# include <cppad/cppad.hpp>
# include <cassert>

namespace {
     // return the vector x that solves the following linear system
     //     a[0] * x[0] + a[1] * x[1] = b[0]
     //     a[2] * x[0] + a[3] * x[1] = b[1]
     // in a way that will record pivot operations on the AD<double> tape
     typedef CPPAD_TESTVECTOR(CppAD::AD<double>) Vector;
     Vector Solve(const Vector &a , const Vector &b)
     {     using namespace CppAD;
          assert(a.size() == 4 && b.size() == 2);

          // copy the vector b into the VecAD object B
          VecAD<double> B(2);
          AD<double>    u;
          for(u = 0; u < 2; u += 1.)
               B[u] = b[ Integer(u) ];

          // copy the matrix a into the VecAD object A
          VecAD<double> A(4);
          for(u = 0; u < 4; u += 1.)
               A[u] = a [ Integer(u) ];

          // tape AD operation sequence that determines the row of A
          // with maximum absolute element in column zero
          AD<double> zero(0), one(1);
          AD<double> rmax = CondExpGt(fabs(a[0]), fabs(a[2]), zero, one);

          // divide row rmax by A(rmax, 0)
          A[rmax * 2 + 1]  = A[rmax * 2 + 1] / A[rmax * 2 + 0];
          B[rmax]          = B[rmax]         / A[rmax * 2 + 0];
          A[rmax * 2 + 0]  = one;

          // subtract A(other,0) times row A(rmax, *) from row A(other,*)
          AD<double> other   = one - rmax;
          A[other * 2 + 1]   = A[other * 2 + 1]
                             - A[other * 2 + 0] * A[rmax * 2 + 1];
          B[other]           = B[other]
                             - A[other * 2 + 0] * B[rmax];
          A[other * 2 + 0] = zero;

          // back substitute to compute the solution vector x.
          // Note that the columns of A correspond to rows of x.
          // Also note that A[rmax * 2 + 0] is equal to one.
          CPPAD_TESTVECTOR(AD<double>) x(2);
          x[1] = B[other] / A[other * 2 + 1];
          x[0] = B[rmax] - A[rmax * 2 + 1] * x[1];

          return x;
     }
}

bool vec_ad(void)
{     bool ok = true;

     using CppAD::AD;
     using CppAD::NearEqual;
     double eps99 = 99.0 * std::numeric_limits<double>::epsilon();

     // domain space vector
     size_t n = 4;
     CPPAD_TESTVECTOR(double)       x(n);
     CPPAD_TESTVECTOR(AD<double>) X(n);
     // 2 * identity matrix (rmax in Solve will be 0)
     X[0] = x[0] = 2.; X[1] = x[1] = 0.;
     X[2] = x[2] = 0.; X[3] = x[3] = 2.;

     // declare independent variables and start tape recording
     CppAD::Independent(X);

     // define the vector b
     CPPAD_TESTVECTOR(double)       b(2);
     CPPAD_TESTVECTOR(AD<double>) B(2);
     B[0] = b[0] = 0.;
     B[1] = b[1] = 1.;

     // range space vector solves X * Y = b
     size_t m = 2;
     CPPAD_TESTVECTOR(AD<double>) Y(m);
     Y = Solve(X, B);

     // create f: X -> Y and stop tape recording
     CppAD::ADFun<double> f(X, Y);

     // By Cramer's rule:
     // y[0] = [ b[0] * x[3] - x[1] * b[1] ] / [ x[0] * x[3] - x[1] * x[2] ]
     // y[1] = [ x[0] * b[1] - b[0] * x[2] ] / [ x[0] * x[3] - x[1] * x[2] ]

     double den   = x[0] * x[3] - x[1] * x[2];
     double dsq   = den * den;
     double num0  = b[0] * x[3] - x[1] * b[1];
     double num1  = x[0] * b[1] - b[0] * x[2];

     // check value
     ok &= NearEqual(Y[0] , num0 / den, eps99, eps99);
     ok &= NearEqual(Y[1] , num1 / den, eps99, eps99);

     // forward computation of partials w.r.t. x[0]
     CPPAD_TESTVECTOR(double) dx(n);
     CPPAD_TESTVECTOR(double) dy(m);
     dx[0] = 1.; dx[1] = 0.;
     dx[2] = 0.; dx[3] = 0.;
     dy    = f.Forward(1, dx);
     ok &= NearEqual(dy[0], 0.         - num0 * x[3] / dsq, eps99, eps99);
     ok &= NearEqual(dy[1], b[1] / den - num1 * x[3] / dsq, eps99, eps99);

     // compute the solution for a new x matrix such that pivioting
     // on the original rmax row would divide by zero
     CPPAD_TESTVECTOR(double) y(m);
     x[0] = 0.; x[1] = 2.;
     x[2] = 2.; x[3] = 0.;

     // new values for Cramer's rule
     den   = x[0] * x[3] - x[1] * x[2];
     dsq   = den * den;
     num0  = b[0] * x[3] - x[1] * b[1];
     num1  = x[0] * b[1] - b[0] * x[2];

     // check values
     y    = f.Forward(0, x);
     ok &= NearEqual(y[0] , num0 / den, eps99, eps99);
     ok &= NearEqual(y[1] , num1 / den, eps99, eps99);

     // forward computation of partials w.r.t. x[1]
     dx[0] = 0.; dx[1] = 1.;
     dx[2] = 0.; dx[3] = 0.;
     dy    = f.Forward(1, dx);
     ok   &= NearEqual(dy[0],-b[1] / den + num0 * x[2] / dsq, eps99, eps99);
     ok   &= NearEqual(dy[1], 0.         + num1 * x[2] / dsq, eps99, eps99);

     // reverse computation of derivative of y[0] w.r.t x
     CPPAD_TESTVECTOR(double) w(m);
     CPPAD_TESTVECTOR(double) dw(n);
     w[0] = 1.; w[1] = 0.;
     dw   = f.Reverse(1, w);
     ok  &= NearEqual(dw[0], 0.         - num0 * x[3] / dsq, eps99, eps99);
     ok  &= NearEqual(dw[1],-b[1] / den + num0 * x[2] / dsq, eps99, eps99);
     ok  &= NearEqual(dw[2], 0.         + num0 * x[1] / dsq, eps99, eps99);
     ok  &= NearEqual(dw[3], b[0] / den - num0 * x[0] / dsq, eps99, eps99);

     return ok;
}