Solving a Nonlinear Equation with Many Independent Variables By Using Microsoft Excel Plus VBA

Typically, we are able to solve system of linear or nonlinear equations which are a set of simultaneous equations (SE). Definitely, solving of a linear SE is very easy while we have to use Newton's method to solve nonlinear SEs. The common case for both of them is, to generate an equation for each variable. It means, we can solve an equation with three variables, if we have three simultaneous equations or solving of four variables needs to find a system of four simultaneous equations and so on.

Can we solve a nonlinear equation with many variables? Yes. In the special conditions, the answer is 

positive.

The purpose of this article is to present some examples which show us possibility to solve a nonlinear equation with many variables where we have a good estimation for limited domain and range of variables. The method applied is the same method stated in article of "The Generating New Probability Theorems" posted on link:

 http://emfps.blogspot.nl/2016/10/the-generating-new-probability-theorems.html

The experienced physicists and engineers have usually the true speculation of domain and range for the variables while they need to obtain precise amounts for the variables. Therefore, this method can be useful for them.

At the first, I start by a nonlinear equation with three variables then four variables and finally five 
variables.

Example (1):

One of professional people asked me a question in math and statistics group of social media as follows:

"I need some help to interpret a stock regression:

SZ is high or low size of the company

BM is high or low book to market size

R is the return of the stock

R = 5% + 6%Bm +2%SZ – 2%*BM*SZ

My question is whether I should short high size and high book to market stocks (stocks that have both characteristics)?

I analyzed his problem by using above method as follows:

Here is my analysis:

1. To reach the maximum R, you should stay SZ the constant in low size and then if you increase BM, you will reach the maximum R.

2. If you increase both of them (SZ and BM), you will significantly decrease R.

3. The most important thing is about R = 0.11 because in this case, it does not take any difference. In fact, above analysis does not work for R = 0.11

4. If 0< SZ and BM <1, then maximum amount of R will be always equal to 0.11 (Rmax = 0.11)

I think that this is really a magic formula.

Example (2)

This is an example about financial and risk management.

As you know, the basic theory which links risk and return rate for all assets is, the Capital Asset Pricing Model. The equation of CAPM is as follows:

Now, suppose you want to invest on an asset in which your expected return rate (required return) is equal 12%. The question is: What are the alternatives or scenarios for three independent variables of risk – free, beta and market return?

Here, by applying the method stated in this article, I have obtained 21 answers for three independent variables as follows:

Example (3): The equation of State for an Ideal Gas

Let me tell you an example about the equation of state for an ideal gas.

We have:

P.V = N.KB.T

Suppose we have a constant volume (V) equal to 0.03 m3

 If Boltzmann’s constant (KB) is equal to 1.38E-23 J/K, what are the answers for P and N and T?

According to my method, I found 9 answers which are as follows:

:Where

T = temperature (K) and P = pressure (Pa) and N = number of molecules

Example (4): Solve Circle Equation

  When we open a calculus book, we can see the signs and footprints of Pythagoras (582 B.C – 496 B.C) everywhere. Therefore, let me start by solving of circle equation for a limited domain and range as follows:

Consider the circle equation with below domain:     

If 

           x^2 + y^2 = r^2

x, y ϵ N,            x, y ≤ 100

Then the range will be

r ϵ N,            r ≤ 141

 Now, I apply above method and get all results of "x, y" for "r" in given range. All number of results to "r" has been presented on below graph

In this case, total sum of possible answers is equal to 126. 

For instance, above graph shows us, if r = 25, 50, 75, 100 then the number of results for "x, y" are equal to 4 and if r = 65, 85 then the number of results for "x, y" are equal to 8. The results are as follows 

 Now, consider the circle equation with below domain

    

If 

           x^2 + y^2 = r^2

    0 0 0 x, y ϵ N,            x, y ≤ 1

Then the range will be

 4 r ϵ N,            r ≤ 14 1

 If I apply above method, I will generate all results of "x, y" for "r" in given range. All number of results to "r" has been presented on below graph:

In this case, total sum of possible answers is equal to 2068. 

For instance, above graph shows us, if r = 325, 425, 650,725,850,925,975 then the number of results for "x, y" are equal to 14. The results are as follows: 

Note: All researchers and individual people, who are interested in having this model, don’t hesitate to send their request to below addresses:

Email: soleimani_gh@hotmail.com

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