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.

Formula for Newton Forward Interpolation:

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.

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 Forward Differences

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

Step 2: Calculate v

Using the formula $v = \frac{x - x_0}{h}$ :

\text{Given } x = 2.5, x_0 = 1, h = 1

v = \frac{2.5 - 1}{1} = 1.5

Step 3: Apply the 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

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

Step-by-step Term Evaluation:

\text{First term = } 2

\text{Second term: } 1.5 \cdot 3 = 4.5

\text{Third term: } \frac{1.5(0.5)}{2} \cdot 2 = 0.75

\text{Fourth term: } 0

Conclusion

P(2.5) = 2 + 4.5 + 0.75 + 0 = 7.25

Therefore, using Newton Forward Interpolation, the estimated value of $y$ at $x = 2.5$ is 7.25.

Newton Forward Interpolation Calculator

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Newton Forward Interpolation Method

Formula for Newton Forward Interpolation:

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.