using Plots
# gr()   # use GR as backend

p1 = plot(err, yscale=:log10,  label="err_norm")  # notice :log10 instead of log10

savefig("myplot.png") # Saves the CURRENT_PLOT as a .png
savefig(p, "myplot.pdf") # Saves the plot from p as a .pdf vector graphic

plot!(...) # http://docs.juliaplots.org/latest/tutorial/#Changing-the-Plotting-Series
# or
y = rand(10, 4)
plot(x, y, layout = (4, 1))
# or
p1 = plot(x, y) # Make a line plot
p2 = scatter(x, y) # Make a scatter plot
p3 = plot(x, y, xlabel = "This one is labelled", lw = 3, title = "Subtitle")
p4 = histogram(x, y) # Four histograms each with 10 points? Why not!
plot(p1, p2, p3, p4, layout = (2, 2), legend = false)

# and more

pip install sympy

# Pkg.add("SymPy")
# # run this in 
using LinearAlgebra  # get norm
using SymPy # we will get symbolic norm function automatically

latex = sympy.latex # define a function we can use these function **latex** to generate latex code

@vars x ϵ                     # 1
x, ϵ = symbols("x ϵ")      # 2
x, ϵ = Sym("x ϵ")           # 3

A= Sym[1 -1; 3 4; 0 2]
A = [Sym(1) -1; 3 4; 0 2]     ## using an annotation to ensure the type. Alternatively, through promotion, just a single symbolic object will result in the same:

A = [1//11 3]  # sometimes can use built in Rational type in Julia

# The following construction should be avoid
# Sym[1 -1; 3 4; 0 2] |> typeof                          # 3x1
# sympy.Matrix([1 2 3;2 3 4;3 4 5 ]) |> typeof     # 3x2
# sympy.Matrix([1 2 3;2 3 4;3 4 5 ])    # . As shown, it is better to avoid the sympy.Matrix constructor when possible, and when not, only pass in a symbolic array created through Julia's array semantics.

A=[1 2 3 3;2 3 4 5;3 4 5 7; 4 5 6 11]

# want the 1st row
A[1,:] ## give a column vector !!!! 
##4-element Array{Int64,1}:
## 1
## 2
## 3
## 3

A[:,1] * A[1,:]     # gives error, dimension does not match
# The right way to do it
A[:,1] * A[1,:]'        # transpose manually

# you'd better add .|> simplify everywhere
x^2 / x .|> simplify  # remember the dot
# without the dot, the performance degrade a lot.

# Notice that += -= /= does not fit very well with .|> and |>
# for example
A[:,I] /= norm(z) .|> simplify       # does not work well. Use A[:,I] = A[:,I] / norm(z) .|> simplify

plot(x->x^2 - 2x)        # just like fplot MATLAB
# x -> x^2-2x    <--- return a function. This function serves as a parameter in plot
# see https://docs.julialang.org/en/v1/manual/functions/#man-anonymous-functions
plot([sin cos])             # plot sin and cos in the same canvas
plot(x->x^2 - 2x, 0, 4)

@show a = 1+2
# a |> display  # use |>
# print( ...)

I # capital i

A'(b - c) # just don't do that 
d  = b - c
A'd # work

norm(A) # Frobenius norm
opnorm(A, 2) # l2 norm

#pwd()
# current directory /home/user/Desktop/
using DelimitedFiles
# data are in /home/user/Desktop/data_problem/
datadir="data_problem/"

# if we want
# /home/user/Desktop/data_problem/data.txt
# /home/user/Desktop/data_problem/label.txt
A = readdlm(datadir * "data.txt") 
y = readdlm(datadir * "label.txt");
# function *   <---- Concatenate strings and/or characters, producing a String.