Intro to Julia
Julia Intro for QNumerics Summer School by Raye Kimmerer.
Other resources
What scientists must know about hardware to write fast code
Linear Algebra
using LinearAlgebra
using SparseArrays
using InteractiveUtils
A = rand(100,100)100×100 Matrix{Float64}:
0.689877 0.442898 0.337051 … 0.918166 0.931666 0.433926
0.765819 0.904232 0.894542 0.152635 0.186052 0.0370883
0.908652 0.586324 0.814448 0.28505 0.501902 0.986995
0.628829 0.65289 0.960936 0.460261 0.76252 0.844714
0.377951 0.182027 0.875066 0.936151 0.58245 0.125421
0.93019 0.945633 0.519722 … 0.921993 0.604815 0.721738
0.432319 0.359336 0.478482 0.569015 0.261859 0.177263
0.208684 0.471807 0.861173 0.444832 0.218652 0.61552
0.991222 0.248672 0.25356 0.978568 0.0260299 0.0491302
0.697716 0.594812 0.681405 0.938027 0.738349 0.0676542
⋮ ⋱
0.789347 0.413027 0.0628976 0.0384678 0.957868 0.759315
0.337977 0.408965 0.648698 0.316743 0.303481 0.711143
0.572835 0.755868 0.119204 0.316745 0.443184 0.429889
0.255512 0.559662 0.257568 0.978429 0.216569 0.913379
0.181709 0.0309443 0.00711086 … 0.985471 0.416796 0.420845
0.534554 0.331107 0.0664834 0.933714 0.568428 0.068777
0.0728202 0.889997 0.513793 0.276813 0.875836 0.979584
0.983905 0.0848033 0.9529 0.145434 0.872911 0.80546
0.374269 0.899537 0.757362 0.0246773 0.364742 0.0248242Transpose and other operations are non-allocating, and instead create a wrapper of the original array. This prevents unecessary allocation and allows more opportunity to dispatch on array types.
B = transpose(A)100×100 transpose(::Matrix{Float64}) with eltype Float64:
0.689877 0.765819 0.908652 … 0.0728202 0.983905 0.374269
0.442898 0.904232 0.586324 0.889997 0.0848033 0.899537
0.337051 0.894542 0.814448 0.513793 0.9529 0.757362
0.839381 0.778814 0.825507 0.0597193 0.37992 0.245574
0.237912 0.233105 0.44183 0.852397 0.790491 0.632232
0.0861786 0.635229 0.912296 … 0.578696 0.343724 0.420249
0.0163067 0.0807648 0.753481 0.436245 0.863603 0.0628934
0.875406 0.453432 0.753933 0.722678 0.539864 0.375855
0.581084 0.216641 0.100293 0.48841 0.795991 0.258537
0.0997436 0.278478 0.590193 0.931255 0.48242 0.494541
⋮ ⋱
0.026961 0.901935 0.388902 0.354084 0.541073 0.521093
0.747279 0.216566 0.474953 0.0675228 0.760942 0.232325
0.930372 0.309718 0.291894 0.934471 0.844834 0.405043
0.907233 0.555821 0.760701 0.0429174 0.285344 0.828271
0.482254 0.521344 0.68237 … 0.196166 0.741738 0.668487
0.0578316 0.849002 0.899415 0.736748 0.113914 0.860721
0.918166 0.152635 0.28505 0.276813 0.145434 0.0246773
0.931666 0.186052 0.501902 0.875836 0.872911 0.364742
0.433926 0.0370883 0.986995 0.979584 0.80546 0.0248242C = copy(B)100×100 Matrix{Float64}:
0.689877 0.765819 0.908652 … 0.0728202 0.983905 0.374269
0.442898 0.904232 0.586324 0.889997 0.0848033 0.899537
0.337051 0.894542 0.814448 0.513793 0.9529 0.757362
0.839381 0.778814 0.825507 0.0597193 0.37992 0.245574
0.237912 0.233105 0.44183 0.852397 0.790491 0.632232
0.0861786 0.635229 0.912296 … 0.578696 0.343724 0.420249
0.0163067 0.0807648 0.753481 0.436245 0.863603 0.0628934
0.875406 0.453432 0.753933 0.722678 0.539864 0.375855
0.581084 0.216641 0.100293 0.48841 0.795991 0.258537
0.0997436 0.278478 0.590193 0.931255 0.48242 0.494541
⋮ ⋱
0.026961 0.901935 0.388902 0.354084 0.541073 0.521093
0.747279 0.216566 0.474953 0.0675228 0.760942 0.232325
0.930372 0.309718 0.291894 0.934471 0.844834 0.405043
0.907233 0.555821 0.760701 0.0429174 0.285344 0.828271
0.482254 0.521344 0.68237 … 0.196166 0.741738 0.668487
0.0578316 0.849002 0.899415 0.736748 0.113914 0.860721
0.918166 0.152635 0.28505 0.276813 0.145434 0.0246773
0.931666 0.186052 0.501902 0.875836 0.872911 0.364742
0.433926 0.0370883 0.986995 0.979584 0.80546 0.0248242When we copy the array it allocates and is no longer a wrapper
Diagonal(A)100×100 LinearAlgebra.Diagonal{Float64, Vector{Float64}}:
0.689877 ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅
⋅ 0.904232 ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ 0.814448 ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ 0.170114 ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋮ ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ 0.276813 ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ 0.872911 ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.0248242subtypes(LinearAlgebra.Factorization)21-element Vector{Any}:
LinearAlgebra.AdjointFactorization
LinearAlgebra.BunchKaufman
LinearAlgebra.Cholesky
LinearAlgebra.CholeskyPivoted
LinearAlgebra.Eigen
LinearAlgebra.GeneralizedEigen
LinearAlgebra.GeneralizedSVD
LinearAlgebra.GeneralizedSchur
LinearAlgebra.Hessenberg
LinearAlgebra.LDLt
⋮
LinearAlgebra.QR
LinearAlgebra.QRCompactWY
LinearAlgebra.QRPivoted
LinearAlgebra.SVD
LinearAlgebra.Schur
LinearAlgebra.TransposeFactorization
SparseArrays.CHOLMOD.Factor
SparseArrays.SPQR.QRSparse
SparseArrays.UMFPACK.UmfpackLUissymmetric(A*A')trueHow do we communicate this to the dispatch system?
D = Symmetric(A*A')100×100 LinearAlgebra.Symmetric{Float64, Matrix{Float64}}:
36.9767 26.9739 28.2335 27.2151 … 25.9279 24.2946 27.1502 23.8749
26.9739 33.7431 26.9764 24.8684 21.7457 22.2812 24.5202 25.6285
28.2335 26.9764 34.3579 25.2187 22.0134 23.5851 25.069 23.3198
27.2151 24.8684 25.2187 31.805 20.7167 22.7103 26.3492 22.4928
28.2948 28.2536 27.6541 23.691 22.7564 23.1357 26.6059 24.2253
27.4578 26.4579 25.6641 26.418 … 21.9073 21.7889 23.3432 23.7683
26.4034 25.6938 25.3785 23.7964 22.1252 21.0764 24.0081 23.3219
29.3402 28.0172 28.1854 25.3263 23.47 23.8132 26.6657 24.9575
24.0686 23.7854 23.4124 21.2582 19.8021 19.134 21.8775 21.8944
27.9855 25.7246 26.2124 25.5573 23.5242 22.8471 25.6106 22.403
⋮ ⋱
28.8021 26.6635 27.3892 26.0997 24.5376 24.0559 28.2537 25.6707
25.3282 25.2905 25.1838 23.1105 21.9496 22.1394 24.7355 22.7307
24.0841 24.2343 24.7356 23.4886 20.1049 22.4576 23.7177 21.3846
28.1161 26.0753 27.4174 25.4254 24.6418 24.8304 26.3101 23.2327
26.7324 24.3896 24.7577 23.5723 … 21.986 22.5309 24.979 22.7549
25.9279 21.7457 22.0134 20.7167 28.5783 19.845 22.5999 19.972
24.2946 22.2812 23.5851 22.7103 19.845 29.4085 22.2209 21.0628
27.1502 24.5202 25.069 26.3492 22.5999 22.2209 32.6584 22.9931
23.8749 25.6285 23.3198 22.4928 19.972 21.0628 22.9931 29.5572Now we can dispatch on the fact that this matrix is symmetric, allowing us to select more efficient algorithms for this case.
Symmetric(A)100×100 LinearAlgebra.Symmetric{Float64, Matrix{Float64}}:
0.689877 0.442898 0.337051 … 0.918166 0.931666 0.433926
0.442898 0.904232 0.894542 0.152635 0.186052 0.0370883
0.337051 0.894542 0.814448 0.28505 0.501902 0.986995
0.839381 0.778814 0.825507 0.460261 0.76252 0.844714
0.237912 0.233105 0.44183 0.936151 0.58245 0.125421
0.0861786 0.635229 0.912296 … 0.921993 0.604815 0.721738
0.0163067 0.0807648 0.753481 0.569015 0.261859 0.177263
0.875406 0.453432 0.753933 0.444832 0.218652 0.61552
0.581084 0.216641 0.100293 0.978568 0.0260299 0.0491302
0.0997436 0.278478 0.590193 0.938027 0.738349 0.0676542
⋮ ⋱
0.026961 0.901935 0.388902 0.0384678 0.957868 0.759315
0.747279 0.216566 0.474953 0.316743 0.303481 0.711143
0.930372 0.309718 0.291894 0.316745 0.443184 0.429889
0.907233 0.555821 0.760701 0.978429 0.216569 0.913379
0.482254 0.521344 0.68237 … 0.985471 0.416796 0.420845
0.0578316 0.849002 0.899415 0.933714 0.568428 0.068777
0.918166 0.152635 0.28505 0.276813 0.875836 0.979584
0.931666 0.186052 0.501902 0.875836 0.872911 0.80546
0.433926 0.0370883 0.986995 0.979584 0.80546 0.0248242This is 'casting' to a symmetric matrix, it will happily create a symmetric matrix out of a non-symmetric one.
Sparse Arrays
A = sprand(100,100,0.1)100×100 SparseArrays.SparseMatrixCSC{Float64, Int64} with 961 stored entries:
⎡⠢⡈⠀⡀⠄⠃⠠⠠⠀⠀⠀⠄⠀⠔⠀⡂⠬⠀⠄⠀⣀⠄⠠⡠⠀⠁⠈⠀⠄⠀⢅⠠⠈⠁⠰⠀⠀⠀⠑⠀⎤
⎢⠁⠅⠒⢈⠄⠄⠈⡊⠀⠠⠀⠴⠀⠀⠈⡀⠀⠄⠀⠤⢀⠈⣀⡠⠁⠀⠀⠠⠀⠀⠢⠀⠠⠀⠈⠠⠀⡢⠔⠀⎥
⎢⠀⡱⠄⠥⢀⠈⠀⡀⠀⢤⠠⡄⡐⡨⡠⠄⠀⠂⠄⢁⡒⠄⠀⠰⠨⡄⣰⠀⠄⠅⠐⠀⡠⠂⢀⢀⠍⠀⠀⠀⎥
⎢⢐⠌⠅⠀⢅⠁⠈⠀⠢⠖⠍⠄⠡⠅⠄⠃⠐⠂⡅⠄⠥⡂⠀⠠⠠⢠⢔⡀⠀⠆⠀⠰⠀⠄⠠⠐⢤⠃⠀⠄⎥
⎢⠁⠇⠀⠃⢀⠀⠀⠁⠐⠄⠀⡠⢀⠄⠀⠦⢀⠁⠀⡀⠀⠀⠄⠀⠀⠀⠂⠇⠀⠀⠨⠂⠐⠄⠀⠀⠄⠄⠈⠐⎥
⎢⠀⠠⠀⠠⠀⠄⠀⠈⠀⠄⠈⢜⠠⠄⠀⠄⡄⠀⢀⡄⡠⠆⠁⢀⢤⠈⠈⢆⠄⠄⠄⠄⠀⠁⠈⢔⡥⠄⢀⠈⎥
⎢⢀⠀⠌⠃⠀⠄⠀⠅⠡⠤⠪⠢⠈⢀⠂⣀⠀⠍⠘⠵⠐⡠⠸⢤⠌⠀⠐⠀⠰⠀⡄⠠⠰⠀⢀⠈⢠⠠⠰⠀⎥
⎢⠀⠔⠆⡀⢀⡤⠠⠄⠠⠀⠁⡄⠐⠩⠀⡢⠀⠔⠀⠔⠀⠄⠠⠅⠐⠅⠔⠁⠀⠀⠀⢀⢂⠀⠥⠀⠠⡀⠈⠨⎥
⎢⠀⠄⠀⡀⠀⡑⠀⠡⠊⡥⠀⡊⠡⠀⠈⠈⠀⠆⠀⠅⠀⠠⡉⢀⠰⠢⢱⠤⠄⠧⠡⠀⠠⠀⡀⠀⠀⡠⠌⠀⎥
⎢⠁⠁⠀⠌⠠⠀⠜⠀⠀⠄⠰⡈⣀⠍⡀⠡⠠⠅⠀⡁⠠⠑⡀⠄⠘⠔⠠⠀⠡⠢⠨⢀⠠⠀⠌⠜⠀⠂⠴⠤⎥
⎢⠀⠁⠀⡄⠄⠀⠀⠀⠀⠢⠐⠀⠀⠀⠀⠒⢀⠄⠠⡀⠐⠁⠀⠘⠄⠀⠠⢀⠤⠠⡨⠈⠀⠐⠀⠆⠀⠀⡘⠀⎥
⎢⠀⠀⠀⠂⢆⠖⠐⠒⠠⠈⢀⠀⠁⠀⠂⠎⡀⠠⡀⠅⠁⠢⠀⠔⠐⡐⠀⠀⡌⠀⢐⠠⠐⠈⢐⠒⠐⠀⠈⠠⎥
⎢⢐⠆⠄⡀⢈⠃⠀⠀⡀⠂⠀⠀⠀⠒⠠⠒⢀⠢⠄⠐⠀⠘⠘⠀⠢⠂⠈⠈⠠⠀⢂⠀⠃⠁⢀⠀⠐⠄⠎⠀⎥
⎢⠀⠑⠀⠀⠂⢠⠆⠠⠄⠀⠀⡆⠥⠄⢔⢀⠀⠂⠈⠀⠈⠘⠀⠀⠉⠀⠀⡀⢅⢃⠠⡀⠠⠄⠀⠑⢓⢁⠊⠂⎥
⎢⠤⢃⠀⠀⠐⠐⠃⡀⠁⠔⠁⠀⠐⠀⠀⠢⠁⠂⠐⠁⡀⠐⠑⠀⠢⡃⡉⠀⠀⠀⠈⢘⠠⢀⠂⢈⡨⠙⠋⠀⎥
⎢⡃⠀⠈⠀⠈⢈⠀⠂⠁⠀⠀⠂⠀⠀⡠⠂⠄⠀⠐⠀⠁⠂⡐⠘⠀⢀⢡⠂⠀⠀⢀⣐⠑⠀⠰⠈⠡⠠⢀⡂⎥
⎢⠌⠔⠀⢁⠐⠠⠀⡠⠀⠒⠂⠀⠂⡂⠠⠀⠐⠐⠁⠀⢈⡁⠉⠁⠉⠀⡠⠀⢀⠂⠀⡁⢀⠁⠀⠂⠐⡀⡐⠂⎥
⎢⠐⠀⢀⠀⠡⠂⢂⡀⠀⠐⠂⢢⠀⠄⠀⡐⠀⠀⠒⠠⠈⠂⠀⠀⠈⠋⠰⠀⠐⠚⠈⠂⢈⠀⡸⢉⠤⠊⠀⠀⎥
⎢⠂⠂⠐⡠⢀⠀⠀⠈⠀⣁⠡⠁⠀⠀⠠⠦⠀⠀⠀⠑⡀⠠⠲⠀⠀⢀⠀⠀⠀⠀⢐⠂⠁⠀⠎⠀⢬⠀⠀⠂⎥
⎣⠀⠀⠁⠠⠐⠀⠀⠀⠂⠅⢰⠄⠀⠀⡄⠐⠂⠁⠐⡀⠠⠀⠂⠐⠌⠁⢨⠀⢀⢈⠀⠄⠀⠀⠔⠘⡐⠘⠐⡐⎦