Using Two Levels of AD: Example and Test

two_levels.py

@(@\newcommand{\B}[1]{{\bf #1}} \newcommand{\R}[1]{{\rm #1}}@)@ Using Two Levels of AD: Example and Test
Purpose
This example is intended to demonstrate how a_float and a2float can be used together to compute derivatives of functions that are defined in terms of derivatives of other functions.

F(u)
For this example, the function @(@ F : \B{R}^2 \rightarrow \B{R} @)@ is defined by @[@ F(u) = u_0^2 + u_1^2 @]@ It follows that @[@ \begin{array}{rcl} \partial_{u(0)} F(u) = 2 * u_0 \\ \partial_{u(1)} F(u) = 2 * u_1 \end{array} @]@

G(x)
For this example, the function @(@ G : \B{R}^2 \rightarrow \B{R} @)@ is defined by @[@ G(x) = x_1 * \partial_{u(0)} F(x_0 , 1) + x_0 * \partial_{u(1)} F(x_0, 1) @]@ where @(@ \partial{u(j)} F(a, b) @)@ denotes the partial of @(@ F @)@ with respect to @(@ u_j @)@ and evaluated at @(@ u = (a, b) @)@. It follows that @[@ \begin{array}{rcl} G (x) & = & 2 * x_1 * x_0 + 2 * x_0 \\ \partial_{x(0)} G (x) & = & 2 * x_1 + 2 \\ \partial_{x(1)} G (x) & = & 2 * x_0 \end{array} @]@


 

from pycppad import *
def pycppad_test_two_levels():
  # start recording a_float operations
  x   = numpy.array( [ 2. , 3. ] )
  a_x = independent(x)

  # start recording a2float operations
  a_u = numpy.array( [a_x[0]  , ad(1) ] )
  a2u = independent(a_u)

  # stop a2float recording and store operations if f
  a2v = numpy.array( [ a2u[0] * a2u[0] + a2u[1] * a2u[1] ] )
  a_f = adfun(a2u, a2v)              # F(u0, u1) = u0 * u0 + u1 * u1

  # evaluate the gradient of F
  a_J = a_f.jacobian(a_u)

  # stop a_float recording and store operations in g
  a_y = numpy.array( [ a_x[1] * a_J[0,0] + a_x[0] * a_J[0,1] ] )
  g   = adfun(a_x, a_y)  # G(x0, x1) = x1 * F_u0(x0, 1) + x0 * F_u1(x0, 1)
  
  # evaluate the gradient of G
  J   = g.jacobian(x)

  assert J[0,0] == 2. * x[1] + 2
  assert J[0,1] == 2. * x[0]

Input File: example/two_levels.py