Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Just asking...
#8
Oy, I'm not so good at explaining things, but I'll try anyway.

When I first tried to interpolate b^^x for non-integer x (and for easy cases like 2^^x or 3^^x), I thought of the following:
For example, 3^^0=1, 3^^1=3, 3^^2=27 - at first, the values appear to grow linearly, then exponentially. So I came up with the idea of a "flexible arithmetic-geometric mean". I expected the function to grow like a linear function ax+b at one point x and like a geometric series a^x*b at x+1, but that didn't turn out to be the case. That was about three years ago.

Playing around with numbers and maths in general being my hobby, I hit upon the following last year: all the "common" mean values (arithmetic, geometric, quadratic and harmonic mean) can be expressed in the form [(a(1)^n+a(2)^n+...+a(m)^n)/m]^(1/n). For n=1, this is the arithmetic mean, for n=2 the quadratic, for n=-1 the harmonic, and, by analytic continuation or whatever you may call it, for n=0 the geometric mean. I assumed that, with such a flexible mean value calculation, it just had to work. All I had to do was figure out the appropriate parameter n for a given interval [x ... x+1]. But lately I started doubting this assumption as well.

Darn, now I begin to see what you meant ... if this formula works between x and x+1, why shouldn't it also work beyond this interval? Seems I'm being still too much of an amateur here.

bo198214 Wrote:Polynomials are not complicated to calculate with either, are they? Wink

Polynomials are not bad, really. I'm just not familiar with making polynomials with degree >= 2 out of given values.
Reply


Messages In This Thread
Just asking... - by martin - 07/14/2008, 09:49 PM
RE: Just asking... - by bo198214 - 07/14/2008, 10:53 PM
RE: Just asking... - by martin - 07/15/2008, 11:40 AM
RE: Just asking... - by bo198214 - 07/16/2008, 08:00 PM
RE: Just asking... - by martin - 07/16/2008, 09:52 PM
RE: Just asking... - by bo198214 - 07/16/2008, 10:36 PM
RE: Just asking... - by bo198214 - 07/16/2008, 10:52 PM
RE: Just asking... - by martin - 07/17/2008, 11:28 AM
RE: Just asking... - by bo198214 - 07/18/2008, 01:48 PM
RE: Just asking... - by martin - 07/18/2008, 06:37 PM
RE: Just asking... - by bo198214 - 07/18/2008, 10:02 PM
RE: Just asking... - by Gottfried - 07/19/2008, 03:48 AM
RE: Just asking... - by Ivars - 07/19/2008, 07:04 AM
RE: Just asking... - by martin - 07/19/2008, 09:03 AM
RE: Just asking... - by Ivars - 07/19/2008, 07:45 PM
RE: Just asking... - by andydude - 07/21/2008, 04:05 AM
RE: Just asking... - by martin - 07/21/2008, 06:31 PM
RE: Just asking... - by Ivars - 07/22/2008, 06:07 AM



Users browsing this thread: 1 Guest(s)