Select Solution with real (positive, etc) coefficients - ignoring variables

Question

I often need to solve equations for numerical coefficients while preserving numerous variables (e.g. to preserve scaling relations). Thus I end up with a set of solutions like

$$ \left\{-\frac{(0.156113\, +0.0384783 i) d^{6/13} f^{6/13} m^{16/13} r^{9/13}}{g^{9/13} L^{6/13}}, -\frac{(0.156113\, -0.0384783 i) d^{6/13} f^{6/13} m^{16/13} r^{9/13}}{g^{9/13} L^{6/13}}, [...] , \frac{0.160785 d^{6/13} f^{6/13} m^{16/13} r^{9/13}}{g^{9/13} L^{6/13}}\right\} $$

where [...] is possible lots more, ugly solutions --- while I just want the last one (for example).

How can I select only the solution with the type of numerical coefficient I want (e.g. Real, or real & positive, etc)?

Solve[eq,var,dom] doesn't seem to work with 'inexact coefficients', and trying things like 'Select[...]' seem to have the same problem.

I've found I can brute force proper selection if I just replace every variable with unity (i.e. {Sols} /.f->1 /.d->1 /.m->1 $...$ But that realls sucks...

Any help would be appreciated, thanks!

Answer 1

You can use Trace to pick out the coefficients. There may be a more elegant way but this seems to work.

sol={(1.233-0.23I)a^(1/2) b^(3/2),(1.233-0.343I)a^(1/2) b^(3/2),   
        (234.234)a^(1/2) b^(3/2)};
coeff=Map[TraceScan[Identity,#][[1]]&,sol];
rule=Select[coeff,( Im[#]<0)&];      (* Select only negative Im coefficients *) 
Extract[sol,Position[coeff,#]&/@rule]

{{(1.233 -0.23 I) Sqrt[a] b^(3/2)},{(1.233 -0.343 I) Sqrt[a] b^(3/2)}}

Answer 2

I am sure there will be some input that will break the following method, but here is a way to use Pick or Select:

Let

sols = {-(((0.25` + 0.5` I) x^2 y^3)/z), ((3 + I) x^3 y)/w^2, 3 - 4 I, (5.` x^5 Sqrt[y])/z^3, 2, -2.5`, 4 x^2};

First, note that you can get the rules to replace variables with 1s using

thrd = Thread[Variables[sols] -> 1]
(* ==>  {w -> 1, x -> 1, y -> 1, z -> 1} *)

and use this in to get the "coefficients" in a single step:

sols /. thrd
(* ==> {-0.25 - 0.5 I, 3 + I, 3 - 4 I, 5., 2, -2.5, 4}  *) 

Now, you can use Select as follows:

Select[sols, myConditions[#] &@(# /. thrd) &]

where myConditions is any pure function specifying your criteria.

Examples:

Select[sols, (Element[#, Integers]) &@(# /. thrd) &]
(*==> {2, 4 x^2} *)
Select[sols, (Element[#, Complexes] && Re[#] > 2) &@(# /. thrd) &]
(* ==> {((3 + I) x^3 y)/w^2, 3 - 4 I, (5. x^5 Sqrt[y])/z^3, 4 x^2} *)
Select[sols, (Element[#, Complexes] && Im[#] < 0) &@(# /. thrd) &]
(* ==> {-(((0.25 + 0.5 I) x^2 y^3)/z), 3 - 4 I} *)

Alternatively, you get the same results using Pick as follows:

Pick[sols, Element[#, Integers] & /@ (sols/.thrd)]
Pick[sols, (Element[#, Complexes] && Re[#] > 2) & /@ (sols/.thrd)]
Pick[sols, (Element[#, Complexes] && Im[#] < 0) & /@ (sols/.thrd)]