using Plots
using PyFormattedStrings

T = [ 500.0,  650,  800,  950, 1100, 1250, 1400, 1550, 1700, 1850, 2000]
R = [5.06898064e-06, 9.49169460e-05, 5.91159348e-04, 2.78345192e-03,
     4.88059454e-03, 1.62306334e-02, 3.57378985e-02, 2.30324885e-02,
     8.40528066e-02, 1.07758145e-01, 6.39353585e-02];

##################### SOLUTION

m = length(T)

#--------- form y and x data

y = log.(R)
x = 1 ./T

#--------- form vector b and matrix A, along with bhat and Ahat

b = copy(y)

f_1 = ones(m)
f_2 = -x
A = [f_1 f_2]

bhat = A' * b
Ahat = A' * A

#--------- solve for η and recover parameters

η = Ahat \ bhat

lnk = η[1]
E   = η[2]

k = exp(lnk)

println(f"Optimal parameters: k={k:.3f}, E={E:.0f}")

#-------- plot the model curve versus the data points

TT = LinRange(minimum(T), maximum(T), 1000)   # lots of T
RR = k*exp.(-E./TT)                           # model R

scatter(T, R, label="data")
plot!(TT,RR, label="fitted model", lw=2)
plot!(xlabel="T", ylabel="R")
plot!(legend_foreground_color=nothing)

x = T
y = R

#----------------

b = y
f_1 = x.^4
f_2 = x.^3
f_3 = x.^2
f_4 = copy(x)
f_5 = ones(length(x))

A = [f_1  f_2  f_3  f_4  f_5]

bhat = A' * b
Ahat = A' * A

#----------------

η = Ahat \ bhat

a,b,c,d,e = η

#----------------

TT = LinRange(minimum(T), maximum(T), 1000)   # lots of T
RR = a*TT.^4 .+ b*TT.^3 .+ c*TT.^2 .+ d*TT .+ e

scatter(T, R, label="data")
plot!(TT,RR, label="fitted model", lw=2)
plot!(xlabel="T", ylabel="R")
plot!(legend_foreground_color=nothing)