Add docs page

docs/make.jl

@@ -12,7 +12,8 @@ makedocs(modules = [DataInterpolations],
     linkcheck = true,
     format = Documenter.HTML(assets = ["assets/favicon.ico"],
         canonical = "https://docs.sciml.ai/DataInterpolations/stable/"),
-    pages = ["index.md", "Methods" => "methods.md",
+    pages = ["index.md", "Interpolation methods" => "methods.md",
+        "Extrapolation methods" => "extrapolation_methods.md",
         "Interface" => "interface.md", "Using with Symbolics/ModelingToolkit" => "symbolics.md",
         "Manual" => "manual.md", "Inverting Integrals" => "inverting_integrals.md"])
 

docs/src/extrapolation_methods.md

@@ -0,0 +1,56 @@
+# Extrapolation methods
+
+We will use the following interpolation to demonstrate the various extrapolation methods.
+
+```@example tutorial
+using DataInterpolations, Plots
+
+u = [0.86, 0.65, 0.44, 0.76, 0.73]
+t = [0.0022, 0.68, 1.41, 2.22, 2.46]
+t_eval_down = range(-1, first(t), length = 25)
+t_eval_up = range(last(t), 3.5, length = 25)
+A = QuadraticSpline(u, t)
+plot(A)
+```
+
+Extrapolation behavior can be set for `t` beyond the data in the negative and positive direction separately with the `extrapolation_down` and `extrapolation_up` keywords of the interpolation constructors respectively.
+
+## `ExtrapolationType.none`
+
+This extrapolation type will throw an error when the input `t` is beyond the data in the specified direction.
+
+## `ExtrapolationType.constant`
+
+This extrapolation type extends the interpolation with the boundary values of the data `u`.
+
+```@example tutorial
+A = QuadraticSpline(u, t; extrapolation_down = ExtrapolationType.constant,
+    extrapolation_up = ExtrapolationType.constant)
+plot(A)
+plot!(t_eval_down, A.(t_eval_down); label = "extrapolation down")
+plot!(t_eval_up, A.(t_eval_up); label = "extrapolation up")
+```
+
+## `ExtrapolationType.linear`
+
+This extrapolation type extends the interpolation with a linear continuation of the interpolation, making it $C^1$ smooth at the data boundaries.
+
+```@example tutorial
+A = QuadraticSpline(u, t; extrapolation_down = ExtrapolationType.linear,
+    extrapolation_up = ExtrapolationType.linear)
+plot(A)
+plot!(t_eval_down, A.(t_eval_down); label = "extrapolation down")
+plot!(t_eval_up, A.(t_eval_up); label = "extrapolation up")
+```
+
+## `ExtrapolationType.extension`
+
+This extrapolation type extends the interpolation with a continuation of the expression for the interpolation at the boundary intervals for maximum smoothness.
+
+```@example tutorial
+A = QuadraticSpline(u, t; extrapolation_down = ExtrapolationType.extension,
+    extrapolation_up = ExtrapolationType.extension)
+plot(A)
+plot!(t_eval_down, A.(t_eval_down); label = "extrapolation down")
+plot!(t_eval_up, A.(t_eval_up); label = "extrapolation up")
+```

docs/src/interface.md

@@ -17,15 +17,15 @@ t = [0.0, 62.25, 109.66, 162.66, 205.8, 252.3]
 
 All interpolation methods return an object from which we can compute the value of the dependent variable at any time point.
 
-We will use the `CubicSpline` method for demonstration, but the API is the same for all the methods. We can also pass the `extrapolate = true` keyword if we want to allow the interpolation to go beyond the range of the timepoints. The default value is `extrapolate = false`.
+We will use the `CubicSpline` method for demonstration, but the API is the same for all the methods. We can also pass the `extrapolation_up = ExtrapolationType.extension` keyword if we want to allow the interpolation to go beyond the range of the timepoints in the positive `t` direction. The default value is `extrapolation_up = ExtrapolationType.none`. For more information on extrapolation see [Extrapolation methods](extrapolation_methods.md).
 
 ```@example interface
 A1 = CubicSpline(u, t)
 
 # For interpolation do, A(t)
 A1(100.0)
 
-A2 = CubicSpline(u, t; extrapolate = true)
+A2 = CubicSpline(u, t; extrapolation_up = ExtrapolationType.extension)
 
 # Extrapolation
 A2(300.0)

src/interpolation_methods.jl

@@ -13,7 +13,8 @@ function _extrapolate_down(A, t)
     if extrapolation_down == ExtrapolationType.none
         throw(DownExtrapolationError())
     elseif extrapolation_down == ExtrapolationType.constant
-        first(A.u) + zero(eltype(A.p.slope)) * t
+        slope = derivative(A, first(A.t))
+        first(A.u) + slope * zero(t)
     elseif extrapolation_down == ExtrapolationType.linear
         slope = derivative(A, first(A.t))
         first(A.u) + slope * (t - first(A.t))
@@ -28,7 +29,8 @@ function _extrapolate_up(A, t)
     if extrapolation_up == ExtrapolationType.none
         throw(UpExtrapolationError())
     elseif extrapolation_up == ExtrapolationType.constant
-        last(A.u) + zero(eltype(A.p.slope)) * t
+        slope = derivative(A, last(A.t))
+        last(A.u) + slope * zero(t)
     elseif extrapolation_up == ExtrapolationType.linear
         slope = derivative(A, last(A.t))
         last(A.u) + slope * (t - last(A.t))