Gauss Backward Interpolation Method

Gauss Backward Interpolation is used to interpolate a value close to the middle or end of the data set. The method uses backward central differences to create an interpolation polynomial.

Formula for Gauss Backward Interpolation:

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

Where:

$y_p$ is the interpolated value at $x_p$
$y_0$ is the initial value corresponding to $x_0$
$\Delta^n y$ are the backward differences of the function values.
$p = \frac{x-x_0}{h}$ , where $h$ is the uniform difference between $x$ values.

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: Choose Initial Point

Take the point closest to $x$ as $x_0 = 1951$ , with $h = 10$ .

Step 2: Calculate p

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

Step 3: Calculate Central Differences

$x$	$p$	$y$	$\Delta y$	$\Delta^2 y$	$\Delta^3 y$	$\Delta^4 y$
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 & 5: Substitute into 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

Conclusion

P(1946) \approx 22.8984

Thus, the interpolated value of $y$ at $x = 1946$ is approximately $22.8984$ .

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Gauss Backward Interpolation Method

Gauss Backward Interpolation is used to interpolate a value close to the middle or end of the data set. The method uses backward central differences to create an interpolation polynomial.

Formula for Gauss Backward Interpolation:

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

Where:

$y_p$ is the interpolated value at $x_p$
$y_0$ is the initial value corresponding to $x_0$
$\Delta^n y$ are the backward differences of the function values.
$p = \frac{x-x_0}{h}$ , where $h$ is the uniform difference between $x$ values.