Nemo: a computer algebra package for Julia
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1 TU Kaiserslautern July 12, 2017
2 Introducing A computer algebra package for the Julia programming language.
3 Background 2006 David Harvey and I began the Flint C library for fast arithmetic 2010 Fredrik Johansson, Sebastian Pancratz and I began Flint 2, complete rewrite of Flint 2013 Claus Fieker and I began Antic C library for number eld element arithmetic 2014 Claus Fieker, Tommy Hofmann, Fredrik Johansson and I began Nemo generics 2015 Oleksandr Motsak and I began Singular.jl package in Julia 2015 Tommy Hofmann and Claus Fieker began Hecke algebraic number theory package 2016 e7.6m grant for computer algebra, will fund OSCAR computer algebra system
4 New features in Flint LGPL license
5 New features in Flint LGPL license Exception system
6 New features in Flint LGPL license Exception system Quadratic sieve factorisation
7 New features in Flint LGPL license Exception system Quadratic sieve factorisation Elliptic curve factorisation
8 New features in Flint LGPL license Exception system Quadratic sieve factorisation Elliptic curve factorisation APRCL primality test
9 New features in Flint LGPL license Exception system Quadratic sieve factorisation Elliptic curve factorisation APRCL primality test Parallelised FFT
10 New features in Flint LGPL license Exception system Quadratic sieve factorisation Elliptic curve factorisation APRCL primality test Parallelised FFT Howell form
11 New features in Flint LGPL license Exception system Quadratic sieve factorisation Elliptic curve factorisation APRCL primality test Parallelised FFT Howell form Characteristic and minimal polynomial
12 New features in Flint LGPL license Exception system Quadratic sieve factorisation Elliptic curve factorisation APRCL primality test Parallelised FFT Howell form Characteristic and minimal polynomial van Hoeij factorisation for Z[x]
13 New features in Flint LGPL license Exception system Quadratic sieve factorisation Elliptic curve factorisation APRCL primality test Parallelised FFT Howell form Characteristic and minimal polynomial van Hoeij factorisation for Z[x] Multivariate polynomial arithmetic Z[x, y, z,...]
14 Integer factorisation : Quadratic sieve Table : Quadratic sieve timings Digits Pari/GP Flint (1 core) Flint (4 cores)
15 APRCL primality test timings
16 FFT: Integer and polynomial multiplication Table : FFT timings Words 1 core 4 cores 8 cores 110k 0.07s 0.05s 0.05s 360k 0.3s s 1.3m 1.1s 0.4s 0.3s 4.6m 4.5s 1.5s 1.0s 26m 28s 9s 6s 120m 140s 48s 33s 500m 800s 240s 150s
17 Characteristic and minimal polynomial Table : Charpoly and minpoly timings Op Sage 6.9 Pari Magma Giac Flint Charpoly 0.2s 0.6s 0.06s 0.06s 0.04s Minpoly 0.07s >160 hrs 0.05s 0.06s 0.04s for matrix over Z with entries in [ 20, 20] and minpoly of degree 40.
18 Multivariate multiplication Table : Dense Fateman multiply bench n Sage Singular Magma Giac Flint s 0.001s 0.002s s s s 0.18s 0.12s 0.006s 0.004s 15 10s 5.6s 1.9s 0.11s 0.045s 20 76s 66s 16s 0.62s 0.50s s 410s 98s 2.8s 2.3s s 1500s 440s 14s 10s 4 variables
19 Multivariate multiplication Table : Sparse Pearce multiply bench n Sage Singular Magma Giac Flint s 0.003s 0.006s 0.004s 0.002s s 0.08s 0.08s 0.03s 0.02s 8 2.0s 1.4s 0.68s 0.28s 0.12s 10 11s 9.5s 3.0s 1.5s 0.55s 12 57s 38s 11s 4.8s 2.5s s 160s 37s 14s 11s s 510s 94s 39s 26s 5 variables
20 Exact quotient Table : Dense quotient only n Sage Singular Magma Giac Flint s 0.003s 0.002s s s s 0.11s 0.16s s s 15 28s 1.5s 3.5s s s 19s 35s 0.25s 0.15s s 130s 210s 1.1s 0.93s s 830s 6.0s 3.6s 4 variables
21 Exact quotient Table : Sparse quotient only n Sage Singular Magma Giac Flint s 0.02s 0.005s 0.001s s 6 107s 0.24s 0.17s 0.014s 0.010s 8 1.6s 3.1s 0.12s 0.068s s 27s 0.93s 0.34s 12 26s 140s 2.5s 1.7s 14 98s 630s 8.0s 6.7s s 2300s 22s 19s 5 variables
22 Divisibility testing Table : Dense divisibility test with quotient n Sage Singular Magma Giac Flint s 0.006s 0.002s 0.001s s s 0.56s 0.16s 0.05s 0.020s 15 28s 15s 3.3s 0.15s 0.054s s 150s 31s 0.90s 0.48s s 840s 200s 4.1s 2.3s s 830s 21s 11s 4 variables, returns quotient
23 Divisibility testing Table : Sparse divisibility test with quotient n Sage Singular Magma Giac Flint s 0.03s 0.005s 0.002s 0.002s 6 107s 0.54s 0.17s 0.024s 0.021s 8 6.6s 3.1s 0.19s 0.15s 10 38s 27s 1.3s 0.69s s 140s 4.3s 2.8s s 630s 14s 9.0s s 2300s 40s 26s 5 variables, returns quotient
24 Language for generic programming In 2010 I started looking for a language for generic programming:
25 Language for generic programming In 2010 I started looking for a language for generic programming: Support for Windows, Linux, Mac
26 Language for generic programming In 2010 I started looking for a language for generic programming: Support for Windows, Linux, Mac 64 bit integers and double precision oats
27 Language for generic programming In 2010 I started looking for a language for generic programming: Support for Windows, Linux, Mac 64 bit integers and double precision oats Console/REPL mode
28 Language for generic programming In 2010 I started looking for a language for generic programming: Support for Windows, Linux, Mac 64 bit integers and double precision oats Console/REPL mode Operator overloading
29 Language for generic programming In 2010 I started looking for a language for generic programming: Support for Windows, Linux, Mac 64 bit integers and double precision oats Console/REPL mode Operator overloading Fast generics and metaprogramming
30 Language for generic programming In 2010 I started looking for a language for generic programming: Support for Windows, Linux, Mac 64 bit integers and double precision oats Console/REPL mode Operator overloading Fast generics and metaprogramming Maintained and popular
31 Language for generic programming In 2010 I started looking for a language for generic programming: Support for Windows, Linux, Mac 64 bit integers and double precision oats Console/REPL mode Operator overloading Fast generics and metaprogramming Maintained and popular Open source
32 Language for generic programming In 2010 I started looking for a language for generic programming: Support for Windows, Linux, Mac 64 bit integers and double precision oats Console/REPL mode Operator overloading Fast generics and metaprogramming Maintained and popular Open source Imperative syntax
33 Language for generic programming In 2010 I started looking for a language for generic programming: Support for Windows, Linux, Mac 64 bit integers and double precision oats Console/REPL mode Operator overloading Fast generics and metaprogramming Maintained and popular Open source Imperative syntax Garbage collected
34 Language for generic programming In 2010 I started looking for a language for generic programming: Support for Windows, Linux, Mac 64 bit integers and double precision oats Console/REPL mode Operator overloading Fast generics and metaprogramming Maintained and popular Open source Imperative syntax Garbage collected Easy/ecient C interop
35 Ecient generics
36 Ecient generics
37
38 JIT compilation : near C performance. Designed by mathematically minded people. Open Source (MIT License). Actively developed since Supports Windows, OSX, Linux, BSD. Friendly C/Python-like (imperative) syntax.
39 Julia is polymorphic: gcd ( a : : I n t, b : : I n t ) gcd ( a : : B i g I n t, b : : B i g I n t ) gcd {T <: F i e l d }( a : : P o l y {T}, b : : P o l y {T})
40 Julia is polymorphic: gcd ( a : : I n t, b : : I n t ) gcd ( a : : B i g I n t, b : : B i g I n t ) gcd {T <: F i e l d }( a : : P o l y {T}, b : : P o l y {T}) Julia supports multimethods: ( a : : I n t, b : : M a t r i x { I n t }) ( a : : M a t r i x { I n t }, b : : I n t )
41 Julia is polymorphic: gcd ( a : : I n t, b : : I n t ) gcd ( a : : B i g I n t, b : : B i g I n t ) gcd {T <: F i e l d }( a : : P o l y {T}, b : : P o l y {T}) Julia supports multimethods: ( a : : I n t, b : : M a t r i x { I n t }) ( a : : M a t r i x { I n t }, b : : I n t ) Julia supports triangular dispatch: {T <: Q u o t i e n t R i n g, S <: P o l y {T}}( x : : T, y : : S )
42 Julia supports coercion in a natural way: +{T <: Domain }( a : : L a u r e n t {T}, b : : S e r i e s { F r a c t i o n F i e l d {T}})
43 Julia supports coercion in a natural way: +{T <: Domain }( a : : L a u r e n t {T}, b : : S e r i e s { F r a c t i o n F i e l d {T}}) Julia supports: Custom array indexing Custom printing of objects Custom promotion rules and conversions
44 Julia supports coercion in a natural way: +{T <: Domain }( a : : L a u r e n t {T}, b : : S e r i e s { F r a c t i o n F i e l d {T}}) Julia supports: Custom array indexing Custom printing of objects Custom promotion rules and conversions Coming soon in Julia: Traits
45
46 Interfaces to C libraries: Flint
47 Interfaces to C libraries: Flint Arb
48 Interfaces to C libraries: Flint Arb Antic
49 Interfaces to C libraries: Flint Arb Antic Singular kernel (via Singular.jl)
50 Interfaces to C libraries: Flint Arb Antic Singular kernel (via Singular.jl) Generic algorithms: Residue rings
51 Interfaces to C libraries: Flint Arb Antic Singular kernel (via Singular.jl) Generic algorithms: Residue rings Fraction elds
52 Interfaces to C libraries: Flint Arb Antic Singular kernel (via Singular.jl) Generic algorithms: Residue rings Fraction elds Dense univariate polynomials
53 Interfaces to C libraries: Flint Arb Antic Singular kernel (via Singular.jl) Generic algorithms: Residue rings Fraction elds Dense univariate polynomials Sparse distributed multivariate polynomials
54 Interfaces to C libraries: Flint Arb Antic Singular kernel (via Singular.jl) Generic algorithms: Residue rings Fraction elds Dense univariate polynomials Sparse distributed multivariate polynomials Dense linear algebra
55 Interfaces to C libraries: Flint Arb Antic Singular kernel (via Singular.jl) Generic algorithms: Residue rings Fraction elds Dense univariate polynomials Sparse distributed multivariate polynomials Dense linear algebra Power series : absolute and relative
56 Interfaces to C libraries: Flint Arb Antic Singular kernel (via Singular.jl) Generic algorithms: Residue rings Fraction elds Dense univariate polynomials Sparse distributed multivariate polynomials Dense linear algebra Power series : absolute and relative Permutation groups
57 Hecke Orders and ideals in absolute number elds Fast ideal and element arithmetic in absolute number elds Veried computations with approximations using interval arithmetic whenever necessary (e.g. computation with embeddings or residue computation of Dedekind zeta functions) Sparse linear algebra over Z Class and unit group computation Pseudo-Hermite normal form for modules over Dedekind domains Beginnings of class eld theory and relative extensions
58 Future plans: Global function elds Class formations and more Galois cohomology Galois module structure Noncommutative algebras and orders
59 Thank You
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