Added Linear() extrapolation method (#93)

* linear extrapolation

* Update SplineExtrapolations.jl

* finite diffs -> AD forward diff

* Revert "finite diffs -> AD forward diff"

This reverts commit aa9b33e.

* interpolation example now working, sorted out Project.toml issue

* using ForwardDiff in extrapolations module

hopefully this fixes the CI error

* edits

removed unnecessary ForwardDiff inclusions, added Linear to extrapolation.md, removed CairoMakie from Project.toml

Project.toml

@@ -7,6 +7,7 @@ version = "0.17.3"
 ArrayLayouts = "4c555306-a7a7-4459-81d9-ec55ddd5c99a"
 BandedMatrices = "aae01518-5342-5314-be14-df237901396f"
 FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838"
+ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 PrecompileTools = "aea7be01-6a6a-4083-8856-8a6e6704d82a"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
@@ -20,6 +21,7 @@ StaticArraysCore = "1e83bf80-4336-4d27-bf5d-d5a4f845583c"
 ArrayLayouts = "1"
 BandedMatrices = "0.17, 1"
 FastGaussQuadrature = "0.5, 1"
+ForwardDiff = "0.10"
 LinearAlgebra = "1.8"
 PrecompileTools = "1.0"
 Random = "1.8"

docs/src/extrapolation.md

@@ -21,6 +21,7 @@ extrapolate
 
 ```@docs
 Flat
+Linear
 Smooth
 ```
 

examples/interpolation.jl

@@ -96,7 +96,7 @@ current_figure()  # hide
 #
 # First, we generate and interpolate new data:
 
-xs = 0:0.2:1
+xs = 0.2:0.2:1.2
 ys = 2 * cospi.(xs)
 S = interpolate(xs, ys, BSplineOrder(4))
 
@@ -110,6 +110,7 @@ S(-0.32)
 # Here we compare both [`Flat`](@ref) and [`Smooth`](@ref) methods:
 
 E_flat   = extrapolate(S, Flat())
+E_linear = extrapolate(S, Linear())
 E_smooth = extrapolate(S, Smooth())
 
 #
@@ -119,6 +120,7 @@ ax = Axis(fig[1, 1])
 scatter!(ax, xs, ys; label = "Data", color = :black)
 lines!(ax, -0.5..1.5, S; label = "No extrapolation", linewidth = 2)
 lines!(ax, -0.5..1.5, E_smooth; label = "Smooth", linestyle = :dash, linewidth = 2)
+lines!(ax, -0.5..1.5, E_linear; label = "Linear", linestyle = :dashdot, linewidth = 2)
 lines!(ax, -0.5..1.5, E_flat; label = "Flat", linestyle = :dot, linewidth = 2)
 axislegend(ax)
 fig

src/BSplineKit.jl

@@ -2,7 +2,7 @@ module BSplineKit
 
 using Reexport
 using PrecompileTools
-using LinearAlgebra: LinearAlgebra  # needed for docs
+using LinearAlgebra: LinearAlgebra # needed for docs
 
 include("BandedTensors/BandedTensors.jl")
 @reexport using .BandedTensors

src/SplineExtrapolations/SplineExtrapolations.jl

@@ -4,10 +4,12 @@ export
     SplineExtrapolation,
     extrapolate,
     Flat,
+    Linear,
     Smooth
 
 using ..BSplines
 using ..Splines
+using ForwardDiff
 
 """
     AbstractExtrapolationMethod
@@ -30,6 +32,26 @@ function extrapolate_at_point(::Flat, S::Spline, x)
     S(x′)
 end
 
+"""
+    Linear <: AbstractExtrapolationMethod
+
+Represents a linear extrapolation: splines values extend linearly beyond the left and right boundaries.
+"""
+struct Linear <: AbstractExtrapolationMethod end
+
+function extrapolate_at_point(::Linear, S::Spline, x)
+    a, b = boundaries(basis(S))
+    if x < a
+        slope = ForwardDiff.derivative(S, a)
+        S(a) + slope * (x - a)
+    elseif x > b
+        slope = ForwardDiff.derivative(S, b)
+        S(b) + slope * (x - b)
+    else
+        S(x)
+    end
+end
+
 """
     Smooth <: AbstractExtrapolationMethod