From 710fa31a9d1801ae288b1a677eed8f1c298b72e8 Mon Sep 17 00:00:00 2001 From: Giannis Tsagkaropoulos <98717569+GiannisTsagkaropoulos@users.noreply.github.com> Date: Thu, 28 Sep 2023 23:13:05 +0300 Subject: [PATCH] Delete notebooks/Polynomial_Interpolation.jl --- notebooks/Polynomial_Interpolation.jl | 1434 ------------------------- 1 file changed, 1434 deletions(-) delete mode 100644 notebooks/Polynomial_Interpolation.jl diff --git a/notebooks/Polynomial_Interpolation.jl b/notebooks/Polynomial_Interpolation.jl deleted file mode 100644 index af2d4d7..0000000 --- a/notebooks/Polynomial_Interpolation.jl +++ /dev/null @@ -1,1434 +0,0 @@ -### A Pluto.jl notebook ### -# v0.19.9 - -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 - -# ╔═╡ d36515a4-1850-11ee-27b1-e5007a081e3a -begin - using PlutoUI, Random, Polynomials, Plots, LinearAlgebra,PlotlyBase - plotly() -end - -# ╔═╡ 8e4180cb-0b86-4167-a889-d60a8a6f8a53 -TableOfContents(title="📚 Table of Contents", aside=true) - -# ╔═╡ c46916c9-b9e7-4545-9393-bc32b168df03 -md""" -# Interpolation Polynomials - -Given a set of $n + 1$ data points -$(x_0, y_0), \ldots, (x_n, y_n)$ with no two $x_{j}$ the same, a polynomial function $p(x)$ is said to interpolate the data if - -$$p(x_{j})=y_{j},\ \forall j\in \{0,1,\dotsc ,n\}.$$ -""" - -# ╔═╡ eeb5dcf5-d240-4373-9fa0-325d4bdb2dfe -md""" -## Uniqueness of interpolation polynomial - -We consider the elementary Lagrange polynomials - -$$L_i(x) = \prod_{\substack{j=0 \\ j\neq i}}^n \cfrac{x-x_j}{x_i-x_j}, i=0,1, \ldots, n.$$ - -It's true that - -$$L_i(x_k)= \delta_{ik} = - \begin{cases} - 1, & \text{if } i=k,\\ - 0, & \text{if } i\neq k - \end{cases}$$ - -Next, we define the following polynomial. - -$$p(x) = \sum_{i=0}^n f(x_i)L_i(x)$$ - -We can check that - -$$p(x_k) = \sum_{i=0}^n f(x_i)L_i(x_k)= \sum_{i=0}^n f(x_i)\delta_{ik}(x_k)= f(x_k), k=0,1, \ldots, n.$$ - -Let us assume that there is another polynomial $p^*$, such that - -$$p^*(𝑥_𝑖)=f(x_𝑖), 𝑖=0,1, \ldots,𝑛.$$ - -Then the polynomial $d=p-p^*$ has degree of at most $𝑛$ but has 𝑛 zeros and - -$$d(x_i)=p(x_i)-p^*(x_i)=f(x_i)-f(x_i)=0, i=0,1,\ldots, n \Longleftrightarrow p = p^*$$ - -""" - -# ╔═╡ 2f130ec1-0d55-4c10-a209-bd06f5d1072e -md""" -## Lagrange Interpolation -""" - -# ╔═╡ ca2fc65f-ae5e-4679-a140-a922b84e7c9a -md""" -We suppose that we've got a $2\times n$ matrix with points of $f$. Let it be $IntP$. -We want to compute the Lagrange polynomial of $f$ at a given $x$. - -The function below does exactly that! -""" - -# ╔═╡ 63fc147c-220e-4201-897c-a5a97c6837df -function LagrangeInterpolation(IntP,x) -# Initialization -# Suppose that IntP has a 2 x n dimension. -X = IntP[1,:] -Y = IntP[2,:] -n = size(IntP)[2] -polynomial_value = 0 ; -## -for i=1:n - # Initialization - numerator = 1 ; #of lagrange polynomial - denominator = 1 ; #of lagrange polynomial - # Main process - for j=1:n - if j!=i - numerator = numerator * (x-X[j]) - denominator = denominator * (X[i]-X[j]) - end - end - lagrange = numerator / denominator - polynomial_value = polynomial_value + Y[i] * lagrange; - end - return(polynomial_value) -end - -# ╔═╡ a35f8971-e176-4d1d-b184-61000da218cc -Show(MIME"text/html"(),""" -

- Define the function f(x) that we want to interpolate by changing the code block below:
- -

""") - -# ╔═╡ 9631d6fe-3350-4687-9f17-cf7dbf872461 -function f(x) - return sin(x) #1/(1+25*x^2); -end - -# ╔═╡ bbb09656-3cde-4016-a97d-3d62e523046f -Show(MIME"text/html"(),""" -

- You can manually select absiccas of points by enabling and using the code block below:
- -

""") - -# ╔═╡ 91ba7d55-c046-4e68-9384-5a8afb4b5508 -Show(MIME"text/html"(),""" -

- You can either do it by choosing random absiccas by enabling and using the code below(use only one method at a time):
- -

""") - -# ╔═╡ 3fc7030b-9b18-4262-a407-e6764edef676 -Show(MIME"text/html"(),""" -

- Select the interval [a,b] for absiccas generation:
-

""") - -# ╔═╡ f348b593-b464-4e93-90d5-9fb99abe2c11 -Show(MIME"text/html"(),""" -

- First select a: -

""") - -# ╔═╡ 58ca6a08-5637-4ef8-8f54-04c72e84e44c -@bind a html""" -""" - -# ╔═╡ ee0eccad-402a-4e15-bfa5-b65e8920362d -Show(MIME"text/html"(),""" -

- Now select b: -

""") - -# ╔═╡ 4b382fb3-0083-487e-b596-58fe2d6ae040 -@bind b html""" -""" - -# ╔═╡ 715f1ff3-7281-4992-95ac-66661178d7c2 -Show(MIME"text/html"(),""" -

- How many points of function "f " do you want to calculate?
-

""") - -# ╔═╡ 21c246b8-44a1-498c-8192-20da834967f9 -@bind f_points html""" -""" - -# ╔═╡ 1298a540-3246-4778-abf7-60430b1990b2 -Show(MIME"text/html"(),""" -

- Here is a graph of the function we want to interpolate:
- -

""") - -# ╔═╡ e8ef6743-d3d6-4ee6-a4c5-9e8f9ea5379f -begin - # Generate random points - Random.seed!(125) - #absiccas=collect(range(a, stop=b, length=f_points)) # to generate partition of x-axis - absiccas = rand(Float64, f_points) .* (b - a) .+ a - X = sort(absiccas) - Y = f.(X) - #Plotting - plot(X,Y) - xlabel!("x") - ylabel!("y") - plot(X, Y, title="Function f graph", linewidth=1) -end - -# ╔═╡ cee176bf-32c2-4b89-9a84-da760f516619 -Show(MIME"text/html"(),""" -

- How many interpolation points do you want to calculate? -

""") - -# ╔═╡ ac754649-72ec-4d1e-a959-9f5b26cb19c1 -@bind lagrange_points html""" -""" - -# ╔═╡ 215df692-5c1a-4b10-98a7-b9dd860e2344 -begin - IntP = vcat(X',Y') ; - Lagrange = zeros(2,lagrange_points) ; - Lagrange[1,:] = collect(range(a, stop=b, length=lagrange_points)) - for i=1:lagrange_points - Lagrange[2,i] = LagrangeInterpolation(IntP,Lagrange[1,i]) ; - end - plot(Lagrange[1,:],Lagrange[2,:], title="Lagrange Interpolation graph", linewidth=1) -end - -# ╔═╡ a84b9394-85eb-449f-9a57-0969c861e8b1 -md""" -### Precision of Lagrange interpolation -""" - -# ╔═╡ b60a5e4c-c7a6-4787-853e-369e3a9dd871 -begin - plot(X, Y, label="Function f", linewidth=1) - plot!(Lagrange[1,:], Lagrange[2,:], label="Lagrange", linewidth=1) - plot!(title="F vs Lagrange Interpolation graph") -end - -# ╔═╡ f55954ea-aefa-4b55-8c13-e72c3d58b7b2 -md""" -## Newton interpolation polynomial - -Using the basis - -$$ -\begin{aligned} -& 1,\\ -& x-x_0,\\ -& (x-x_0)(x-x_1),\\ -& (x-x_0)(x-x_1)(x-x_2),\\ -& \qquad \vdots\\ -& (x-x_0)(x-x_1)\cdots (x-x_{n-1}) -\end{aligned}$$ - -the polynomial can be written as - -$$ -p_n(x)=\alpha_0 + \alpha_1(x-x_0)+\alpha_2(x-x_0)(x-x_1)+\cdots +\alpha_n(x-x_0)(x-x_1)\cdots (x-x_{n-1}).$$ - - -""" - -# ╔═╡ 79c220ad-6cb3-4c92-ab9e-94882d32030b -md""" -We intend to create a function that given -- the interpolation points of a function, -- an absicca, $z_0$, -it should return the value $p_n(z_0)$. -""" - -# ╔═╡ 6377bad9-8bf5-47e0-8606-fe183fe5d8a1 -md"""We will be using divided differences of $f$ to determine the coefficients $\alpha_0, \alpha_1, \ldots, \alpha_n$ and thus the value of the interpolation polynomial for a given $z_0$. -""" - -# ╔═╡ 886b0a46-5388-4cfd-9e1b-7fa190f76574 -function NewtonInterpolation(IntP,z0) - n = size(IntP)[2] - DivDiff = zeros(n,n) - DivDiff[:,1]= IntP[2,:] - for j=2:n - for i=j:n - DivDiff[i,j] = (DivDiff[i,j-1]-DivDiff[i-1,j-1]) / (IntP[1,i]-IntP[1,i-j+1]); - end - end - # Now the coefficients of p are in the main diagonal of DivDiff -#Initialization for main process - a_coefficients = diag(DivDiff) - X = z0 .- IntP[1,:]#[z0-x_1, z0-x_2,z0-x_3,...,z0-x_n] - newton_ordinate = a_coefficients[1]; -#Main process - for j = 2:n - newton_ordinate = newton_ordinate + a_coefficients[j]*prod(X[1:j-1]); - end - return(newton_ordinate) -end - -# ╔═╡ a9ff8b57-3c43-4475-8428-933392e4b3ab -Show(MIME"text/html"(),""" -

- Do you want to match number of absiccas used for Newton and Lagrange interpolation? (If not, select how many absiccas do you want to use calculate Newton polynomial) -

""") - -# ╔═╡ 59b63956-b9a9-4350-8fe3-bf19f5da3671 -@bind match_styles CheckBox(default=true) - -# ╔═╡ d5f45a1e-5d7a-4cc5-a5c2-084275a9ac72 -begin - if match_styles - newton_points = lagrange_points - Show(MIME"text/html"(),""" -

- We will be using $newton_points points to calculate the Newton polynomial -

""") - else - @bind newton_points html""" - """ - end -end - - -# ╔═╡ 08ed7b9d-b8f6-47f7-b3e0-690cc535a743 -Show(MIME"text/html"(),""" -

- We will be using $lagrange_points points to calculate the Lagrange polynomial and we will be using $newton_points points to calculate the Newton polynomial! -

""") - -# ╔═╡ 73879254-de99-4d51-8eca-ae7b4e2aea9b -begin - Newton = zeros(2,newton_points) ; - Newton[1,:] = collect(range(a, stop=b, length=newton_points)) - for i=1:newton_points - Newton[2,i] = NewtonInterpolation(IntP,Newton[1,i]) - end - plot(Newton[1,:], Newton[2,:], label="Newton", linewidth=1) - plot!(Lagrange[1,:], Lagrange[2,:], label="Lagrange", linewidth=1) - plot!(X, Y, label="Function f", linewidth=1) - plot!(title="Newton vs Lagrange Interpolation 🤺") -end - - -# ╔═╡ 00000000-0000-0000-0000-000000000001 -PLUTO_PROJECT_TOML_CONTENTS = """ -[deps] -LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" -PlotlyBase = "a03496cd-edff-5a9b-9e67-9cda94a718b5" -Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" -PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" -Polynomials = "f27b6e38-b328-58d1-80ce-0feddd5e7a45" -Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" - -[compat] -PlotlyBase = "~0.8.19" -Plots = "~1.38.16" -PlutoUI = "~0.7.51" -Polynomials = "~3.2.13" -""" - -# ╔═╡ 00000000-0000-0000-0000-000000000002 -PLUTO_MANIFEST_TOML_CONTENTS = """ -# This file is machine-generated - 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