#J, 222

*247 - recursion + numerical*

    NB. multiple lines for visibility
    f=:[:-(]$:~[:}.[-%&{.*],#&0@l)`[@.(0>l=:-&#)
    r=:1={.
    s=:[:=/+/@(2=&*/\])@(,.]*_1^i.@#)@({.@>@(([;];f;j;f f j=.]f f)([:}:(i._5)*])))@|.
    u=:0,0([(]*_1^r@k)>:@]^:(1 1-:k=:[:*(0 p.~[)*(p.(,-)))^:_)~]
    h=:{.@((n@],]{~[:r[:*p.*(p.n=:{.+-:@-~/))^:_ u)`('n'"_)@.s@|.

### Methods

* Using Sturm's theorem to find number of roots.

 `f` is polynomial division (recursive)

 `s` is Sturm's theorem output: `0` for no real roots, `1` otherwise. It evaluates Sturm's polynomials `([;];f;j;f f j=.]f f)` and checks their behaviour at +/- infinity.


* Using bisection method to avoid divisions by zero, local maxima etc.

 `u` scans from 0 to +/- infinity to find an appropriate interval

 `((n@],]{~[:r[:*p.*(p.n=:{.+-:@-~/))^:_ u)` evaluates the polynomial at the edges and midpoint then halfs the interval accordingly



### Verification of results

Using J's buildin polynomial root finder as `realRoots`:

    ti,>([;h;realRoots) each samples
    ┌──────────────┬────────┬─────────────────┐
    │a b c d e     │output  │all real roots   │
    ├──────────────┼────────┼─────────────────┤
    │_1 2 3 4 5    │_1.11029│3.37192 _1.11029 │
    ├──────────────┼────────┼─────────────────┤
    │_1 2 3 4 _5   │0.728727│3.18248 0.728727 │
    ├──────────────┼────────┼─────────────────┤
    │1 2 3 4 5     │n       │                 │
    ├──────────────┼────────┼─────────────────┤
    │1 1 2 _1 _1   │0.728262│0.728262 _0.52236│
    ├──────────────┼────────┼─────────────────┤
    │1 2 3 4 0     │0       │_1.65063 0       │
    ├──────────────┼────────┼─────────────────┤
    │1 2 0 0 1     │_1      │_1.83929 _1      │
    ├──────────────┼────────┼─────────────────┤
    │_1 _1 _1 _1 _1│n       │                 │
    └──────────────┴────────┴─────────────────┘