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What's next in Julia
1. What’s next in Jul_
Jiahao Chen
MIT Computer Science and Artificial Intelligence Laboratory
julialang.org
Alan EdelmanJeff Bezanson Stefan Karpinski Viral B. Shah
2. What’s the big deal about Julia ?
julialang.org/benchmarks
3. It bridges the divide between computer science
and computational science
What’s the big deal about Julia ?
4. It bridges the divide between computer science
and computational science
What’s the big deal about Julia ?
data abstraction
performance
5. 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?
6. It’s a programming language designed for technical computing
What’s the big deal about Julia ?
7. It’s a programming language designed for technical computing
What’s the big deal about Julia ?
The key ingredients
Multi-methods (multiple dispatch)
Dataflow type inference
together allow for cost-efficient data abstraction
15. 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
class-based OO
!
classes are more
fundamental
than methods
16. Object-oriented programming with multi-methods
What can I do with/to a thing?
top up
pay fare
lose
buy
generic
function
objectsmethods
17. Object-oriented programming with multi-methods
What can I do with/to a thing?
top up
pay fare
lose
buy
generic
function
objectsmethods
multimethods
!
relationships between
objects and functions
18. Multi-methods with type hierarchy
top up
pay fare
lose
buy
generic
function
objectsmethods
rechargeable
subway
pass
single-use
subway
ticket
is a subtype of
subway
ticket
abstract object
21. The Julia codebase is compact
5,774 lines of Scheme
32,707 lines of C
59727 lines of Julia
!
+LLVM, BLAS, LAPACK, SuiteSparse, ARPACK,
Rmath, GMP, MPFR, FFTW,…
22. Data types as a lattice
Dana Scott, Data types as lattices, SIAM J. Comput. 5: 522-87. 1976
Real
Number
FloatingPoint Rational
Complex
Float64 BigFloat…
…
Integer
is a parameter of
is a subtype of
Signed BigInt
Any
Int64…
…
23. Signed
Multiple dispatch with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…
24. Signed
Multiple dispatch with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…
25. Signed
Multiple dispatch with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…
26. Signed
Multiple dispatch 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
27. Signed
Multiple dispatch 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
28. Signed
Multiple dispatch 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
29. Signed
Multiple dispatch with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…
30. Signed
Multiple dispatch with type lattice traversal
Real
Number
FloatingPoint Rational
Float64 BigFloat…
Integer
is a parameter of
is a subtype of
BigInt
Any
Int64…
…
36. Type lattice of arrays
DenseArray{T, N}
AbstractArray{T, N}
Array{T, N}
…
is a subtype of
Any
Array{T,1}===Vector{T}
Array{T,2}===Matrix{T}
A single parametric type defines many types
of matrices
Matrix{Float64}, Matrix{Int64},
Matrix{BigFloat}, Matrix{Complex128},
Matrix{Rational}, Matrix{Quaternion{Float64}},…
37. Type lattice of arrays
DenseArray{T, N}
AbstractArray{T, N}
Array{T, N}
…
is a subtype of
Any
Array{T,1}===Vector{T}
Array{T,2}===Matrix{T}
A single parametric type defines many types
of matrices
Matrix{Float64}, Matrix{Int64},
Matrix{BigFloat}, Matrix{Complex128},
Matrix{Rational}, Matrix{Quaternion{Float64}},…
38. Type lattice of arrays
DenseArray{T, N}
AbstractArray{T, N}
Array{T, N}
…
is a subtype of
Any
Array{T,1}===Vector{T}
Array{T,2}===Matrix{T}
rotation matrices
structured matrices
distributed arrays
A single parametric type defines many types
of matrices
Matrix{Float64}, Matrix{Int64},
Matrix{BigFloat}, Matrix{Complex128},
Matrix{Rational}, Matrix{Quaternion{Float64}},…
39. Multi-methods for linear algebra
What can I do with/to a thing?
find eigenvalues and eigenvectors
find singular values
find singular values and vectors
find eigenvalues
generic
function
objectsmethods
general matrix
symmetric tridiagonal matrix
bidiagonal matrix
Methods can take advantage of special matrix structures
eigvals
eigfact
svdvals
svdfact
Matrix
SymTridiagonal
Bidiagonal
41. So how does this help us with linear algebra?
Multi-method dispatch on special matrix types
42. So how does this help us with linear algebra?
Multi-method dispatch on special matrix types
43. So how does this help us with linear algebra?
Multi-method dispatch on special matrix types
stev!{T<:BlasFloat} calls sgestv
dgestv
cgestv
zgestv
and handles workspace
memory allocation
44. So how does this help us with linear algebra?
Multi-method dispatch with generic fallbacks
Matrix operations on general rings
45. So how does this help us with linear algebra?
Multi-method dispatch with generic fallbacks
Matrix operations on general rings textbook algorithm
46. So how does this help us with linear algebra?
Multi-method dispatch with generic fallbacks
Matrix operations on general rings
48. Iterative algorithms as iterators
for item in iterable
#body
end
!
#is equivalent to
!
state = start(iterable)
while !done(iterable, state)
item, state = next(iterable, state)
# body
end
53. immutable cg_hs #iterative solver
K :: KrylovSpace #wraps A, v0, k
t :: Terminator #termination criteria
end
immutable cg_hs_state
r :: Vector #residual
p :: Vector #search direction
rnormsq :: Float64 #Squared norm of previous residual
iter :: Int #iteration count
end
start(a::cg_hs) = cg_hs_state(a.K.v0,
zeros(size(a.K.v0,1)), Inf, 0)
function next(a::cg_hs, s::cg_hs_state)
rnormsq = dot(s.r, s.r)
p = s.r + (rnormsq/s.rnormsq)*s.p
Ap = a.K.A*p
α = rnormsq / dot(p, Ap)
α*p, cg_hs_state(s.r-α*Ap, p, rnormsq, s.iter+1)
end
done(a::cg_hs, s::cg_hs_state) = done(a.t, s)
!
K = method(KrylovSpace(A, b, k), Terminator())
x += reduce(+, K) # for dx in K; x += dx; end
Hestenes-Stiefel CG
54. immutable cg_hs #iterative solver
K :: KrylovSpace #wraps A, v0, k
t :: Terminator #termination criteria
end
immutable cg_hs_state
r :: Vector #residual
p :: Vector #search direction
rnormsq :: Float64 #Squared norm of previous residual
iter :: Int #iteration count
end
start(a::cg_hs) = cg_hs_state(a.K.v0,
zeros(size(a.K.v0,1)), Inf, 0)
function next(a::cg_hs, s::cg_hs_state)
rnormsq = dot(s.r, s.r)
p = s.r + (rnormsq/s.rnormsq)*s.p
Ap = a.K.A*p
α = rnormsq / dot(p, Ap)
α*p, cg_hs_state(s.r-α*Ap, p, rnormsq, s.iter+1)
end
done(a::cg_hs, s::cg_hs_state) = done(a.t, s)
!
K = method(KrylovSpace(A, b, k), Terminator())
x += reduce(+, K) # for dx in K; x += dx; end
Hestenes-Stiefel CG
Abstracts out termination check and solution update steps
61. JuMP: writing simple DSLs in Julia
Iain Dunning Miles Lubin
MIT
Operations Research
62. Andreas N. Jensen
U. Copenhagen
Economics
Alan EdelmanJeff Bezanson Stefan Karpinski Viral B. Shah
Carlo Baldassi
Poly. Torino
Neuroscience
Tim E. Holy
WUSTL
Anatomy
Douglas M. Bates
Wisconsin-Madison
Statistics
Steven G. Johnson
MIT
Mathematics
