From 213c4e851816a5db78c7597dcb1e28748793cc17 Mon Sep 17 00:00:00 2001 From: jakob Date: Sun, 2 Jul 2023 20:28:19 +0200 Subject: [PATCH 01/16] Initial commit --- LICENSE | 21 +++++++++++++++++++++ README.md | 1 + 2 files changed, 22 insertions(+) create mode 100644 LICENSE create mode 100644 README.md diff --git a/LICENSE b/LICENSE new file mode 100644 index 0000000..4bbcfd1 --- /dev/null +++ b/LICENSE @@ -0,0 +1,21 @@ +MIT License + +Copyright (c) 2023 jakob + +Permission is hereby granted, free of charge, to any person obtaining a copy +of this software and associated documentation files (the "Software"), to deal +in the Software without restriction, including without limitation the rights +to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +copies of the Software, and to permit persons to whom the Software is +furnished to do so, subject to the following conditions: + +The above copyright notice and this permission notice shall be included in all +copies or substantial portions of the Software. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE +SOFTWARE. diff --git a/README.md b/README.md new file mode 100644 index 0000000..85dc0d9 --- /dev/null +++ b/README.md @@ -0,0 +1 @@ +# Interpolation_with_the_barycentric_formula.jl \ No newline at end of file From 306151a81f075e0c537387aaba438fb8f891f524 Mon Sep 17 00:00:00 2001 From: lovc21 Date: Sun, 2 Jul 2023 20:57:08 +0200 Subject: [PATCH 02/16] Docs commit --- .github/workflows/Documenter.yml | 24 ++++++++++++++++++++++++ .github/workflows/Runtests.yml | 22 ++++++++++++++++++++++ 2 files changed, 46 insertions(+) create mode 100644 .github/workflows/Documenter.yml create mode 100644 .github/workflows/Runtests.yml diff --git a/.github/workflows/Documenter.yml b/.github/workflows/Documenter.yml new file mode 100644 index 0000000..d07b446 --- /dev/null +++ b/.github/workflows/Documenter.yml @@ -0,0 +1,24 @@ +name: Documenter +on: + push: + branches: + - master + tags: '*' + pull_request: +jobs: + build: + runs-on: ubuntu-latest # [ubuntu-latest, windows-latest, macOS-latest] + steps: + - uses: actions/checkout@v2 + - uses: julia-actions/setup-julia@latest + with: + version: '1.8.5' + - name: Check Git installation + run: git --version + - name: Install dependencies + run: julia --project=docs/ -e 'using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()' + - name: Build and deploy + env: + GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} # For authentication with GitHub Actions token + DOCUMENTER_KEY: ${{ secrets.DOCUMENTER_KEY }} # For authentication with SSH deploy key + run: julia --project=docs/ docs/make.jl \ No newline at end of file diff --git a/.github/workflows/Runtests.yml b/.github/workflows/Runtests.yml new file mode 100644 index 0000000..420897e --- /dev/null +++ b/.github/workflows/Runtests.yml @@ -0,0 +1,22 @@ +name: Runtests +on: + push: + pull_request: +jobs: + test: + runs-on: ${{ matrix.os }} + strategy: + matrix: + julia-version: ['1.8.5'] + julia-arch: [x64] + os: [windows-latest] # [ubuntu-latest, windows-latest, macOS-latest] + steps: + - uses: actions/checkout@v2 + - uses: julia-actions/setup-julia@latest + with: + version: ${{ matrix.julia-version }} + - uses: julia-actions/julia-buildpkg@latest + - name: Run tests + uses: julia-actions/julia-runtest@latest + with: + flags: '--color=yes --verbosity=verbose' \ No newline at end of file From 2d6b165341013057b539cc2f0b26db913b8b48ca Mon Sep 17 00:00:00 2001 From: lovc21 Date: Sun, 2 Jul 2023 20:58:15 +0200 Subject: [PATCH 03/16] Docs commit --- docs/make.jl | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/docs/make.jl b/docs/make.jl index 21e35f7..60dd105 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -10,6 +10,6 @@ makedocs( # Documenter can also automatically deploy documentation to gh-pages. # See "Hosting Documentation" and deploydocs() in the Documenter manual # for more information. -#=deploydocs( - repo = "" -)=# +deploydocs( + repo = "github.com/lovc21/Interpolation_with_the_barycentric_formula.jl.git" +) From 7ca7985f1d31ff71cfb5ccbaadea0a824172a509 Mon Sep 17 00:00:00 2001 From: lovc21 Date: Sun, 2 Jul 2023 21:15:49 +0200 Subject: [PATCH 04/16] make.jl fix commit --- docs/make.jl | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/docs/make.jl b/docs/make.jl index 60dd105..7c503a0 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -11,5 +11,7 @@ makedocs( # See "Hosting Documentation" and deploydocs() in the Documenter manual # for more information. deploydocs( - repo = "github.com/lovc21/Interpolation_with_the_barycentric_formula.jl.git" + repo = "github.com/lovc21/Interpolation_with_the_barycentric_formula.jl.git", + push_preview = true, + devbranch = "main", ) From a9f8967b1ab6273288c51944272dc6f05ae4be1f Mon Sep 17 00:00:00 2001 From: lovc21 Date: Sun, 2 Jul 2023 21:22:43 +0200 Subject: [PATCH 05/16] make.jl fix commit --- .github/workflows/Documenter.yml | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/.github/workflows/Documenter.yml b/.github/workflows/Documenter.yml index d07b446..7117054 100644 --- a/.github/workflows/Documenter.yml +++ b/.github/workflows/Documenter.yml @@ -1,24 +1,24 @@ name: Documenter + on: push: branches: - master tags: '*' pull_request: + jobs: build: - runs-on: ubuntu-latest # [ubuntu-latest, windows-latest, macOS-latest] + runs-on: ubuntu-latest steps: - uses: actions/checkout@v2 - - uses: julia-actions/setup-julia@latest + - uses: julia-actions/setup-julia@v1 with: - version: '1.8.5' - - name: Check Git installation - run: git --version + version: '1' # Use the latest Julia 1.x release - name: Install dependencies run: julia --project=docs/ -e 'using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()' - name: Build and deploy env: - GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} # For authentication with GitHub Actions token - DOCUMENTER_KEY: ${{ secrets.DOCUMENTER_KEY }} # For authentication with SSH deploy key + GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} + DOCUMENTER_KEY: ${{ secrets.DOCUMENTER_KEY }} run: julia --project=docs/ docs/make.jl \ No newline at end of file From b6fa846107ca1eb9c16bdee92b610e24972841a6 Mon Sep 17 00:00:00 2001 From: lovc21 Date: Sun, 2 Jul 2023 21:25:42 +0200 Subject: [PATCH 06/16] Documanter fix --- .github/workflows/Documenter.yml | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/.github/workflows/Documenter.yml b/.github/workflows/Documenter.yml index 7117054..38562be 100644 --- a/.github/workflows/Documenter.yml +++ b/.github/workflows/Documenter.yml @@ -1,24 +1,24 @@ name: Documenter - on: push: branches: - - master + - main tags: '*' pull_request: - jobs: build: - runs-on: ubuntu-latest + runs-on: ubuntu-latest # [ubuntu-latest, windows-latest, macOS-latest] steps: - uses: actions/checkout@v2 - - uses: julia-actions/setup-julia@v1 + - uses: julia-actions/setup-julia@latest with: - version: '1' # Use the latest Julia 1.x release + version: '1.8.5' + - name: Check Git installation + run: git --version - name: Install dependencies run: julia --project=docs/ -e 'using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()' - name: Build and deploy env: - GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} - DOCUMENTER_KEY: ${{ secrets.DOCUMENTER_KEY }} + GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} # For authentication with GitHub Actions token + DOCUMENTER_KEY: ${{ secrets.DOCUMENTER_KEY }} # For authentication with SSH deploy key run: julia --project=docs/ docs/make.jl \ No newline at end of file From 1740a92446c9589e2d253366c2aa73a1088f08cf Mon Sep 17 00:00:00 2001 From: lovc21 Date: Sun, 2 Jul 2023 21:54:35 +0200 Subject: [PATCH 07/16] add title to readme --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 85dc0d9..719d544 100644 --- a/README.md +++ b/README.md @@ -1 +1 @@ -# Interpolation_with_the_barycentric_formula.jl \ No newline at end of file +# Interpolation using Barycentric Formula \ No newline at end of file From 6ba88c90d8a5da8bfe0010d3bb27adefffe11864 Mon Sep 17 00:00:00 2001 From: lovc21 Date: Wed, 12 Jul 2023 18:40:58 +0200 Subject: [PATCH 08/16] add Interpolation --- Project.toml | 1 + README.md | 5 +- Scripts/script.jl | 25 +++++++ ...erpolation_with_the_barycentric_formula.jl | 75 ++++++++++++++++++- test/runtests.jl | 55 +++++++++++++- 5 files changed, 157 insertions(+), 4 deletions(-) create mode 100644 Scripts/script.jl diff --git a/Project.toml b/Project.toml index cddf415..97b06d0 100644 --- a/Project.toml +++ b/Project.toml @@ -4,4 +4,5 @@ authors = ["lovc21 "] version = "0.1.0" [deps] +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" diff --git a/README.md b/README.md index 719d544..b1e8987 100644 --- a/README.md +++ b/README.md @@ -1 +1,4 @@ -# Interpolation using Barycentric Formula \ No newline at end of file +# Interpolation using Barycentric Formula + +[![Documenter](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Documenter.yml/badge.svg)](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Documenter.yml) +[![Runtests](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Runtests.yml/badge.svg)](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Runtests.yml) \ No newline at end of file diff --git a/Scripts/script.jl b/Scripts/script.jl new file mode 100644 index 0000000..31a3d15 --- /dev/null +++ b/Scripts/script.jl @@ -0,0 +1,25 @@ +using Interpolation + +f1(x) = exp.(-x.^2) +f2(x) = sin.(x) +f3(x) = abs.(x.^2 - 2 .* x) + +functions = [f1, f2, f3] +intervals = [(-1, 1), (0, 10), (1, 3)] +names = ["exp(-x^2)", "sin(x)", "|x^2 - 2x|"] + +for (f, interval, name) in zip(functions, intervals, names) + println("Interpolating function $name on interval $interval...") + a, b = interval + degree = 0 + error = Inf + while error > 1e-6 && degree < 100 + degree += 1 + x, w = chebyshev_points_weights(degree) + y = f(transform(x, a, b)) + x_test = range(-1, stop=1, length=100) + y_interpolated = barycentric_interpolation(x, y, w, x_test) + error = interpolation_error(f, y_interpolated, x_test, a, b) + end + println(" Achieved error $error with polynomial degree $degree") +end \ No newline at end of file diff --git a/src/Interpolation_with_the_barycentric_formula.jl b/src/Interpolation_with_the_barycentric_formula.jl index d7a0232..a3373e4 100644 --- a/src/Interpolation_with_the_barycentric_formula.jl +++ b/src/Interpolation_with_the_barycentric_formula.jl @@ -1,5 +1,76 @@ -module Interpolation_with_the_barycentric_formula +module InterpolationModule -greet() = print("Hello World!") +using LinearAlgebra + +function chebyshev_points_weights(n) + i = 0:n + x = cos.(π * i / n) # Chebyshev points + w = ones(n+1) # weights + w[1] = 0.5 + w[end] = 0.5 + w = w .* (-1) .^ i + return x, w +end + +function barycentric_interpolation(x, y, w, x0) + y0 = zeros(length(x0)) + for (j, x0j) in enumerate(x0) + numerator = 0.0 + denominator = 0.0 + for (xi, yi, wi) in zip(x, y, w) + if abs(x0j - xi) < 1e-10 + y0[j] = yi + break + else + r = wi / (x0j - xi) + numerator += r * yi + denominator += r + end + end + if y0[j] == 0.0 + y0[j] = numerator / denominator + end + end + return y0 +end + +function transform(x, a, b) + return a .+ (x .+ 1) * (b - a) / 2 +end + +function inverse_transform(x, a, b) + return 2 * (x .- a) / (b - a) .- 1 +end + +f1(x) = exp.(-x.^2) +f2(x) = sin.(x) +f3(x) = abs.(x.^2 - 2 .* x) + +function interpolation_error(f, y_interpolated, x, a, b) + y_true = f(transform(x, a, b)) + return maximum(abs.(y_true .- y_interpolated)) +end + +functions = [f1, f2, f3] +intervals = [(-1, 1), (0, 10), (1, 3)] +names = ["exp(-x^2)", "sin(x)", "|x^2 - 2x|"] + +for (f, interval, name) in zip(functions, intervals, names) + println("Interpolating function $name on interval $interval...") + a, b = interval + degree = 0 + error = Inf + while error > 1e-6 && degree < 100 + degree += 1 + x, w = chebyshev_points_weights(degree) + y = f(transform(x, a, b)) + x_test = range(-1, stop=1, length=100) + y_interpolated = barycentric_interpolation(x, y, w, x_test) + error = interpolation_error(f, y_interpolated, x_test, a, b) + end + println(" Achieved error $error with polynomial degree $degree") +end + +export chebyshev_points_weights, barycentric_interpolation, transform, inverse_transform, f1, f2, f3, interpolation_error end # module Interpolation_with_the_barycentric_formula diff --git a/test/runtests.jl b/test/runtests.jl index 5b84ae8..9428802 100644 --- a/test/runtests.jl +++ b/test/runtests.jl @@ -1,5 +1,58 @@ using Test +using InterpolationModule -@test true +# Test 1: Chebyshev points and weights +@testset "Test 1: Chebyshev points and weights" begin + x, w = chebyshev_points_weights(5) + @test length(x) == 6 + @test length(w) == 6 + @test x[1] == 1 + @test w[1] == 0.5 +end + +# Test 2: Barycentric interpolation +@testset "Test 2: Barycentric interpolation" begin + x, w = chebyshev_points_weights(5) + y = f1(transform(x, -1, 1)) + x0 = range(-1, stop=1, length=10) + y0 = barycentric_interpolation(x, y, w, x0) + @test length(y0) == 10 + @test y0[1] == y[1] +end + +# Test 3: Transform and inverse transform +@testset "Test 3: Transform and inverse transform" begin + x = range(-1, stop=1, length=10) + x_transformed = transform(x, 0, 10) + x_inverse_transformed = inverse_transform(x_transformed, 0, 10) + @test x_transformed[1] == 0 + @test x_inverse_transformed[1] == -1 +end + +# Test 4: Interpolation error +@testset "Test 4: Interpolation error" begin + x, w = chebyshev_points_weights(5) + y = f1(transform(x, -1, 1)) + x0 = range(-1, stop=1, length=10) + y0 = barycentric_interpolation(x, y, w, x0) + error = interpolation_error(f1, y0, x0, -1, 1) + @test error < 1 +end + +# Test 5: Interpolation of functions +@testset "Test 5: Interpolation of functions" begin + functions = [f1, f2, f3] + intervals = [(-1, 1), (0, 10), (1, 3)] + for (f, interval) in zip(functions, intervals) + degree = 10 + x, w = chebyshev_points_weights(degree) + y = f(transform(x, interval[1], interval[2])) + x0 = range(-1, stop=1, length=10) + y0 = barycentric_interpolation(x, y, w, x0) + error = interpolation_error(f, y0, x0, interval[1], interval[2]) + @test error < 1 + end +end + From f4c524e6dca9b32a8fdf542513349595c7092b69 Mon Sep 17 00:00:00 2001 From: lovc21 Date: Wed, 12 Jul 2023 18:46:44 +0200 Subject: [PATCH 09/16] fix funcion name --- .vscode/settings.json | 2 ++ Scripts/script.jl | 2 +- ...erpolation_with_the_barycentric_formula.jl | 32 +++---------------- test/runtests.jl | 2 +- 4 files changed, 8 insertions(+), 30 deletions(-) create mode 100644 .vscode/settings.json diff --git a/.vscode/settings.json b/.vscode/settings.json new file mode 100644 index 0000000..7a73a41 --- /dev/null +++ b/.vscode/settings.json @@ -0,0 +1,2 @@ +{ +} \ No newline at end of file diff --git a/Scripts/script.jl b/Scripts/script.jl index 31a3d15..695de11 100644 --- a/Scripts/script.jl +++ b/Scripts/script.jl @@ -1,4 +1,4 @@ -using Interpolation +using Interpolation_with_the_barycentric_formula f1(x) = exp.(-x.^2) f2(x) = sin.(x) diff --git a/src/Interpolation_with_the_barycentric_formula.jl b/src/Interpolation_with_the_barycentric_formula.jl index a3373e4..d836a9e 100644 --- a/src/Interpolation_with_the_barycentric_formula.jl +++ b/src/Interpolation_with_the_barycentric_formula.jl @@ -1,7 +1,9 @@ -module InterpolationModule +module Interpolation_with_the_barycentric_formula using LinearAlgebra +export chebyshev_points_weights, barycentric_interpolation, transform, inverse_transform, interpolation_error + function chebyshev_points_weights(n) i = 0:n x = cos.(π * i / n) # Chebyshev points @@ -42,35 +44,9 @@ function inverse_transform(x, a, b) return 2 * (x .- a) / (b - a) .- 1 end -f1(x) = exp.(-x.^2) -f2(x) = sin.(x) -f3(x) = abs.(x.^2 - 2 .* x) - function interpolation_error(f, y_interpolated, x, a, b) y_true = f(transform(x, a, b)) return maximum(abs.(y_true .- y_interpolated)) end -functions = [f1, f2, f3] -intervals = [(-1, 1), (0, 10), (1, 3)] -names = ["exp(-x^2)", "sin(x)", "|x^2 - 2x|"] - -for (f, interval, name) in zip(functions, intervals, names) - println("Interpolating function $name on interval $interval...") - a, b = interval - degree = 0 - error = Inf - while error > 1e-6 && degree < 100 - degree += 1 - x, w = chebyshev_points_weights(degree) - y = f(transform(x, a, b)) - x_test = range(-1, stop=1, length=100) - y_interpolated = barycentric_interpolation(x, y, w, x_test) - error = interpolation_error(f, y_interpolated, x_test, a, b) - end - println(" Achieved error $error with polynomial degree $degree") -end - -export chebyshev_points_weights, barycentric_interpolation, transform, inverse_transform, f1, f2, f3, interpolation_error - -end # module Interpolation_with_the_barycentric_formula +end # module Interpolation_with_the_barycentric_formula \ No newline at end of file diff --git a/test/runtests.jl b/test/runtests.jl index 9428802..d5b768f 100644 --- a/test/runtests.jl +++ b/test/runtests.jl @@ -1,5 +1,5 @@ using Test -using InterpolationModule +using Interpolation_with_the_barycentric_formula # Test 1: Chebyshev points and weights From dbacaf4a18fe0c04bdcf1ee9b00da02a0491547b Mon Sep 17 00:00:00 2001 From: lovc21 Date: Wed, 2 Aug 2023 00:04:38 +0200 Subject: [PATCH 10/16] fix script --- README.md | 7 ++++++- Scripts/script.jl | 9 +++++---- 2 files changed, 11 insertions(+), 5 deletions(-) diff --git a/README.md b/README.md index b1e8987..cd5e818 100644 --- a/README.md +++ b/README.md @@ -1,4 +1,9 @@ # Interpolation using Barycentric Formula [![Documenter](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Documenter.yml/badge.svg)](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Documenter.yml) -[![Runtests](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Runtests.yml/badge.svg)](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Runtests.yml) \ No newline at end of file +[![Runtests](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Runtests.yml/badge.svg)](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Runtests.yml) + +In this repository, you can finde the code for Homework 3 of the Numerical Methods course.The code is written in Julia, and the main implementation can be found in the file `src/Interpolation_with_the_barycentric_formula.jl`. The code is tested using the file `test/runtests.jl`, and it is documented using the file docs/make.jl, you can test the code for the specific by running the `Scripts/script.jl` file. + +To run the code, it is necessary to have Julia installed on your computer. Once downloaded, you can run the code for the following equations: e^{-x^2} on \([-1,1]\) , sin(x)/x on \([0,10]\) and |x^2-2x| on \([1,3]\).To run the code, you can simply start the script `Scripts/script.jl`.The script will compute the interpolation for the three equations and will return the reuslts to the precision of 1e-6. + diff --git a/Scripts/script.jl b/Scripts/script.jl index 695de11..65d75dd 100644 --- a/Scripts/script.jl +++ b/Scripts/script.jl @@ -1,7 +1,9 @@ using Interpolation_with_the_barycentric_formula +precision = 1e-6 + f1(x) = exp.(-x.^2) -f2(x) = sin.(x) +f2(x) = sin.(x) f3(x) = abs.(x.^2 - 2 .* x) functions = [f1, f2, f3] @@ -9,11 +11,10 @@ intervals = [(-1, 1), (0, 10), (1, 3)] names = ["exp(-x^2)", "sin(x)", "|x^2 - 2x|"] for (f, interval, name) in zip(functions, intervals, names) - println("Interpolating function $name on interval $interval...") a, b = interval degree = 0 error = Inf - while error > 1e-6 && degree < 100 + while error > precision && degree < 100 degree += 1 x, w = chebyshev_points_weights(degree) y = f(transform(x, a, b)) @@ -21,5 +22,5 @@ for (f, interval, name) in zip(functions, intervals, names) y_interpolated = barycentric_interpolation(x, y, w, x_test) error = interpolation_error(f, y_interpolated, x_test, a, b) end - println(" Achieved error $error with polynomial degree $degree") + println(" Achieved error $error with polynomial degree $degree for function $name.") end \ No newline at end of file From 4b80783ba3593141a6cb8ae80034d7cecdf330ef Mon Sep 17 00:00:00 2001 From: lovc21 Date: Sat, 5 Aug 2023 01:52:26 +0200 Subject: [PATCH 11/16] fix tags onm readme --- README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index cd5e818..864480b 100644 --- a/README.md +++ b/README.md @@ -1,7 +1,7 @@ # Interpolation using Barycentric Formula -[![Documenter](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Documenter.yml/badge.svg)](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Documenter.yml) -[![Runtests](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Runtests.yml/badge.svg)](https://github.com/lovc21/Methode_of_conjugate_gradients.lj/actions/workflows/Runtests.yml) +[![Documenter](https://github.com/lovc21/Interpolation_with_the_barycentric_formula.jl/actions/workflows/Documenter.yml/badge.svg)](https://github.com/lovc21/Interpolation_with_the_barycentric_formula.jl/actions/workflows/Documenter.yml) +[![Runtests](https://github.com/lovc21/Interpolation_with_the_barycentric_formula.jl/actions/workflows/Runtests.yml/badge.svg)](https://github.com/lovc21/Interpolation_with_the_barycentric_formula.jlin/actions/workflows/Runtests.yml) In this repository, you can finde the code for Homework 3 of the Numerical Methods course.The code is written in Julia, and the main implementation can be found in the file `src/Interpolation_with_the_barycentric_formula.jl`. The code is tested using the file `test/runtests.jl`, and it is documented using the file docs/make.jl, you can test the code for the specific by running the `Scripts/script.jl` file. From b0bbd0f48e64b0e7ed766f2a3b391a5238eb63d1 Mon Sep 17 00:00:00 2001 From: lovc21 Date: Tue, 15 Aug 2023 02:14:44 +0200 Subject: [PATCH 12/16] full rewrite of the code (test,script and the Interpolation_with_the_barycentric_formula) --- .vscode/settings.json | 9 + Project.toml | 1 + README.md | 18 +- Scripts/Ploting.jl | 19 ++ Scripts/script.jl | 45 ++-- docs/src/index.md | 4 +- images/ezgif-4-53e4254853.gif | Bin 0 -> 349844 bytes images/ezgif-4-cfcfa1457c.gif | Bin 0 -> 272553 bytes images/ezgif-4-e9a28c1c2b.gif | Bin 0 -> 248052 bytes ...erpolation_with_the_barycentric_formula.jl | 226 +++++++++++++++--- test/runtests.jl | 80 +++---- 11 files changed, 303 insertions(+), 99 deletions(-) create mode 100644 Scripts/Ploting.jl create mode 100644 images/ezgif-4-53e4254853.gif create mode 100644 images/ezgif-4-cfcfa1457c.gif create mode 100644 images/ezgif-4-e9a28c1c2b.gif diff --git a/.vscode/settings.json b/.vscode/settings.json index 7a73a41..be0b405 100644 --- a/.vscode/settings.json +++ b/.vscode/settings.json @@ -1,2 +1,11 @@ { + "spellright.language": [ + "en", + "sl" + ], + "spellright.documentTypes": [ + "markdown", + "latex", + "plaintext" + ] } \ No newline at end of file diff --git a/Project.toml b/Project.toml index 97b06d0..9d8569d 100644 --- a/Project.toml +++ b/Project.toml @@ -5,4 +5,5 @@ version = "0.1.0" [deps] LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" diff --git a/README.md b/README.md index 864480b..4cc78bb 100644 --- a/README.md +++ b/README.md @@ -5,5 +5,21 @@ In this repository, you can finde the code for Homework 3 of the Numerical Methods course.The code is written in Julia, and the main implementation can be found in the file `src/Interpolation_with_the_barycentric_formula.jl`. The code is tested using the file `test/runtests.jl`, and it is documented using the file docs/make.jl, you can test the code for the specific by running the `Scripts/script.jl` file. -To run the code, it is necessary to have Julia installed on your computer. Once downloaded, you can run the code for the following equations: e^{-x^2} on \([-1,1]\) , sin(x)/x on \([0,10]\) and |x^2-2x| on \([1,3]\).To run the code, you can simply start the script `Scripts/script.jl`.The script will compute the interpolation for the three equations and will return the reuslts to the precision of 1e-6. +To run the code, it is necessary to have Julia installed on your computer. Once downloaded, you can run the code for the following equations: e^(-x^2) on \([-1,1]\) , sin(x)/x on \([0,10]\) and |x^2-2x| on \([1,3]\).To run the code, you can simply start the script `Scripts/script.jl`.The script will compute the interpolation for the three equations and will return the reuslts to the precision of 1e-6. +Here are the results of the interpolation for the three equations for the nodes 1 ,3 and 5 the number so that the error is less than 1e-6: + +1. e^(-x^2) +

+ +

+ +2. sin(x)/x +

+ +

+ +3. |x^2-2x| +

+ +

\ No newline at end of file diff --git a/Scripts/Ploting.jl b/Scripts/Ploting.jl new file mode 100644 index 0000000..d051fac --- /dev/null +++ b/Scripts/Ploting.jl @@ -0,0 +1,19 @@ + +using Plots + +x1 = -1:0.01:1 +x2 = 0.01:0.1:10 +x3 = 1:0.01:3 + +f1(x) = exp(-x^2) +f2(x) = sin(x) / x +#f2(x) = x == 0 ? 1 : sin(x)/x +f3(x) = abs(x^2 - 2x) + +# Plot the functions +plot(x1, f1.(x1), label="e^{-x^2}", legend=:topright) + +plot(x2, f2.(x2), label="sin(x)/x") + +plot(x3, f3.(x3), label="|x^2-2x|") + diff --git a/Scripts/script.jl b/Scripts/script.jl index 65d75dd..f77b636 100644 --- a/Scripts/script.jl +++ b/Scripts/script.jl @@ -1,26 +1,39 @@ using Interpolation_with_the_barycentric_formula precision = 1e-6 +n = 10 f1(x) = exp.(-x.^2) -f2(x) = sin.(x) +#f2(x) = sin.(x) ./ x to ne dela +f2(x) = x == 0 ? 1 : sin(x)/x # to dela f3(x) = abs.(x.^2 - 2 .* x) functions = [f1, f2, f3] intervals = [(-1, 1), (0, 10), (1, 3)] -names = ["exp(-x^2)", "sin(x)", "|x^2 - 2x|"] - -for (f, interval, name) in zip(functions, intervals, names) - a, b = interval - degree = 0 - error = Inf - while error > precision && degree < 100 - degree += 1 - x, w = chebyshev_points_weights(degree) - y = f(transform(x, a, b)) - x_test = range(-1, stop=1, length=100) - y_interpolated = barycentric_interpolation(x, y, w, x_test) - error = interpolation_error(f, y_interpolated, x_test, a, b) +names = ["exp(-x^2)", "sin(x)/x", "|x^2 - 2x|"] + + +# Calculate and print the optimal degree for each function +for (i, f) in enumerate(functions) + degree = get_optimal_degree(f, intervals[i][1], intervals[i][2], n, precision) + println("Optimal degree for $(names[i]) on [$(intervals[i][1]), $(intervals[i][2])]: $degree") +end + +# Compute the maximum error for each function at the optimal degrees and print the results +degrees = [10, 13, 1567] +for (i, f) in enumerate(functions) + error = compute_max_error(f, intervals[i][1], intervals[i][2], degrees[i]) + println("Maximum error for $(names[i]) on [$(intervals[i][1]), $(intervals[i][2])]: $error") +end + +# Display the plots for each function for increasing n values up to their max degrees +for (i, f) in enumerate(functions) + for n in [1, 3, degrees[i]] + p = plot_interpolation(f, intervals[i][1], intervals[i][2], n, names[i]) + display(p) + sleep(2) end - println(" Achieved error $error with polynomial degree $degree for function $name.") -end \ No newline at end of file +end + + + diff --git a/docs/src/index.md b/docs/src/index.md index 3498c49..bbfad83 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -1,3 +1,3 @@ -# Interpolation_with_the_barycentric_formula.jl +# Interpolation with the barycentric formula + -Documentation for Interpolation_with_the_barycentric_formula.jl diff --git a/images/ezgif-4-53e4254853.gif b/images/ezgif-4-53e4254853.gif new file mode 100644 index 0000000000000000000000000000000000000000..33393ed1fbd8fcca05e16a702e50faa3c59d710a GIT binary patch literal 349844 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zp<0w+q??e3jh4tL%ps0*fdtICEmE+6XBUd|(Z*bzE`Ze_873Vn_jzM_v@e~a+dH!i zHx>9SUuX*C_r+mNAtksF=Y{RpDgOlLE-j41a!V92U_+8@oP1t|LaBT~_NU*_kbvQy zc7}XmUP;uXH)FfNHzn)y0@-Lz$vg{5zNvWo3VCc#NV4pMw)eA(6(<&50HI&h5Gj%{ zQrTdB@rxR=D?<)w_T#F;eBm~-EZ5af>SP%3+&ev){>JLaHP8?rcGlnf9dZh?$gfY} zMJQ?01T){cGDb@akPGZX)0P{b(wIQaOjJo{|;kOAYJuEO~z1 zM?cVnl9>C}E91J5z!#n&fY|?`VAm$c)mU^WJZ27BU-g)C9D4iU+~~1S?E^-%fehFS z)hhLu`EECAtaZa%?a6s(x=&iGZ_(PzYXicftq}F3bh`J2_`UPi$Y-GGOJ;k^>fl%W zH}$nK$@Tq=*N{$0z`mCE0!ooo; exp(-x^2) + result1 = get_optimal_degree(f1, -1, 1, 1, 1e-6) + @test result1 == 10 + + f2 = x -> x == 0 ? 1 : sin(x)/x + result2 = get_optimal_degree(f2, 0, 10, 1, 1e-6) + @test result2 == 13 + + f3 = x -> abs(x^2 - 2 * x) + result3 = get_optimal_degree(f3, 1, 3, 1, 1e-6) + @test result3 == 1567 end -# Test 5: Interpolation of functions -@testset "Test 5: Interpolation of functions" begin - functions = [f1, f2, f3] - intervals = [(-1, 1), (0, 10), (1, 3)] - for (f, interval) in zip(functions, intervals) - degree = 10 - x, w = chebyshev_points_weights(degree) - y = f(transform(x, interval[1], interval[2])) - x0 = range(-1, stop=1, length=10) - y0 = barycentric_interpolation(x, y, w, x0) - error = interpolation_error(f, y0, x0, interval[1], interval[2]) - @test error < 1 - end +# Test 5: Compute Max Error +@testset "Test 5: Compute Max Error" begin + f1 = x -> exp(-x^2) + result1 = compute_max_error(f1, -1, 1, 10) + @test 0 <= result1 <= 1e-6 end - From 23f52b30e411bb8e57aa8dc8f1b952895904afd2 Mon Sep 17 00:00:00 2001 From: lovc21 Date: Tue, 15 Aug 2023 02:20:06 +0200 Subject: [PATCH 13/16] fix runtest.yml --- .github/workflows/Runtests.yml | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/.github/workflows/Runtests.yml b/.github/workflows/Runtests.yml index 420897e..fa844a5 100644 --- a/.github/workflows/Runtests.yml +++ b/.github/workflows/Runtests.yml @@ -12,11 +12,10 @@ jobs: os: [windows-latest] # [ubuntu-latest, windows-latest, macOS-latest] steps: - uses: actions/checkout@v2 - - uses: julia-actions/setup-julia@latest + - uses: julia-actions/setup-julia@v1 with: version: ${{ matrix.julia-version }} - uses: julia-actions/julia-buildpkg@latest - name: Run tests uses: julia-actions/julia-runtest@latest - with: - flags: '--color=yes --verbosity=verbose' \ No newline at end of file + \ No newline at end of file From d93ad96c1facfe6de2cb257b82a6edb10ee502e4 Mon Sep 17 00:00:00 2001 From: lovc21 Date: Tue, 15 Aug 2023 13:49:15 +0200 Subject: [PATCH 14/16] fix runtest.yml 2 --- .github/workflows/Runtests.yml | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/.github/workflows/Runtests.yml b/.github/workflows/Runtests.yml index fa844a5..769ca3e 100644 --- a/.github/workflows/Runtests.yml +++ b/.github/workflows/Runtests.yml @@ -9,7 +9,7 @@ jobs: matrix: julia-version: ['1.8.5'] julia-arch: [x64] - os: [windows-latest] # [ubuntu-latest, windows-latest, macOS-latest] + os: [ubuntu-latest] # [ubuntu-latest, windows-latest, macOS-latest] steps: - uses: actions/checkout@v2 - uses: julia-actions/setup-julia@v1 @@ -18,4 +18,4 @@ jobs: - uses: julia-actions/julia-buildpkg@latest - name: Run tests uses: julia-actions/julia-runtest@latest - \ No newline at end of file + \ No newline at end of file From eb16a81f29ff36a22e2daf266229578ba23ca8b3 Mon Sep 17 00:00:00 2001 From: lovc21 Date: Wed, 16 Aug 2023 00:35:09 +0200 Subject: [PATCH 15/16] add documntacion --- README.md | 2 +- docs/src/images/ezgif-4-53e4254853.gif | Bin 0 -> 349844 bytes docs/src/images/ezgif-4-cfcfa1457c.gif | Bin 0 -> 272553 bytes docs/src/images/ezgif-4-e9a28c1c2b.gif | Bin 0 -> 248052 bytes docs/src/index.md | 157 +++++++++++++++++++++++++ 5 files changed, 158 insertions(+), 1 deletion(-) create mode 100644 docs/src/images/ezgif-4-53e4254853.gif create mode 100644 docs/src/images/ezgif-4-cfcfa1457c.gif create mode 100644 docs/src/images/ezgif-4-e9a28c1c2b.gif diff --git a/README.md b/README.md index 4cc78bb..9dddbab 100644 --- a/README.md +++ b/README.md @@ -7,7 +7,7 @@ In this repository, you can finde the code for Homework 3 of the Numerical Metho To run the code, it is necessary to have Julia installed on your computer. Once downloaded, you can run the code for the following equations: e^(-x^2) on \([-1,1]\) , sin(x)/x on \([0,10]\) and |x^2-2x| on \([1,3]\).To run the code, you can simply start the script `Scripts/script.jl`.The script will compute the interpolation for the three equations and will return the reuslts to the precision of 1e-6. -Here are the results of the interpolation for the three equations for the nodes 1 ,3 and 5 the number so that the error is less than 1e-6: +Here are the results of the interpolation for the three equations for the nodes 1 ,3 the number so that the error is less than 1e-6: 1. e^(-x^2)

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zp<0w+q??e3jh4tL%ps0*fdtICEmE+6XBUd|(Z*bzE`Ze_873Vn_jzM_v@e~a+dH!i zHx>9SUuX*C_r+mNAtksF=Y{RpDgOlLE-j41a!V92U_+8@oP1t|LaBT~_NU*_kbvQy zc7}XmUP;uXH)FfNHzn)y0@-Lz$vg{5zNvWo3VCc#NV4pMw)eA(6(<&50HI&h5Gj%{ zQrTdB@rxR=D?<)w_T#F;eBm~-EZ5af>SP%3+&ev){>JLaHP8?rcGlnf9dZh?$gfY} zMJQ?01T){cGDb@akPGZX)0P{b(wIQaOjJo{|;kOAYJuEO~z1 zM?cVnl9>C}E91J5z!#n&fY|?`VAm$c)mU^WJZ27BU-g)C9D4iU+~~1S?E^-%fehFS z)hhLu`EECAtaZa%?a6s(x=&iGZ_(PzYXicftq}F3bh`J2_`UPi$Y-GGOJ;k^>fl%W zH}$nK$@Tq=*N{$0z`mCE0!ooo; + + + + + + + +And for the Chebyshev points on the interval [a, b]: + + + + + +
+ +
+ +### Barycentric formula +The barycentric formula is a method for numerical interpolation. It is based on the Lagrange interpolation formula. The Lagrange interpolation formula is the following: + + + + + +
+ +
+ +But this formula is numerically unstable, so we use the barycentric formula instead. The barycentric formula is the following: + + + + + +
+ +
+ +This formula offers a numerically stable way to interpolate a function and it is also faster than the Lagrange interpolation formula. + +### The results of the interpolation +The following pictures show the results of the interpolation of the given functions with the barycentric formula and Chebyshev points. + +1. e^(-x^2) +

+ +

+ +2. sin(x)/x +

+ +

+ +3. |x^2-2x| +

+ +

+ +## Code documentation +First we calculate the Chebyshev nodes using the second formula for the Chebyshev points. We use the following function to calculate the Chebyshev nodes: + +```julia +function chebyshev_nodes(n, a, b) + + # Check the nodes calculation + nodes = [cos(π * (i / n)) for i in 0:n] + + # Check the mapping to the interval + mapped_nodes = [(a + b) / 2 + (b - a) / 2 * x for x in nodes] + + return mapped_nodes +end +``` +Next we calculate the weights for the barycentric formula. We use the following function to calculate the weights: + +```julia +function chebyshev_weights(n) + weights = [Float64((-1)^i) for i in 0:n] + weights[1] *= 0.5 + weights[end] *= 0.5 + return weights +end +``` +After that we can caculate the barycentric interpolation using the barycentric formula. We use the following function to calculate the barycentric interpolation: +```julia +function barycentric_interpolation(nodes,values, weights, x) + if x in nodes + return values[nodes .== x][1] + else + # Calculate the numerator + numerator = sum(values[i] * weights[i] ./ (x .- nodes[i]) for i in 1:length(nodes)) + # Calculate the denominator + denominator = sum(weights[i] ./ (x .- nodes[i]) for i in 1:length(nodes)) + end + + return numerator / denominator +end +``` + +After we have the barycentric interpolation we can run the code so that we are in the exaptible level of precision. We use the following function to run the code: +```julia +function get_optimal_degree(func, a, b,n,precision) + while true + nodes = chebyshev_nodes(n, a, b) + values = [func(node) for node in nodes] + weights = chebyshev_weights(n) + + test_points = range(a, stop=b, length=1000) + max_error = 0.0 + + for x in test_points + interpolated_value = barycentric_interpolation(nodes, values, weights, x) + error = abs(func(x) - interpolated_value) + max_error = max(max_error, error) + end + + if max_error < precision + return n + end + + n += 1 + + end +end +``` + +Finally we can plot the results of the interpolation. We use the following function to plot the results: +```julia +function plot_interpolation(f, a, b, n, name) + nodes = chebyshev_nodes(n, a, b) + values = [f(node) for node in nodes] + weights = chebyshev_weights(n) + + x_vals = range(a, stop=b, length=1000) + y_true = [f(x) for x in x_vals] + y_interp = [barycentric_interpolation(nodes, values, weights, x) for x in x_vals] + + p = plot(x_vals, y_true, label="Original $name", lw=2) + plot!(p, x_vals, y_interp, label="Interpolated $name n=$n", linestyle=:dash, lw=2) + title!("Interpolation of $name with degree $n") + xlabel!("x") + ylabel!("f(x)") + return p +end +``` \ No newline at end of file From 4f04200e68358ecea17e4721a189b0fe1c88eeef Mon Sep 17 00:00:00 2001 From: lovc21 Date: Wed, 16 Aug 2023 00:37:53 +0200 Subject: [PATCH 16/16] update the readme --- README.md | 3 +++ 1 file changed, 3 insertions(+) diff --git a/README.md b/README.md index 9dddbab..0126a17 100644 --- a/README.md +++ b/README.md @@ -7,6 +7,9 @@ In this repository, you can finde the code for Homework 3 of the Numerical Metho To run the code, it is necessary to have Julia installed on your computer. Once downloaded, you can run the code for the following equations: e^(-x^2) on \([-1,1]\) , sin(x)/x on \([0,10]\) and |x^2-2x| on \([1,3]\).To run the code, you can simply start the script `Scripts/script.jl`.The script will compute the interpolation for the three equations and will return the reuslts to the precision of 1e-6. +The code and the Mathematical backround is documented using Documenter.jl. To see the documentation, you can run the docs/make.jl file. The documentation is also available online at [documentation](https://lovc21.github.io/Interpolation_with_the_barycentric_formula.jl/dev/). + +--- Here are the results of the interpolation for the three equations for the nodes 1 ,3 the number so that the error is less than 1e-6: 1. e^(-x^2)