fix: Lagrange interpolations derivative

src/derivatives.jl

@@ -37,74 +37,70 @@
 
 function derivative(A::LagrangeInterpolation{<:AbstractVector}, t::Number)
     ((t < A.t[1] || t > A.t[end]) && !A.extrapolate) && throw(ExtrapolationError())
-    idxs = findRequiredIdxs(A, t)
-    if A.t[idxs[1]] == t
-        return zero(A.u[idxs[1]])
-    end
-    G = zero(A.u[1])
-    F = zero(A.t[1])
-    DG = zero(A.u[1])
-    DF = zero(A.t[1])
-    tmp = G
-    for i in 1:length(idxs)
-        if isnan(A.bcache[idxs[i]])
+    der = zero(A.u[1])
+    for j in eachindex(A.t)
+        tmp = zero(A.t[1])
+        if isnan(A.bcache[j])
             mult = one(A.t[1])
-            for j in 1:(i - 1)
-                mult *= (A.t[idxs[i]] - A.t[idxs[j]])
+            for i in 1:(j - 1)
+                mult *= (A.t[j] - A.t[i])
             end
-            for j in (i + 1):length(idxs)
-                mult *= (A.t[idxs[i]] - A.t[idxs[j]])
+            for i in (j + 1):length(A.t)
+                mult *= (A.t[j] - A.t[i])
             end
-            A.bcache[idxs[i]] = mult
+            A.bcache[j] = mult
         else
-            mult = A.bcache[idxs[i]]
+            mult = A.bcache[j]
+        end
+        for l in eachindex(A.t)
+            if l != j
+                k = one(A.t[1])
+                for m in eachindex(A.t)
+                    if m != j && m != l
+                        k *= (t - A.t[m])
+                    end
+                end
+                k *= inv(mult)
+                tmp += k
+            end
         end
-        wi = inv(mult)
-        tti = t - A.t[idxs[i]]
-        tmp = wi / (t - A.t[idxs[i]])
-        g = tmp * A.u[idxs[i]]
-        G += g
-        DG -= g / (t - A.t[idxs[i]])
-        F += tmp
-        DF -= tmp / (t - A.t[idxs[i]])
+        der += A.u[j]*tmp
     end
-    (DG * F - G * DF) / (F^2)
+    der
 end
 
 function derivative(A::LagrangeInterpolation{<:AbstractMatrix}, t::Number)
     ((t < A.t[1] || t > A.t[end]) && !A.extrapolate) && throw(ExtrapolationError())
-    idxs = findRequiredIdxs(A, t)
-    if A.t[idxs[1]] == t
-        return zero(A.u[:, idxs[1]])
-    end
-    G = zero(A.u[:, 1])
-    F = zero(A.t[1])
-    DG = zero(A.u[:, 1])
-    DF = zero(A.t[1])
-    tmp = G
-    for i in 1:length(idxs)
-        if isnan(A.bcache[idxs[i]])
+    der = zero(A.u[:, 1])
+    for j in eachindex(A.t)
+        tmp = zero(A.t[1])
+        if isnan(A.bcache[j])
             mult = one(A.t[1])
-            for j in 1:(i - 1)
-                mult *= (A.t[idxs[i]] - A.t[idxs[j]])
+            for i in 1:(j - 1)
+                mult *= (A.t[j] - A.t[i])
             end
-            for j in (i + 1):length(idxs)
-                mult *= (A.t[idxs[i]] - A.t[idxs[j]])
+            for i in (j + 1):length(A.t)
+                mult *= (A.t[j] - A.t[i])
             end
-            A.bcache[idxs[i]] = mult
+            A.bcache[j] = mult
         else
-            mult = A.bcache[idxs[i]]
+            mult = A.bcache[j]
+        end
+        for l in eachindex(A.t)
+            if l != j
+                k = one(A.t[1])
+                for m in eachindex(A.t)
+                    if m != j && m != l
+                        k *= (t - A.t[m])
+                    end
+                end
+                k *= inv(mult)
+                tmp += k
+            end
         end
-        wi = inv(mult)
-        tti = t - A.t[idxs[i]]
-        tmp = wi / (t - A.t[idxs[i]])
-        g = tmp * A.u[:, idxs[i]]
-        @. G += g
-        @. DG -= g / (t - A.t[idxs[i]])
-        F += tmp
-        DF -= tmp / (t - A.t[idxs[i]])
+        @. der += A.u[:, j]*tmp
     end
-    @. (DG * F - G * DF) / (F^2)
+    der
 end
 
 derivative(A::LagrangeInterpolation{<:AbstractVector}, t::Number, i) = derivative(A, t), i

test/derivative_tests.jl

@@ -1,25 +1,53 @@
 using DataInterpolations, Test
 using FiniteDifferences
 using DataInterpolations: derivative
-# using Symbolics
+using Symbolics
 
 function test_derivatives(method, u, t, args...; name::String)
     func = method(u, t, args...; extrapolate = true)
-    trange = range(minimum(t) - 5.0, maximum(t) + 5.0, length = 32)
+    trange = collect(range(minimum(t) - 5.0, maximum(t) + 5.0, step = 0.1))
+    trange_exclude = filter(x -> !in(x, t), trange)
     @testset "$name" begin
-        for t in trange
-            cdiff = central_fdm(5, 1; geom = true)(_t -> func(_t), t)
-            adiff = derivative(func, t)
+        # Rest of the points
+        for _t in trange_exclude
+            cdiff = central_fdm(5, 1; geom = true)(func, _t)
+            adiff = derivative(func, _t)
             @test isapprox(cdiff, adiff, atol = 1e-8)
         end
+
+        # Interpolation time points
+        for _t in t[2:end-1]
+            fdiff = if func isa BSplineInterpolation || func isa BSplineApprox
+                forward_fdm(5, 1; geom = true)(func, _t)
+            else
+                backward_fdm(5, 1; geom = true)(func, _t)
+            end
+            adiff = derivative(func, _t)
+            @test isapprox(fdiff, adiff, atol = 1e-8)
+        end
+
+        # t = t0
+        fdiff = forward_fdm(5, 1; geom = true)(func, t[1])
+        adiff = derivative(func, t[1])
+        if func isa BSplineInterpolation || func isa BSplineApprox
+            # Bug in BSplines
+            @test_broken isapprox(fdiff, adiff, atol = 1e-8)
+        else
+            @test isapprox(fdiff, adiff, atol = 1e-8)
+        end
+
+        # t = tend
+        fdiff = backward_fdm(5, 1; geom = true)(func, t[end])
+        adiff = derivative(func, t[end])
+        @test isapprox(fdiff, adiff, atol = 1e-8)
     end
     func = method(u, t, args...)
     @test_throws DataInterpolations.ExtrapolationError derivative(func, t[1] - 1.0)
     @test_throws DataInterpolations.ExtrapolationError derivative(func, t[end] + 1.0)
 end
 
 @testset "Linear Interpolation" begin
-    u = 2.0collect(1:10)
+    u = vcat(collect(1:5), 2*collect(6:10))
     t = 1.0collect(1:10)
     test_derivatives(LinearInterpolation, u, t; name = "Linear Interpolation (Vector)")
     u = vcat(2.0collect(1:10)', 3.0collect(1:10)')