double piecewise_linear_product_integral ( double a, double b, int f_num, 
  double f_x[], double f_v[], int g_num, double g_x[], double g_v[] );
double piecewise_linear_product_quad ( double a, double b, int f_num, 
  double f_x[], double f_v[], int g_num, double g_x[], double g_v[], 
  int quad_num );
double r8_abs ( double x );
double r8_epsilon ( );
double r8_max ( double x, double y );
double r8_min ( double x, double y );
void r8vec_bracket3 ( int n, double t[], double tval, int *left );
void timestamp ( );