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pluto_newton.jl
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pluto_newton.jl
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### A Pluto.jl notebook ###
# v0.19.36
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ 5ffc5ec0-f91f-11ec-0552-37ef5f25102d
begin
using Symbolics
using ForwardDiff
using Plots
using PlutoUI
using LaTeXStrings
using ForwardDiff: jacobian
end
# ╔═╡ 2dc43448-3cf4-475f-8b61-d16c3594d9e3
md"""
## Newton's Method
Define a sequance $\{x_n\}$ by
$x_{n+1}:=x_n-\frac{f(x_n)}{f^\prime(x_n)}, \quad n=0,1,2,\dotsc$
so that $x_n \rightarrow c$ ($n \rightarrow \infty$) and $f(c)=0$.
"""
# ╔═╡ 76cb2ec2-fc76-4ad5-8d57-777395755f7f
md"""
#### Example 1 $f(x)=x^2-9$
"""
# ╔═╡ 3c743ad1-c58f-4358-95ab-6a8bb851d93a
begin
nbsp = html" "
end
# ╔═╡ 8b7193bf-cff2-4476-9ac1-1f6ff081fd0c
md"""
n = $(@bind n2 Slider(0:10, show_value=true, default=0))
$nbsp $nbsp
x₀ = $(@bind x02 Slider(-10:10, show_value=true, default=6))
"""
# ╔═╡ 1d24debe-6898-483e-b9c3-ecc0ede4f7c1
md"""
#### Example 2 $f(x)=x^3/5-4x+1$
"""
# ╔═╡ 30affe0b-b5e2-4fac-b79a-500e2b6ba47c
md"""
n = $(@bind n Slider(0:10, show_value=true, default=0))
$nbsp $nbsp
x₀ = $(@bind x0 Slider(-10:0.5:10, show_value=true, default=6))
"""
# ╔═╡ 38181bb5-d97a-47fe-8513-f6cfbf7bdf85
# ╔═╡ 7bd2ec5e-7532-4cca-912f-3da6acdd6783
md"""
#### Example 3 $f(x)=-x^3+3x$
"""
# ╔═╡ d969748a-10d5-4d61-97c9-5f8e9ac52681
md"""
n = $(@bind n3 Slider(0:10, show_value=true, default=0))
$nbsp $nbsp
x₀ = $(@bind x03 Slider(-2:0.1:2, show_value=true, default=6))
"""
# ╔═╡ 4699a0c5-47d8-425c-bc73-52495b366d49
# ╔═╡ 49027c98-8c18-456b-922a-adcc33eb034f
function newton1D(f, x0)
f′(x) = ForwardDiff.derivative(f, x) # \prime<TAB>
x0 = 37.0 # starting point
sequence = [x0]
x = x0
for i in 1:10
x -= f(x) / f′(x)
end
return x
end
# ╔═╡ bb7d5554-f731-4fb8-9bd8-6d1093864891
function newton2D_step(T, x)
J = ForwardDiff.jacobian(T, x) # should use StaticVectors
δ = J \ T(x) # J^(-1) * T(x)
return x - δ
end
# ╔═╡ df735acb-07bd-4692-8dec-cb067cfc2423
"Looks for x such that f(x) = y, i.e. f(x) - y = 0"
function inverse(f, y, x0=[0, 0])
return newton2D(x -> f(x) - y, x0)
end
# ╔═╡ 8b1db0d9-8d46-4fee-989d-4dd0f563b97a
straight(x0, y0, x, m) = y0 + m * (x - x0)
# ╔═╡ 15d89b4b-101f-4520-a1c7-f4644f8ddbc9
function standard_Newton(f, n, x_range, x0, ymin=-10, ymax=10)
f′ = x -> ForwardDiff.derivative(f, x)
p = plot(f, x_range, lw=3, ylim=(ymin, ymax), legend=:false, size=(400, 300))
hline!([0.0], c="magenta", lw=3, ls=:dash)
scatter!([x0], [0], c="green", ann=(x0, -5, L"x_0", 10))
for i in 1:n
plot!([x0, x0], [0, f(x0)], c=:gray, alpha=0.5)
scatter!([x0], [f(x0)], c=:red)
m = f′(x0)
plot!(x_range, [straight(x0, f(x0), x, m) for x in x_range],
c=:blue, alpha=0.5, ls=:dash, lw=2)
x1 = x0 - f(x0) / m
scatter!([x1], [0], c="green", ann=(x1, -5, L"x_%$i", 10))
x0 = x1
end
p |> as_svg
end
# ╔═╡ ab1c3d69-0499-4fad-b2a8-5360ed7339e1
let
f(x) = x^2 - 9
standard_Newton(f, n2, -10:0.01:10, x02, -15, 70)
end
# ╔═╡ 0af86953-7034-473e-8ab1-e84e1def532d
let
f(x) = x^3/5 - 4x + 1
standard_Newton(f, n, -10:0.01:10, x0, -70, 70)
end
# ╔═╡ 94023f0c-61ab-48d9-9595-ee8b0ea0aaaf
let
f(x) = -x^3+3*x
standard_Newton(f, n3, -3:0.01:3, x03, -5, 5)
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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