// [[Rcpp::depends(BH)]]
// [[Rcpp::depends(RcppEigen)]]
// [[Rcpp::depends(StanHeaders)]]
#include <stan/math.hpp>
#include <Rcpp.h>
#include <RcppEigen.h>
#include <unsupported/Eigen/AutoDiff>

// [[Rcpp::export]]
double test_Jacobian(Eigen::MatrixXd A) {
  stan::math::matrix_v A_var = stan::math::to_var(A);
  typedef Eigen::AutoDiffScalar<stan::math::vector_v> ADScalar;
  typedef Eigen::Matrix<ADScalar,Eigen::Dynamic,Eigen::Dynamic> ADMatrix;
  ADMatrix A_(A.rows(), A.cols());
  for (int i = 0; i < A.size(); i++) A_(i).value() = A_var(i);
  const int derivative_num = A.size();
  int derivative_idx = 0;
  for (int i = 0; i < A.size(); i++) {
    A_(i).derivatives() = 
      Eigen::VectorXd::Unit(derivative_num, derivative_idx);
    derivative_idx++;
  }
  ADMatrix A_inv = A_.inverse();
  stan::math::matrix_v J(derivative_num, derivative_num);
  int pos = 0;
  for (int i = 0; i < A.rows(); i++) for (int j = 0; j < A.cols(); j++) {
    J.col(pos++) = A_inv(i, j).derivatives();
  }
  stan::math::var log_abs_det_J = stan::math::log_determinant(J);
  return log_abs_det_J.val();
}