A = [4.  -1.; 1.  2.]

b = [1., 2.]

x = A\b   # solve A*x = b

A*x

i = 103491;
typeof(i)

a = 1. + 3im
b = complex(2., 1.)
a/b

n = 17
c = 's'
s = "tip "*string(n)*": string"*string(c)*" are concatenated with *"

log(-1.)

log(complex(-1))

using LinearAlgebra
λ, Ψ = eigen(A)  # eigenvalue decomposition, with two separate return values

A⁻¹= inv(A)

c = rand();
if c <= 0.49999
    println("heads");
elseif c >= 0.50001
    println("tails");
else
    println("side");
end

(rand() <= 0.5 ? "heads" : "tails")

for i = 1:10
    print(i, " ");
end

p = 0;
while p < 3 && p > -5
    p += rand([-1,1]);
    println(p, "  ", abs(p), "  ", sign(p));
end

y = zeros(size(A,2));
for i = 1:size(A,1)
    for j = 1:size(A,2)
        y[i] += A[i,j] * x[j];
    end
end
y

x = range(0., stop=1., length=20);
x.^4 .* exp.(-x.^2/2)

A = [1. 2. ; 3. 4.]

A[1,:]

A[:,1] = [5, 6];
A

arr1 = Array{Array{Float64,1},2}(undef, 2, 2)

arr1[1,1] = Array{Float64,1}();
arr1[1,2] = Float64[];
arr1[2,1] = Array{Float64,1}([1., 2.]);
arr1[2,2] = Float64[2., 3.];
arr1

arr2 = [1, 1., "1", [1.]]

a = [1., 1.];
b = a;  # b references the same array as a
c = copy(a); # c is a copy of a
b[1] = 2.; 
println("a = ", a, "\nb = ", b, "\nc = ", c)

t = (1, 2)

t = 1, 2

x, y = 1, 2;   # same as (x,y) = (1,2)
println("x = ", x, ", y = ", y);

y, x = x, y;  # swap x and y
println("x = ", x, ", y = ", y);

MixedSet = Set();
push!(MixedSet,"abc");
push!(MixedSet,π);
push!(MixedSet,9);
MixedSet

π in MixedSet

SparseIntSet = Set{Int}([-2411, 1022981,9]);
push!(SparseIntSet, 3);
for i in SparseIntSet
    if i ∈ MixedSet
        println(i);
    end
end

DenseIntSet = BitSet([1,3,5,7,11])  # implemented by bit vectors, for non-sparse sets

union(SparseIntSet,DenseIntSet)  # analogously: intersect, symdiff, ...

D = Dict{Tuple{Int,Int},Float64}((1,2)=>π, (1,0)=>ℯ);
D[(9,9)] = 0.;
D

(1,1) ∈ keys(D)

D[(1,0)]

A = [1., 2., 3.];
for a ∈ A
    print(a, "  ");
end
println("");
B = BitSet([13, 5, 2, 9]);
for b ∈ B
    print(b, "  ");
end    

typeof(1:10)

typeof(0:0.1:1)

collect(0:0.1:1)

for (i, x) in enumerate(B)
    println(i, ":  ", x);
end

V = [1, 2, 3]; W = [4, 5, 6];
for (x,y) in zip(V, W)
    println(x, ", ", y);
end

[n^2 for n=1:5]

[1.0/(i+j) for i = 1:3, j = 1:6]

sum(i for i=1:10)

Dict(x => sin(π*x) for x in 0:.5:2.)

function f(x,y)
    y*cos(x), y*sin(x)
end
f(π/3, 2)

function newtonstep(f::Array, Df::Array, x::Array)
    println("using vector definition");
    return x - Df\f
end

function newtonstep(f::Number, Df::Number, x::Number)
    println("using scalar definition");
    return x - f/Df
end

newtonstep([1,1], [2. -1.; -1. 2.], [0.,0.])

newtonstep(1, 2., 0)

f(x) = x^2;
g = x->x^3;
f(2), g(2)

h = x->(cos(x),sin(x));
h.([π, π/3, π/8])

map(t->t[1], [h(i*π/10) for i = 0:10])

SomeSet = Set([1, 2]);
empty!(SomeSet)

function setone!(v)
    fill!(v, 1.);
end
A = [2. 2.; 2. 2.];
setone!(A[:,1]);
A

setone!(view(A,:,1));
A

abstract type RoundShape end

mutable struct Ellipse <: RoundShape
    center::Tuple{Float64,Float64}
    A::Real
    B::Real
end

mutable struct Circle <: RoundShape
    center::Tuple{Float64,Float64}
    r::Real
end

getcenter(S::RoundShape) = S.center;

area(S::Ellipse) = π * S.A * S.B;
area(S::Circle) = π * S.r^2;

Ell = Ellipse((1., 0.), 2, 1);
getcenter(Ell), area(Ell)

@time s = 0; for i=1:1e6 s += i^3; end; print("result: ", s);

@time s = sum(map(x->x^3, 1:1e6)); print("result: ", s);

@code_native sin(π/2)^2

using Printf
println(@sprintf "iteration %d: error %e, step size %e" 139 2.34232131e-5 1.290137e-6)

using Pkg
Pkg.add("StaticArrays");

using StaticArrays

Pkg.add("Revise");
using Revise

Pkg.add("PyPlot");
using PyPlot
x = range(1e-2,stop=1,length=10000);
plot(x, sin.(1.0./x));

Pkg.add("SymPy");
using SymPy
x, y = symbols("x, y", real=true);
diff(x^y*exp(y*exp(x*y)) - x*y + 1, y)

rand([1, 4, 16])

rand("abc")

rand(3,3,3)

rand(Int, 2, 2, 2)

using Random
A = zeros(5)
rand!(view(A,1:3))
A

rng = MersenneTwister(5312)  
rand!(rng, A)   # similar effect as calling seed!(5312), but this does not change global random number generator

bitrand(3, 3, 3)

G = randn(1000000);

using Plots; gr()
histogram(G, nbins=200)