1. ## Interpolation of adjusted square roots...emergency

We've got a project where we're required to interpolate. We've got 10 values, square roots of 10 to 1. These roots are then adjusted by adding 50% to root of 10, 40% to root of 9, 30 % to root of 8, 20% to root of 7 and 10% to root of 6. Then again, 10% to root of 5, 20% to root of 4, 30% to root of 3, 40% to root of 2 and 50% to root of 1.

It looks like this:
3.16 + 50%=4.74
3 + 40%=4.2
2.83 + 30%=3.68
2.65 + 20%=3.17
2.45 + 10%=2.69
2.24 + 10%=2.46
2 + 20%=2.4
1.73 + 30%=2.25
1.41 + 40%=1.98
1 + 50%=1.5

Now we take cumulative totals and the final total is 29.08.
Now, if 10 is 4.74, 20 is 8.94(sum of 4.74 and 4.2)....90 is 27.58(sum of 1st 9 items or say 29.08-1.5)...

Can I interpolate and find values of say 3, 27, 67, 83 etc. now that we can values of all multiples of 10?

I actually tried using Newton's Approximation Formula but it either didn't work all the time or it seems it is giving approximate values which is of little use.

2. Well its basically this:

x sqrt(x) %inc y sum z
--- ------- ---- ---- ----- ----
10 3.16 50% 4.74 4.74 10
9 3.00 40% 4.20 8.94 20
8 2.83 30% 3.68 12.62 30
7 2.65 20% 3.17 15.79 40
6 2.45 10% 2.69 18.48 50
5 2.24 10% 2.46 20.94 60
4 2.00 20% 2.40 23.34 70
3 1.73 30% 2.25 25.59 80
2 1.41 40% 1.98 27.57 90
1 1.00 50% 1.50 29.07 100

So now I want to find values of say when z is equal to 3, 25, 47,94 etc. Is Interpolation possible? How? And note you cannot do something like the following which is linear Interpolation which is just not accurate.

z sum
--- -----
20 8.94
27 ?
30 12.62

The approximation would be

8.94 + (27-20)/(30-20)*(12.62-8.94) = 8.94 + 0.7*3.68 = 11.516

This is simply 7/10 of the way from 8.95 to 12.62.

Obviously this is not correct. The value for z=27 would be something more than 11.516.