# Define τ(y,x) on vectors

τ(y::Vector, x::Vector) = minimum(y./x)

# Example
y = [10,5,6,9]
x = [1,2,3,4]
τ(y,x)

all(y.≥2x), all(y.≥1.99x),all(y.≥2.01x)  # check these by hand

# Example
A= [ 1 2 3;4 5 6; 7 8 9]
y = [0, .1, .0]
A * y # any one positive entry multiplies an entire positive column of A

# An example
n = 6
A = rand(n,n)
x = rand(n)
[τ(A^k*x, A^(k-1)*x) for k=1:7] # This sequence will be increasing, but to an eig limit.

# Pkg.add("JuMP")
using JuMP
# Pkg.add("Ipopt")  (On my mac, this worked with 0.6.2 but not 0.6.0)
using Ipopt

A = rand(7,7)

n=size(A,1)

m = Model(solver=IpoptSolver(print_level=2))
@variable(m, t);         @objective(m, Max, t)

@variable(m, x[1:n]>=0); @constraint(m, sum(x)==1)
@variable(m, y[1:n]);    @constraint(m, y .== A*x)

@NLconstraint(m, [i=1:n], t <= y[i]/x[i])  # nonlinear constraint

status = solve(m)
x = getvalue.(x)
t = getobjectivevalue(m)

norm(A*x-t*x) #  demonstrate we have found an eigenpair through optimization

A = rand(5,5)

eigvals(A)

Λ,X=eig(A);x=X[:,2];λ=Λ[2]

norm(A*x-λ*x)

τ(A*abs.(x),abs.(x)) - abs(λ) # This is non-negative