Separate functionality in compute_bwd_err

[jarlebring]

Currently, the function compute_bwd_theta_exponential does two things 1: computes the backward error function bnd_rel_bwd_err 2: Computes the intersection of eps()/2 with bnd_rel_bwd_err using fzero.

I would prefer to have this in several functions:

• compute_bnd_rel_bwd_err(:exp, graph, ; ... )
• compute_bwd_err_theta(:exp, ; bnd_rel_err=compute_bnd_rel_bwd_err(:exp, graph, ; ... ))

The reason is that the fzero call often fails and in those situations one would be interested in the function. When fzero fails it throws an error so the information about the function is not available.

Besides: The fzero often takes many iteartions. Instead of running fzero on eps()/2=bnd_rel_bwd_err(x), we can run it on log(eps()/2) = log(bnd_rel_bwd_err(x)). It should be faster considering how the bwd_err- function behaves. This feature would be natural to incorporate in compute_bwd_err_theta eg as a kwarg.

[mfasi]

I've split compute_bwd_theta_exponential into ``compute_bnd_rel_bwd_err_exp(graph; ...)andcompute_bwd_err_theta(;bnd_rel_err=compute_bnd_rel_bwd_err_exp(graph; ...))` in cc43276 and I've added the `use_log` kwarg to the latter in 8872636. The suffix `_exp` in the function names is due to the fact that I don't understand is the use of the symbol `:exp` in the argument list. I don't think it's meant to be use for dispatch, as it's a symbol and not a type.

[jarlebring]

Nice.

The reason I added :exp in the argument list and removed it from the function name was it seems cleaner wrt future functionality, and less hard-coded than in a function name. Since we only support :exp at the moment, the function would start with checking if fun==:exp and throw an error if this is not the case.

It would have been nicer to handle this with dispatch (as you suggest) but I don't see how to do that without adding considerable amounts of code.

[jarlebring]

Ah... and the function compute_bwd_theta_exp is already independent of function whose backward error we analyze. So it can be renamed compute_bwd_theta without the use of :exp. (Okay the default kwarg refers to the exponential but that can remain so even if we rename the function.)

[jarlebring]

Looks good to me!