NAG Library Manual, Mark 27.2
Interfaces:  FL   CL   CPP   AD 

NAG AD Library Introduction
Example description
/* nag::ad::e04fc Tangent over Tangent Example Program.
 */

#include <dco.hpp>
#include <iostream>
#include <nagad.h>

std::stringstream filecontent("0.14  1.0 15.0  1.0\
 0.18  2.0 14.0  2.0\
 0.22  3.0 13.0  3.0\
 0.25  4.0 12.0  4.0\
 0.29  5.0 11.0  5.0\
 0.32  6.0 10.0  6.0\
 0.35  7.0  9.0  7.0\
 0.39  8.0  8.0  8.0\
 0.37  9.0  7.0  7.0\
 0.58 10.0  6.0  6.0\
 0.73 11.0  5.0  5.0\
 0.96 12.0  4.0  4.0\
 1.34 13.0  3.0  3.0\
 2.10 14.0  2.0  2.0\
 4.39 15.0  1.0  1.0");

// Function which calls NAG AD Library routines.
template <typename T>
void func(std::vector<T> &y,
          std::vector<T> &t,
          std::vector<T> &x,
          std::vector<T> &fvec,
          T &             fsumsq);

// Driver with the tangent calls.
// Computes the minimum of the sum of squares of m nonlinear functions, the
// solution point and the corresponding residuals. Also, computes the sum of
// Hessian elements of fsumsq w.r.t. inputs y and t, and the sum of Hessian
// elements of x w.r.t. inputs y and t.
void driver(const std::vector<double> &yv,
            std::vector<double> &      tv,
            std::vector<double> &      xv,
            std::vector<double> &      fvecv,
            double &                   fsumsqv,
            double &                   d2f_dall2,
            double &                   d2x_dall2);

// Evaluates the residuals.
template <typename T>
void NAG_CALL lsqfun(void *&        ad_handle,
                     Integer &      iflag,
                     const Integer &m,
                     const Integer &n,
                     const T        xc[],
                     T              fvec[],
                     Integer        iuser[],
                     T              ruser[]);

int main(void)
{
  std::cout << " nag::ad::e04fc Tangent over Tangent Example Program Results\n";

  // Problem dimensions
  const Integer       m = 15, n = 3, nt = 3;
  std::vector<double> yv(m), tv(m * nt);
  for (int i = 0; i < m; i++)
    {
      filecontent >> yv[i];
      for (int j = 0; j < nt; j++)
        {
          Integer k = j * m + i;
          filecontent >> tv[k];
        }
    }

  std::vector<double> xv(n), fvecv(m);

  // Sum of squares of the residuals at the computed point xv
  double fsumsqv;

  // Sum of Hessian elements of sum of squares fsumsqv with respect to
  // parameters y, t1, t2 and t3
  double d2f_dall2;
  // Sum of Hessian elements of x with respect to parameters y, t1, t2 and t3
  double d2x_dall2;

  // Call driver
  driver(yv, tv, xv, fvecv, fsumsqv, d2f_dall2, d2x_dall2);

  // Primal results
  std::cout.setf(std::ios::scientific, std::ios::floatfield);
  std::cout.precision(12);
  std::cout << "\n Sum of squares = ";
  std::cout.width(20);
  std::cout << fsumsqv;
  std::cout << "\n Solution point = ";
  for (int i = 0; i < n; i++)
    {
      std::cout.width(20);
      std::cout << xv[i];
    }
  std::cout << std::endl;

  std::cout << "\n Residuals :\n";
  for (int i = 0; i < m; i++)
    {
      std::cout.width(20);
      std::cout << fvecv[i] << std::endl;
    }
  std::cout << std::endl;

  std::cout << "\n Derivatives calculated: Second order tangents\n";

  // Print derivatives of fsumsq
  std::cout
      << "\n Sum of Hessian elements of sum of squares fsumsq w.r.t. parameters y and t:\n";
  std::cout << " sum_ij [d2fsumsq/dall2]_ij = " << d2f_dall2 << std::endl;

  // Print derivatives of solution points
  std::cout
      << "\n Sum of Hessian elements of solution points x w.r.t. parameters y and t:\n";
  std::cout << " sum_ij [d2x/dall2]_ij = " << d2x_dall2 << std::endl;

  return 0;
}

// Driver with the tangent calls.
// Computes the minimum of the sum of squares of m nonlinear functions, the
// solution point and the corresponding residuals. Also, computes the sum of
// Hessian elements of fsumsq w.r.t. inputs y and t, and the sum of Hessian
// elements of x w.r.t. inputs y and t.
void driver(const std::vector<double> &yv,
            std::vector<double> &      tv,
            std::vector<double> &      xv,
            std::vector<double> &      fvecv,
            double &                   fsumsqv,
            double &                   d2f_dall2,
            double &                   d2x_dall2)
{
  using T = dco::gt1s<dco::gt1s<double>::type>::type;

  // Problem dimensions
  const Integer m = fvecv.size(), n = xv.size();
  const Integer nt = n;

  // AD routine arguments
  std::vector<T> y(m), t(m * nt), x(n), fvec(m);
  T              fsumsq;
  for (int i = 0; i < m; i++)
    {
      y[i] = yv[i];
    }
  for (int i = 0; i < m * nt; i++)
    {
      t[i] = tv[i];
    }

  dco::derivative(dco::value(y)) = std::vector<double>(y.size(), 1.0);
  dco::value(dco::derivative(y)) = std::vector<double>(y.size(), 1.0);
  dco::derivative(dco::value(t)) = std::vector<double>(t.size(), 1.0);
  dco::value(dco::derivative(t)) = std::vector<double>(t.size(), 1.0);

  // Call the NAG AD Lib functions
  func(y, t, x, fvec, fsumsq);

  // Sum of squares
  fsumsqv = dco::passive_value(fsumsq);
  // Solution point
  xv = dco::passive_value(x);
  // Residuals
  fvecv = dco::passive_value(fvec);

  // Get sum of Hessian elements of fsumsq w.r.t. y and t
  d2f_dall2 = dco::derivative(dco::derivative(fsumsq));

  // Get sum of Hessian elements of solution points x w.r.t. y and t
  d2x_dall2 = 0;
  for (int i = 0; i < n; i++)
    {
      d2x_dall2 += dco::derivative(dco::derivative(x[i]));
    }
}

// Function which calls NAG AD Library routines.
template <typename T>
void func(std::vector<T> &y,
          std::vector<T> &t,
          std::vector<T> &x,
          std::vector<T> &fvec,
          T &             fsumsq)
{
  // Problem dimensions
  const Integer m = fvec.size(), n = x.size();
  const Integer ldfjac = m, nt = n, ldv = n;

  // All additional data accessed in the callback MUST be in ruser.
  // Pack the parameters [y, t1, t2, t3] into the columns of ruser
  std::vector<T> ruser(m * nt + m);
  T *            _y = &ruser[0];
  T *            _t = &ruser[m];

  for (int i = 0; i < m; i++)
    {
      _y[i] = y[i];
    }

  for (int i = 0; i < m * nt; i++)
    {
      _t[i] = t[i];
    }

  // Initial guess of the position of the minimum
  dco::passive_value(x[0]) = 0.5;
  for (int i = 1; i < n; i++)
    {
      dco::passive_value(x[i]) = dco::passive_value(x[i - 1]) + 0.5;
    }

  std::vector<T> s(n), v(ldv * n), fjac(m * n);
  Integer        niter, nf;

  Integer iprint = -1;
  Integer maxcal = 400 * n;
  T       eta    = 0.5;
  T       xtol   = 10.0 * sqrt(X02AJC);
  T       stepmx = 10.0;

  // Create AD configuration data object
  Integer ifail     = 0;
  void *  ad_handle = 0;
  nag::ad::x10aa(ad_handle, ifail);
  // Routine for computing the minimum of the sum of squares of m nonlinear
  // functions.
  ifail = 0;
  nag::ad::e04fc(ad_handle, m, n, lsqfun<T>, nullptr, iprint, maxcal, eta, xtol,
                 stepmx, x.data(), fsumsq, fvec.data(), fjac.data(), ldfjac,
                 s.data(), v.data(), ldv, niter, nf, 0, nullptr, ruser.size(),
                 ruser.data(), ifail);

  // Remove computational data object
  ifail = 0;
  nag::ad::x10ab(ad_handle, ifail);
}

// Evaluates the residuals.
template <typename T>
void NAG_CALL lsqfun(void *&        ad_handle,
                     Integer &      iflag,
                     const Integer &m,
                     const Integer &n,
                     const T        xc[],
                     T              fvec[],
                     Integer        iuser[],
                     T              ruser[])
{
  const T *y  = ruser;
  const T *t1 = ruser + m;
  const T *t2 = ruser + 2 * m;
  const T *t3 = ruser + 3 * m;
  const T  x1 = xc[0], x2 = xc[1], x3 = xc[2];

  for (int i = 0; i < m; i++)
    {
      T di = x2 * t2[i] + x3 * t3[i];
      T yi = x1 + t1[i] / di;
      // The values of the residuals
      fvec[i] = yi - y[i];
    }
}