GiNaC. Symbolic computation with C++ Jens Vollinga. Institut für Physik Universität Mainz. GiNaC p.1/13

Transcription

1 GiNaC Symbolic computation with C++ Jens Vollinga Institut für Physik Universität Mainz GiNaC p.1/13

2 Outline C++ library Developed since 1999 CAS in high energy physics Fundamentals of GiNaC Features of GiNaC Applications GiNaC p.2/13

3 CAS in high energy physics Perturbation theory Feynman rules Feynman diagrams Dirac algebra and traces Loop integration Summation of diagrams Phase space integration Checks... GiNaC p.3/13

4 CAS in high energy physics Perturbation theory Feynman rules Feynman diagrams Dirac algebra and traces Loop integration Summation of diagrams Phase space integration Checks... Combinatorics Algebra Data management Numerics I/O GiNaC p.3/13

5 CAS in high energy physics Perturbation theory Feynman rules Feynman diagrams Dirac algebra and traces Loop integration Summation of diagrams Phase space integration Checks... Combinatorics (Maple, C, Perl) Algebra (Maple, FORM) Data management (C, BASH, Perl) Numerics (FORTRAN, C) I/O (C, Tcl/Tk) GiNaC p.3/13

6 CAS in physics Taming COMPLEXITY Software Engineering (coding, social, technology,... ) GiNaC p.4/13

7 CAS in physics Taming COMPLEXITY Software Engineering (coding, social, technology,... ) What if you are a C++ programmer? Combinatorics Algebra Data management Numerics I/O GiNaC p.4/13

8 CAS in physics Taming COMPLEXITY Software Engineering (coding, social, technology,... ) What if you are a C++ programmer? Combinatorics Algebra Data management Numerics I/O GiNaC p.4/13

9 CAS in physics Taming COMPLEXITY Software Engineering (coding, social, technology,... ) What if you are a C++ programmer? Combinatorics Algebra Data management Numerics I/O GiNaC p.4/13

10 CAS in physics Taming COMPLEXITY Software Engineering (coding, social, technology,... ) What if you are a C++ programmer? Combinatorics Algebra Data management Numerics I/O GiNaC p.4/13

11 CAS in physics Taming COMPLEXITY Software Engineering (coding, social, technology,... ) What if you are a C++ programmer? Combinatorics Algebra GiNaC Data management Numerics I/O GiNaC p.4/13

12 Fundamentals of GiNaC Write a C++ program: #include <iostream> using namespace std; #include <ginac/ginac.h> using namespace GiNaC; int main() { symbol x("x"); ex result = Li(2,x).diff(x); cout << result << endl; return 0; } GiNaC p.5/13

13 Fundamentals of GiNaC Write a C++ program: #include <iostream> using namespace std; #include <ginac/ginac.h> using namespace GiNaC; int main() { symbol x("x"); ex result = Li(2,x).diff(x); cout << result << endl; return 0; } Library header GiNaC p.5/13

14 Fundamentals of GiNaC Write a C++ program: #include <iostream> using namespace std; #include <ginac/ginac.h> using namespace GiNaC; int main() { New data types from GiNaC symbol x("x"); ex result = Li(2,x).diff(x); cout << result << endl; return 0; } GiNaC p.5/13

15 Fundamentals of GiNaC Write a C++ program: #include <iostream> using namespace std; #include <ginac/ginac.h> using namespace GiNaC; int main() { symbol x("x"); ex result = Li(2,x).diff(x); cout << result << endl; return 0; } Algebra in C++! GiNaC p.5/13

16 Fundamentals of GiNaC Compile the program: $ g++ -o myprg myprg.cpp -lginac Run it: $./myprg -log(1-x)*xˆ(-1) GiNaC p.6/13

17 Fundamentals of GiNaC Container for arbitrary algebraic expressions ex GiNaC p.7/13

18 Fundamentals of GiNaC Container for arbitrary algebraic expressions ex After declaration ex myexpr; or as a function parameter for example ex nloopfct(ex momentum) {... } GiNaC p.7/13

19 Fundamentals of GiNaC Container for arbitrary algebraic expressions ex After declaration ex myexpr; or as a function parameter for example ex nloopfct(ex momentum) {... } use it like C++ built-in types: momentum = sqrt(p) * pow(mu, 2) - d; myexpr = sin(x).series(x, 10); cout << myexpr << endl; if (myexpr.has(x)) y = myexpr; GiNaC p.7/13

20 Fundamentals of GiNaC Container for arbitrary algebraic expressions ex After declaration ex myexpr; or as a function parameter for example ex nloopfct(ex momentum) {... } use it like C++ built-in types: momentum = sqrt(p) * pow(mu, 2) - d; myexpr = sin(x).series(x, 10); cout << myexpr << endl; if (myexpr.has(x)) y = myexpr; GiNaC p.7/13

21 Fundamentals of GiNaC Symbols symbol Declaration: symbol x("x"); symbol eps("\epsilon") GiNaC p.8/13

22 Fundamentals of GiNaC Symbols symbol Declaration: symbol x("x"); symbol eps("\epsilon") What else? Numbers 1.34, 3/4, 2i,... Mathematical functions Matrices Algebras... GiNaC p.8/13

23 Features of GiNaC Arbitrary symbolic expressions GiNaC p.9/13

24 Features of GiNaC Arbitrary symbolic expressions Complex arithmetics with arbitrary precision GiNaC p.9/13

25 Features of GiNaC Arbitrary symbolic expressions Complex arithmetics with arbitrary precision Operations on rational functions, e.g. x2 y 2 (x+y) 2 x y x+y GiNaC p.9/13

26 Features of GiNaC Arbitrary symbolic expressions Complex arithmetics with arbitrary precision Operations on rational functions, e.g. x2 y 2 (x+y) 2 x y x+y Symbolic derivation and series expansion GiNaC p.9/13

27 Features of GiNaC Arbitrary symbolic expressions Complex arithmetics with arbitrary precision Operations on rational functions, e.g. x2 y 2 (x+y) 2 x y x+y Symbolic derivation and series expansion Matrices, vectors, linear equation systems GiNaC p.9/13

28 Features of GiNaC Arbitrary symbolic expressions Complex arithmetics with arbitrary precision Operations on rational functions, e.g. x2 y 2 (x+y) 2 x y x+y Symbolic derivation and series expansion Matrices, vectors, linear equation systems In-/Output of expressions in various formats (text, binary, LaTeX,... ) GiNaC p.9/13

29 Features of GiNaC Arbitrary symbolic expressions Complex arithmetics with arbitrary precision Operations on rational functions, e.g. x2 y 2 (x+y) 2 x y x+y Symbolic derivation and series expansion Matrices, vectors, linear equation systems In-/Output of expressions in various formats (text, binary, LaTeX,... ) New functions and classes can easily be added GiNaC p.9/13

30 Features of GiNaC Functions Γ, B, ψ, ( n k),... sin, cos, tan, sinh,..., exp, log, ζ, S, H, Li,... GiNaC p.10/13

31 Features of GiNaC Functions Γ, B, ψ, ( n k),... sin, cos, tan, sinh,..., exp, log, ζ, S, H, Li,... Objects with indices: p µ, A ij Special algebras: Clifford, SU(3) GiNaC p.10/13

32 Features of GiNaC Functions Γ, B, ψ, ( n k),... sin, cos, tan, sinh,..., exp, log, ζ, S, H, Li,... Objects with indices: p µ, A ij Special algebras: Clifford, SU(3) Automatic code generation ( numeric integration) GiNaC p.10/13

33 Features of GiNaC Functions Γ, B, ψ, ( n k),... sin, cos, tan, sinh,..., exp, log, ζ, S, H, Li,... Objects with indices: p µ, A ij Special algebras: Clifford, SU(3) Automatic code generation ( numeric integration) Good documentation, open source (GPL) GiNaC p.10/13

34 Applications with GiNaC Loop calculations: xloops nestedsums gtybalt feelfem, PURRS, MBDyn Publications On the Invariance of Residues of Feynman Graphs I. Bierenbaum, R. Kreckel, D. Kreimer, J. Math. Phys (2002) The Massless Two-Loop Two-Point Function I. Bierenbaum, S. Weinzierl, Eur. Phys. J. C32:67-78 (2003) The Electroweak Standard Model in the Axial Gauge C. Dams, R. Kleiss, Eur. Phys. J. C34: (2004) An Example of Clifford Algebras Calculations with GiNaC V. Kisil, [arxiv:cs.ms/ ] GiNaC p.11/13

35 Summary GiNaC a C++ library for symbolic computation Complete computational framework in C++ Features of GiNaC GiNaC p.12/13

36 Summary GiNaC a C++ library for symbolic computation Complete computational framework in C++ Features of GiNaC Developed since 1999 Version 1.0 in 2001, current version Technically matured and well documented Active development going on (better factorization, more special functions, improved interface to MC integration) GiNaC p.12/13

37 GiNaC p.13/13