CoCalc Public Filesjulia-1.4.ipynb
Author: Harald Schilly
Views : 449
Compute Environment: Ubuntu 18.04 (Deprecated)

# Julia 1.4

https://docs.julialang.org/en/v1.4/NEWS/#

In [10]:
VERSION

v"1.4.2"
In [2]:
println("Hello 1.4")

Hello 1.4
In [3]:
using FFTW
using LinearAlgebra
a = 2 * rand(10) .- 1
norm(ifft(fft(a)) - a)

0.0
In [4]:
using PyCall
# pygui(true)
using PyPlot
import Random:seed!; seed!(99)
import LinearAlgebra:norm

function activate(x,W,b)
return 1 ./ (1 .+ exp.(-(W*x+b)))
end

x1 = [0.1,0.3,0.1,0.6,0.4,0.6,0.5,0.9,0.4,0.7]
x2 = [0.1,0.4,0.5,0.9,0.2,0.3,0.6,0.2,0.4,0.6]
y = [ones(1,5) zeros(1,5) ; zeros(1,5) ones(1,5)]

W2 = 0.5*randn(2,2); W3 = 0.5*randn(3,2); W4 = 0.5*randn(2,3);
b2 = 0.5*randn(2,1); b3 = 0.5*randn(3,1); b4 = 0.5*randn(2,1);

In [5]:
PyPlot.plot(x1, x2)

1-element Array{PyObject,1}: PyObject <matplotlib.lines.Line2D object at 0x7f321011f358>
In [6]:
using Pkg

In [7]:
for i = 1:10
println(i)
end

1 2 3 4 5 6 7 8 9 10
In [8]:
using NLopt

if length(grad) > 0
end
return sqrt(x[2])
end

function myconstraint(x::Vector, grad::Vector, a, b)
if length(grad) > 0
grad[1] = 3a * (a*x[1] + b)^2
end
(a*x[1] + b)^3 - x[2]
end

opt = Opt(:LD_MMA, 2)
opt.lower_bounds = [-Inf, 0.]
opt.xtol_rel = 1e-4

opt.min_objective = myfunc
inequality_constraint!(opt, (x,g) -> myconstraint(x,g,2,0), 1e-8)
inequality_constraint!(opt, (x,g) -> myconstraint(x,g,-1,1), 1e-8)

(minf,minx,ret) = optimize(opt, [1.234, 5.678])
numevals = opt.numevals # the number of function evaluations
println("got $minf at$minx after $numevals iterations (returned$ret)")

got 0.5443310477213124 at [0.3333333342139688, 0.29629628951338166] after 11 iterations (returned XTOL_REACHED)
In [ ]:


In [9]:
using Printf

s = 0
for i = [1 2 5 100 -1 5]
s = s + i
@printf("i = %4d  →  s = %4d\n", i, s)
end

i = 1 → s = 1 i = 2 → s = 3 i = 5 → s = 8 i = 100 → s = 108 i = -1 → s = 107 i = 5 → s = 112
In [2]:
using Pkg

 Resolving package versions... Updating ~/.julia/environment/v1.4/Project.toml [no changes] Updating ~/.julia/environment/v1.4/Manifest.toml [no changes] 
In [3]:
# Import libraries.
using Turing, StatsPlots, Random

Failed to precompile Turing [fce5fe82-541a-59a6-adf8-730c64b5f9a0] to /home/user/.julia/compiled/v1.4/Turing/gm4QC_G6IaO.ji.  Stacktrace:  [1] error(::String) at ./error.jl:33  [2] compilecache(::Base.PkgId, ::String) at ./loading.jl:1272  [3] _require(::Base.PkgId) at ./loading.jl:1029  [4] require(::Base.PkgId) at ./loading.jl:927  [5] require(::Module, ::Symbol) at ./loading.jl:922  [6] top-level scope at In[3]:1 
In [20]:
# Set the true probability of heads in a coin.
p_true = 0.5

# Iterate from having seen 0 observations to 100 observations.
Ns = 0:100;

# Draw data from a Bernoulli distribution, i.e. draw heads or tails.
Random.seed!(12)
data = rand(Bernoulli(p_true), last(Ns))

# Declare our Turing model.
@model coinflip(y) = begin
# Our prior belief about the probability of heads in a coin.
p ~ Beta(1, 1)

# The number of observations.
N = length(y)
for n in 1:N
# Heads or tails of a coin are drawn from a Bernoulli distribution.
y[n] ~ Bernoulli(p)
end
end;

# Settings of the Hamiltonian Monte Carlo (HMC) sampler.
iterations = 1000
ϵ = 0.05
τ = 10

# Start sampling.
chain = sample(coinflip(data), HMC(ϵ, τ), iterations);

# Plot a summary of the sampling process for the parameter p, i.e. the probability of heads in a coin.
histogram(chain[:p])

InterruptException:  Stacktrace:  [1] _include_from_serialized(::String, ::Array{Any,1}) at ./loading.jl:681  [2] _require_search_from_serialized(::Base.PkgId, ::String) at ./loading.jl:781  [3] _tryrequire_from_serialized(::Base.PkgId, ::UInt64, ::String) at ./loading.jl:712  [4] _require_search_from_serialized(::Base.PkgId, ::String) at ./loading.jl:770  [5] _tryrequire_from_serialized(::Base.PkgId, ::UInt64, ::Nothing) at ./loading.jl:712  [6] _require_from_serialized(::String) at ./loading.jl:743  [7] _require(::Base.PkgId) at ./loading.jl:1039  [8] require(::Base.PkgId) at ./loading.jl:927  [9] require(::Module, ::Symbol) at ./loading.jl:922  [10] top-level scope at In[20]:1 
In [4]:
using Compat

In [5]:
using PyPlot
x = range(0, stop = 4*pi, length=1000)
y = sin.(3*x + 1.5*cos.(2*x))

plot(x, y, color="red", linewidth=2.0, linestyle="--")

1-element Array{PyCall.PyObject,1}: PyObject <matplotlib.lines.Line2D object at 0x7f08fa1982e8>
In [10]:
using SymPy
x = symbols("x")      # or   @vars x, Sym("x"), or  Sym(:x)
y = sin(10 * pi*x)
y(1), y(2.2), y(123456)

(-1.22464679914735e-15, 4.41120439947682e-15, 6.39055017337619e-11)
In [11]:
simplify(diff(sqrt(2 + exp(x)) / (1 + x - cos(x)^2), x))

$\begin{equation*}\frac{\left(x + \sin^{2}{\left(x \right)}\right) e^{x} - 2 \left(e^{x} + 2\right) \left(\sin{\left(2 x \right)} + 1\right)}{2 \left(x + \sin^{2}{\left(x \right)}\right)^{2} \sqrt{e^{x} + 2}}\end{equation*}$
In [12]:
using Test

foo(x) = length(x)^2

α = 1

@testset "foo" begin
@test α == 2-1
@test π ≈ 3.14 atol=0.01
@test isapprox(9.81, 10, atol=0.2)
@test foo("foo") == 9
end

Test Summary: | Pass Total foo | 4 4 
Test.DefaultTestSet("foo", Any[], 4, false)
In [ ]:
using SpecialMatrices

In [ ]:
using ApproxFun


## HomotopyContinuation

In [ ]:
using HomotopyContinuation, LinearAlgebra, DynamicPolynomials

c² = 2

@polyvar z[1:3, 1:6]
z_vec = vec(z)[1:17] # the 17 variables in a vector
Z = [zeros(3) z[:,1:5] [z[1,6]; z[2,6]; 0] [√c²; 0; 0]] # the eight points in a matrix

# define the functions for cyclooctane:
F1 = [(Z[:, i] - Z[:, i+1]) ⋅ (Z[:, i] - Z[:, i+1]) - c² for i in 1:7]
F2 = [(Z[:, i] - Z[:, i+2]) ⋅ (Z[:, i] - Z[:, i+2]) - 8c²/3 for i in 1:6]
F3 = (Z[:, 7] - Z[:, 1]) ⋅ (Z[:, 7] - Z[:, 1]) - 8c²/3
F4 = (Z[:, 8] - Z[:, 2]) ⋅ (Z[:, 8] - Z[:, 2]) - 8c²/3
f = [F1; F2; F3; F4]


In [2]:
n = 2 # dimension of the cyclooctane variety
N = 17 # ambient dimension
@polyvar Aᵥ[1:n, 1:N] bᵥ[1:n] # variables for the linear equations
p = [vec(Aᵥ); bᵥ] # parameters
F = [f; Aᵥ * z_vec - bᵥ] # the polynomial system we have to solve

# now we solve one particular instance for A,b complex. we use this as start system
A₀ = randn(ComplexF64, n, N)
b₀ = randn(ComplexF64, n)
p₀ = [vec(A₀); b₀]

F₀ = [subs(Fᵢ, p => [vec(A₀); b₀]) for Fᵢ in F]
complex_result = solve(F₀)
S_p₀ = solutions(complex_result)

Tracking 32768 paths... 100%|████████████████████████████████████████| Time: 0:02:32 # paths tracked: 32768 # non-singular solutions (real): 1408 (0) # singular solutions (real): 0 (0) # total solutions (real): 1408 (0) 
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In [3]:
tracker = pathtracker(F; parameters=p, generic_parameters=p₀)

# we compute 100 random intersections
data = [randn(n*N+n) for _ in 1:100]
Ω = map(data) do pp
# We want to store all solutions. Create an empty array.
S_p = similar(S_p₀, 0)
for s in S_p₀
result = track(tracker, s; target_parameters=pp, details=:minimal)
# check that the tracking was successfull and that we have a real solution
if is_success(result) && is_real(result)
# only store the solutions
push!(S_p, solution(result))
end
end
# return an array of type Array{Float64}
# (and not Array{ComplexF64})
real.(S_p)
end
Ω = vcat(Ω...)

794-element Array{Array{Float64,1},1}: [-0.47140452079103135, -1.0450657970165176, 0.8280188751979082, -1.4072716179482827, -1.8276293465988476, 0.1126538863121203, -0.8376968482498709, -2.2386216369390524, -1.114810947808969, 0.40168704888186707, -2.8742312373669674, -0.870009004649992, 1.4568131830371633, -1.9849387596762313, -1.1796341492122016, 1.8856180831641267, -1.3333333333333335] [-0.47140452079103135, -0.8787951153546268, 1.0027446948284624, -1.5699566040480526, -1.616862376833217, 0.5043070980270642, -1.1150400951914357, -2.85830851549161, 0.0024480991241107356, 0.2768074075497607, -2.9749202829205923, 0.2241736610473671, 0.9710783993831666, -2.1566719393661917, -0.6969438331786075, 1.8856180831641267, -1.3333333333333335] [-0.4714045207910314, -0.7946734805177347, -1.0706408534796377, 0.25094213821647227, -0.47990298318034214, -2.245006570889111, 1.4959527579877956, 0.09154072488973561, -1.89372229118997, 2.531459755327646, -0.8340187087968385, -2.1603024727001405, 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-2.1305654718076577, 2.238923877543447, 1.2574522711091476, -2.2809538385042036, 2.8459299601484576, 1.4938118131985325, -1.0256937927193406, 1.8856180831641267, 1.3333333333333335] [-0.4714045207910314, -0.36145578243137416, -1.2834046497986131, 0.17940674679681626, 0.4238418331552316, -2.263074159855291, 1.5573080600224904, 0.1060482378370949, -2.282953842592591, 2.3165071899826963, 1.2991158548857844, -2.268589225736365, 2.8875430848502717, 1.479099351867025, -0.9873695148408486, 1.885618083164127, 1.3333333333333333] [-0.47140452079103135, 0.5688017280450942, 1.205919720358985, 0.5977128843453564, 0.6779325040459827, 2.1252012048695135, 1.8268372586051689, 0.5826892079122182, 1.4322495563816835, 2.6120256632155607, -0.44511280121441993, 2.004178047804024, 2.436053346594702, -1.6387250796453514, 1.2663952392742992, 1.885618083164127, -1.3333333333333333] [-0.47140452079103135, 1.01949229960036, 0.8593097397524063, 0.4316252259992146, 1.1841663907997846, 1.9351441694392368, 1.627661339317607, 0.48210822114837515, 1.6583555460026036, 1.557007062856779, -0.8114597072614212, 2.225532756235336, 2.121345375684943, -1.7499911498072265, 1.3307249556283274, 1.885618083164127, -1.3333333333333333] [-0.4714045207910314, -0.3319457266763361, -1.2913519320150868, -1.1636789711744073, 0.7730066623061634, -1.8389250352883042, -0.2399382336337838, 1.6519958612844385, -2.4505466813142074, 0.43716421875665873, 2.392379002822633, -1.4538696660816486, 1.7467379744631204, 1.8824348666505366, -1.2958394273658649, 1.885618083164127, 1.3333333333333333] ⋮ [-0.47140452079103135, -1.2853862909988014, -0.3543442686006406, -1.154243984951937, -1.8565385995716481, 0.7445257449115834, -0.35991563623008005, -2.879022773747076, 1.3133566406772785, 0.6575314149834368, -3.2472149558150063, 0.40273267051122896, 0.8805576408434317, -2.18867586046699, -0.5081758244532483, 1.8856180831641267, -1.3333333333333335] [-0.47140452079103135, -1.331970902557534, 0.06026020675244135, -0.24866558258536722, -1.8526866816721754, 1.356116079443554, 0.1639434382464032, -3.2017746379204057, 1.257548933628023, 0.849603921829774, -3.396995004067007, 0.03617318886566478, 0.9463332722875073, -2.1654206629515245, -0.6522075742778064, 1.885618083164127, -1.3333333333333333] [-0.47140452079103135, 0.9320244950983979, 0.9534716137957935, -0.24176879652005476, 0.4310655026500103, 2.255895324430562, 0.12209530444594109, -0.9333608168502115, 2.178799704040771, -0.23418087515015426, -1.4452596279691334, 0.9095373627466455, 0.8401672166594262, -2.2029560318847454, 0.38831539743967297, 1.8856180831641267, -1.3333333333333335] [-0.47140452079103135, 0.3679173693238242, 1.2815672386291774, 0.6244321855761877, 0.5092826715975187, 2.1642663744004276, 1.35218173878464, -0.7024428590980487, 2.2101120510280303, 0.5163116058573611, -1.7698625448089045, 1.8076989107889583, 0.880053542638232, -2.188854086096631, 0.506876904611235, 1.8856180831641267, -1.3333333333333335] [-0.47140452079103135, 0.8992220279562192, 0.9844681417984447, -0.36940639328957986, 0.2974265106095656, 2.2601791346521045, -0.2974969423649234, -1.1074046953571175, 2.1143082547277396, -0.2995808043087642, -1.4393209965358302, 0.7395983847801785, 0.8547910717990976, -2.1977857183165703, 0.4360249382054165, 1.8856180831641267, -1.3333333333333335] [-0.4714045207910314, 0.8613176413096886, 1.0177964917145728, 0.12683490797560132, 0.5073924416390924, 2.2493997309548326, 0.8146520248827888, -0.7197007799260432, 2.1039698878056985, 0.027345694779926544, -1.6140083353630406, 1.342154464708827, 0.8267112507634773, -2.2077134342509668, 0.33789903730921056, 1.8856180831641267, -1.3333333333333335] [-0.4714045207910314, 0.8712797093877704, 1.0092816483950058, -0.43554722286892456, 0.2089773317170862, 2.258309196016548, -0.5722910285252211, -1.184177076602339, 2.0572571475445525, 0.7087823260572257, -1.7712678186017947, 1.9381961512552355, 0.9198157188404517, -2.1747960338829686, 0.599304861193794, 1.885618083164127, -1.3333333333333333] [-0.4714045207910314, 0.469041544066437, 1.2481096937839842, 0.5205290772709177, 0.2621468369827796, 2.2346502744073273, 0.39629547925267644, -1.0444505583051868, 2.761308854276814, 1.3818037363130589, -1.8800986613869521, 2.1864473726295777, 1.059946716920792, -2.1252522443844453, 0.831343029853193, 1.8856180831641267, -1.3333333333333335] [-0.47140452079103135, -0.8786811225498685, 1.0028445854928274, -0.07185062875982146, -0.40213126979484587, 2.272985099452554, -0.30997838203832173, 0.9897303041854314, 2.3505514065431945, -0.14735821610210365, 1.5678791324987185, 1.0702004335866186, 1.1618566113395445, 2.089221655677693, 0.9512108998774896, 1.8856180831641267, 1.3333333333333333] [-0.4714045207910314, -0.924192810319554, 0.9610647361813994, -0.8111061496177753, -0.23664128857047134, 2.1492884980684344, 0.2681040423276839, 0.5890630990015677, 2.541101751985818, 0.34378838053421523, 1.706979912189246, 1.6782379853676186, 1.685959470668116, 1.9039233127424469, 1.2785004942721274, 1.885618083164127, 1.3333333333333333] [-0.4714045207910314, -0.44933198582759326, 1.2553400114271869, 0.4497336631353765, -0.08227642507109022, 2.2636924604401742, 0.7239562680491822, 1.3019590142501025, 2.1704497047883162, -0.05397967596205814, 1.8678579001534423, 1.1338330360294633, 0.7866715253305094, 2.221869614936215, 0.05310121750928835, 1.8856180831641267, 1.3333333333333335] [-0.47140452079103135, -0.8400429977026833, 1.035425294161036, -0.024580865906231788, -0.3464959763419161, 2.283127165258978, 0.9988763672529918, 0.6067800831313921, 2.0738425592099214, 0.42189699558027854, 1.8459559727803165, 1.71116099202328, 0.8464829524921471, 2.2007230820669967, 0.40967450981856396, 1.8856180831641267, 1.3333333333333335]
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using JuMP
using Ipopt
m = Model(with_optimizer(Ipopt.Optimizer, print_level=0))
@variable(m, 0 <= x <= 2 )
@variable(m, 0 <= y <= 30 )
@objective(m, Min, x*x+ 2x*y + y*y )
@constraint(m, x + y >= 1 )
optimize!(m)
println(termination_status(m))
println("| x = ", JuMP.value(x), "| y = ", JuMP.value(y))

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