— Function: int gsl_deriv_central (const gsl_function * f, double x, double h, double * result, double * abserr)

This function computes the numerical derivative of the function f at the point x using an adaptive central difference algorithm with a step-size of h. The derivative is returned in result and an estimate of its absolute error is returned in abserr.

The initial value of h is used to estimate an optimal step-size, based on the scaling of the truncation error and round-off error in the derivative calculation. The derivative is computed using a 5-point rule for equally spaced abscissae at x-h, x-h/2, x, x+h/2, x+h, with an error estimate taken from the difference between the 5-point rule and the corresponding 3-point rule x-h, x, x+h. Note that the value of the function at x does not contribute to the derivative calculation, so only 4-points are actually used.

— Function: int gsl_deriv_forward (const gsl_function * f, double x, double h, double * result, double * abserr)

This function computes the numerical derivative of the function f at the point x using an adaptive forward difference algorithm with a step-size of h. The function is evaluated only at points greater than x, and never at x itself. The derivative is returned in result and an estimate of its absolute error is returned in abserr. This function should be used if f(x) has a discontinuity at x, or is undefined for values less than x.

The initial value of h is used to estimate an optimal step-size, based on the scaling of the truncation error and round-off error in the derivative calculation. The derivative at x is computed using an “open” 4-point rule for equally spaced abscissae at x+h/4, x+h/2, x+3h/4, x+h, with an error estimate taken from the difference between the 4-point rule and the corresponding 2-point rule x+h/2, x+h.

— Function: int gsl_deriv_backward (const gsl_function * f, double x, double h, double * result, double * abserr)

This function computes the numerical derivative of the function f at the point x using an adaptive backward difference algorithm with a step-size of h. The function is evaluated only at points less than x, and never at x itself. The derivative is returned in result and an estimate of its absolute error is returned in abserr. This function should be used if f(x) has a discontinuity at x, or is undefined for values greater than x.

This function is equivalent to calling gsl_deriv_forward with a negative step-size.