Numerical derivatives should now be calculated using the `GSL::Deriv.forward, GSL::Deriv.central`

and `GSL::Deriv.backward`

methods, which accept a step-size argument in addition to the position x. The original `GSL::Diff`

methods (without the step-size) are deprecated.

`GSL::Deriv.central(`

`f`,`x`,`h`=`1e`-`8`)`GSL::Function#deriv_central(`

`x`,`h`=`1e`-`8`)-
These methods compute the numerical derivative of the function

`f`at the point`x`using an adaptive central difference algorithm with a step-size of`h`. If a scalar`x`is given, the derivative and an estimate of its absolute error are returned as an array, [`result, abserr, status`]. If a vector/matrix/array`x`is given, an array of two elements are returned, [`result, abserr`], here each them is also a vector/matrix/array of the same dimension of`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 is computed using a 5-point rule for equally spaced abscissae at x-h, x-h/2, x, x+h/2, x, 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. `GSL::Deriv.forward(`

`f`,`x`,`h`=`1e`-`8`)`GSL::Function#deriv_forward(`

`x`,`h`=`1e`-`8`)-
These methods compute 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 and an estimate of its absolute error are returned as an array, [`result, abserr`]. These methods 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. `GSL::Deriv.backward(`

`f`,`x`,`h`)`GSL::Function#deriv_backward(`

`x`,`h`)-
These methods compute 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 and an estimate of its absolute error are returned as an array, [`result, abserr`]. This function should be used if f(x) has a discontinuity at`x`, or is undefined for values greater than`x`.These methods are equivalent to calling the method

`forward`

with a negative step-size.

`GSL::Diff.central(`

`f`,`x`)`GSL::Function#diff_central(`

`x`)- These compute the numerical derivative of the function
`f`( GSL::Function object) at the point`x`using an adaptive central difference algorithm. The result is returned as an array which contains the derivative and an estimate of its absolute error. `GSL::Diff.forward(`

`f`,`x`)`GSL::Function#diff_forward(`

`x`)- These compute the numerical derivative of the function at the point x using an adaptive forward difference algorithm.
`GSL::Diff.backward(`

`f`,`x`)`GSL::Function#diff_backward(`

`x`)- These compute the numerical derivative of the function at the point x using an adaptive backward difference algorithm.

#!/usr/bin/env ruby require "gsl" f = Function.new { |x| pow(x, 1.5) } printf ("f(x) = x^(3/2)\n"); x = 2.0 h = 1e-8 result, abserr = f.deriv_central(x, h) printf("x = 2.0\n"); printf("f'(x) = %.10f +/- %.10f\n", result, abserr); printf("exact = %.10f\n\n", 1.5 * Math::sqrt(2.0)); x = 0.0 result, abserr = Deriv.forward(f, x, h) # equivalent to f.deriv_forward(x, h) printf("x = 0.0\n"); printf("f'(x) = %.10f +/- %.10f\n", result, abserr); printf("exact = %.10f\n", 0.0);

The results are

f(x) = x^(3/2) x = 2.0 f'(x) = 2.1213203120 +/- 0.0000004064 exact = 2.1213203436 x = 0.0 f'(x) = 0.0000000160 +/- 0.0000000339 exact = 0.0000000000