double he_double_product_integral ( int i, int j );
double he_triple_product_integral ( int i, int j, int k );
void pce_ode_hermite ( double ti, double tf, int nt, double ui, int np, 
  double alpha_mu, double alpha_sigma, double t[], double u[] );
double r8_abs ( double x );
double r8_factorial ( int n );
void timestamp ( );