Diffuse filter and smoother

• harmonize tolerance criteria for diffuse likelihood to diffuse smoother
• avoid overwriting global restrict var list in smoother
• fix numerical issues in schur_statespace_transformation

@rattoma I understand the that tolerance criteria for diffuse filter and diffuse smoother should be the same. But looking at the diffuse smoother routines the tolerance parameter crit1 is set equal to 1e-8 (not 1e-6 as you wrote in commit 511ee537). Also why should we use a hard coded value? We could use options_.kalman_tol (or add a new option if the default value of kaman_tol is too small as you suggest). Also, looking at the diffuse filter code (but this not in your commit), it is not clear to me why we should use crit1 when testing the rank of Pinf and kalman_tolwhen testing the condition number of Finf.

@stepan-a. Concerning the hardcoded crit1 I would very much like to add a new option specific for diffuse step, with default 1.e-8. Concerning the tolerance for Finf:

• in the multivariate diffuse filter [which by the way is almost never used in practice, since usually we have a mixture of unit roots and stationary variables in observation equations], you are right that we should always use crit1, also for testing Finf: I did a mistake there.
• in the univariate filter, in turn, it makes sense to use kalman_tol for Finf since an individual diagonal term can become small enough to need a tighter tolerance, especially for models with a lot of unit roots [i.e. the univariate diffuse algorithm while killing unit roots one by one can lead to small enough non-zero diagonal terms]. Viceversa, the much bigger Pinfmatrix requires the courser crit1to avoid numerical noise affecting its rank calculations [I am talking about matrix sizes hundreds x hundreds: for small models this discussion is irrelevant]

By the way, I was the one introducing crit1in the smoother a long time ago, which made the diffuse steps work OK for so many years [in the smoother the noise issues are bigger than for the filter, due to the larger number of zero eigenvalues, I address this issue in commit 0336d5e7].

@rattoma Ok, I understand now. I will change the tolerance parameter in the multivariate diffuse filter, and add an option. I am short of ideas for the name, would kalman_smoother_tol be a good name for you?

I also noticed that the hard coded values for crit1 are different in the multivariate and univariate diffuse filter... Is this difference intentional? Should we have two options (for instance kaman_smoother_tol.univariate and kaman_smoother_tol.multivariate)?

for the option name I would use diffuse_kalman_tol since it applies to both filter and smoother.

for the difference in crit1between multi and univ folters I was just sloppy sorry.

thanks a lotg

@rattoma Ok, I do that.

Merged manually (fixed bug in 363c4a61 and added new option).