\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

atomic_two_eigen_mat_inv.cpp

View page source

Atomic Eigen Matrix Inverse: Example and Test

Description

The ADFun function object f for this example is

\[\begin{split}f(x) = \left( \begin{array}{cc} x_0 & 0 \\ 0 & x_1 \end{array} \right)^{-1} \left( \begin{array}{c} 0 \\ x_2 \end{array} \right) = \left( \begin{array}{c} 0 \\ x_2 / x_1 ) \end{array} \right)\end{split}\]

Class Definition

This example uses the file atomic_two_eigen_mat_inv.hpp which defines matrix multiply as a atomic_two operation.

Use Atomic Function

# include <cppad/cppad.hpp>
# include <cppad/example/atomic_two/eigen_mat_inv.hpp>
# include <cppad/example/atomic_two/eigen_mat_mul.hpp>

bool eigen_mat_inv(void)
{
   typedef double                                   scalar;
   typedef CppAD::AD<scalar>                        ad_scalar;
   typedef atomic_eigen_mat_inv<scalar>::ad_matrix  ad_matrix;
   //
   bool ok    = true;
   scalar eps = 10. * std::numeric_limits<scalar>::epsilon();
   using CppAD::NearEqual;
   //

Constructor

   // -------------------------------------------------------------------
   // object that multiplies matrices
   atomic_eigen_mat_mul<scalar> mat_mul;
   // -------------------------------------------------------------------
   // object that computes inverse of a square matrix
   atomic_eigen_mat_inv<scalar> mat_inv;
   // -------------------------------------------------------------------
   // declare independent variable vector x
   size_t n = 3;
   CPPAD_TESTVECTOR(ad_scalar) ad_x(n);
   for(size_t j = 0; j < n; j++)
      ad_x[j] = ad_scalar(j + 1);
   CppAD::Independent(ad_x);
   // -------------------------------------------------------------------
   // left = [ x[0]  0    ]
   //        [ 0     x[1] ]
   size_t nr_left  = 2;
   ad_matrix ad_left(nr_left, nr_left);
   ad_left(0, 0) = ad_x[0];
   ad_left(0, 1) = ad_scalar(0.0);
   ad_left(1, 0) = ad_scalar(0.0);
   ad_left(1, 1) = ad_x[1];
   // -------------------------------------------------------------------
   // right = [ 0 , x[2] ]^T
   size_t nc_right = 1;
   ad_matrix ad_right(nr_left, nc_right);
   ad_right(0, 0) = ad_scalar(0.0);
   ad_right(1, 0) = ad_x[2];
   // -------------------------------------------------------------------
   // use atomic operation to compute left^{-1}
   ad_matrix ad_left_inv = mat_inv.op(ad_left);
   // use atomic operation to multiply left^{-1} * right
   ad_matrix ad_result   = mat_mul.op(ad_left_inv, ad_right);
   // -------------------------------------------------------------------
   // declare the dependent variable vector y
   size_t m = 2;
   CPPAD_TESTVECTOR(ad_scalar) ad_y(2);
   for(size_t i = 0; i < m; i++)
      ad_y[i] = ad_result( long(i), 0);
   CppAD::ADFun<scalar> f(ad_x, ad_y);
   // -------------------------------------------------------------------
   // check zero order forward mode
   CPPAD_TESTVECTOR(scalar) x(n), y(m);
   for(size_t i = 0; i < n; i++)
      x[i] = scalar(i + 2);
   y   = f.Forward(0, x);
   ok &= NearEqual(y[0], 0.0,          eps, eps);
   ok &= NearEqual(y[1], x[2] / x[1],  eps, eps);
   // -------------------------------------------------------------------
   // check first order forward mode
   CPPAD_TESTVECTOR(scalar) x1(n), y1(m);
   x1[0] = 1.0;
   x1[1] = 0.0;
   x1[2] = 0.0;
   y1    = f.Forward(1, x1);
   ok   &= NearEqual(y1[0], 0.0,        eps, eps);
   ok   &= NearEqual(y1[1], 0.0,        eps, eps);
   x1[0] = 0.0;
   x1[1] = 0.0;
   x1[2] = 1.0;
   y1    = f.Forward(1, x1);
   ok   &= NearEqual(y1[0], 0.0,        eps, eps);
   ok   &= NearEqual(y1[1], 1.0 / x[1], eps, eps);
   x1[0] = 0.0;
   x1[1] = 1.0;
   x1[2] = 0.0;
   y1    = f.Forward(1, x1);
   ok   &= NearEqual(y1[0], 0.0,                  eps, eps);
   ok   &= NearEqual(y1[1], - x[2] / (x[1]*x[1]), eps, eps);
   // -------------------------------------------------------------------
   // check second order forward mode
   CPPAD_TESTVECTOR(scalar) x2(n), y2(m);
   x2[0] = 0.0;
   x2[1] = 0.0;
   x2[2] = 0.0;
   scalar  f1_x1_x1 = 2.0 * x[2] / (x[1] * x[1] * x[1] );
   y2    = f.Forward(2, x2);
   ok   &= NearEqual(y2[0], 0.0,            eps, eps);
   ok   &= NearEqual(y2[1], f1_x1_x1 / 2.0, eps, eps);
   // -------------------------------------------------------------------
   // check first order reverse
   CPPAD_TESTVECTOR(scalar) w(m), d1w(n);
   w[0] = 1.0;
   w[1] = 0.0;
   d1w  = f.Reverse(1, w);
   ok  &= NearEqual(d1w[0], 0.0, eps, eps);
   ok  &= NearEqual(d1w[1], 0.0, eps, eps);
   ok  &= NearEqual(d1w[2], 0.0, eps, eps);
   w[0] = 0.0;
   w[1] = 1.0;
   d1w  = f.Reverse(1, w);
   ok  &= NearEqual(d1w[0], 0.0,                  eps, eps);
   ok  &= NearEqual(d1w[1], - x[2] / (x[1]*x[1]), eps, eps);
   ok  &= NearEqual(d1w[2], 1.0 / x[1],           eps, eps);
   // -------------------------------------------------------------------
   // check second order reverse
   CPPAD_TESTVECTOR(scalar) d2w(2 * n);
   d2w  = f.Reverse(2, w);
   // partial f_1 w.r.t x_0
   ok  &= NearEqual(d2w[0 * 2 + 0], 0.0,                  eps, eps);
   // partial f_1 w.r.t x_1
   ok  &= NearEqual(d2w[1 * 2 + 0], - x[2] / (x[1]*x[1]), eps, eps);
   // partial f_1 w.r.t x_2
   ok  &= NearEqual(d2w[2 * 2 + 0], 1.0 / x[1],           eps, eps);
   // partial f_1 w.r.t x_1, x_0
   ok  &= NearEqual(d2w[0 * 2 + 1], 0.0,                  eps, eps);
   // partial f_1 w.r.t x_1, x_1
   ok  &= NearEqual(d2w[1 * 2 + 1], f1_x1_x1,             eps, eps);
   // partial f_1 w.r.t x_1, x_2
   ok  &= NearEqual(d2w[2 * 2 + 1], - 1.0 / (x[1]*x[1]),  eps, eps);
   // -------------------------------------------------------------------
   return ok;
}