An Introduction to Open-Source Mathematics Software
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1 1 An Introduction to Open-Source Mathematics Software (Specifically: GAP,, Macaulay2 and Sage) Jason B. Hill Slow Pitch Colloquium University of Colorado March 17, 2010
2 2 Outline 1 Introduction 2 GAP - Groups, Algorithms, Programming 3 4 Macaulay2 5 Sage
3 3 Introduction Why use these programs?
4 3 Introduction Why use these programs? When you think of mathematics software, something like Mathematica, Wolfram Alpha or Maple may come to mind.
5 Introduction 3 Why use these programs? When you think of mathematics software, something like Mathematica, Wolfram Alpha or Maple may come to mind. These all have a rich GUI = Graphical User Interface.
6 Introduction 3 Why use these programs? When you think of mathematics software, something like Mathematica, Wolfram Alpha or Maple may come to mind. These all have a rich GUI = Graphical User Interface. They can also perform some amazing calculations.
7 Introduction 3 Why use these programs? When you think of mathematics software, something like Mathematica, Wolfram Alpha or Maple may come to mind. These all have a rich GUI = Graphical User Interface. They can also perform some amazing calculations. Unfortunately, they are too often exploited for trivial calculations.
8 Introduction Why use these programs?
9 Introduction Why use these programs? The programs presented here differ slightly.
10 Introduction Why use these programs? The programs presented here differ slightly. All of them are free.
11 Introduction Why use these programs? The programs presented here differ slightly. All of them are free. All of them are open-source.
12 Introduction Why use these programs? The programs presented here differ slightly. All of them are free. All of them are open-source. They are developed and maintained by professional mathematicians, people who need to do serious (nontrivial) calculations.
13 Introduction Why use these programs? The programs presented here differ slightly. All of them are free. All of them are open-source. They are developed and maintained by professional mathematicians, people who need to do serious (nontrivial) calculations. All of the programs presented here are algebraic programs. There are also statistical, numerical,... programs that fall under the free/open-source category. I m just not familiar with them.
14 5 Introduction Terminal based versus GUI
15 5 Introduction Terminal based versus GUI All of the programs shown in this talk are terminal native.
16 Introduction 5 Terminal based versus GUI All of the programs shown in this talk are terminal native. Most of them have some sort of GUI that can be added.
17 Introduction 5 Terminal based versus GUI All of the programs shown in this talk are terminal native. Most of them have some sort of GUI that can be added. For example, Sage Notebook has become a standard Sage interface.
18 Introduction 5 Terminal based versus GUI All of the programs shown in this talk are terminal native. Most of them have some sort of GUI that can be added. For example, Sage Notebook has become a standard Sage interface. For serious computation, the terminal is faster.
19 6 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
20 6 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming Purpose Mostly: Group Theory Developers University of St. Andrews, RWTH Aachen, Technische Universität Braunschweig, Colorado State University Stable Release December 2008 Structure Written in C Internally like Python Download or repositories OS Source + Linux, OS-X, Windows, BSD, Solaris
21 7 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming GAP is installed on euclid.
22 7 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming GAP is installed on euclid. There are two ways to access euclid:
23 7 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming GAP is installed on euclid. There are two ways to access euclid: From one of the Math Dept office desktops, open a terminal.
24 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming GAP is installed on euclid. There are two ways to access euclid: From one of the Math Dept office desktops, open a terminal. Open a terminal anywhere online and SSH to euclid.
25 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming GAP is installed on euclid. There are two ways to access euclid: From one of the Math Dept office desktops, open a terminal. Open a terminal anywhere online and SSH to euclid. Once on euclid (or any Math Dept desktop) the prompt looks like: euclid: >
26 GAP - Groups, Algorithms, Programming 7 GAP Groups, Algorithms, Programming GAP is installed on euclid. There are two ways to access euclid: From one of the Math Dept office desktops, open a terminal. Open a terminal anywhere online and SSH to euclid. Once on euclid (or any Math Dept desktop) the prompt looks like: euclid: > Type gap.sh to start GAP euclid: >gap.sh
27 GAP - Groups, Algorithms, Programming 7 GAP Groups, Algorithms, Programming GAP is installed on euclid. There are two ways to access euclid: From one of the Math Dept office desktops, open a terminal. Open a terminal anywhere online and SSH to euclid. Once on euclid (or any Math Dept desktop) the prompt looks like: euclid: > Type gap.sh to start GAP euclid: >gap.sh Note: Installing GAP on another computer may change this: hilljb@tarski: $ gap
28 8 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
29 9 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming All (or most) GAP commands end in a semi-colon.
30 9 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming All (or most) GAP commands end in a semi-colon. 2+2;
31 9 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming All (or most) GAP commands end in a semi-colon. 2+2; 65*3;
32 9 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming All (or most) GAP commands end in a semi-colon. 2+2; 65*3; GAP commands are case-sensitive.
33 9 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming All (or most) GAP commands end in a semi-colon. 2+2; 65*3; GAP commands are case-sensitive. Factorial(72);
34 9 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming All (or most) GAP commands end in a semi-colon. 2+2; 65*3; GAP commands are case-sensitive. Factorial(72); IsPrime( );
35 GAP - Groups, Algorithms, Programming 9 GAP Groups, Algorithms, Programming All (or most) GAP commands end in a semi-colon. 2+2; 65*3; GAP commands are case-sensitive. Factorial(72); IsPrime( ); It is useful to know how to stop GAP once started. gap> quit;
36 GAP - Groups, Algorithms, Programming 9 GAP Groups, Algorithms, Programming All (or most) GAP commands end in a semi-colon. 2+2; 65*3; GAP commands are case-sensitive. Factorial(72); IsPrime( ); It is useful to know how to stop GAP once started. gap> quit; Often, your input doesn t match a function s defintion. Type Ctrl+D when this happens.
37 10 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
38 11 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
39 12 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
40 13 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
41 14 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
42 15 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming GAP has several ways to construct groups.
43 GAP - Groups, Algorithms, Programming 15 GAP Groups, Algorithms, Programming GAP has several ways to construct groups. Many groups it is already familiar with: gap> G:=SymmetricGroup(8); Sym( [ ] ) gap> G; Sym( [ ] )
44 GAP - Groups, Algorithms, Programming 15 GAP Groups, Algorithms, Programming GAP has several ways to construct groups. Many groups it is already familiar with: gap> G:=SymmetricGroup(8); Sym( [ ] ) gap> G; Sym( [ ] ) Or you can specify your own: gap> H:=Group([(1,2),(1,2,3,4,5,6,7,8)]); Group([ (1,2), (1,2,3,4,5,6,7,8) ])
45 16 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming In short, GAP can represent the same group in several ways.
46 GAP - Groups, Algorithms, Programming 16 GAP Groups, Algorithms, Programming In short, GAP can represent the same group in several ways. gap> F1 := FreeGroup( "a", "b" ); <free group on the generators [ a, b ]>
47 GAP - Groups, Algorithms, Programming 16 GAP Groups, Algorithms, Programming In short, GAP can represent the same group in several ways. gap> F1 := FreeGroup( "a", "b" ); <free group on the generators [ a, b ]> gap> A5 := F1 / [ F1.1 2, F1.2 3, (F1.1*F1.2) 5 ]; <fp group on the generators [ a, b ]>
48 GAP - Groups, Algorithms, Programming 16 GAP Groups, Algorithms, Programming In short, GAP can represent the same group in several ways. gap> F1 := FreeGroup( "a", "b" ); <free group on the generators [ a, b ]> gap> A5 := F1 / [ F1.1 2, F1.2 3, (F1.1*F1.2) 5 ]; <fp group on the generators [ a, b ]> gap> G := AlternatingGroup(5); Alt( [ ] )
49 GAP - Groups, Algorithms, Programming 16 GAP Groups, Algorithms, Programming In short, GAP can represent the same group in several ways. gap> F1 := FreeGroup( "a", "b" ); <free group on the generators [ a, b ]> gap> A5 := F1 / [ F1.1 2, F1.2 3, (F1.1*F1.2) 5 ]; <fp group on the generators [ a, b ]> gap> G := AlternatingGroup(5); Alt( [ ] ) gap> IsomorphismGroups( A5, G ); [ a*b 2 *a*b 3 *a*b, b ] -> [ (2,4)(3,5), (1,2,3) ] gap>
50 GAP - Groups, Algorithms, Programming 17 GAP Groups, Algorithms, Programming GAP documentation is given by searching within GAP: If you want to find out how to compute an orbit, type?orbit
51 18 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
52 19 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming Then, typing?1 gives...
53 20 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
54 21 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming GAP works well for large groups. Consider the permutation group on faces of a Rubik s cube.
55 22 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
56 23 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
57 24 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming Some other calculations we may be interested in...
58 25 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
59 26 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
60 27 GAP - Groups, Algorithms, Programming GAP Groups, Algorithms, Programming
61 28
62 28 Purpose Mostly: Number Theory & Elliptic Curves Developers Université Bordeaux Stable Release February 2010 Structure PARI is a C library gp is a terminal calculator Download OS Source + Linux, OS-X, Windows, BSD, Solaris
63 29
64 29 PARI is a C library with an emphasis on efficiency
65 29 PARI is a C library with an emphasis on efficiency It does no calculations itself.
66 29 PARI is a C library with an emphasis on efficiency It does no calculations itself. But the libraries/functions it contains are amazing.
67 29 PARI is a C library with an emphasis on efficiency It does no calculations itself. But the libraries/functions it contains are amazing. GP is a terminal calculator based on PARI
68 29 PARI is a C library with an emphasis on efficiency It does no calculations itself. But the libraries/functions it contains are amazing. GP is a terminal calculator based on PARI Less efficient than compiling with PARI, still very efficient
69 29 PARI is a C library with an emphasis on efficiency It does no calculations itself. But the libraries/functions it contains are amazing. GP is a terminal calculator based on PARI Less efficient than compiling with PARI, still very efficient A little awkward. The output is less user friendly.
70 29 PARI is a C library with an emphasis on efficiency It does no calculations itself. But the libraries/functions it contains are amazing. GP is a terminal calculator based on PARI Less efficient than compiling with PARI, still very efficient A little awkward. The output is less user friendly. The entire system is rigorously typed.
71 29 PARI is a C library with an emphasis on efficiency It does no calculations itself. But the libraries/functions it contains are amazing. GP is a terminal calculator based on PARI Less efficient than compiling with PARI, still very efficient A little awkward. The output is less user friendly. The entire system is rigorously typed. Light programming is possible in GP, for anything else use C/PARI
72 30
73 31 To start the GP calculator on euclid, euclid: >gp To exit the GP calculator on euclid,? quit
74 32
75 33
76 34 The documentation in GP is built-in and in PDF form:
77 34 The documentation in GP is built-in and in PDF form: Type? for the help dialog
78 34 The documentation in GP is built-in and in PDF form: Type? for the help dialog Type?function for information about function
79 34 The documentation in GP is built-in and in PDF form: Type? for the help dialog Type?function for information about function In my experience, PARI is better documented externally.
80 35
81 36 Some algebraic number theory calculations in GP:
82 36 Some algebraic number theory calculations in GP: Create a polynomial / find characteristic polynomials.
83 36 Some algebraic number theory calculations in GP: Create a polynomial / find characteristic polynomials. Check if that polynomial is reducible.
84 36 Some algebraic number theory calculations in GP: Create a polynomial / find characteristic polynomials. Check if that polynomial is reducible. Initialize a number field using the roots of a polynomial.
85 36 Some algebraic number theory calculations in GP: Create a polynomial / find characteristic polynomials. Check if that polynomial is reducible. Initialize a number field using the roots of a polynomial. Find the discriminant
86 36 Some algebraic number theory calculations in GP: Create a polynomial / find characteristic polynomials. Check if that polynomial is reducible. Initialize a number field using the roots of a polynomial. Find the discriminant... or an integral basis
87 36 Some algebraic number theory calculations in GP: Create a polynomial / find characteristic polynomials. Check if that polynomial is reducible. Initialize a number field using the roots of a polynomial. Find the discriminant... or an integral basis... or the regulator
88 36 Some algebraic number theory calculations in GP: Create a polynomial / find characteristic polynomials. Check if that polynomial is reducible. Initialize a number field using the roots of a polynomial. Find the discriminant... or an integral basis... or the regulator... or a generating set for the unit group of the integer ring.
89 37
90 38
91 39
92 40
93 41
94 Other calculations in GP:
95 Other calculations in GP: Elliptic Curves Sum points on a curve Compute L-function coefficients. Find Z-basis for free parts of Mordell-Weil groups. Compute heights
96 Other calculations in GP: Elliptic Curves Sum points on a curve Compute L-function coefficients. Find Z-basis for free parts of Mordell-Weil groups. Compute heights Matrix calculations Solving large systems of equations. Various calculations
97 43
98 Macaulay2 Macaulay2
99 Macaulay2 Macaulay2 Purpose Mostly: Commutative Algebra & Algebraic Geometry Developers Daniel Grayson Michael Stillman David Eisenbud Stable Release November 2009 Structure Good question Download OS Source + Linux, BSD, OS-X, Windows
100 45 Macaulay2 Macaulay2 Macaulay2 is not on euclid.
101 Macaulay2 45 Macaulay2 Macaulay2 is not on euclid. But, it is free and you can download it if you need it.
102 Macaulay2 45 Macaulay2 Macaulay2 is not on euclid. But, it is free and you can download it if you need it. The notation is slightly odd, but the documentation on the Macaulay2 website does a pretty good job of showing examples.
103 Macaulay2 45 Macaulay2 Macaulay2 is not on euclid. But, it is free and you can download it if you need it. The notation is slightly odd, but the documentation on the Macaulay2 website does a pretty good job of showing examples. Macaulay2 is considered a more specific piece of software, dealing with: finite fields, polynomial rings, graded/multigraded polynomial rings, quotient rings, factorization, exterior algebras, symmetric algebras, tensor products, associative algebras, etc.
104 46 Macaulay2 Macaulay2
105 Macaulay2 Macaulay2 We could...
106 Macaulay2 Macaulay2 We could... Generate random homogeneous elements of our polynomial ring...
107 Macaulay2 47 Macaulay2 We could... Generate random homogeneous elements of our polynomial ring... Form a basis of the subspace of ring elements of a given degree..
108 Macaulay2 47 Macaulay2 We could... Generate random homogeneous elements of our polynomial ring... Form a basis of the subspace of ring elements of a given degree.. Construct a polynomial ring over our polynomial ring...
109 Macaulay2 47 Macaulay2 We could... Generate random homogeneous elements of our polynomial ring... Form a basis of the subspace of ring elements of a given degree.. Construct a polynomial ring over our polynomial ring... Examine ideals and factorization...
110 48 Macaulay2 Macaulay2
111 49 Macaulay2 Macaulay2
112 50 Macaulay2 Macaulay2
113 51 Macaulay2 Macaulay2
114 52 Sage Sage
115 52 Sage Sage Purpose Algebra, Combinatorics, Numerical, Calculus Developers William Stein + others Stable Release January 2010 Structure Written in Python Interface varies Download OS Source + Linux, OS-X, Solaris, LiveCD
116 53 Sage Sage Sage is largely a combination of other open-source packages.
117 Sage 53 Sage Sage is largely a combination of other open-source packages. Sage includes most of PARI and GAP, for instance.
118 Sage 53 Sage Sage is largely a combination of other open-source packages. Sage includes most of PARI and GAP, for instance. Sage will also interface with non-open-source software: Mathematica, Magma, Maple.
119 Sage 53 Sage Sage is largely a combination of other open-source packages. Sage includes most of PARI and GAP, for instance. Sage will also interface with non-open-source software: Mathematica, Magma, Maple. As a result, it is capable of A LOT.
120 54 Sage Sage
121 Sage 55 Sage Sage is installed on euclid. Initialize Sage with the command: euclid: >sage Exit Sage with the command: sage: quit
122 56 Sage Sage
123 57 Sage Sage Sage can do group theory
124 58 Sage Sage
125 59 Sage Sage Sage can do calculus
126 60 Sage Sage
127 61 Sage Sage Sage can do more advanced mathematics
128 62 Sage Sage
129 Sage 63 Sage Sage notebook is a web-based GUI for Sage.
130 64 Sage Sage
131 Sage 65 The End
Open source software and Sage 1
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