Newton Forward Interpolation Method

Newton Forward Interpolation is used to estimate the value of a function at a given point when the data points are tabulated at equal intervals. This method is particularly useful when you want to interpolate a value near the beginning of the data set. It utilizes forward differences to form the interpolation polynomial.

The formula for Newton Forward Interpolation is:

P(x) = y_0 + v \cdot \Delta y_0 + \frac{v(v-1)}{2!} \cdot \Delta^2 y_0 + \frac{v(v-1)(v-2)}{3!} \cdot \Delta^3 y_0 + \frac{v(v-1)(v-2)(v-3)}{4!} \cdot \Delta^4 y_0 + \cdots

where:

$v = \frac{x - x_0}{h}$
$x_0$ is the first value of $x$ in the data.
$h$ is the uniform difference between the $x$ values (where $h = x_1 - x_0$ ).
$\Delta y_0, \Delta^2 y_0, \dots$ are the forward differences.

This method is efficient for interpolating at or near the beginning of the data set.

Example of Newton Forward Interpolation

Let's say we are given the following data points:

$x$	$y$
1	2
2	5
3	10
4	17

We are tasked with finding $y$ where $x = 2.5$

Step 1: Calculate the forward differences for the $y$ values.

$x$	$y$	${\Delta y}$	${\Delta^2 y}$	${\Delta^3 y}$
1	2	3	2	0
2	5	5	2
3	10	7
4	17

Step 2: Using the formula $v = \frac{x - x_0}{h}$ values.

\text{Given } x = 2.5 \text{ and } x_0 = 1 \text{ with } h = 1

v = \frac{x - x_0}{h}

v = \frac{2.5 - 1}{1}

v = 1.5

Step 3: Apply the Newton Forward Interpolation formula:

P(x) = y_0 + v \cdot \Delta y_0 + \frac{v(v-1)}{2!} \cdot \Delta^2 y_{0} + \frac{v(v-1)(v-2)}{3!} \cdot \Delta^3 y_0

Substitute the known values:

P(2.5) = 2 + 1.5 \cdot 3 + \frac{1.5(1.5-1)}{2!} \cdot 2 + \frac{1.5(.5-1)(1.5-2)}{3!} \cdot (0)

Step-by-step:

$\ ext{First term = } 2$
$\ ext{Second term: } 1.5 \cdot 3 = 4.5$
$\ ext{Third term: } \ rac{1.5(1.5-1)}{2!} \cdot 2 = 0.75$
$\ ext{Fourth term: } \ rac{1.5(.5-1)(1.5-2)}{3!} \cdot (0) = 0$

Adding them together:

P(33)= 2+(4.5)+(0.75)+(0)

Thus, the interpolated value of $y$ at $x =2.5$ is approximately $7.25$

Newton Forward Interpolation Calculator

Newton Forward Interpolation Method

Example of Newton Forward Interpolation

Let's say we are given the following data points:

Step-by-step:

Newton Forward Interpolation Calculator

Plot: