Implement second derivative for Hermite splines

docs/pyplots/plot_b1.py

@@ -6,7 +6,7 @@
 
 b1 = splinebox.basis_functions.B1()
 
-t = np.linspace(-2, 2, 100)
+t = np.linspace(-2, 2, 1000)
 
 b1_0th = b1.eval(t)
 b1_1st = b1.eval(t, derivative=1)

docs/pyplots/plot_b2.py

@@ -6,7 +6,7 @@
 
 b2 = splinebox.basis_functions.B2()
 
-t = np.linspace(-2, 2, 100)
+t = np.linspace(-2, 2, 1000)
 
 b2_0th = b2.eval(t)
 b2_1st = b2.eval(t, derivative=1)

docs/pyplots/plot_b3.py

@@ -6,7 +6,7 @@
 
 b3 = splinebox.basis_functions.B3()
 
-t = np.linspace(-3, 3, 100)
+t = np.linspace(-3, 3, 1000)
 
 b3_0th = b3.eval(t)
 b3_1st = b3.eval(t, derivative=1)

docs/pyplots/plot_catmullrom.py

@@ -6,7 +6,7 @@
 
 basis = splinebox.basis_functions.CatmullRom()
 
-t = np.linspace(-3, 3, 100)
+t = np.linspace(-3, 3, 1000)
 
 basis_0th = basis.eval(t)
 basis_1st = basis.eval(t, derivative=1)

docs/pyplots/plot_cubichermite.py

@@ -2,26 +2,31 @@
 import numpy as np
 import splinebox.basis_functions
 
-fig, axes = plt.subplots(2, 2, sharex=True)
+fig, axes = plt.subplots(3, 2, sharex=True)
 
 basis = splinebox.basis_functions.CubicHermite()
 
-t = np.linspace(-3, 3, 100)
+t = np.linspace(-3, 3, 1000)
 
 basis_0th = basis.eval(t)
 basis_1st = basis.eval(t, derivative=1)
+basis_2nd = basis.eval(t, derivative=2)
 
 fig.suptitle("Cubic Hermite basis function and its derivatives")
 axes[0][0].plot(t, basis_0th[0], label=r"$\Phi_1(t)$")
 axes[0][0].legend()
 axes[1][0].plot(t, basis_1st[0], label=r"$\frac{d\Phi_1}{dt}(t)$")
 axes[1][0].legend()
-axes[1][0].set_xlabel(r"$t$")
+axes[2][0].plot(t, basis_2nd[0], label=r"$\frac{d^2\Phi_1}{dt^2}(t)$")
+axes[2][0].legend()
+axes[2][0].set_xlabel(r"$t$")
 axes[0][1].plot(t, basis_0th[1], label=r"$\Phi_2(t)$")
 axes[0][1].legend()
 axes[1][1].plot(t, basis_1st[1], label=r"$\frac{d\Phi_2}{dt}(t)$")
 axes[1][1].legend()
-axes[1][1].set_xlabel(r"$t$")
+axes[2][1].plot(t, basis_2nd[1], label=r"$\frac{d^2\Phi_2}{dt^2}(t)$")
+axes[2][1].legend()
+axes[2][1].set_xlabel(r"$t$")
 plt.tight_layout()
 
 plt.show()

docs/pyplots/plot_exponential.py

@@ -7,7 +7,7 @@
 M = 5
 basis = splinebox.basis_functions.Exponential(M=M)
 
-t = np.linspace(-3, 3, 100)
+t = np.linspace(-3, 3, 1000)
 
 basis_0th = basis.eval(t)
 basis_1st = basis.eval(t, derivative=1)

docs/pyplots/plot_exponentialhermite.py

@@ -2,27 +2,32 @@
 import numpy as np
 import splinebox.basis_functions
 
-fig, axes = plt.subplots(2, 2, sharex=True)
+fig, axes = plt.subplots(3, 2, sharex=True)
 
 M = 5
 basis = splinebox.basis_functions.ExponentialHermite(M=M)
 
-t = np.linspace(-3, 3, 100)
+t = np.linspace(-3, 3, 1000)
 
 basis_0th = basis.eval(t)
 basis_1st = basis.eval(t, derivative=1)
+basis_2nd = basis.eval(t, derivative=2)
 
 fig.suptitle(f"Exponential Hermite basis function and its derivatives for $M={M}$")
 axes[0][0].plot(t, basis_0th[0], label=r"$\Phi_1(t)$")
 axes[0][0].legend()
 axes[1][0].plot(t, basis_1st[0], label=r"$\frac{d\Phi_1}{dt}(t)$")
 axes[1][0].legend()
-axes[1][0].set_xlabel(r"$t$")
+axes[2][0].plot(t, basis_2nd[0], label=r"$\frac{d^2\Phi_1}{dt^2}(t)$")
+axes[2][0].legend()
+axes[2][0].set_xlabel(r"$t$")
 axes[0][1].plot(t, basis_0th[1], label=r"$\Phi_2(t)$")
 axes[0][1].legend()
 axes[1][1].plot(t, basis_1st[1], label=r"$\frac{d\Phi_2}{dt}(t)$")
 axes[1][1].legend()
-axes[1][1].set_xlabel(r"$t$")
+axes[2][1].plot(t, basis_2nd[1], label=r"$\frac{d^2\Phi_2}{dt^2}(t)$")
+axes[2][1].legend()
+axes[2][1].set_xlabel(r"$t$")
 plt.tight_layout()
 
 plt.show()

docs/pyplots/plot_no_padding.py

@@ -11,7 +11,7 @@
 
 spline.control_points = control_points
 
-t = np.linspace(0, spline.M - 1, 100)
+t = np.linspace(0, spline.M - 1, 1000)
 vals = spline.eval(t)
 
 plt.figure(figsize=(6, 3))

docs/pyplots/plot_padding.py

@@ -10,7 +10,7 @@
 
 spline.control_points = control_points
 
-t = np.linspace(0, spline.M - 1, 100)
+t = np.linspace(0, spline.M - 1, 1000)
 vals = spline.eval(t)
 
 plt.figure(figsize=(6, 3))

src/splinebox/basis_functions.py

@@ -739,9 +739,28 @@ def h32prime(t):  # pragma: no cover
             val = 3 * t * t + 4 * t + 1
         return val
 
+    def _derivative_2(self, t):
+        return np.array([self.h31primeprime(t), self.h32primeprime(t)])
+
     @staticmethod
-    def _derivative_2(t):
-        raise RuntimeError("CubicHermite isn't twice differentiable.")
+    @numba.vectorize([numba.float64(numba.float64)], nopython=True, cache=True)
+    def h31primeprime(t):  # pragma: no cover
+        val = 0
+        if t >= 0 and t <= 1:
+            val = 12 * t - 6
+        elif t < 0 and t >= -1:
+            val = -12 * t - 6
+        return val
+
+    @staticmethod
+    @numba.vectorize([numba.float64(numba.float64)], nopython=True, cache=True)
+    def h32primeprime(t):  # pragma: no cover
+        val = 0
+        if t >= 0 and t <= 1:
+            val = 6 * t - 4
+        elif t < 0 and t >= -1:
+            val = 6 * t + 4
+        return val
 
     def h31_autocorrelation(self, i, j, M):  # pragma: no cover
         """
@@ -972,9 +991,40 @@ def _g2prime(t, M):
         val = _g2prime(t, M) if t >= 0 else _g2prime(-t, M)
         return val
 
+    def _derivative_2(self, t):
+        return np.array([self._he31primeprime(t, self.M), self._he32primeprime(t, self.M)])
+
+    @staticmethod
+    @numba.vectorize([numba.float64(numba.float64, numba.float64)], nopython=True, cache=True)
+    def _he31primeprime(t, M):  # pragma: no cover
+        def _g1primeprime(t, M):
+            val = 0
+            if t >= 0 and t <= 1:
+                alpha = np.pi / M
+                denom = (alpha * np.cos(alpha)) - np.sin(alpha)
+                num = 2 * alpha**2 * np.sin(alpha - (2 * alpha * t))
+                val = num / denom
+            return val
+
+        val = _g1primeprime(t, M) if t >= 0 else _g1primeprime(-t, M)
+        return val
+
     @staticmethod
-    def _derivative_2(x):
-        raise RuntimeError("ExponentialHermite isn't twice differentiable.")
+    @numba.vectorize([numba.float64(numba.float64, numba.float64)], nopython=True, cache=True)
+    def _he32primeprime(t, M):  # pragma: no cover
+        def _g2primeprime(t, M):
+            val = 0
+            if t >= 0 and t <= 1:
+                alpha = np.pi / M
+                denom = ((alpha * np.cos(alpha)) - np.sin(alpha)) * 8 * alpha * np.sin(alpha)
+                num = +(8 * alpha**2 * np.sin(alpha) * np.cos(2 * alpha * (t - 0.5))) - (
+                    8 * alpha**3 * np.cos(2 * alpha * (t - 1))
+                )
+                val = num / denom
+            return val
+
+        val = _g2primeprime(t, M) if t >= 0 else -1 * _g2primeprime(-t, M)
+        return val
 
     @staticmethod
     def filter_symmetric(s):

tests/conftest.py

@@ -201,11 +201,7 @@ def _not_differentiable_twice(obj):
         basis_function = obj.basis_function if is_spline(obj) else obj
         return isinstance(
             basis_function,
-            (
-                splinebox.basis_functions.B1,
-                splinebox.basis_functions.CubicHermite,
-                splinebox.basis_functions.ExponentialHermite,
-            ),
+            (splinebox.basis_functions.B1,),
         )
 
     return _not_differentiable_twice