This document discusses Julia's use of multimethod dispatch and generic functions to provide both data abstraction and high performance. Julia bridges the divide between computer science and computational science by allowing users to write generic code that can also achieve native machine performance when applicable. Its type system and multimethods allow methods to be specialized for different argument types, enabling both abstraction and optimized implementations.

1. Multimethods for abstraction and
performance
Jiahao Chen
Research Scientist
MIT CSAIL
jiahao.github.io
GitHub: @jiahao

2. People like to reinvent programming languages

4. Alan Edelman Andreas Noack Xianyi Zhang Jarrett Revels
Oscar BlumbergDavid Sanders
The
Julia Lab
at MIT
Simon Danisch
Jiahao Chen
Weijian Zhang
U. Manchester
Jake Bolewski
USAP
Shashi GowdaAmit Murthy Tanmay Mohapatra
Eka Palamadai
Collaborators
Joey Huchette
Isaac Virshup
Steven Johnson
MIT Mathematics
Yes Sian Ng Miles Lubin Iain Dunning
Jon Malmaud Simon Kornblith
Yichao Yu
Harvard
Jeremy Kepner
Lincoln Labs
Stavros
Papadopoulos
Intel Labs
Nikos
Patsopoulos
Brigham Woman’s
Hospital
Pete Szolovits
CSAIL
Alex Townsend
MIT Mathematics
Jack Poulson
Stanford

5. Mike Innes
Julia Computing
Summer of
Code alums
Keno Fischer
Julia Computing
Jameson Nash
Julia Computing
Simon Danisch
Julia Lab
Shashi Gowda
Julia Lab
Leah Hanson
Stripe
John Myles White
Facebook
Jarrett Revels
Julia Lab
2013
2014
2015
Jacob Quinn
Domo
Kenta Sato
U. Tokyo
Rohit Varkey Thankachan
Nat’l Inst. Tech. Karnataka
Simon Danish
Julia Lab
David Gold
U. Washington
Keno FischerJameson NashStefan KarpinskiJeff Bezanson Viral B. Shah
Alums at
Mike Innes

6. 445 contributors to Julia
726 package authors
as of 2016-03-15

7. https://github.com/jiahao/ijulia-notebooks
The world of
808 packages, 726 authors
445 contributors to julia repo
6,841 stargazers
549 watchers

8. What’s the big deal about Julia ?
A high level language
with C-like speed
julialang.org/benchmarks

9. What’s the big deal about Julia ?
Normalized average lines of code (shortest = 0, longest = 1)
Execution time (C = 1)
https://groups.google.com/d/msg/julia-users/BYRAeQJuvTw/O7VK7-vp1EEJ
fast and expressive

10. A simple example

11. It bridges the divide between computer science
and computational science
What’s the big deal about Julia ?

12. It bridges the divide between computer science
and computational science
What’s the big deal about Julia ?
data abstraction
performance

13. It bridges the divide between computer science
and computational science
What’s the big deal about Julia ?
data abstraction
performance
What if you didn’t have to choose between
data abstraction and performance?

14. Thesis:
Users want to write generic code.
Users also want native machine performance (when available).
Julia’s generic function system with multimethods (multiple
dispatch) lets you have both.

15. Is Julia object oriented?
Yes, but maybe not in a way you’re familiar with.*
Generic functions** like in CLOS,
unlike C++ classes.
*G. Castagna, Object Oriented Programming: A Unified Foundation, http://link.springer.com/book/10.1007/978-1-4612-4138-6
**L. G. DeMichiel and R. P. Gabriel, §3.2 of The Common Lisp Object System: An Overview, http://www.dreamsongs.com/Files/ECOOP.pdf

16. Object-oriented programming with classes
What can I do with/to a thing?

17. methods objects
Object-oriented programming with classes
What can I do with/to a thing?

18. methods objects
Object-oriented programming with classes
What can I do with/to a thing?

19. methods objects
Object-oriented programming with classes
What can I do with/to a thing?
top up
pay fare
lose
buy

20. methods objects
Object-oriented programming with classes
What can I do with/to a thing?
top up
pay fare
lose
buy
top up
pay fare
lose
buy

21. methods objects
Object-oriented programming with classes
What can I do with/to a thing?
top up
pay fare
lose
buy
top up
pay fare
lose
buy

22. methods objects
Object-oriented programming with classes
What can I do with/to a thing?
top up
pay fare
lose
buy
top up
pay fare
lose
buy
pay fare
lose
buy

23. methods objects
Object-oriented programming with classes
What can I do with/to a thing?
top up
pay fare
lose
buy
classes are more
fundamental
than methods
top up
pay fare
lose
buy
pay fare
lose
buy

24. OOP with classes multimethods
What can I do with/to a thing?
top up
pay fare
lose
buy
generic
function
objectsmethods
multimethods
relationships between
objects and functions

25. Multimethods for linear algebra
What can I do with/to a thing?
compute spectral factorization
compute singular values
compute singular values and vectors
compute eigenvalues
generic
function
objectsmethods
Methods can take advantage of special matrix structures
eigvals
eigfact
svdvals
svdfact
Matrix
SymTridiagonal
Bidiagonal

26. Data types as a lattice
D. Scott, “Data types as lattices”, SIAM J. Comput. 5: 522-87, 1976, doi:10.1137/0205037
Real
Number
FloatingPoint Rational
Complex
Float64 BigFloat…
…
Integer
is a parameter of
is a subtype of
Signed BigInt
Any
Int64…
…

27. Signed
Multimethods with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…

28. Signed
Multimethods with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…

29. Signed
Multimethods with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…

30. Signed
Multimethods with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…
no method here
try supertype
super(Int64) = Signed

31. Signed
Multimethods with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…
no method here
try supertype
super(Int64) = Signed
no method here either
super(Signed) = Integer

32. Signed
Multimethods with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…
no method here
try supertype
super(Int64) = Signed
no method here either
super(Signed) = Integer
found a method

33. Signed
Multimethods with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…

34. Signed
Multimethods with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…

42. easy to call external C
functions, e.g. CLAPACK
sstev, dstev…

43. So how does this help us with linear algebra?
Multi-method dispatch with generic fallbacks
Matrix operations on general rings

44. So how does this help us with linear algebra?
Multi-method dispatch with generic fallbacks
Matrix operations on general rings textbook algorithm

45. So how does this help us with linear algebra?
Multi-method dispatch with generic fallbacks
Matrix operations on general rings

46. Native parallelism constructs

47. Native parallelism constructs

48. Native parallelism constructs

49. JuMP: a domain specific language
Iain Dunning Miles Lubin
MIT
Operations Research
Joey HuchetteYes Sian Ng

50. http://github.com/JuliaLang/IJulia.jl
http://blog.jupyter.org/2016/02/16/jupyterdays-boston-2016/

52. In summary,
Types helps users express scientific computations
and helps the compiler specialize code for performance
Other advanced features for genericness and performance:
metaprogramming with macros and generated functions,
parametric polymorphism, native parallel computing, …
try Julia today!
juliabox.org
JuliaCon - June 21-25 at MIT CSAIL
juliacon.org

53. Other references
B. Liskov, The Power of Abstraction, Turing Award lecture,
OOPSLA’09. https://www.youtube.com/watch?
v=qAKrMdUycb8
J. Sammet, Programming Languages: History and
Fundamentals, Prentice-Hall, 1969.

54. Are classes really that bad?
Yes, when you break their abstraction.
http://okmij.org/ftp/Computation/Subtyping/

55. The circle-ellipse problem (aka the square-rectangle problem)

56. The circle-ellipse problem (aka the square-rectangle problem)
Is a circle an ellipse?

57. The circle-ellipse problem (aka the square-rectangle problem)
Is a circle an ellipse?
Yes: it is a special case of an ellipse with
degenerate foci
zero eccentricity
etc.

58. The circle-ellipse problem (aka the square-rectangle problem)
Is a circle an ellipse?
Yes: it is a special case of an ellipse with
degenerate foci
zero eccentricity
etc.
No: it makes no sense to change the second focus of a circle.

59. The circle-ellipse problem
#Program A
foo = Ellipse(r1 = 0.5, r2 = 1.0)
get_r2(foo)

60. The circle-ellipse problem
#Program A
foo = Ellipse(r1 = 0.5, r2 = 1.0)
get_r2(foo)
Circle(r = 0.5)

61. The circle-ellipse problem
#Program A
foo = Ellipse(r1 = 0.5, r2 = 1.0)
get_r2(foo)
Circle(r = 0.5)
You can make this program work: define
get_r2(x::Circle) = x.r

62. #Program B
foo = Ellipse()
set_r1(foo, 0.5)
set_r2(foo, 1.0)
get_r1(foo)*get_r2(foo) == 0.5
The circle-ellipse problem
#Program A
foo = Ellipse(r1 = 0.5, r2 = 1.0)
get_r2(foo)
Circle(r = 0.5)
You can make this program work: define
get_r2(x::Circle) = x.r

63. #Program B
foo = Ellipse()
set_r1(foo, 0.5)
set_r2(foo, 1.0)
get_r1(foo)*get_r2(foo) == 0.5
The circle-ellipse problem
#Program A
foo = Ellipse(r1 = 0.5, r2 = 1.0)
get_r2(foo)
Circle()
Circle(r = 0.5)
You can make this program work: define
get_r2(x::Circle) = x.r

64. #Program B
foo = Ellipse()
set_r1(foo, 0.5)
set_r2(foo, 1.0)
get_r1(foo)*get_r2(foo) == 0.5
The circle-ellipse problem
#Program A
foo = Ellipse(r1 = 0.5, r2 = 1.0)
get_r2(foo)
Circle()
Circle(r = 0.5)
You can’t substitute a Circle for an Ellipse in this program.
You can make this program work: define
get_r2(x::Circle) = x.r

65. #Program B
foo = Ellipse()
set_r1(foo, 0.5)
set_r2(foo, 1.0)
get_r1(foo)*get_r2(foo) == 0.5
The circle-ellipse problem
#Program A
foo = Ellipse(r1 = 0.5, r2 = 1.0)
get_r2(foo)
Circle()
Circle(r = 0.5)
You can’t substitute a Circle for an Ellipse in this program.
You can make this program work: define
get_r2(x::Circle) = x.r
If you can mutate objects (“covariant context”)*, then circles
are not ellipses because of the Liskov substitution
principle** (one of the SOLID principles).
*G. Castagna, Covariance and contravariance : conflict without a cause.
In Object Oriented Programming: A Unified Foundation, http://link.springer.com/book/10.1007/978-1-4612-4138-6
**B. Liskov, Keynote address - data abstraction and hierarchy, OOPSLA '87 Proceedings Addendum, 17-34.
B. Liskov and J. M. Wing. "A Behavioral Notion of Subtyping," ACM Trans. Programming Languages and Systems 16(6):1811-1841, Nov 1994.

66. n-ary operations
mylist = ListOfNumbers();
mylist.push!(1.0);

67. n-ary operations
Methods are associated with classes.
mylist = ListOfNumbers();
mylist.push!(1.0);

68. n-ary operations
Methods are associated with classes.
mylist = ListOfNumbers();
mylist.push!(1.0);
A unary operation on a class can be implemented as a
method of that class.
The abstraction of classes can break down for functions
involving multiple classes.

69. n-ary operations
Neither. Matrix + DiagonalMatrix needs to access private
fields, thus breaking encapsulation*.
*A loaded term; see D. J. Armstrong, The quarks of object-oriented development, Commun. ACM 49 (2): 123-8, Feb 2006, doi:10.1145/1113034.1113040

70. n-ary operations
Adding a diagonal matrix and an ordinary (dense) matrix:
is it a method of Matrix or DiagonalMatrix?
Neither. Matrix + DiagonalMatrix needs to access private
fields, thus breaking encapsulation*.
*A loaded term; see D. J. Armstrong, The quarks of object-oriented development, Commun. ACM 49 (2): 123-8, Feb 2006, doi:10.1145/1113034.1113040

71. n-ary operations
Adding a diagonal matrix and an ordinary (dense) matrix:
is it a method of Matrix or DiagonalMatrix?
class Matrix {
private:
int m, n;
float *data;
// …
friend Matrix operator+(Matrix M, DiagonalMatrix D); //non-member function
};
class DiagonalMatrix{
private:
int n;
//store only the diagonals
float *diagonaldata;
// …
friend Matrix operator+(Matrix M, DiagonalMatrix D); //non-member function
};
Neither. Matrix + DiagonalMatrix needs to access private
fields, thus breaking encapsulation*.
*A loaded term; see D. J. Armstrong, The quarks of object-oriented development, Commun. ACM 49 (2): 123-8, Feb 2006, doi:10.1145/1113034.1113040