Software that outputs a general polynomial formula for a finite series

In summary, there are various tools and methods available for finding general polynomial formulas for finite sequences, including recursive formulas and the Pouffe's Inverter. These methods involve plugging in the given values and solving linear equations for the coefficients of the polynomial. However, it is important to note that there are also limitations and considerations to take into account when using these tools.
  • #1
Helicobacter
158
0
Is there a tool that spits out a general polynomial formula for an input consisting of a finite sequence, e.g.:

you type in
1 ,3 , 5, 7

the program processes the input and spits out
(n+2), 7>=n>=1
 
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  • #2
Helicobacter said:
Is there a tool that spits out a general polynomial formula for an input consisting of a finite sequence, e.g.:

you type in
1 ,3 , 5, 7

the program processes the input and spits out
(n+2), 7>=n>=1
I would think that a polynomial formula would be more like 2n-1 or 2n+1. If you are looking for a general recursive formula that is another animal. There is a website that provides that also, but I don't have the link offhand. Also it wouldn't make much sense to spit out the lower and higher boundary points of your entry.
 
  • #3
Here. I was bored.
When given these inputs, the outputs are:

0 1 4 9 16
n^2 , n=0..4​
80 5 30
50n^2 - 125n + 80 , n=0..2​
4 3 2 1
- n + 4 , n=0..3​

C source only.
 

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  • #4
I'm bored but not yet intrigued, so I didn't look at Dodo's boring zip file. I assume he encoded Newton's forward difference formula, http://mathworld.wolfram.com/NewtonsForwardDifferenceFormula.html.

I'm bored because I'm supposed to be writing a proposal. If I let myself become sufficiently intrigued I will have to stay up all night to finish the proposal. On top of this, I have an 8:30 AM meeting tomorrow that I can't skip.
 
  • #5
i think this is what you're asking, i learned this in linear algebra. it's not the program but it's a method from which a program shouldn't be hard to come up with. i haven't done this in a while, and my notation is going to be messy.

you have a finite sequence of n elements A1, A2,...An. Then you consider them as ordered pairs (1,A1), (2,A2),...(n,An). To get a general n-1th degree polynomial

y = C1 + C2x +...+Cnx^(n-1) and then plug in some values and solve the linear equation.

to do your sequence for example 1,3,5,7. put them in ordered pairs (1,1) (2,3) (3,5) (4,7).

1 = C1 + C2(1) + C3(1) + C4(1)
3 = C1 + C2(2) + C3(4) + C4(8)
5 = C1 + C2(3) + C3(9) + C4(27
7 = C1 + C2(4) + C3(16) + C4(64)

4 equations, 4 unknowns, and i ended up with the polynomial

y = 2x - 1 x= 4,3,2,1.

just in case it isn't clear, Ck is the coefficient of x^(k-1).

if i understood, that's what you're looking for.
 
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  • #6
I wrote such a program and put it online. I originally developed this method based on my independent analysis of series in high school in the early 70s. As others have mentioned it does rely on the forward distance (although, at the time, I had no idea that's what it was called).

It takes about 20 lines of code to implement (including documentation) in PHP.

http://bentley.110mb.com/?test:series [Broken]
 
Last edited by a moderator:
  • #7

1. What is a polynomial formula?

A polynomial formula is a mathematical expression that consists of variables, coefficients, and exponents combined using basic arithmetic operations such as addition, subtraction, multiplication, and division. It is typically written in the form of ax^n + bx^(n-1) + ... + k, where a, b, and k are constants and x is the variable.

2. What is a finite series?

A finite series is a sequence of numbers or terms that has a finite number of terms. It is often represented in the form of a sum, where each term is added to the previous term. For example, 1+2+3+4+5 is a finite series with 5 terms.

3. How does software output a general polynomial formula for a finite series?

The software uses algorithms and mathematical calculations to analyze the given finite series and determine the pattern or relationship between the terms. It then uses this information to generate a general polynomial formula that can be used to find any term in the series.

4. What are the benefits of using software to output a general polynomial formula?

Using software allows for quick and accurate calculation of a general polynomial formula, which would otherwise be time-consuming and prone to human error. It also allows for the handling of more complex series with a larger number of terms.

5. Can this software output a general polynomial formula for any type of finite series?

The capabilities of the software depend on its programming and algorithms. Some software may be limited to specific types of series, while others may be able to handle a wider range of series. It is important to check the software's specifications and capabilities before use.

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