07/17/2008, 11:28 AM
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.
Polynomials are not bad, really. I'm just not familiar with making polynomials with degree >= 2 out of given values.
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?
Polynomials are not bad, really. I'm just not familiar with making polynomials with degree >= 2 out of given values.