Introduction to GAP. (Groups, Algorithms, Programming) Henry Hutasoit, SSE RWTH Aachen Seminar Language in Scientific Computing

Transcription

1 Introduction to GAP (Groups, Algorithms, Programming) Henry Hutasoit, SSE RWTH Aachen Seminar Language in Scientific Computing

2 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 2

3 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 3

4 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 4

5 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 5

6 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 6

7 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 7

8 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 8

9 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 9

10 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 10

11 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 11

12 What is GAP? GAP is: 12

13 What is GAP? GAP is: a system for computational discrete algebra (Computational Group Theory) that provides: 13

14 What is GAP? GAP is: a system for computational discrete algebra (Computational Group Theory) that provides: a language; 14

15 What is GAP? GAP is: a system for computational discrete algebra (Computational Group Theory) that provides: a language; a library; 15

16 What is GAP? GAP is: a system for computational discrete algebra (Computational Group Theory) that provides: a language; a library; a data library 16

17 What is GAP? GAP is: a system for computational discrete algebra (Computational Group Theory) that provides: a language; a library; a data library 17

18 What is GAP? GAP is: a system for computational discrete algebra (Computational Group Theory) that provides: a language; a library; a data library A symbolic computation/ computer algebra/ algebraic computation tool 18

19 What is GAP? Capabilities: Forming lists, records, domains Handling objects and elements, Booleans, orderings, mappings, and relations Computing integers, rational numbers, cyclotomic numbers(gaussian numbers), finite fields, p-adic numbers, polynomials, rational functions, group elements, permutation, matrices, abstract words etc 19

20 Symbolic Mathematics/ Computing vs Numerical Mathematics/ Computing Symbolic Computing Definition: Wikipedia: scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects 20

21 Symbolic Mathematics/ Computing vs Numerical Mathematics/ Computing Symbolic Computing Definition: Wikipedia: scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects Numerical Computing Definition: Wikipedia: the study of algorithms that use numerical approximation for the problems of mathematical analysis 21

22 Symbolic Mathematics/ Computing vs Numerical Mathematics/ Computing Symbolic Computing Definition: Wikipedia: scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects Numerical Computing Definition: Wikipedia: the study of algorithms that use numerical approximation for the problems of mathematical analysis What can be considered as of symbolic computing: symbolic mathematical operation (integration, summation, solution of differential equation,..) Polynomial factorisation etc 22

23 Symbolic Mathematics/ Computing vs Numerical Mathematics/ Computing Symbolic Computing Definition: Wikipedia: scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects Numerical Computing Definition: Wikipedia: the study of algorithms that use numerical approximation for the problems of mathematical analysis What can be considered as of symbolic computing: symbolic mathematical operation (integration, summation, solution of differential equation,..) Polynomial factorisation etc Good for teaching mathematics! 23

24 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 24

25 Group theory 25

26 Group theory Studies algebraic structures known as group 26

27 Group theory Studies algebraic structures known as group Groups like: rings, fields, vector spaces with their operations and axioms 27

28 Group theory Studies algebraic structures known as group Groups like: rings, fields, vector spaces with their operations and axioms More focused disciplines: Linear algebraic groups, Lie groups 28

29 Group theory Studies algebraic structures known as group Groups like: rings, fields, vector spaces with their operations and axioms More focused disciplines: Linear algebraic groups, Lie groups Applications in: Chemistry; Physics; Cryptography, etc 29

30 Group theory Studies algebraic structures known as group Groups like: rings, fields, vector spaces with their operations and axioms More focused disciplines: Linear algebraic groups, Lie groups Applications in: Chemistry; Physics; Cryptography, etc 30

31 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 31

32 History of GAP (1) 1986: developed first by Lehrstuhl D für Mathematik, RWTH Aachen: Joachim Neubüser, Johannes Meier, Alice Niemayer, Werner Nickel, Martin Schönert 1988: first publicly released version: GAP : development in CIRCA, St Andrews, Scotland, UK 2008: awarded the ACM/SIGSAM Richard Dimick Jenks Memorial Prize for Excellence in Software Engineering applied to Computer Algebra 32

33 History of GAP (2) 2012: Latest version GAP Originally supported by DFG (since 1992) Developed now by a group of researchers in: 33

34 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 34

35 The GAP systems 35

36 The GAP systems Language GAP: C language 36

37 The GAP systems Language GAP: C language Packages: ca 100 packages, act as algorithms or methods or libraries 37

38 The GAP systems Language GAP: C language Packages: ca 100 packages, act as algorithms or methods or libraries Data Library: 38

39 The GAP systems Language GAP: C language Packages: ca 100 packages, act as algorithms or methods or libraries Data Library Interactive Environment (GUI) 39

40 The GAP systems Language GAP: C language Packages: ca 100 packages, act as algorithms or methods or libraries Data Library Interactive Environment (GUI) Documentation 40

41 The GAP systems Language GAP: C language Packages: ca 100 packages, act as algorithms or methods or libraries Data Library: Interactive Environment (GUI) Documentation The kernel is written in C! 41

42 GAP codes libraries GMP Kernel C codes 42

43 Kernel C codes 43

44 Implemented in GAP language Kernel C codes 44

45 Implemented in GAP language Interactive environment in developing and using GAP Kernel C codes 45

46 Implemented in GAP language Interactive environment in developing and using GAP Kernel C codes Memory management 46

47 Implemented in GAP language Interactive environment in developing and using GAP Kernel C codes fast versions of time critical operations for various data types Memory management 47

48 GAP codes 48

49 GAP codes Pascal dialect using streams, lists, and records automatic memory management, garbage collection! 49

50 GAP codes Pascal dialect using streams, lists, and records automatic memory management, garbage collection! 50

51 GAP codes Pascal dialect using streams, lists, and records automatic memory management, garbage collection! 51

52 GMP: GNU Multiple Precision Arithmetic Library 52

53 GMP: GNU Multiple Precision Arithmetic Library GMP 53

54 GMP: GNU Multiple Precision Arithmetic Library GMP Floating Points 54

55 GMP: GNU Multiple Precision Arithmetic Library GMP Floating Points 55

56 GMP: GNU Multiple Precision Arithmetic Library GMP Floating Points 56

57 GMP: GNU Multiple Precision Arithmetic Library free library GMP Floating Points 57

58 GMP: GNU Multiple Precision Arithmetic Library free library basic interface is for C, but also supports C#, Ocaml, Perl, PHP, Python GMP Floating Points 58

59 GMP: GNU Multiple Precision Arithmetic Library free library basic interface is for C, but also supports C#, Ocaml, Perl, PHP, Python GMP Floating Points 59

60 libraries Support mathematical operations: basic algebra, group theory, etc 60

61 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 61

62 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 62

63 Bibliography overview, installation, documentation, downloads, Ogilvie, J. F., Teaching and doing Mathematics with Symbolic Computation, Symbolic Computation (An Editorial), Journal of Symbolic Computation, Volume 1, Number 1-6,

64 Outline What is GAP? Symbolic computation Group theory History The GAP System Demo & Functionality Bibliography Q & A 64

65 Q & A 65

66 Thanks to Prof. Dr. Paolo Bientinessi and his team for giving opportunity to present this seminar Danan Wicaksono, M.Sc for introducing the importance of symbolic computing Dr. Frank Lübeck for providing further information on GAP Prof. Dr. Klaus Lux for providing further information on GAP 66