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The open source ecosystem for technical computing Alan Edelman Mathematics Computer Science & AI Laboratories May 15, 2014

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2/30 How should Move technical computing forward? A vision of the future octave

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3/30 The obvious… High level technical computing languages traditionally Consist of libraries of functionality Packaged for user convenience Open source or proprietary We could all compete, or we could benefit from each other’s great work!

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4/30 Some Examples

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5/30 We used to talk about “consumer reports” For numerical software Especially optimization

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6/30 Ipython is wonderful! And not only just for python

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7/30 From the Python World: MATPLOTLIB

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8/30 Scilab calling Python

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9/30 Moving Technical Computing Forward Goal: Bring Computer Science and Computational Science Together Stereotypes: Computer Scientists don’t take derivatives but computer scientists are great at abstractions Computational Scientists can’t declare types but computational scientists are great at simulations We want to bridge the gap between computer science and computational science

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10/30 Living the life as a Computer Scientist, Computational Scientist, and Mathematician

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methodsobjects 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 15

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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 16

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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 17

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generic function objectsmethods rechargeable subway pass single-use subway ticket is a subtype of subway ticket top up pay fare lose buy abstract object Multi-methods with type hierarchy 18

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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 19

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So how does this help us with linear algebra? 20

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21/30 “Disconnected” Technical Computing World Desktop ProductivitySupercomputer Power Cloud Potential 21/32

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22/30 “Disconnected” Technical Computing World Desktop ProductivitySupercomputer Power Cloud Potential Easy on Humans Non-Expert Users Interface to vast libraries Desktop Performance Deep Chasm to cross The Village The Market The Availability The Raw Power Hard on Humans Software 22/32

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23/30 The “Two Language” Problem As long as the developers’ language is harder than the users language, technical computing will always be hindered

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24/30 Looking for Abstractions

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25/30 Parallel Prefix 1,2,3,4,5,… 1,3,6,10,15,…

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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 26

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Conjugate gradients (Hestenes-Stiefel) http://en.wikipedia.org/wiki/Conjugate_gradient_method Can we write this as an iterator? state = start(iterable) done(iterable, state) 27

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Conjugate gradients (Hestenes-Stiefel) http://en.wikipedia.org/wiki/Conjugate_gradient_method state = start(iterable) done(iterable, state) state, dx = next(iterable, state) 28

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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 29

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30/30 Conclusions… Open source computing allows us to benefit from each other’s work But Even More importantly allows us to grow the very design base

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