Derivatives are tested using a Taylor expansion of **f**
around a given point . The following Taylor series is formulated
at where
is a scalar:

where and are the gradient and Hessian,
respectively, evaluated at .
If only the gradient routines are tested, the
second-order Taylor term ` YHY`
is set to zero, and the truncation error is
.
Our test is performed by computing this Taylor approximation at
smaller and smaller values of and checking
whether the truncation errors are as expected:
and
if the approximation is correct up to
the gradient and Hessian terms, respectively.
At every step
we half and test if indeed the
truncation errors decrease as they should
(i.e., if the error corresponding to is ,
the error for should be if the gradient
is correct, and if the Hessian is also correct.)