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)GSL::Diff.forward(f, x)GSL::Function#diff_forward(x)GSL::Diff.backward(f, x)GSL::Function#diff_backward(x)#!/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