Improve cubic interpolation (#204)

lib/scholar/interpolation/cubic_spline.ex

@@ -52,8 +52,8 @@ defmodule Scholar.Interpolation.CubicSpline do
       %Scholar.Interpolation.CubicSpline{
         coefficients: Nx.tensor(
           [
-            [0.0, 1.500000238418579, -3.500000238418579, 2.0],
-            [0.0, 1.5, -0.4999999403953552, 0.0]
+            [0.0, 1.5, -3.5, 2.0],
+            [0.0, 1.5, -0.5, 0.0]
           ]
         ),
         x: Nx.tensor(
@@ -110,7 +110,7 @@ defmodule Scholar.Interpolation.CubicSpline do
 
           b = Nx.stack([2 * slope[0], 3 * (dx[0] * slope[1] + dx[1] * slope[0]), 2 * slope[1]])
 
-          Nx.LinAlg.solve(a, b)
+          tridiagonal_solve(a, b)
 
         {_, :not_a_knot} ->
           up_diag =
@@ -151,7 +151,7 @@ defmodule Scholar.Interpolation.CubicSpline do
               Nx.new_axis(b_n, 0)
             ])
 
-          Nx.LinAlg.solve(a, b)
+          tridiagonal_solve(a, b)
 
         _ ->
           up_diag =
@@ -190,7 +190,7 @@ defmodule Scholar.Interpolation.CubicSpline do
               Nx.new_axis(b_n, 0)
             ])
 
-          Nx.LinAlg.solve(a, b)
+          tridiagonal_solve(a, b)
       end
 
     t = (s[0..-2//1] + s[1..-1//1] - 2 * slope) / dx
@@ -291,4 +291,46 @@ defmodule Scholar.Interpolation.CubicSpline do
 
     Nx.reshape(result, original_shape)
   end
+
+  defnp tridiagonal_solve(a, b) do
+    n = Nx.size(b)
+    w = Nx.broadcast(0, {n - 1})
+    p = g = Nx.broadcast(0, {n})
+    i = Nx.take_diagonal(a, offset: -1)
+    j = Nx.take_diagonal(a)
+    k = Nx.take_diagonal(a, offset: 1)
+
+    w_0 = k[0] / j[0]
+    g_0 = b[0] / j[0]
+    w = Nx.indexed_put(w, Nx.new_axis(0, 0), w_0)
+    g = Nx.indexed_put(g, Nx.new_axis(0, 0), g_0)
+
+    {{w, g}, _} =
+      while {{w, g}, {index = 1, i, j, k, b}}, index < n do
+        w =
+          if index < n - 1 do
+            w_i = k[index] / (j[index] - i[index - 1] * w[index - 1])
+            Nx.indexed_put(w, Nx.new_axis(index, 0), w_i)
+          else
+            w
+          end
+
+        g_i = (b[index] - i[index - 1] * g[index - 1]) / (j[index] - i[index - 1] * w[index - 1])
+        g = Nx.indexed_put(g, Nx.new_axis(index, 0), g_i)
+
+        {{w, g}, {index + 1, i, j, k, b}}
+      end
+
+    p = Nx.indexed_put(p, Nx.new_axis(n - 1, 0), g[n - 1])
+
+    {p, _} =
+      while {p, {index = n - 1, g, w}}, index > 0 do
+        p_i = g[index - 1] - w[index - 1] * p[index]
+        p = Nx.indexed_put(p, Nx.new_axis(index - 1, 0), p_i)
+
+        {p, {index - 1, g, w}}
+      end
+
+    p
+  end
 end