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[Bug-gsl] diff.c (numerical differentiation)

From: spasmous
Subject: [Bug-gsl] diff.c (numerical differentiation)
Date: Sun, 6 Apr 2008 13:29:41 -0700

In several places in the diff.c code there are lines like the following:

  h = sqrt (GSL_SQRT_DBL_EPSILON / (2.0 * a2));
  *result = (GSL_FN_EVAL (f, x + h) - GSL_FN_EVAL (f, x)) / h;

There is an assumption here that the difference between (x+h) and x is
exactly identical to h, which is true only in infinite precision. In finite
precision there can be a small error introduced because x+h may not be
exactly representable in finite precision and so a nearby number is used
(rounded or truncated). It is very easy to remove this error in the
following way:

  h = sqrt (GSL_SQRT_DBL_EPSILON / (2.0 * a2));
  h = (x+h)-x; // ensure difference between (x+h) and x is exactly h
  *result = (GSL_FN_EVAL (f, x + h) - GSL_FN_EVAL (f, x)) / h;

The issue is explained in detail here -

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