Gauss's Backward Interpolation Formula in Numerical Methods

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Gauss's Backward Interpolation Formula is an interpolation method used to estimate the value of a function near the end of the data range. This technique is particularly effective when data points are equally spaced and interpolation is required at the latter portion of the dataset.

Understanding Backward Differences

Before learning about Gauss's Backward Interpolation Formula, it's important to understand the concept of backward differences. Backward differences are calculated by subtracting the preceding function values. The first-order backward difference is represented as:

\nabla f(x_n) = f(x_n) - f(x_{n-1})

Higher-order backward differences are calculated by applying the backward difference operation iteratively.

\nabla^2 f(x_n) = \nabla f(x_n) - \nabla f(x_{n-1})

Gauss's Backward Interpolation Formula

Gauss's Backward Interpolation Formula helps us estimate the value of the function $f(x)$ at a given point $x$ using known data points, particularly near the end of the dataset. The formula is:

P(x) = f(x_n) + p \nabla f(x_n) + \frac{p(p+1)}{2!} \nabla^2 f(x_n) + \frac{p(p+1)(p+2)}{3!} \nabla^3 f(x_n) + \dots

Where:

$p = \frac{x - x_n}{h}$ ,
$h$ is the interval between consecutive data points,
$\nabla f(x_n)$ is the first backward difference,
$\nabla^2 f(x_n)$ is the second backward 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 $f(0.45)$ using Gauss's Backward Interpolation Formula.

Step 1: Calculate Backward Differences

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

\nabla f(0.5) = f(0.5) - f(0.4) = 1.6487 - 1.4918 = 0.1569

The second backward difference:

\nabla^2 f(0.5) = \nabla f(0.5) - \nabla f(0.4) = 0.1569 - 0.1285 = 0.0284

Similarly, calculate higher-order backward differences if needed.

Step 2: Apply Gauss's Backward Formula

Using the formula, we calculate $f(0.45)$ . Here, $p = \frac{0.45 - 0.5}{0.1} = -0.5$ :

P(0.45) = f(0.5) + p \nabla f(0.5) + \frac{p(p+1)}{2!} \nabla^2 f(0.5)

P(0.45) = 1.6487 - 0.5(0.1569) + \frac{-0.5(-0.5 + 1)}{2}(0.0284)

P(0.45) = 1.6487 - 0.0785 + 0.00355

P(0.45) \approx 1.57375

Applications of Gauss's Backward Interpolation Formula

Gauss's Backward Interpolation Formula is commonly used in fields like numerical analysis, engineering, and computer science for:

Interpolating values near the end of the dataset.
Estimating function values for equally spaced data points.
Reducing computational effort when working with data near the endpoint.

Conclusion

Gauss's Backward Interpolation Formula is an effective method for estimating function values near the end of a dataset when the data points are equally spaced. Using backward differences, this method provides accurate results with fewer computations.