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lines 7-182 of file: example/atomic_two/eigen_mat_inv.cpp
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{xrst_begin atomic_two_eigen_mat_inv.cpp app}

Atomic Eigen Matrix Inverse: Example and Test
#############################################

Description
***********
The :ref:`ADFun-name` function object *f* for this example is

.. math::

   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)

{xrst_toc_hidden
   include/cppad/example/atomic_two/eigen_mat_inv.hpp
}

Class Definition
****************
This example uses the file :ref:`atomic_two_eigen_mat_inv.hpp-name`
which defines matrix multiply as a :ref:`atomic_two-name` operation.

Use Atomic Function
*******************
{xrst_spell_off}
{xrst_code cpp} */
# 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;
   //
/* {xrst_code}
{xrst_spell_on}
Constructor
===========
{xrst_spell_off}
{xrst_code cpp} */
   // -------------------------------------------------------------------
   // 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;
}
/* {xrst_code}
{xrst_spell_on}

{xrst_end atomic_two_eigen_mat_inv.cpp}