Horner's method for Radau as well as derivative of interpolation poly…

…nomial.

scipy_dae/integrate/_dae/radau.py

@@ -494,7 +494,6 @@ def _step_impl(self):
                 Z0 = np.zeros((s, y.shape[0]))
             else:
                 Z0 = self.sol(t + h * C)[0].T - y
-            Z0 = np.zeros((s, y.shape[0]))
             scale = atol + np.abs(y) * rtol
 
             converged = False
@@ -701,38 +700,33 @@ def _call_impl(self, t):
         x = (t - self.t_old) / self.h
         x = np.atleast_1d(x)
 
-        # factors for nterpolation polynomial and its derivative
-        # p = np.tile(x, (self.order + 1, 1))
-        # p = np.cumprod(p, axis=0)
+        # factors for interpolation polynomial and its derivative
         c = np.arange(1, self.order + 2)[:, None]
         p = x**c
         dp = (c / self.h) * (x**(c - 1))
 
-        # Here we don't multiply by h, not a mistake.
+        # 1. compute collocation polynomial for y and yp
         y = np.dot(self.Q, p)
         yp = np.dot(self.Qp, p)
         y += self.y_old[:, None]
         yp += self.yp_old[:, None]
-        if t.ndim == 0:
-            y = np.squeeze(y)
-            yp = np.squeeze(yp)
 
-        # compute derivative of interpolation polynomial
-        yp = np.dot(self.Q, dp)
+        # # 2. compute derivative of interpolation polynomial for y
+        # yp = np.dot(self.Q, dp)
 
-        # # compute both values by Horner's method:
-        # # https://cut-the-knot.org/Curriculum/Calculus/HornerMethod.shtml
-        # # https://math.stackexchange.com/questions/2139142/how-does-horner-method-evaluate-the-derivative-of-a-function
+        # # 3. compute both values by Horner's rule
         # y = np.zeros_like(y)
-        # y += self.Q[:, -1][:, None]
         # yp = np.zeros_like(y)
-        # for i in range(self.order + 1, 1, -1):
+        # for i in range(self.order, -1, -1):
+        #     y = self.Q[:, i][:, None] + y * x[None, :]
         #     yp = y + yp * x[None, :]
-        #     y = self.Q[:, i - 1][:, None] + y * x[None, :]
-
-        # # y += self.y_old[:, None]
+        # y = self.y_old[:, None] + y * x[None, :]
         # yp /= self.h
 
+        if t.ndim == 0:
+            y = np.squeeze(y)
+            yp = np.squeeze(yp)
+
         return y, yp
 
 

scipy_dae/integrate/_dae/tests/test_dae.py

@@ -186,7 +186,7 @@ def test_integration_rational(vectorized, method, t_span, jac):
     e = compute_error(yc, yc_true, rtol, atol)
     assert_(np.all(e < 5))
 
-    assert_allclose(res.sol(res.t)[0], res.y, rtol=1e-15, atol=1e-15)
+    assert_allclose(res.sol(res.t)[0], res.y, rtol=1e-14, atol=1e-14)
 
 
 parameters_stiff = ["BDF", "Radau"]