Newton's Divided Difference Formula in Numerical Methods

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Newton's Divided Difference Formula is a powerful interpolation technique used to estimate the value of a function at any point given a set of known data points. This method can be applied even when the given data points are unequally spaced, which makes it more versatile compared to Newton’s Forward and Backward Difference methods.

The basic idea is to create a polynomial that fits the given data points, and the formula is expressed using divided differences.

Understanding Divided Differences

Divided differences are central to Newton's Divided Difference Formula. They provide a recursive way to compute coefficients of the interpolation polynomial. The divided difference of two points $(x_0, f(x_0))$ and $(x_1, f(x_1))$ is given by:

f[x_0, x_1] = \frac{f(x_1) - f(x_0)}{x_1 - x_0}

Higher-order divided differences are defined as follows:

f[x_0, x_1, x_2] = \frac{f[x_1, x_2] - f[x_0, x_1]}{x_2 - x_0}

Newton's Divided Difference Formula

The general form of Newton's Divided Difference Formula is:

P(x) = f(x_0) + (x - x_0) f[x_0, x_1] + (x - x_0)(x - x_1) f[x_0, x_1, x_2] + \dots + (x - x_0)(x - x_1)\cdots(x - x_{n-1}) f[x_0, x_1, \dots, x_n]

Where:

$f[x_0, x_1]$ is the first divided difference.
$f[x_0, x_1, x_2]$ is the second divided difference.
And so on for higher-order divided differences.

Divided Difference Table

A divided difference table is constructed to make the process of calculating the polynomial easier. It contains the known values of $x$ and their corresponding function values, along with columns for each divided difference.

Let's go through an example to clarify this process.

Example:

Consider the following data points:

$x$	$f(x)$
1.0	1.000
1.3	1.091
1.6	1.232
1.9	1.433
2.2	1.698

We want to estimate $f(1.5)$ .

Step 1: Construct the Divided Difference Table

The divided differences are computed as follows:

$x$	$f(x)$	$f[x_0, x_1]$	$f[x_0, x_1, x_2]$	$f[x_0, x_1, x_2, x_3]$	$f[x_0, x_1, x_2, x_3, x_4]$
1.0	1.000	0.303	0.042	-0.001	0.001
1.3	1.091	0.470	0.049	0.002
1.6	1.232	0.670	0.053
1.9	1.433	0.883
2.2	1.698

Step 2: Apply the Newton Divided Difference Formula

Using the formula, we calculate $f(1.5)$ :

P(1.5) = 1.000 + (1.5 - 1.0)(0.303) + (1.5 - 1.0)(1.5 - 1.3)(0.042) + (1.5 - 1.0)(1.5 - 1.3)(1.5 - 1.6)(-0.001)

Substituting the values:

P(1.5) = 1.000 + 0.1515 + (0.2)(0.042) + (0.2)(-0.1)(-0.001)

P(1.5) = 1.000 + 0.1515 + 0.0084 + 0.00002

Final result:

P(1.5) \approx 1.15992

Applications of Newton's Divided Difference Formula

Newton's Divided Difference Formula is widely used in various fields of numerical analysis and computer science for:

Interpolating values from unevenly spaced data points.
Calculating polynomial approximations.
Handling data points added at irregular intervals.

Conclusion

Newton's Divided Difference Formula is a flexible and efficient method for polynomial interpolation, especially when dealing with unevenly spaced data points. By constructing a divided difference table and using the recursive formula, accurate function values can be estimated.