WIP: allow generic real number types (higher precision) where possible

[chriscoey]

fixes #13

we are sometimes limited by what Julia linear algebra methods are already defined, eg we cannot use qr (in preprocessing) with sparse non-Float64 matrices because the external SPQR package that is called seems to only do Float64

for default tolerances, I currently choose them depending on the real number type used:

        tol_rel_opt = max(1e-12, 1e-2 * cbrt(eps(T))),
        tol_abs_opt = tol_rel_opt,
        tol_feas = tol_rel_opt,

so far it seems this strikes a nice balance for Float32 and Float64 at least.

TODO

• generalize other linear system solvers
• allow other cones to use generic reals
• update remaining native tests
• modify cholesky calls for generic types (maybe want to define various methods in one file and use those elsewhere to make changing the methods everywhere easy)

[mtanneau]

tol_rel_opt = max(1e-12, 1e-2 * cbrt(eps(T)))

Any particular reason for using the cubic root (as opposed to, say, square root)?

Also, do we have to tighten numerical tolerances when using higher precision?
Obviously yes if the point is to solve with high precision. But, if one uses quad precision to get rid of some rounding errors, then shouldn't the "usual" Float64 tolerances suffice?

[chriscoey]

I tried sqrt first but found that cbrt with the 1e2 factor struck a better balance for Float32 and Float64 and BigFloat IMO. These are just default tolerances - the user can set whatever tolerance they like. I expect the default tolerances here will change in future after significantly more testing.

[chriscoey]

One thing that will be cool to try in future is fast hardware Float128 built into POWER9 systems.

[mtanneau]

Something to be aware of as well: BLAS and LAPACK only support real (and complex) Float32 and Float64 values.

So, beyond the lack of hardware support, most (dense) linear algebra operations will probably dispatch the generic methods. Expect a (even higher) performance hit for going higher precision...

[chriscoey]

I'm well aware