Gauss's Forward Interpolation Formula in Numerical Methods

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Gauss's Forward Interpolation Formula is an interpolation technique used to estimate the value of a function near the beginning of the data range. It is particularly useful when data points are equidistant. This formula is an advancement of Newton’s Forward Difference Formula, focusing on central points.

Understanding Forward Differences

Before diving into Gauss's Forward Interpolation Formula, we need to understand the concept of forward differences. Forward differences are calculated by subtracting consecutive function values. The first-order forward difference is represented as:

\Delta f(x_0) = f(x_1) - f(x_0)

Higher-order forward differences are computed by applying the forward difference operation successively.

\Delta^2 f(x_0) = \Delta f(x_1) - \Delta f(x_0)

Gauss's Forward Interpolation Formula

Gauss's Forward Interpolation Formula allows us to estimate the value of the function $f(x)$ for a point $x$ using known data points. It is given by the following formula:

P(x) = f(x_0) + p \Delta f(x_0) + \frac{p(p - 1)}{2!} \Delta^2 f(x_0) + \frac{p(p - 1)(p - 2)}{3!} \Delta^3 f(x_0) + \dots

Where:

$p = \frac{x - x_0}{h}$ ,
$h$ is the interval between consecutive data points,
$\Delta f(x_0)$ is the first forward difference,
$\Delta^2 f(x_0)$ is the second forward difference, and so on.

Example:

Consider the following data points:

$x$	$f(x)$
0.1	1.1052
0.2	1.2214
0.3	1.3499
0.4	1.4918
0.5	1.6487

We want to estimate $f(0.25)$ using Gauss's Forward Interpolation Formula.

Step 1: Calculate Forward Differences

First, we compute the forward differences. Using the given data, the first forward difference is calculated as:

\Delta f(0.1) = f(0.2) - f(0.1) = 1.2214 - 1.1052 = 0.1162

The second forward difference:

\Delta^2 f(0.1) = \Delta f(0.2) - \Delta f(0.1) = 0.1285 - 0.1162 = 0.0123

Similarly, calculate higher-order forward differences if needed.

Step 2: Apply Gauss's Forward Formula

Using the formula, we calculate $f(0.25)$ . Here, $p = \frac{0.25 - 0.1}{0.1} = 1.5$ :

P(0.25) = f(0.1) + p \Delta f(0.1) + \frac{p(p-1)}{2!} \Delta^2 f(0.1)

P(0.25) = 1.1052 + 1.5(0.1162) + \frac{1.5(1.5 - 1)}{2} (0.0123)

P(0.25) = 1.1052 + 0.1743 + 0.0046

P(0.25) \approx 1.2841

Applications of Gauss's Forward Interpolation Formula

Gauss's Forward Interpolation Formula is widely used in numerical analysis, computer science, and engineering for:

Interpolating values near the beginning of the dataset.
Estimating function values for evenly spaced data points.
Providing accurate results with fewer calculations for central points.

Conclusion

Gauss's Forward Interpolation Formula is an effective method for estimating function values when the data points are equally spaced and interpolation is required near the start of the dataset. By leveraging forward differences, this method ensures high accuracy and is computationally efficient.