refactor: use BaryCentric Formula for Lagrange Interpolation

src/derivatives.jl

@@ -32,70 +32,62 @@ end
 
 function _derivative(A::LagrangeInterpolation{<:AbstractVector}, t::Number)
     ((t < A.t[1] || t > A.t[end]) && !A.extrapolate) && throw(ExtrapolationError())
-    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 i in 1:(j - 1)
-                mult *= (A.t[j] - A.t[i])
-            end
-            for i in (j + 1):length(A.t)
-                mult *= (A.t[j] - A.t[i])
-            end
-            A.bcache[j] = mult
-        else
-            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
+    if all(isnan.(A.derp.w))
+        for i in eachindex(A.u)
+            deru1 = zero(A.du[1])
+            deru2 = zero(A.t.parent[1])
+            for j in eachindex(A.t)
+                i == j && continue
+                val = (A.p.w[j] / A.p.w[i]) / (A.t[i] - A.t[j])
+                deru1 += val * A.u[j]
+                deru2 += val
             end
+            A.du[i] = deru1 - deru2 * A.u[i]
         end
-        der += A.u[j] * tmp
+        der_temp_p = lagrange_parameter_cache(A.du, A.t.parent)
+        A.derp.w .= der_temp_p.w
+        A.derp.wu .= der_temp_p.wu
+    end
+    idx = _searchsortedfirst(A.t, t)
+    !isnothing(idx) && return A.du[idx]
+    N = zero(A.du[1])
+    D = zero(A.t[1])
+    for i in 1:(A.n + 1)
+        ti = t - A.t[i]
+        N += (A.derp.wu[i]) / ti
+        D += (A.derp.w[i]) / ti
     end
-    der
+    N / D
 end
 
 function _derivative(A::LagrangeInterpolation{<:AbstractMatrix}, t::Number)
     ((t < A.t[1] || t > A.t[end]) && !A.extrapolate) && throw(ExtrapolationError())
-    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 i in 1:(j - 1)
-                mult *= (A.t[j] - A.t[i])
-            end
-            for i in (j + 1):length(A.t)
-                mult *= (A.t[j] - A.t[i])
-            end
-            A.bcache[j] = mult
-        else
-            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
+    if all(isnan.(A.derp.w))
+        for i in 1:size(A.u, 2)
+            deru1 = zero(A.du[:, 1])
+            deru2 = zero(A.t.parent[1])
+            for j in eachindex(A.t)
+                i == j && continue
+                val = (A.p.w[j] / A.p.w[i]) / (A.t[i] - A.t[j])
+                @. deru1 += val * A.u[:, j]
+                deru2 += val
             end
+            @. A.du[:, i] = deru1 - deru2 * A.u[:, i]
         end
-        der += A.u[:, j] * tmp
+        der_temp_p = lagrange_parameter_cache(A.du, A.t.parent)
+        A.derp.w .= der_temp_p.w
+        A.derp.wu .= der_temp_p.wu
+    end
+    idx = _searchsortedfirst(A.t, t)
+    !isnothing(idx) && return A.du[:, idx]
+    N = zeros(promote_type(eltype(A.u), eltype(t)), length(A.u[:, 1]))
+    D = zero(A.t[1])
+    for i in 1:(A.n + 1)
+        ti = t - A.t[i]
+        @. N += (A.derp.wu[:, i]) / ti
+        D += (A.derp.w[i]) / ti
     end
-    der
+    N / D
 end
 
 function _derivative(A::LagrangeInterpolation{<:AbstractVector}, t::Number, idx)

src/interpolation_caches.jl

@@ -94,25 +94,24 @@ It is the method of interpolation using Lagrange polynomials of (k-1)th order pa
   - `extrapolate`: boolean value to allow extrapolation. Defaults to `false`.
   - `safetycopy`: boolean value to make a copy of `u` and `t`. Defaults to `true`.
 """
-struct LagrangeInterpolation{uType, tType, T, bcacheType} <:
+struct LagrangeInterpolation{uType, tType, duType, T} <:
        AbstractInterpolation{T}
     u::uType
     t::tType
     n::Int
-    bcache::bcacheType
-    idxs::Vector{Int}
+    p::LagrangeParameterCache
+    du::duType
+    derp::LagrangeParameterCache
     extrapolate::Bool
     idx_prev::Base.RefValue{Int}
     safetycopy::Bool
-    function LagrangeInterpolation(u, t, n, extrapolate, safetycopy)
-        bcache = zeros(eltype(u[1]), n + 1)
-        idxs = zeros(Int, n + 1)
-        fill!(bcache, NaN)
-        new{typeof(u), typeof(t), eltype(u), typeof(bcache)}(u,
+    function LagrangeInterpolation(u, t, n, p, du, derp, extrapolate, safetycopy)
+        new{typeof(u), typeof(t), typeof(du), eltype(u)}(u,
             t,
             n,
-            bcache,
-            idxs,
+            p,
+            du,
+            derp,
             extrapolate,
             Ref(1),
             safetycopy
@@ -126,7 +125,12 @@ function LagrangeInterpolation(
     if n != length(t) - 1
         error("Currently only n=length(t) - 1 is supported")
     end
-    LagrangeInterpolation(u, t, n, extrapolate, safetycopy)
+    p = lagrange_parameter_cache(u.parent, t.parent)
+    derpw = similar(t.parent)
+    derpw .= NaN
+    derp = LagrangeParameterCache(derpw, similar(u.parent))
+    du = !(u.parent[1] isa AbstractVector || u.parent[1] isa AbstractMatrix) ? similar(u.parent) : similar.(u.parent)
+    LagrangeInterpolation(u, t, n, p, du, derp, extrapolate, safetycopy)
 end
 
 """

src/interpolation_methods.jl

@@ -58,61 +58,29 @@ end
 
 # Lagrange Interpolation
 function _interpolate(A::LagrangeInterpolation{<:AbstractVector}, t::Number, iguess)
-    idx = get_idx(A.t, t, iguess)
-    findRequiredIdxs!(A, t, idx)
-    if A.t[A.idxs[1]] == t
-        return A.u[A.idxs[1]], idx
-    end
+    idx = _searchsortedfirst(A.t, t)
+    !isnothing(idx) && return A.u[idx], idx
     N = zero(A.u[1])
     D = zero(A.t[1])
-    tmp = N
-    for i in 1:length(A.idxs)
-        if isnan(A.bcache[A.idxs[i]])
-            mult = one(A.t[1])
-            for j in 1:(i - 1)
-                mult *= (A.t[A.idxs[i]] - A.t[A.idxs[j]])
-            end
-            for j in (i + 1):length(A.idxs)
-                mult *= (A.t[A.idxs[i]] - A.t[A.idxs[j]])
-            end
-            A.bcache[A.idxs[i]] = mult
-        else
-            mult = A.bcache[A.idxs[i]]
-        end
-        tmp = inv((t - A.t[A.idxs[i]]) * mult)
-        D += tmp
-        N += (tmp * A.u[A.idxs[i]])
+    for i in 1:(A.n + 1)
+        ti = t - A.t[i]
+        N += (A.p.wu[i]) / ti
+        D += (A.p.w[i]) / ti
     end
-    N / D, idx
+    N / D, iguess
 end
 
 function _interpolate(A::LagrangeInterpolation{<:AbstractMatrix}, t::Number, iguess)
-    idx = get_idx(A.t, t, iguess)
-    findRequiredIdxs!(A, t, idx)
-    if A.t[A.idxs[1]] == t
-        return A.u[:, A.idxs[1]], idx
-    end
+    idx = _searchsortedfirst(A.t, t)
+    !isnothing(idx) && return A.u[:, idx], idx
     N = zero(A.u[:, 1])
     D = zero(A.t[1])
-    tmp = D
-    for i in 1:length(A.idxs)
-        if isnan(A.bcache[A.idxs[i]])
-            mult = one(A.t[1])
-            for j in 1:(i - 1)
-                mult *= (A.t[A.idxs[i]] - A.t[A.idxs[j]])
-            end
-            for j in (i + 1):length(A.idxs)
-                mult *= (A.t[A.idxs[i]] - A.t[A.idxs[j]])
-            end
-            A.bcache[A.idxs[i]] = mult
-        else
-            mult = A.bcache[A.idxs[i]]
-        end
-        tmp = inv((t - A.t[A.idxs[i]]) * mult)
-        D += tmp
-        @. N += (tmp * A.u[:, A.idxs[i]])
+    for i in 1:(A.n + 1)
+        ti = t - A.t[i]
+        @. N += (A.p.wu[:, i]) / ti
+        D += (A.p.w[i]) / ti
     end
-    N / D, idx
+    N / D, iguess
 end
 
 function _interpolate(A::AkimaInterpolation{<:AbstractVector}, t::Number, iguess)

src/interpolation_utils.jl

@@ -1,32 +1,10 @@
-function findRequiredIdxs!(A::LagrangeInterpolation, t, idx)
-    n = length(A.t) - 1
-    i_min, idx_min, idx_max = if t == A.t[idx]
-        A.idxs[1] = idx
-        2, idx, idx
+function _searchsortedfirst(ts, t)
+    idx = searchsortedfirst(ts, t)
+    if idx > lastindex(ts) || ts[idx] != t
+        return nothing
     else
-        1, idx + 1, idx
+        return idx
     end
-    for i in i_min:(n + 1)
-        if idx_min == 1
-            A.idxs[i:end] .= range(idx_max + 1, idx_max + (n + 2 - i))
-            break
-        elseif idx_max == length(A.t)
-            A.idxs[i:end] .= (idx_min - 1):-1:(idx_min - (n + 2 - i))
-            break
-        else
-            left_diff = abs(t - A.t[idx_min - 1])
-            right_diff = abs(t - A.t[idx_max + 1])
-            left_expand = left_diff <= right_diff
-        end
-        if left_expand
-            idx_min -= 1
-            A.idxs[i] = idx_min
-        else
-            idx_max += 1
-            A.idxs[i] = idx_max
-        end
-    end
-    return idx
 end
 
 function spline_coefficients!(N, d, k, u::Number)

src/parameter_caches.jl

@@ -50,6 +50,30 @@ function quadratic_interpolation_parameters(u, t, idx)
     return l₀, l₁, l₂
 end
 
+struct LagrangeParameterCache{wType, wuType}
+    w::wType
+    wu::wuType
+end
+
+function lagrange_parameter_cache(u, t)
+    w = similar(t)
+    wu = similar(u)
+    for i in eachindex(w)
+        mul = one(eltype(t))
+        for j in eachindex(t)
+            i != j && (mul *= (t[i] - t[j]))
+        end
+        w[i] = inv(mul)
+        val = u isa Matrix ? w[i] .* u[:, i] : w[i] * u[i]
+        if u isa Matrix
+            wu[:, i] .= val
+        else
+            wu[i] = val
+        end
+    end
+    return LagrangeParameterCache(w, wu)
+end
+
 struct QuadraticSplineParameterCache{pType}
     σ::pType
 end

test/interpolation_tests.jl

@@ -323,7 +323,7 @@ end
     t = [1.0, 2.0, 3.0]
     A = LagrangeInterpolation(u, t; extrapolate = true)
     @test A(0.0) == 0.0
-    @test A(4.0) == 16.0
+    @test A(4.0) ≈ 16.0
     A = LagrangeInterpolation(u, t)
     @test_throws DataInterpolations.ExtrapolationError A(-1.0)
     @test_throws DataInterpolations.ExtrapolationError A(4.0)