Re: create a list fulfill special requirement by

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• Subject: [mg130641] Re: create a list fulfill special requirement by
• From: Ray Koopman <koopman at sfu.ca>
• Date: Sun, 28 Apr 2013 01:00:39 -0400 (EDT)
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```f[a_,x_] := x/(x + a(1-x)), with a > 0, maps [0,1]->[0,1].
The slope at (0,0) is 1/a, the slope at (1,1) is a.
Try f[a,Range[0,13]/13]*3.5+2.5 with a = 10 or so.

On Fri, Apr 26, 2013 @ 08:10 PM, Joug Raw <jougraw at gmail.com> wrote:

>
> I want to generate a list of number bewteen 2.5 and 6 with 14 elements
> totally.
>
> The number of the list should fulfill such requirement that it will be very
> tight at the beginning then loos at the end. That means the numbers started
> from 2.5 will be more closed then the number at around 6. Something like
> {2.5, 2.52, 2.55, 2.60, 2.71, 2.80, 2.95, 3.1...4.2, 5, 6}
>
> I tried exponential as well as power function for generating this list with
> the following code:
>
> 1. list = With[{a = 3},
>   Array[a^# &, 14, {x, y}] /.
>    Join[Solve[a^x == 2.5, x][[-1]], Solve[a^y == 6., y][[-1]]]]
>
> which fit the list points with exponential function
>
> 2. list = With[{a = 5},
>   Array[#^a &, 14, {x, y}] /.
>    Join[Solve[x^a == 2.5, x][[-1]], Solve[y^a == 6., y][[-1]]]]
>
> which fit the list points with power function
>
> The 1st one gives me output
> {2.5, 2.67416, 2.86045, 3.05972, 3.27286, 3.50086, 3.74474, 4.00561, \
> 4.28466, 4.58314, 4.90242, 5.24393, 5.60924, 6.}
>
> The 2nd one gives me output
> {2.5, 2.6895, 2.89031, 3.10295, 3.32792, 3.56575, 3.81699, 4.0822, \
> 4.36195, 4.65683, 4.96744, 5.29441, 5.63838, 6.}
>
> They are indeed tight at the end closed to 2.5 and lose at the end near 6, *
> BUT* it is not enough for me, what I need is VERY tight and VERY lose at
> the two ends such as {2.5, 2.52, 2.6, 2.69...
> and ...3.0, 4.2, 6.0} and in between the transition should be gradually and
> smoothly done.
>
> I tired to adjust the number "a" in the exponential function and the power
> function in my codes. Even if I gave it a very big value for a, this dose
> not really change so much the tight and the lose of the numbers at both
> ends.
>
> What function should I use to generate the list as I wished?
>
> And, how to make my codes a bit smarter? The code I used I think is bit
> more complected. Any suggestion that I could bypass the "With" and the "/."
> in my code which still allow me to define directly the two ends{2.5, 6} and
> size of "a" in a more straightforward and easier way?

```

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