How is the Gregory-Newton formula used to construct difference tables?

[gavztheouch]

Hi i need some help understanding the method needed to construct difference tables
so far this is what I've got

table

X -2 -1 0 1 2 3
F(x) -105 -45 -25 -27 -9 95

this is my data the x values are -2,-1,-0,1,2,3
and the f(x) values correspond to the values above

Here is the difference table from the answer


-2...-105
......60
-1...-45.....-40
......20.....18
0...-25.....-22...24
.....-2...42
1...-27.....20...24
.....18.....66
2...-9...86
......104
3...95

What i would like to know is how it is contructed obviously the x values are in the first colum the f(x) in the next then the difference between the F(x) in the following colums down to the end. What i don't understand is the negative terms for example the difference between 60 and 20 is -40?

Cheers
Gav

 

[Navin A S]

To construct a difference table, you need to start with the given x-values and f(x) values. You then need to calculate the differences between the adjacent f(x) values for each row. The first column is the x-values and the second column is the f(x) values. From there, you will enter the difference between each adjacent f(x) value in the appropriate column. For example, let's look at the row for x=-2. The difference between -105 and -45 is 60. So, you would enter 60 in the third column. Then, you would take the difference between -45 and -25 and enter that in the fourth column. You would continue this process until you reach the last row. In the case of negative terms, it is important to remember that a negative number subtracted from another negative number results in a positive number. For example, the difference between -45 and -25 is 20. Since 20 is a positive number, you would enter 20 in the fourth column.

 

[ravenprp]

The Gregory-Newton formula is a method used to construct difference tables, which are used in numerical analysis to calculate approximate values for functions. To understand how the formula is used, let's first define some terms:

- X: This represents the independent variable, or the input values for the function.
- F(x): This represents the dependent variable, or the output values for the function.
- Difference table: This is a table that shows the differences between consecutive values of F(x) for a given set of X values.

To construct a difference table using the Gregory-Newton formula, follow these steps:

1. Write the X values in the first column of the table.
2. Write the F(x) values in the second column of the table.
3. Calculate the first-order differences by subtracting each F(x) value from the one above it. Write these values in the third column of the table.
4. Calculate the second-order differences by subtracting each first-order difference from the one above it. Write these values in the fourth column of the table.
5. Repeat this process until all differences have been calculated and the table is complete.

In your example, the first-order differences are calculated by subtracting each F(x) value from the one above it:

- For -1, the first-order difference is 60-(-105) = 165
- For 0, the first-order difference is -45-60 = -105
- For 1, the first-order difference is -25-(-45) = 20
- For 2, the first-order difference is -27-(-25) = -2
- For 3, the first-order difference is -9-(-27) = 18

The second-order differences are then calculated by subtracting each first-order difference from the one above it:

- For -1, the second-order difference is 165-(-105) = 270
- For 0, the second-order difference is -105-165 = -270
- For 1, the second-order difference is 20-(-105) = 125
- For 2, the second-order difference is -2-20 = -22
- For 3, the second-order difference is 18-(-2) = 20

The negative values in the difference table represent the direction of the change in F(x). For example, a negative value in the first-order differences column indicates that