Arnoldifit (#632)

* update reference; stub for fourier extension

Author: jverzani

src/polynomials/standard-basis/standard-basis.jl

@@ -665,9 +665,7 @@ polynomial evaluations afterwards for higher-degree polynomials.
 
 The two main functions are translations from example code in:
 
-*VANDERMONDE WITH ARNOLDI*;
-PABLO D. BRUBECK, YUJI NAKATSUKASA, AND LLOYD N. TREFETHEN;
-[arXiv:1911.09988](https://people.maths.ox.ac.uk/trefethen/vander_revised.pdf)
+Pablo D. Brubeck, Yuji Nakatsukasa, and Lloyd N. Trefethen, “Vandermonde with Arnoldi”, SIAM Review 63, doi.org/10.1137/19M130100X (2021).
 
 For more details, see also:
 
@@ -744,6 +742,35 @@ function polyvalA(d, H::AbstractMatrix{S}, s::T) where {T, S}
     sum(W[i]*d[i] for i in eachindex(d))
 end
 
+#=
+# Finds a + bim following formula 5.3 of paper, which solves
+# Re(Vₙ) [a₀ ⋯ aₙ] - Im(Vₙ[2:n]) * [b₁ ⋯ bₙ] = [y₁ ⋯ yₘ]
+function polyfit_fourier_extension(x, y, n::Int=length(x) - 1; var=Var(:x))
+    m = length(x)
+    T = eltype(y)
+    Q = ones(T, m, n+1)
+    H = zeros(T, n+1, n)
+
+    q = zeros(T, m)
+
+    @inbounds for k = 1:n
+        q .= x .* Q[:,k]
+        for j in 1:k
+            λ = dot(Q[:,j], q)/m
+            H[j,k] = λ
+            q .-= λ * Q[:,j]
+        end
+        H[k+1,k] = norm(q)/sqrt(m)
+        Q[:,k+1] .= q/H[k+1,k]
+    end
+
+    d = [real(Q) imag(view(Q, :, 2:n+1))] \ y
+    d = d[1:(n+1)] - vcat(0, view(d, (n+2):(2n+1)))
+    ArnoldiFit{eltype(d),Symbol(var)}(d, H)
+end
+=#
+
+
 # Polynomial Interface
 """
     ArnoldiFit