Overview of auto_diff module#
The auto_diff
module provides Fortran derived types that support automatic
differentiation via operator overloading. Users will not generally
need to interact with this module, but it can be used within
run_star_extras to make derivatives easier to calculate (e.g. in the
implicit hooks like other_surface
).
Usage is by writing use auto_diff
at the top of a module or routine.
This imports types such as auto_diff_real_4var_order1
, which supports first-order derivatives
with respect to up to four independent variables.
A variable of this type could be declared via:
type(auto_diff_real_4var_order1) :: x
This variable then holds five fields: x%val
stores the value of x
.
x%d1val1
stores the derivative of x with respect to the first independent
variable. x%d1val2
is the same for the second independent variable, and so on.
All d1val_
fields are initialized to zero when the variable is first set.
Once an auto_diff variable is initialized, all mathematical operations can be performed
as they would be on a real(dp)
variable. auto_diff variables also interoperate with
real(dp)
and integer
types.
So for instance in the following f%d1val1
stores df/dx and f%d1val2
stores df/dy.:
x = 3d0
x%d1val1 = 1d0
y = 2d0
y%d1val2 = 1d0
f = exp(x) * y + x + 4
Similar types are included supporting higher-order and mixed-partial
derivatives. These derivatives are accessed via e.g. d2val1
(\(\partial^2 f/\partial x^2\)), d1val1_d2val2
(\(\partial^3 f/\partial x \partial y^2\)).
An additional special type auto_diff_real_star_order1
provides support
for first-order derivatives accessed using arrays.
This type contains a value (x%val
) and an array of first partial derivatives
with respect to at least as many variables as the solver in MESA/star
.
This type is meant to make it easy to write equations and then, after the fact,
change the basis of independent variables or re-index them.
The current indices are defined in star_data/public/star_data_def.inc
.
E.g., if my_var
is of type auto_diff_real_star_order1
,
then my_var% d1Array(i_lnT_00)
contains the derivative of my_var
with respect to lnT
at the same mesh point.