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

exp_eps_rev2.cpp

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exp_eps: Verify Second Order Reverse Sweep

# include <cstddef>                     // define size_t
# include <cmath>                       // for fabs function
extern bool exp_eps_for0(double *v0);   // computes zero order forward sweep
extern bool exp_eps_for1(double *v1);   // computes first order forward sweep
bool exp_eps_rev2(void)
{  bool ok = true;

   // set the value of v0[j], v1[j] for j = 1 , ... , 7
   double v0[8], v1[8];
   ok &= exp_eps_for0(v0);
   ok &= exp_eps_for1(v1);

   // initial all partial derivatives as zero
   double f_v0[8], f_v1[8];
   size_t j;
   for(j = 0; j < 8; j++)
   {  f_v0[j] = 0.;
      f_v1[j] = 0.;
   }

   // set partial derivative for f_7
   f_v1[7] = 1.;
   ok &= std::fabs( f_v1[7] - 1.  ) <= 1e-10; // partial f_7 w.r.t. v_7^1

   // f_6 = f_7( v_1^0 , ... , v_6^1 , v_4^0 + v_6^0, v_4^1 , v_6^1 )
   f_v0[4] += f_v0[7];
   f_v0[6] += f_v0[7];
   f_v1[4] += f_v1[7];
   f_v1[6] += f_v1[7];
   ok &= std::fabs( f_v0[4] - 0.  ) <= 1e-10; // partial f_6 w.r.t. v_4^0
   ok &= std::fabs( f_v0[6] - 0.  ) <= 1e-10; // partial f_6 w.r.t. v_6^0
   ok &= std::fabs( f_v1[4] - 1.  ) <= 1e-10; // partial f_6 w.r.t. v_4^1
   ok &= std::fabs( f_v1[6] - 1.  ) <= 1e-10; // partial f_6 w.r.t. v_6^1

   // f_5 = f_6( v_1^0 , ... , v_5^1 , v_5^0 / 2 , v_5^1 / 2 )
   f_v0[5] += f_v0[6] / 2.;
   f_v1[5] += f_v1[6] / 2.;
   ok &= std::fabs( f_v0[5] - 0.  ) <= 1e-10; // partial f_5 w.r.t. v_5^0
   ok &= std::fabs( f_v1[5] - 0.5 ) <= 1e-10; // partial f_5 w.r.t. v_5^1

   // f_4 = f_5( v_1^0 , ... , v_4^1 , v_3^0 * v_1^0 ,
   //            v_3^1 * v_1^0 + v_3^0 * v_1^1 )
   f_v0[1] += f_v0[5] * v0[3] + f_v1[5] * v1[3];
   f_v0[3] += f_v0[5] * v0[1] + f_v1[5] * v1[1];
   f_v1[1] += f_v1[5] * v0[3];
   f_v1[3] += f_v1[5] * v0[1];
   ok &= std::fabs( f_v0[1] - 0.5  ) <= 1e-10; // partial f_4 w.r.t. v_1^0
   ok &= std::fabs( f_v0[3] - 0.5  ) <= 1e-10; // partial f_4 w.r.t. v_3^0
   ok &= std::fabs( f_v1[1] - 0.25 ) <= 1e-10; // partial f_4 w.r.t. v_1^1
   ok &= std::fabs( f_v1[3] - 0.25 ) <= 1e-10; // partial f_4 w.r.t. v_3^1

   // f_3 = f_4(  v_1^0 , ... , v_3^1 , 1 + v_3^0 , v_3^1 )
   f_v0[3] += f_v0[4];
   f_v1[3] += f_v1[4];
   ok &= std::fabs( f_v0[3] - 0.5 ) <= 1e-10;  // partial f_3 w.r.t. v_3^0
   ok &= std::fabs( f_v1[3] - 1.25) <= 1e-10;  // partial f_3 w.r.t. v_3^1

   // f_2 = f_3( v_1^0 , ... , v_2^1 , v_2^0 , v_2^1 )
   f_v0[2] += f_v0[3];
   f_v1[2] += f_v1[3];
   ok &= std::fabs( f_v0[2] - 0.5 ) <= 1e-10;  // partial f_2 w.r.t. v_2^0
   ok &= std::fabs( f_v1[2] - 1.25) <= 1e-10;  // partial f_2 w.r.t. v_2^1

   // f_1 = f_2 ( v_1^0 , v_2^0 , v_1^0 , v_2^0 )
   f_v0[1] += f_v0[2];
   f_v1[1] += f_v1[2];
   ok &= std::fabs( f_v0[1] - 1.  ) <= 1e-10;  // partial f_1 w.r.t. v_1^0
   ok &= std::fabs( f_v1[1] - 1.5 ) <= 1e-10;  // partial f_1 w.r.t. v_1^1

   return ok;
}