15. Integration Example // user provided free C function double myFunc ( double x) {… } // wrap the function ROOT::Math::WrappedFunction wFunc(myFunc); // create integrator and integrate ROOT::Math::Integrator ig(wFunc); Double result = ig.Integral(a, b); Double error = ig.Error(); • WrappedFunction can be replaced with any IGenFunction in the Integrator ROOT2005 Workshop, Andràs Zsenei

  16. Interpolation • Linear, polynomial, Akima and Akima periodic interpola-tions ROOT2005 Workshop, Andràs Zsenei

  17. Root Finding • Root finding of one dimensional functions • Bracketing algorithms: bisection, false position, Brent-Dekker • Polishing algorithms (derivatives): Newton, secant, Steffenson ROOT2005 Workshop, Andràs Zsenei

  18. Next Steps • Eliminate duplication • Other parts of the ROOT should use mathcore/mathmore • Moving functionality from TMath into mathcore/mathmore (TMath will remain for backward compatibility) • Implement incomplete gamma function in mathcore (for 2) • A more detailed discussion is needed to finalize function interfaces (signatures) • Prototype version of TF1 using algorithms from mathcore and implementing function interface • Add algorithms for multi-dimensional functions • New additions according to user requests (ex. FFT) ROOT2005 Workshop, Andràs Zsenei

  19. Conclusions • Mathmore is available in ROOT 5.04/00 (--enable-mathmore switch in configure) • Works on all supported platforms • Documentation available at • http://seal.web.cern.ch/seal/MathLibs/MathMore-5_0_4/html/index.html ROOT2005 Workshop, Andràs Zsenei

  20. References • Special and Random number C++ extension proposal • link to C++ extension draft • http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1687.pdf • Statistical functions proposal to C++ standard • http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1069.pdf • MathCore reference doc • http://seal.web.cern.ch/seal/MathLibs/MathCore-5_0_4/html • MathMore reference doc • http://seal.web.cern.ch/seal/MathLibs/MathMore-5_0_4/html • SEAL Math Web site • http:://cern.ch/mathlib • New C++ Minuit Web site • http:://cern.ch/minuit ROOT2005 Workshop, Andràs Zsenei

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