How to use the Newton Divided Difference Calculator
- 1. Enter the dataType the x values (any spacing) and the matching f(x) values.
- 2. Choose a pointEnter the x at which you want to estimate f(x), or leave it blank.
- 3. Read the resultSee the table, Newton's form, the expanded polynomial and the value.
What divided differences are
For data at unequal x values, the first divided difference is f[x₀, x₁] = (f(x₁) − f(x₀))/(x₁ − x₀). Higher ones are built the same way: f[x₀, x₁, x₂] = (f[x₁, x₂] − f[x₀, x₁])/(x₂ − x₀), always dividing by the gap between the outer x values. Arrange them in a triangle; the top entry of each column is a coefficient.
Newton's divided difference formula is P(x) = f(x₀) + (x − x₀)f[x₀, x₁] + (x − x₀)(x − x₁)f[x₀, x₁, x₂] + …. It passes through every data point, and adding a new point only adds one more term.
Worked example
Given f(5) = 150, f(7) = 392, f(11) = 1452, f(13) = 2366, f(17) = 5202, find f(9). First differences: 121, 265, 457, 709. Second: 24, 32, 42. Third: 1, 1. Fourth: 0.
So P(x) = 150 + 121(x − 5) + 24(x − 5)(x − 7) + 1(x − 5)(x − 7)(x − 11). At x = 9: 150 + 484 + 192 − 16 = 810. The polynomial simplifies to x³ + x², which fits all five points exactly.
Related methods
For equally spaced x, the same polynomial comes from the Newton forward difference table. Lagrange's formula gives the identical polynomial in another form: try the Lagrange interpolation calculator.
Frequently asked questions
When should I use divided differences instead of forward differences?
When the x values are not equally spaced. Forward differences need a constant step h; divided differences do not.
Is the answer the same as Lagrange interpolation?
Yes. Through n points there is only one polynomial of degree at most n − 1, so both methods give the same result.
Does the order of the points matter?
No. The table looks different, but the final polynomial and estimate are the same.
How many points can I enter?
From 2 to 12 points, each with a different x value.
Is this calculator free?
Yes. You fill a short form once when you see your first result, and other Uncodemy tools then open without the form. Everything is calculated in your browser.