How to plot Newton's method

[mthelm85]

Don’t know if this is helpful but:

GitHub - mthelm85/newton-raphson-plot

Also, see this:

Visualizing Newton-Raphson

I’ve been studying the Newton-Raphson algorithm and coded up a simple example to show some of its shortcomings with particular functions. I’m posting here simply to share with the community and, as always, I welcome feedback! First is the case where we choose a good starting point and the algorithm quickly converges to find the root: [Newton-Raphson1] Next is a case where the algorithm doesn’t converge and it just oscillates around: [Newton-Raphson2] Finally, this starting point eventually …

[brett_knoss]

Thanks, not what I was thinking,but an interesting approach. I’ll think about it.

[j_verzani]

You might need the master version using Pkg; Pkg.add("SimplePlots#master")

[brett_knoss]

@j_verzani

Pkg; Pkg.add(“SimplePlots#master”)

I’m told that’s not a valid package name. What version of Julia are you using?

[brett_knoss]

@j_verzani

using Pkg; Pkg.add("SimplePlots#master"

now I get

MethodError: no method matching getindex(::typeof(f), ::Int64)

Stacktrace:
 [1] (::SimplePlots.var"#16#17")(::Function) at C:\Users\Brett\.julia\packages\SimplePlots\6y94O\src\plots\plots.jl:16
 [2] plot!(::SimplePlots.SimplePlot, ::Function, ::Vararg{Any,N} where N; kwargs::Base.Iterators.Pairs{Symbol,Tuple{Real,Real},Tuple{Symbol,Symbol},NamedTuple{(:xlim, :ylim),Tuple{Tuple{Float64,Float64},Tuple{Int64,Int64}}}}) at C:\Users\Brett\.julia\packages\SimplePlots\6y94O\src\plots\plots.jl:23
 [3] plot(::Function, ::Vararg{Any,N} where N; kwargs::Base.Iterators.Pairs{Symbol,Tuple{Real,Real},Tuple{Symbol,Symbol},NamedTuple{(:xlim, :ylim),Tuple{Tuple{Float64,Float64},Tuple{Int64,Int64}}}}) at C:\Users\Brett\.julia\packages\SimplePlots\6y94O\src\plots\plots.jl:3
 [4] (::var"#21#22")(::Float64) at .\In[12]:9
 [5] top-level scope at C:\Users\Brett\.julia\packages\SimplePlots\6y94O\src\interact\gui\make_gui_block.jl:13
 [6] top-level scope at C:\Users\Brett\.julia\packages\SimplePlots\6y94O\src\interact\gui.jl:40
 [7] include_string(::Function, ::Module, ::String, ::String) at .\loading.jl:1091

[j_verzani]

Ooops, that Plots syntax is something I have loaded privately. Add these two lines and comment out the offending one.

ts = range(a, b, length=250)
plot(ts, f.(ts), xlim=(a,b), ylim=(-5,5))
#plot(f, a, b, xlim=(a,b), ylim=(-5,5))

[brett_knoss]

@j_verzani

ts = range(a, b, length=250)
plot(ts, f.(ts), xlim=(a,b), ylim=(-5,5))
#plot(f, a, b, xlim=(a,b), ylim=(-5,5))

Back to saying scatter not deffined.

I got an eroor trying to install Simple Plots master, and instead did

(@v1.5) pkg> add SimplePlots#master

[j_verzani]

Not sure, that function is exported here: https://github.com/djsegal/SimplePlots.jl/blob/master/src/plots/plots.jl#L62

I edited the script posted earlier. Try restarting and running that entire script in a cell to make sure all the packages are loading. If that fails, you can do this with Plots and Interact in the same manner, I just thought this would be easier to get going.

[brett_knoss]

still says scatter! not defined

[brett_knoss]

function plot(varargs...; kwargs...)
  simple_plot = SimplePlot(; kwargs...)
  plot!(simple_plot, varargs...; kwargs...)
end

function scatter(varargs...; kwargs...)
  simple_plot = SimplePlot(; kwargs...)
  scatter!(simple_plot, varargs...; kwargs...)
end

it looks like scatter needs to be defined as a function.

[j_verzani]

Okay, I edited the script from an earlier post to use Plots + Interact. SimplePlots was only intended to make things easier. You may need to install Interact, but try the above after a restart of the kernel.

[brett_knoss]

@j_verzani

using Plots, Interact
using ForwardDiff
D(f) = x -> ForwardDiff.derivative(f, float(x))
f(x)  = x^5 - x - 1
a, b = -1.5, 1.5
nsteps = 20
@manipulate for x0=slider(a:0.1:b, value=1.0, label="x0")
    ts = range(a, b, length=251)
    plot(ts, f.(ts), xlim=(a,b), ylim=(-5,5), legend=false)
    plot!(ts, f.(ts)) 
    scatter!([x0],[0])

    xs = Float64[]
    ys = Float64[]
    for i in 1:nsteps
        x1 = x0 - f(x0)/D(f)(x0)
        append!(xs, [x0, x0, x1])
        append!(ys, [0,f(x0), 0])
        x0 = x1
    end

    plot!(xs, ys)

end

I can’t execute it in the REPL. it seems to be missing a bracket

[brett_knoss]

It said I didn’t have Web IO, so I put

using WebIO
using IJulia

but it says unable to load WebIO

[j_verzani]

I’m confused. Interact runs in jupyter, not the REPL. But copy and paste should work without a syntax error.

[brett_knoss]

Interact seems to run in both, the plot doesn’t run .

using WebIO

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

[brett_knoss]

Does makie allow a reaction to pressing a mouse button, or the mouse x and y locations?

[brett_knoss]

It should be AbstractPlotting.scatter! (if I have Plots loaded)

[dpsanders]

Yes it does:

http://makie.juliaplots.org/stable/interaction.html

[brett_knoss]

Cool! I was doing something similar in PyDev, and I would like to do it in Julia.

[j_verzani]

@j_verzani

using AbstractPlotting
using AbstractPlotting.MakieLayout
using GLMakie

using ForwardDiff
D(f) = x -> ForwardDiff.derivative(f, float(x))


function newton(f=x->x^5 - x - 1, x0=1, a=-1, b=1.5, n = 20)

    function add_move!(scene, points, pplot)
        idx = Ref(0); dragstart = Ref(false); startpos = Base.RefValue(Point2f0(0))
        on(events(scene).mousedrag) do drag
            if ispressed(scene, Mouse.left)
                if drag == Mouse.down
                    plot, _idx = mouse_selection(scene)
                    if plot == pplot
                        idx[] = _idx; dragstart[] = true
                        startpos[] = to_world(scene, Point2f0(scene.events.mouseposition[]))
                    end
                elseif drag == Mouse.pressed && dragstart[] && checkbounds(Bool, points[], idx[])
                    pos = to_world(scene, Point2f0(scene.events.mouseposition[]))
                    
                    # we work with components, not vector
                    x,y = pos
                    
                    ptidx = idx[]
                    
                    x = clamp(x, a, b)
                    y = zero(y)
                    points[][idx[]] = [x,y]
                    
                    points[] = points[]
                end
            else
                dragstart[] = false
            end
            return
        end
    end

    start_point = Node(Point2f0[(x0, 0)])
    points = lift(start_point) do x₀
        xs = [x₀[1][1]]
        for i in 1:n
            xᵢ₋₁ = xs[end]
            xᵢ = xᵢ₋₁ - f(xᵢ₋₁) /D(f)(xᵢ₋₁)
            push!(xs, xᵢ)
        end
        xs
    end

    plt_points = lift(points) do pts
        xs = Float64[]
        ys = Float64[]
        for i in 1:length(pts)-1
            pᵢ, pᵢ₊₁ = pts[i], pts[i+1]
            append!(xs, [pᵢ, pᵢ, pᵢ₊₁])
            append!(ys, [0, f(pᵢ), 0])
        end
        (xs, ys)
    end

    scene = Scene()
    lines!(scene, a..b, f, linewidth=5, color=:blue)
    lines!(scene, a..b, zero, color=:red)
    lines!(scene, plt_points, color=:black, linewidth=2)
    scatter!(scene, start_point, color=:red)

    add_move!(scene, start_point, scene[end])

    scene
end

The example I have above in this thread (which needs updating for the newest makie, as the layout is kinda bad) shows how one can drag a point around and have the graph be reactive to that action. Something similar could be done for clicking on points. The particular example above is modified from one of the Makie examples, I think a polygon one.