Any reasonable way to do this??

DeVore

I need to program a way to do the following, with some constraints on how.

Given real numbers X and Y where X < Y, I need to create 10 exponential steps from X to Y which would be graphically represented something like this:

So, lets suppose X is 1.5 and Y is 2.5. I need a simple (non-recursive) equation which spits out 10 exponentially decreasing steps from 1.5 to 2.5 ..

The best I can come up with is something like:
F(t) = X + X(e^rt) where r is the rate and t is the step number....

But that falls down in that is doesnt consider Y at all.
It describes the write sort of arc but its not really what I'm looking for.


I've been looking at this for so long I cant see the wood for the trees I feel...

ZorbaTehZ

If I follow you correctly, then this may work

[latex]F(t)=\log \left(\frac{1}{10} t \left(e^Y-e^X\right) + e^X\right)[/latex]

Edit: the plus has disappeared (on my screen at least), so the function is F(t)=\log \left(\frac{1}{10} t \left(e^Y-e^X\right) + e^X\right).

ray giraffe

To make each step half the size of the previous step do:

[latex]\displaystyle F(t) = X + (Y-X) \left( \frac{1-0.5^t}{1-0.5^{10}} \right)[/latex]

For X=5 and Y=14:

5
9.504398827
11.75659824
12.88269795
13.4457478
13.72727273
13.86803519
13.93841642
13.97360704
13.99120235
14



To make each step a quarter of the previous step do:

[latex]\displaystyle F(t) = X + (Y-X) \left( \frac{1-0.25^t}{1-0.25^{10}} \right)[/latex]

For X=5 and Y=14:

5
11.75000644
13.43750805
13.85938345
13.9648523
13.99121951
13.99781132
13.99945927
13.99987125
13.99997425
14

t runs from 0 to 10, so that's 11 numbers (with 10 gaps)

Ray

DeVore

OFFS!!!! I was so close to that and turned back from it!

Many thanks Ray, that looks exactly like what I need!! And thanks Zorba, thats a very comprehensive answer too!