SymPy implementation Variance Arithmetic

[Chengpu Wang]

Hi, I am a new member of this group.  It is my first time to post in this group.  I love SymPy because I see its most important position in futural numerical calculations when not only value but also uncertainty needs to be answered.  In the attached file of this post, I have present you the first generic statistical solution to track result variance of analytic functions using Taylor expansion (Please See Section 2.8 Analytic Functions). The calculations of results with variance are very different from without, because analytic expressions are necessary to trace dependency of input variables (Please See Section 2.9 Dependence Tracing) and the paper proves that the traditional numerical approaches are not valid anymore (Please see Section 2.10 Traditional Execution and Dependency Problem).  In fact, my paper shows that the current numerical library may have huge numerical errors due to its ignorance of input uncertainty and rounding errors, e.g., the numerical errors between sin(x)/cos(x) and tan(x) can be up to 10^-3 (Please see Section 9.4 Library Errors). Thus, SymPy should not be just a niche, but a generic futural approach in numerical calculations.

I am in the process to adopt SymPy for my open source on this subject (Chengpu0707/VarianceArithmetic: A floating-point arithmetic with value and uncertainty (variance) pair (github.com)), and I wonder if anyone can give me some advice on how to do it.

The attached file is 80-pages long, but I hope that you will enjoy it if you love math (on the college level).  Please also let me know if you find anyway to improve it. 

Regards,
Dr. Chengpu Wang