Re: Maximise polynomial degree N


marmitage@xxxxxxxxx wrote:
I'd like find a,b and n to maximise the following polynomial:

1 + b + (a-b) b^n + (b-1-a) b^n a^(N-n)

where 0 < a < b < 1
and N > 4 is fixed
(n and N are integers)

Ideally I'd like to show that n*>(N/2)
or that for a given division of N into m and N-m then it is optimal to
set n = max(m, N-m) if a and b are optimal also.

I've solved fully numerically but analytically have sunk a lot of time
into it and not got anywhere. If anyone can help, or explain to me why
an analytical solution is evading me, I'd be very grateful.

In a variety of ways, such as taking special case a = b,
although strictly speaking your inequality a < b doesn't
allow this, we obtain the simpler optimization problem:

F(x) = 1 + x - x^N for x in (0,1) [x = a = b]

An analytic solution can be obtained for the maximum
at x = N^(-1/(N-1)), and this independent of the choice
of n. I suspect that this is the answer, but I have not
analyzed it satisfactorily yet.

regards, chip

.

Relevant Pages

  • Re: Maximise polynomial degree N
    ... Fix n, and find a and b in terms of n. ... or that for a given division of N into m and N-m then it is optimal to ... I've solved fully numerically but analytically have sunk a lot of time ... an analytical solution is evading me, ...
    (sci.math)
  • Maximise polynomial degree N
    ... I'd like find a,b and n to maximise the following polynomial: ... or that for a given division of N into m and N-m then it is optimal to ... an analytical solution is evading me, I'd be very grateful. ...
    (sci.math)