#
# = Numerical Differentiation
# The functions described in this chapter compute numerical derivatives by 
# finite differencing. An adaptive algorithm is used to find the best choice 
# of finite difference and to estimate the error in the derivative. 
#
# Contentes:
# 1. {Deriv methods}[link:files/rdoc/diff_rdoc.html#1]
# 1. {Diff methods}[link:files/rdoc/diff_rdoc.html#2]
#
# == {}[link:index.html"name="1] Deriv methods (for GSL 1.4.90 or later)
# Numerical derivatives should now be calculated using the
# <tt>GSL::Deriv.forward, GSL::Deriv.central</tt> and <tt>GSL::Deriv.backward</tt> methods,
# which accept a step-size argument in addition to the position x.  The
# original <tt>GSL::Diff</tt> 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 <tt>f</tt> 
#   at the point <tt>x</tt> using an adaptive central difference algorithm with a 
#   step-size of <tt>h</tt>. If a scalar <tt>x</tt> is given, the derivative and an 
#   estimate of its absolute error are returned as an array, [<tt>result, abserr, status</tt>].
#   If a vector/matrix/array <tt>x</tt> is given, an array of two elements are returned,
#   [<tt>result, abserr</tt>], here each them is also a vector/matrix/array of the same
#   dimension of <tt>x</tt>.
#
#   The initial value of <tt>h</tt> 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 <tt>f</tt> at 
#   the point <tt>x</tt> using an adaptive forward difference algorithm with a step-size 
#   of <tt>h</tt>. The function is evaluated only at points greater than <tt>x</tt>, 
#   and never at <tt>x</tt> itself. The derivative and an estimate of its absolute error 
#   are returned as an array, [<tt>result, abserr</tt>]. 
#   These methods should be used if f(x) has a 
#   discontinuity at <tt>x</tt>, or is undefined for values less than <tt>x</tt>.
#
#   The initial value of <tt>h</tt> 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 <tt>f</tt> at the 
#   point <tt>x</tt> using an adaptive backward difference algorithm with a step-size 
#   of <tt>h</tt>. The function is evaluated only at points less than <tt>x</tt>, 
#   and never at <tt>x</tt> itself. The derivative and an estimate of its absolute error 
#   are returned as an array, [<tt>result, abserr</tt>]. 
#   This function should be used if f(x) has a discontinuity at <tt>x</tt>, 
#   or is undefined for values greater than <tt>x</tt>.
#
#   These methods are equivalent to calling the method <tt>forward</tt> 
#   with a negative step-size.
#
# == {}[link:index.html"name="2] Diff Methods (obsolete)
#
# ---
# * GSL::Diff.central(f, x)
# * GSL::Function#diff_central(x)
#
#   These compute the numerical derivative of the function <tt>f</tt> ( {GSL::Function}[link:files/rdoc/function_rdoc.html] object) at the point <tt>x</tt> 
#   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. 
#
# == {}[link:index.html"name="3] Example
#
#      #!/usr/bin/env ruby
#      require "gsl"
#
#      f = GSL::Function.alloc { |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
#
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