+ return sum( divided_difference(xs[:k+1], ys[:k+1])*lagrange_psi(x, xs[:k])
+ for k in range(len(xs)) )
+
+
+def hermite_coefficient(k, x, xs):
+ """
+ Return the Hermite coefficient h_{k}(x). See Atkinson, p. 160.
+
+ INPUT:
+
+ - ``k`` -- The index of the coefficient.
+
+ - ``x`` -- The symbolic variable to use as the argument of h_{k}.
+
+ - ``xs`` -- The list of points at which the function values are
+ known.
+
+ OUTPUT:
+
+ A symbolic expression.
+
+ """
+ lk = lagrange_coefficient(k, x, xs)
+ return (1 - 2*lk.diff(x)(x=xs[k])*(x - xs[k]))*(lk**2)
+
+
+def hermite_deriv_coefficient(k, x, xs):
+ """
+ Return the Hermite derivative coefficient, \tilde{h}_{k}(x). See
+ Atkinson, p. 160.
+
+ INPUT:
+
+ - ``k`` -- The index of the coefficient.
+
+ - ``x`` -- The symbolic variable to use as the argument of h_{k}.
+
+ - ``xs`` -- The list of points at which the function values are
+ known.
+
+ OUTPUT:
+
+ A symbolic expression.
+
+ """
+ lk = lagrange_coefficient(k, x, xs)
+ return (x - xs[k])*(lk**2)
+
+
+def hermite_interpolant(x, xs, ys, y_primes):
+ """
+ Return the Hermite interpolant `H(x)` such that H(xs[k]) = ys[k]
+ and H'(xs[k]) = y_primes[k] for each k.
+
+ Reference: Atkinson, p. 160.
+
+ INPUT:
+
+ - ``x`` -- The symbolic variable to use as the argument of H(x).
+
+ - ``xs`` -- The list of points at which the function values are
+ known.
+
+ - ``ys`` -- The function values at the `xs`.
+
+ - ``y_primes`` -- The derivatives at the `xs`.
+
+ OUTPUT:
+
+ A symbolic expression.
+
+ SETUP::
+
+ sage: from mjo.interpolation import hermite_interpolant
+
+ TESTS::
+
+ sage: xs = [ 0, pi/6, pi/2 ]
+ sage: ys = list(map(sin, xs))
+ sage: y_primes = list(map(cos, xs))
+ sage: H = hermite_interpolant(x, xs, ys, y_primes)
+ sage: expected = -27/4*(pi - 6*x)*(pi - 2*x)^2*sqrt(3)*x^2/pi^4
+ sage: expected += (5*(pi - 2*x)/pi + 1)*(pi - 6*x)^2*x^2/pi^4
+ sage: expected += 81/2*((pi - 6*x)/pi + 1)*(pi - 2*x)^2*x^2/pi^4
+ sage: expected += (pi - 6*x)^2*(pi - 2*x)^2*x/pi^4
+ sage: bool(H == expected)
+ True
+
+ """
+ s1 = sum( ys[k] * hermite_coefficient(k, x, xs)
+ for k in range(len(xs)) )
+
+ s2 = sum( y_primes[k] * hermite_deriv_coefficient(k, x, xs)
+ for k in range(len(xs)) )
+
+ return (s1 + s2)
+
+
+def lagrange_psi(x, xs):
+ """
+ The function,
+
+ Psi(x) = (x - xs[0])*(x - xs[1])* ... *(x - xs[-1])