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$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$
exp_2: Verify Second Order Reverse Sweep
# include <cstddef>                 // define size_t
# include <cmath>                   // prototype for fabs
extern bool exp_2_for0(double *v0); // computes zero order forward sweep
extern bool exp_2_for1(double *v1); // computes first order forward sweep
bool exp_2_rev2(void)
{     bool ok = true;

// set the value of v0[j], v1[j] for j = 1 , ... , 5
double v0[6], v1[6];
ok &= exp_2_for0(v0);
ok &= exp_2_for1(v1);

// initial all partial derivatives as zero
double f_v0[6], f_v1[6];
size_t j;
for(j = 0; j < 6; j++)
{     f_v0[j] = 0.;
f_v1[j] = 0.;
}

// set partial derivative for f_5
f_v1[5] = 1.;
ok &= std::fabs( f_v1[5] - 1. ) <= 1e-10; // partial f_5 w.r.t v_5^1

// f_4 = f_5( v_1^0 , ... , v_4^1 , v_2^0 + v_4^0 , v_2^1 + v_4^1 )
f_v0[2] += f_v0[5] * 1.;
f_v0[4] += f_v0[5] * 1.;
f_v1[2] += f_v1[5] * 1.;
f_v1[4] += f_v1[5] * 1.;
ok &= std::fabs( f_v0[2] - 0. ) <= 1e-10; // partial f_4 w.r.t. v_2^0
ok &= std::fabs( f_v0[4] - 0. ) <= 1e-10; // partial f_4 w.r.t. v_4^0
ok &= std::fabs( f_v1[2] - 1. ) <= 1e-10; // partial f_4 w.r.t. v_2^1
ok &= std::fabs( f_v1[4] - 1. ) <= 1e-10; // partial f_4 w.r.t. v_4^1

// f_3 = f_4( v_1^0 , ... , v_3^1, v_3^0 / 2 , v_3^1 / 2 )
f_v0[3] += f_v0[4] / 2.;
f_v1[3] += f_v1[4] / 2.;
ok &= std::fabs( f_v0[3] - 0.  ) <= 1e-10; // partial f_3 w.r.t. v_3^0
ok &= std::fabs( f_v1[3] - 0.5 ) <= 1e-10; // partial f_3 w.r.t. v_3^1

// f_2 = f_3(  v_1^0 , ... , v_2^1, v_1^0 * v_1^0 , 2 * v_1^0 * v_1^1 )
f_v0[1] += f_v0[3] * 2. * v0[1];
f_v0[1] += f_v1[3] * 2. * v1[1];
f_v1[1] += f_v1[3] * 2. * v0[1];
ok &= std::fabs( f_v0[1] - 1.  ) <= 1e-10; // partial f_2 w.r.t. v_1^0
ok &= std::fabs( f_v1[1] - 0.5 ) <= 1e-10; // partial f_2 w.r.t. v_1^1

// f_1 = f_2( v_1^0 , v_1^1 , 1 + v_1^0 , v_1^1 )
f_v0[1] += f_v0[2] * 1.;
f_v1[1] += f_v1[2] * 1.;
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;
}

Input File: introduction/exp_2_rev2.cpp