6_A

C#

Link

https://drive.google.com/drive/folders/14va6UOgbm5kpAhRKqGRljkvYOCChsTca?usp=sharing

In this application, I tested 2 cases: arithmetic mean and summation. In both cases the number of values "N" is equal to 100000. For the arithmetic mean, I used values: PI and E, that are defined in Math library in C#. While for summation, 0.1 and 0.3 values are used.

For the arithmetic mean, I compared naive, Knuth's [2] and Kahan's [1, 2] approaches to show that naive algorithm is not stable and losses significance. The same comparison was done in summation test, but only for naive and Kahan's algorithms.

Arithmetic mean:

Note that in this test, Kahan is used for summation and then the result is divided by "N".

PI = 3.1415926535897931

E = 2.7182818284590451

• Naive approach = { 3.1415926535859, 2.71828182845473}

• Error = {0.0000000000038931, 0000000000043151}

• Knuth = { 3.14159265358979, 2.71828182845905}

• Error = {0.0000000000000031, 0000000000000049}

• Kahan = { 3.14159265358979, 2.71828182845905}

• Error = {0.0000000000000031, 0000000000000049}

Sum:

value_1 = 0.1

value_2 = 0.3

• Naive approach = { 10000.0000000188, 29999.9999999506}

• Error = {0.0000000188, 0.0000000494}

• Kahan = { 10000, 30000}

• Error = {0, 0}

We can clearly see the errors that occur when naive approach is used. Whereas both Kahan and Knuth provide more accurate results.

References

1. https://en.wikipedia.org/wiki/Kahan_summation_algorithm

2. https://mlblogblog.wordpress.com/2017/11/22/r2-the-best-algorithm-to-compute-the-online-mean/