In [1]:
using Plots
In [11]:
""" 
Evaluate the polynomial interpolant in Lagrange form.

xi is a vector of length n+1
fi are the values f(xi) for each point xi

x is the value where we compute
p(x)
which is the degree n+1 polynomial interpolating f at each point xi

The values xi must be distinct

Note that x can be a matrix or vector
"""
function lagrange_interp(x, xi, fi)
    f = zeros(size(x))
    for i=1:length(fi)
        elli = ones(size(x))
        for j=1:length(xi)
            if i==j continue; end
            elli = elli.*(x .- xi[j])/(xi[i] - xi[j])
        end
        f = f + fi[i]*elli
    end
    return f
end

# The polynomial
fi = [2,.33,-1];
xi = [1,2,3];
xx = collect(range(0,stop=4,length=100))

plot(xx,lagrange_interp(xx,xi,fi))
scatter!(xi,fi,marker=:diamond)
Out[11]:
In [13]:
lagrange_interp([2],xi,fi)
Out[13]:
1-element Array{Float64,1}:
 3.0
In [17]:
function compute_barycentric_weights(xi)
    ws = zeros(size(xi))
    for i=1:length(xi)
        w = 1
  # TODO, add check if x == xi, and just return fi in that case. 
        for j=1:length(xi)
            if j==i continue; end
            w = w*1 ./ (xi[i] - xi[j]);
        end
        ws[i] = w;
    end
    return ws
end  

function barycentric_interp_with_weights(x,xi,fi,wi)
    fnumer = zeros(size(x))
    fdenom = zeros(size(x))
    for i=1:length(xi)
        fnumer += fi[i] .* wi[i] ./ (x .- xi[i])
        fdenom += wi[i] ./ (x .- xi[i])
    end
    f = fnumer./fdenom
end

function barycentric_interp(x,xi,fi)
    return barycentric_interp_with_weights(x,xi,fi,compute_barycentric_weights(xi))
end

# The polynomial
fi = [2,3,6];
xi = [1,2,3];
xx = collect(range(0,stop=4,length=100));

plot(xx,barycentric_interp(xx,xi,fi))
Out[17]:
In [16]:
barycentric_interp([nextfloat(2.0)],xi,fi)
Out[16]:
1-element Array{Float64,1}:
 3.000000000000001
In [12]:
# show Trefethen's demo
fun(x) = abs.(x) + .5 .* x- x.^2;
n = 10
xi = cos.(pi .* (0:n)./n);
fi = fun(xi)
xx = range(-1,stop=1,length=5000)
plot(xx,barycentric_interp(xx,xi,fi))
scatter!(xi,fi)

UndefVarError: barycentric_interp not defined

Stacktrace:
 [1] top-level scope at In[12]:7
 [2] include_string(::Function, ::Module, ::String, ::String) at ./loading.jl:1091
In [15]:
using Blink,Interact

Unable to load WebIO. Please make sure WebIO works for your Jupyter client. For troubleshooting, please see the WebIO/IJulia documentation.

In [19]:
fun(x) = abs.(x) + .5 .* x- x.^2;

ui = @manipulate for n=10:10:1000
    xi = cos.(pi*(0:n)/n);
    fi = fun(xi)
    xx = collect(range(-1,stop=1,length=5000))
    plot(xx,barycentric_interp(xx,xi,fi))
    plot!(xi,fi,seriestype = :scatter,markersize=1.)
end

w = Window()
body!(w, ui)
Out[19]:
Page(2, WebSocket(server, CONNECTED), Dict{String,Any}("webio" => Blink.AtomShell.var"#24#25"{Blink.AtomShell.WebIOBlinkComm}(Blink.AtomShell.WebIOBlinkComm(Window(2, Electron(Process(`/Users/dgleich/.julia/packages/Blink/u1xcH/deps/Julia.app/Contents/MacOS/Julia /Users/dgleich/.julia/packages/Blink/u1xcH/src/AtomShell/main.js port 8297`, ProcessRunning), Sockets.TCPSocket(RawFD(0x0000003a) active, 0 bytes waiting), Dict{String,Any}("callback" => Blink.var"#1#2"())), Page(#= circular reference @-5 =#), Task (done) @0x00000001103bcd90))),"callback" => Blink.var"#1#2"()), Distributed.Future(1, 1, 2, Some(true)))
┌ Error: An exception occurred while trying to send a WebIO message to a frontend:
│   exception = (Base.IOError("stream is closed or unusable", 0), Union{Ptr{Nothing}, Base.InterpreterIP}[Ptr{Nothing} @0x000000010760dc9d, Ptr{Nothing} @0x0000000107c0f8ce, Ptr{Nothing} @0x00000001076154a3, Ptr{Nothing} @0x0000000149979489, Ptr{Nothing} @0x0000000149979691, Ptr{Nothing} @0x00000001499796c3, Ptr{Nothing} @0x00000001499b91bb, Ptr{Nothing} @0x00000001499b938a, Ptr{Nothing} @0x00000001499b70a7, Ptr{Nothing} @0x00000001499b7a66, Ptr{Nothing} @0x00000001499b7fb4, Ptr{Nothing} @0x00000001499b7fcc, Ptr{Nothing} @0x00000001008bc4aa])
└ @ WebIO /Users/dgleich/.julia/packages/WebIO/nTMDV/src/connection.jl:120
In [18]:
fun(x) = abs.(x) + .5 .* x- x.^2;

ui2 = @manipulate for n=5:5:50
    xi = collect(-1.:(2/n):1.)
    fi = fun(xi)
    xx = collect(range(-1,stop=1,length=5000))
    plot(xx,barycentric_interp(xx,xi,fi))
    plot!(xi,fi,seriestype = :scatter,markersize=1.)
    ylims!(-1.,1.)
end

w2 = Window()
body!(w2, ui2)
Out[18]:
Page(1, WebSocket(server, CONNECTED), Dict{String,Any}("webio" => Blink.AtomShell.var"#24#25"{Blink.AtomShell.WebIOBlinkComm}(Blink.AtomShell.WebIOBlinkComm(Window(1, Electron(Process(`/Users/dgleich/.julia/packages/Blink/u1xcH/deps/Julia.app/Contents/MacOS/Julia /Users/dgleich/.julia/packages/Blink/u1xcH/src/AtomShell/main.js port 8297`, ProcessRunning), Sockets.TCPSocket(RawFD(0x0000003a) active, 0 bytes waiting), Dict{String,Any}("callback" => Blink.var"#1#2"())), Page(#= circular reference @-5 =#), Task (done) @0x000000012a529210))),"callback" => Blink.var"#1#2"()), Distributed.Future(1, 1, 1, Some(true)))