Split difference method to determine the polynomial function that models the first few given terms of a sequence

When looking at a sequence of numbers, one that can be defined by a polynomial function of a natural number degree, one most commonly would use a difference table to find the degree followed by a system of equations to find the equation that models the sequence. This method can prove to be very time consuming as solving a system of equations can become tedious at higher degrees. Alternately, some people would use a method where they use the difference table to find the leading coefficient in addition to the degree to give the first term. Then they would subtract this term from the function and repeat this process. However, this can be unnecessarily complicated as this method requires one to create a difference table numerous times only to need the last difference. This method uses a simple pattern triangle and only the first difference table of the sequence. It is already necessary to create the first difference table and this pattern triangle can be used to improve upon the second method. The pattern triangle allows us to walk through the difference table of the lower degree polynomials quite easily, removing the need for multiple difference table. This method differs from existing methods in that:• It is much faster• It uses a unique pattern triangle.

Now, you should have the following differences for each row: Creating the term zero difference triangle Begin with the triangle: The next step is to add another row. This is done by adding the numbers above it, similar to the rule used in Pascal's Triangle. However, now we will multiply each sum by their own term number. The leftmost number of each row will have the term number of 0. This process must be repeated till there are one plus the degree found in part one number of rows. Like Pascal's Triangle, this pattern triangle is the same every time you use this method.
Repeating this pattern will give you the triangle: Apply and solve Finding the coefficients. The final step is to piece together parts 1 and 2 to determine the function.
To do so, the 0th differences found in part 1 is used. First, the final difference is divided by the last term of the last row of the triangle in part 2. That value will be the leading coefficient. The rest of the numbers of that row is multiplied by that coefficient and subtracted from the other differences respectively. While going up the triangle, this process will be repeated till all of the numbers have been used up.
Example: Begin with the differences found in Part 1.
1 , 8 , 32 , 48 , 24 Divide the last term of the differences by the corresponding term in the pattern triangle and subtract. Repeat until all the numbers in the pattern triangle are used.
Substitute the variables in the equation above with the quotients found earlier in Part 3 (a is the first quotient). Example:

Alternate form of split difference
There is a similar method to split difference which can be found to be easier in some circumstances.

Finding the differences
Repeat the Part 1 process from the standard Split Difference method. However, there is no need to find the 0th term or the 0th differences. Example: Row 4  24  24  Row 3  72  96  120  Row 2  80  152  248  368  Row 1  40  120  272  520  888  Row 0  9  49  169  441  961  1849 Creating the first term difference triangle Begin with the triangle: 1 1 1 Create another row by adding the numbers above times the term number each number holds, starting with left as 1. For example, the second number in the next row would be (1 * 1) + (1 * 2). Once again, create as many rows as the degree plus 1. Like Pascal's Triangle, this pattern triangle is the same every time you use this method. Apply and solve -Here, the method works same as it did in part 1 Finding the coefficients The final step is to piece together parts 1 and 2 to determine the function. To do so, the 0th differences found in part 1 is used. First, the final difference is divided by the last term of the last row of the triangle in part 2. That value will be the leading coefficient. The rest of the numbers of that row is multiplied by that coefficient and subtracted from the other differences respectively. While going up the triangle, this process will be repeated till all of the numbers have been used up.
Example: Begin with the differences found in Part 1.
9 , 40 , 80 , 72 , 24 Divide the last term of the differences by the corresponding term in the pattern triangle and subtract. Repeat until all the numbers in the pattern triangle are used.

Determine the equation
Here, we use the empty polynomial equation of the degree you found in Part 1, as shown below: Substitute the variables in the equation above with the quotients found earlier in Part 3 Example:

System of equations [1]
The current method works by using a system of equations to solve the polynomial. One must set up multiple equations following the structure: They will then substitute x for 1 and y for the first term and so on till they have as many equations as coefficients. Using the elimination method, this method requires you to find the coefficients one by one. However, this method can become very tedious, especially when dealing with higher degree polynomials. It is also very difficult when dealing with non-integer coefficients.
Difference table [2] The Split Difference Method is certainly based off the ideas behind this method. This method requires creating the difference table for the sequence to find the leading coefficient and highest degree. Then, the method calls for subtracting this value from each term in the sequence and drawing the difference table once more to find the next highest degree. This process is repeated until all the terms are found. However, this method is extremely time consuming as it requires one to create multiple difference tables.
Newton's interpolation method [3] This method involves creating a special difference table where the 'difference' between each successive term is found using a special equation. While this method may be fast, it doesn't provide one with a simplified polynomial function, but rather a quite large one that will take too much time to simplify at higher degrees.

Split difference method benefits
This method compared to any previous method runs much faster, especially at higher degree equations. The first step of finding the differences and degree is the same as you would in any other method because it is necessary to determine the degree of the function. The triangle is one that doesn't change based on the scenario and therefore can be memorized for quick use. The only extra calculation occurs in the final step which is comprised of simple arithmetic and very little of it. This minimizes the issue of making an accidental error which tends to be common in the system of equations method. By reusing the difference table that would need to have been made anyways and limiting the method to only one difference table, we get a much faster process than that of the System of Equations and the Difference Table method. Additionally, we get a simplified result unlike that of Newton's Interpolation Method.
x g(x) or 0 Using the previous two coefficients, we can let h( The first term of n for h(x) would be: If this process is repeated until every coefficient is found, the function will only have the leading coefficient. We can call this function z(x).
The first term of n for z(x) would be: Since z(x) is a constant coefficient function, there is only one n to pay attention to -0 . All other columns will be 0.
Thus, we can simplify the summation above to: Now we can define a function, P m (x) which represents a part of the original polynomial, f(x) with degree d, where m represents the degree of the function. The general formula for first difference of each column n for P m (x) would be: Now if we write the first term of each column n for any degree, d we get: Taking the difference between the rows, without the last value in each row, we get: With the last value in each row being: This pattern is similar to the pattern triangle which shows the values as:   1  1  1  1  3  2  1  7  12  6  1 15 50 60 24 The difference between this pattern triangle and the table we have above is that the constants behind each number in the pattern triangle are set as 1.
Simplify the following equation Expand the summation: Group like terms: Expand the combinations: Factor out like terms from each group: Simplify into a summation: Simply the inner combinations: The expression above can be simplified to: Therefore, this follows the pattern of the First Term Difference Triangle

Term zero difference triangle
The first pattern triangle can be derived in a similar way except we start with the x = 0 term instead of x = 1 since the pattern triangle uses the 0th differences rather than the 1st. The difference table would for the sequence would be: Now if we write the first term of each column n for any degree, d we get: The last difference is each row is still equal to that row factorial, which gives the degree of that partial polynomial. Taking the difference between the rows, without the last value in each row, we get: With the last value in each row being: This pattern is similar to the pattern triangle which shows the values as: The difference between this pattern triangle and the table we have above is that the constants behind each number in the pattern triangle are set as 1.
Simplify the following equation: Expand the summation: Distribute: Group like terms: Expand the combinations: Factor out like terms.
Simplify into a summation.
This follows the pattern of the Term Zero Difference Triangle

Proof of application of pattern triangle
From the previous proof, we know that the first term of each column n for any degree is: where the summation gives each term in the pattern triangle. Thus, the following equation: give you the differences of the polynomial function one degree lower (removing the highest degree variable like shown in the proof of the pattern triangle for g(x)). Since A d is the last term of each row divided by n!, the equation above can be simplified to: This equation above means that the first term of each column in the difference table of the function P d-1 (x) can be found by taking the first term of each column in the difference table of the function P d (x) and subtracting the quantity of each respective value in the pattern triangle times the value of the leading coefficient of P d (x), which can also be found by knowing the differences of P d (x).
It is important to understand that there can be an infinite amount number of polynomial equations that can represent the function. For example, if you were given the first two terms of a sequence -0,1 -any equation of form f(x) = (x-1) d will match the function, where d is all real numbers. Therefore, if given the first n terms of a sequence, one can determine the lowest degree polynomial function of maximum degree, n -1, that represents the sequence.

Conclusion
In summary, the "Split Difference" method can be used to determine the equation to fit a given sequence. This method uses a pattern triangle and applies it to the difference table used to find the degree of a polynomial in order to speed up this process of determining a representative polynomial function. The experimental results have shown this method to be fast and very accurate.