I have a case where a variable (a, in this case) is calculated at each loop iteration and stops where the increment of value between two iterations is small enough.
I would like to know of a general way to find the value for that variable in this kind of case, without having to do that "convergence" work using loops.
There I would like to know if the solution is to put everything in equations, or if some tools exist to tackle that.
a = 10
b = 10
diff = 1
while diff > .1:
old_a = a
a += b
diff = 1 - (old_a/a)
print(diff)
The present code produces:
0.5
0.33333333333333337
0.25
0.19999999999999996
0.16666666666666663
0.1428571428571429
0.125
0.11111111111111116
0.09999999999999998
Therefore, it takes 9 iterations to find a relative difference of the value of a between two iterations inferior to 10%.
You have
a_n = a_0 + n * b
and try to find where
1 - (a_(n-1) / a_n)
= 1 - (a_0 + (n--1)*b) / ( a_0 + n * b)
= 1 - (a_0 + n*b -b) / (a_0 + n*b)
= 1 - 1 + b / (a_0 + n*b)
= b / (a_0 + n * b)
< 0.1
That is the same as
(a_0 / b) + n * b / b
= (a_0 / b) + n
> 10
(because 0.1 = 1 / 10 and 1/x > 1/y <=> y > x if x,y != 0)
Since you metion in the comments that your actual problem is more complex: If finding a closed form solution like above is not feasible, look at this wikipedia page about fixed point iteration, which is exactly the kind of problem you try to solve.