Gauss Backward Interpolation Method

Gauss Backward Interpolation is used to interpolate a value close to the beginning of the data set. The method uses Backward differences to create an interpolation polynomial.

P = y_0 + p \Delta y_{-1} + \frac{(p+1)p}{2!} \cdot \Delta^2 y_{-1} + \frac{(p+1)p(p-1)}{3!} \cdot \Delta^3 y_{-2} + \frac{(p+2)(p+1)p(p-1)}{4!} \cdot \Delta^4 y_{-2} +

Where:

$y_p$ is the interpolated value at $x_p$
$y_0$ is the initial value corresponding to $x_0$ its the First value of $x_p$ greater than $x$ given, to interpolate.
$\Delta^n y$ are the Backward differences of the function values taken from the central difference table.
$p = \frac{x-x_0}{h}, h$ is the uniform difference between the $x$ values.
The terms $p,(p-1),(p+1) \dots$ are the factors involving the relative position of 𝑥 with respect to the data points.

Example of Gauss Backward Interpolation

Given the following data points:

x	y
1931	15
1941	20
1951	27
1961	39
1971	52

We are tasked with finding $y_p$ where $x = 1946$ .

Step 1: taking the closest to $x \text{ as } x_0 = 1951 ,$ $h = x_2 - x_1 = 1931 - 1941$

Step 2: Use the formula $p = \frac{x - x_0}{h}$ .

p = \frac{1946 - 1951}{10} = -0.5

Step 3: Calculate the central differences for the $y$ values.

x	p	y	$\Delta y$	$\Delta^2 y$	$\Delta^3 y$	$\Delta^4 y$
1931	-20	15
1941	-10	20	5	2
1951	0	27	7	5	3	7
1961	10	39	12	1	-4
1971	20	52	13

Step 4: Apply the Gauss Backward Interpolation formula:

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

Step 5: Substituting the values into the interpolation formula:

P(1946) = 27 + (-0.5) \cdot 7 + \frac{(-0.5+1)∗(-0.5)}{2!} \cdot 5 + \frac{(-0.5+1)∗(-0.5)∗(-0.5−1)}{3!} \cdot 3 + \frac{(-0.5+2)(0.5+1)∗(-0.5)∗(0.5−1)}{4!} \cdot -7

Step 6: After solving the equation, the interpolated value of $y$ at $x = 1946$ is approximately:

P(1946) \approx 22.8984375

This is the final result of the Gauss Backward Interpolation method.

Gauss Backward Interpolation Calculator

Gauss Backward Interpolation Method

Example of Gauss Backward Interpolation

Gauss Backward Interpolation Calculator

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