Kata

For a certain simulation, a group of students have to produce numbers close to 2 in a certain order. Their instructor guide them explaining an algorithm useful for that task.

An array of integers, arr, is received.

arr = [n1, n2, ....,nk]

The array should be transformed, doing the following steps:

• each value of the array, ni, will be substituted by the value (ni + μ) / μ

μ = (Σ ni) / k (mean of the array)

After doing this process recursively, it will be seen that the final array will trend to this final one: [2.0, 2.0, .....2.0]

Let's start with this one: [12, 7, 23]

array                                               μ                             transformed array                        number of calls
[12, 7, 23]                                        14.0             [1.8571428571428572, 1.5, 2.642857142857143]                1
[1.8571428571428572, 1.5, 2.642857142857143]        2.0             [1.9285714285714286, 1.75, 2.321428571428571]               2
[1.9285714285714286, 1.75, 2.321428571428571]       2.0             [1.9642857142857144, 1.875, 2.1607142857142856]             3
.............................................       ....             ............................................               ...
.............................................       ....             ............................................               ...
[2.0, 2.0, 2.0]                                     2.0             [2.0, 2.0, 2.0]                                            54

After certain number of recursive steps, the difference between 2 and the obtained values is so small that cannot be detected by the programming language system.

In order to estimate the amount of generated values, they need to determine the number of recursive calls for this process.

Do you want to help them?

They need the function simul_close_to2() that receives a certain array, and a minimum absolute error, abs_error. The function will output the inner calls.

The recursion should stop when |arr[i] - arr[i + 1]| ≤ abs_error for all i of the array. For that purpose we need a helper function are_contigElemen_closeEnough() that receives an array and the abs_error and outputs True when this condition is fulfilled and False when not.

Let's see the above case for different absolute errors:

simul_close_to2([12, 7, 23], pow(10, -5)) == 18 # final array [1.9999989100864957, 1.9999961853027344, 2.00000490461077]

simul_close_to2([12, 7, 23], pow(10, -8)) == 28 # final array [1.9999999989356314, 1.9999999962747097, 2.000000004789659]

simul_close_to2([12, 7, 23], pow(10, -12)) == 42 # final array [1.9999999999999352, 1.9999999999997726, 2.000000000000292]

The tests for the function will have the following features:

N < 250 (N, number of tests)
3 ≤ L ≤ 5000 (L, length of the array)
1 ≤ ni ≤ 10000000 (arr = [n1, n2,..., ni, ..., nk])
10^(-17) ≤ abs_error ≤ 0.001

Enjoy it!