Create sum formula of polinomial functions

This algorithm help compute the sum formula of known function.

Usage

Check algorithm work

python .\sumfor.py -h

Output:

usage: sumfor.py [-h] [-f FORMULA] [-v] [-s START] [-t STEP]

optional arguments:
  -h, --help            show this help message and exit
  -f FORMULA, --formula FORMULA
                        Sum formula of function
  -v, --version         show version
  -s START, --start START
                        Start point (default = 1)
  -t STEP, --step STEP  Step (default = 1)

To use the algorithm, use the following syntax:

python .\sumfor.py -f <formula input> -s <start point> -t <step>

The form of the input formula is a dictionary string with keys being the exponent and the values being the coefficient of that exponent. Example input form of $x^3-\frac{2}{3}x^2+1$ is '{0:1,2:-2/3,3:1}' (dictionary's keys order isn't matter).

Example

Example 1

For example, we have to compute the sum formula of the $x^3-2x^2+1$ function from $1$, mean $\sum\limits_{x = 1}^x {({x^3} - {2x^2} + 1)}$.

Running algorithm by the following syntax to find the sum formula.

python .\sumfor.py -f '{3:1,2:-2,0:1}'

Result:

+2/3*x^1-3/4*x^2-1/6*x^3+1/4*x^4

So the sum formula of $x^3-x^2+1$ from $1$ is $\frac{2}{3}x - \frac{3}{4}{x^2} - \frac{1}{6}{x^3} + \frac{1}{4}{x^4}$.

It means: $$\sum\limits_{x = 1}^x {({x^3} - {2x^2} + 1)} = \frac{2}{3}x - \frac{3}{4}{x^2} - \frac{1}{6}{x^3} + \frac{1}{4}{x^4}$$

Example 2

If we need to compute the sum formula of the $x^2+1$ from $-1$, mean $\sum\limits_{x = -1}^x {({x^2} + 1)}$.

Similarly, we execute the following syntax:

   python .\sumfor.py -f '{2:1,0:1}' -s -1

Output:

    3*x^0+7/6*x^1+1/2*x^2+1/3*x^3

Mean:

$$\sum\limits_{x = -1}^x {({x^2} + 1)} = 3 + \frac{7}{6}x + \frac{1}{2}{x^2} + \frac{1}{3}{x^3}$$

Example 3

Were we have to find a Hit Sum Function following:

$$\left. {{S_2}} \right|_{ - 1}^x\left( {{x^3} - 2{x^2} + 1} \right)$$

It means calculating the sum of formula $x^3 - 2x^2 + 1$ from $-1$ to $x$, the step equals 2. We execute the following command:

python .\sumfor.py -f '{3:1,2:-2,0:1}' -s -1 -t 2

Output:

-13/8*x^0-1/6*x^1-1/2*x^2+1/6*x^3+1/8*x^4

That mean:

$$\left. {{S_2}} \right|{{ - 1}}^x\left( {{x^3} - 2{x^2} + 1} \right) = \frac{{ - 13}}{8} - \frac{1}{6}{x^1} - \frac{1}{2}{x^2} + \frac{1}{6}{x^3} + \frac{1}{8}{x^4}$$

Algorithm mechanism

The basic

Base on the Triagle matrix of polinomial sum was created before. Apply a simple rule about the formula for the sum of sum functions:

$$ \sum {\left( {{k_0} + {k_1}{x^1} + \ldots + {k_n}{x^n}} \right)} = {k_0}\sum {{x^0}} + {k_1}\sum {{x^1}} + \ldots + {k_n}\sum {{x^n}} $$

• With $k_0,k_1,\ldots,k_n$ are the coefficients of the corresponding exponent.

• $\sum {{x^0}},\sum {{x^1}},\ldots,\sum {{x^n}}$ can be computed easily in matrix the Triagle matrix.

Order of algorithm execution

• Processing input data
• Determine the largest power exponent
• Multiply coefficients by the corresponding sum formula
• Add the sum formulas together

Introduction to the Hit Sum Function

The Hit Sum Function is just the generality form of the Sigma Sum Function with the additional element is the step.

$$\left. {{S_n}} \right|_a^bf\left( x \right) = f(a) + f(a + n) + f(a + 2n) + \ldots + f(b)$$

With the step equal to 1, Hit Sum Function is Sigma Sum.

$$\left. {{S_1}} \right|a^bf\left( x \right) = \sum\limits{x = a}^b {f(x)}$$

To calculate the Hit Sum Function, it is necessary to use the step conversion rule of the Hit Sum Function to convert it to a Sigma Sum Function.

The step conversion rule is:

$$\left. {{S_n}} \right|a^bf(x) = \left. {{S_m}} \right|{_{a.\frac{m}{n}}}^{^{b.\frac{m}{n}}}f(x.\frac{n}{m})$$

Therefore,

$$\left. {{S_n}} \right|a^bf(x) = \sum\limits{_{\frac{a}{n}}}^{\frac{x}{n}} {f(n.x)}$$

From here we can calculate the rule as mentioned in the basic section and then change the variable $x$ to $\frac{x}{n}$

Application

• Formal Regression: from the value points, regression to the polynomial function that generates those values