I am trying to write some functions to generate a Lagrange interpolating polynomial basis. That is, I want to create a vector where each element is a lagrange polynomial basis function. I have some code below but it does not work. And I am not sure how to set this up to work correctly. I know that there is a package in julia for Lagrange polynomials, but I wanted to write this myself so that I understand them better.

what I would like is to have some vector of functions such that each element corresponds to a lagrange basis function. So if the vector is `\psi`

, then \psi1`should apply the first lagrange basis function at the value of`

x`. Here is some code I am working with, but multiplying the functions to create the polynomial basis functions is proving harder than I through. Some subtle errors in how to handle functions and multiply them, etc. I was hoping someone could help. Here is some code that I am starting with.

```
function lagrange_polynomial(x, i, points)
p(x) = 1;
N = length(points)
for k=1:N
if k != i
p = x -> p(x)*(x - points[k])/(points[i] - points[k])
@show p(x)
end
end
return p
end
function lagrange_polynomials_01(x, N)
h = 1.0/N
points = [(i-1)*h for i=1:N+1]
ψ_array = Array{Function}(undef, N+1);
for i=1:N+1
ψ_array[i] = x -> lagrange_polynomial(x, i, points)
end
return ψ_array, points
end
```

So I should be able to do something like.

```
julia>psi, points = lagrange_polynomials_01(0.4, 2)
julia>psi[1](0.4)
```

But then I get the error:

```
ERROR: StackOverflowError:
Stacktrace:
[1] (::var"#9#11"{Int64, Vector{Float64}, Int64})(x::Float64) (repeats 10823 times)
@ Main /media/hayagriva/Dropbox/sandbox/julia/julia_pde/julia_finite_elements/langetangen/approx1d.jl:145
[2] macro expansion
@ ./show.jl:1128 [inlined]
[3] lagrange_polynomial(x::Float64, i::Int64, points::Vector{Float64})
@ Main /media/hayagriva/Dropbox/sandbox/julia/julia_pde/julia_finite_elements/langetangen/approx1d.jl:146
```

So not sure how to make this run smoothly. Seems like there are some simple function tricks that I am missing. Any help is appreciated.