In GSL, min/brent.c:128 has this for the evaluation of the parabolic fit:

 

  if (fabs (p) < fabs (0.5 q r) && p < q w_lower && p < q w_upper)

 

This matches the Algol60 implementation given by Richard Brent in the book. Unfortunately, there seems to be a transcription (or similar) error and the code should actually be

 

  if (fabs (p) < fabs (0.5 q r) && p > q w_lower && p < q w_upper)

 

which matches the actual text of the book which says that the algorithm takes the golden section path under some conditions, including “x+p/q not in (a,b)” which would translate to:

 

x + p/q <= a || x +p/q >= b …

p <= q (a - x) || p >= q (b - x) …

p > q (a - x) && p < q (b - x)

 

In fact, other authors such as Forsythe have found and corrected the error. Netlib implementation of fmin.f is also corrected (seems to be the same as that in the Forsythe book). The Numerical Recipes and boost implementations are also fixed.

 

Some references for further information:

 

·         http://www.netlib.org/go/fmin.f

·         http://www.nr.com/forum/showthread.php?p=4893

·         Computer Methods for Mathematical Computations – G.E. Forsythe, Michael A. Malcolm, Cleve B. Moler. Prentice Hall. ISBN 0-13-165332-6.