opt over real coeffs

[jarlebring]

At some point we changed from designing graphs with complex coefficients and now all graphs have real coefficients. It would therefore be natural to have the real version of the optimizers. Preliminary tests with this in the simulations seems to be very advantageous.

Example:
At the moment

• A) solve_linsqr minimizes
  min_{c1,c2 in C} ||Ac-b||
  where A in C^{nxm} and b in CC^n. In general A\b will give a complex solution.

We want

• B) Minimizes
  min_{c1,c2 in R} ||Ac-b||
  where A in C^{nxm} and b in CC^n.

This can be achieved by noting that B is equivalent to

• C) Minimize
  min_{c1,c2 in R} ||Atilde*c-btilde||
  where
  Atilde=[real(A);imag(A)] in R^{2nxm} and btilde=[real(b);imag(b)] in R^2n. c=Atilde\btilde will return a real vector

In practice, I suggest this is done by (in solve_linsqr) allowing linlsqr to take values :real_backslash, :real_nrmeq, etc.

[jarlebring]

What about :real_svd?

[eringh]

I'll look into it!

[eringh]

The function solve_linlsqr modifies the input matrix A if it is in BigFloat and we use :svd. It is likely efficient, but surprising that it only modifies the input sometimes. Maybe we should at least call it solve_linlsqr!, i.e., add an exclamation mark, and write about it in the documentation. What do you think?

[jarlebring]

👍 for modifying to exclamationmark.

[eringh]

:real_svd implemented and name changed.

I think that svd (and svd! from GenericSVD.jl) should return real matrices if called with real matrices (such factorization exists), but I cannot find such guarantee in the documentation.

[jarlebring]

I think the return type question can be handled if it appears later.