Newton’s interpolation using divided differences and difference table

Newton's interpolation is a mathematical method used to approximate the values of a function at points between known data points. It relies on constructing an interpolating polynomial that passes through the given data points, allowing us to estimate the function's value at any desired point within the range of the data.

One of the popular techniques for performing Newton's interpolation is using divided differences and constructing a difference table. The steps involved in this method are as follows:

Step 1: Given Data Points Suppose we have a set of data points (x0, y0), (x1, y1), ..., (xn, yn), where xi's are distinct data points, and yi's are the corresponding function values.

Step 2: Compute the Divided Differences Divided differences are the coefficients of the interpolating polynomial. They are computed as follows:

a) Zeroth Divided Differences: f[x0] = y0, f[x1] = y1, ..., f[xn] = yn.

b) First Divided Differences: f[x0, x1] = (f[x1] - f[x0]) / (x1 - x0), f[x1, x2] = (f[x2] - f[x1]) / (x2 - x1), ..., f[xn-1, xn] = (f[xn] - f[xn-1]) / (xn - xn-1).

c) Second Divided Differences: f[x0, x1, x2] = (f[x1, x2] - f[x0, x1]) / (x2 - x0), f[x1, x2, x3] = (f[x2, x3] - f[x1, x2]) / (x3 - x1), ..., f[xn-2, xn-1, xn] = (f[xn-1, xn] - f[xn-2, xn-1]) / (xn - xn-2).

Continue this process to find higher-order divided differences.

Step 3: Construct the Difference Table The difference table organizes the divided differences in a triangular form, making it easier to compute the interpolating polynomial coefficients. The first column contains the x-values, the second column contains the y-values (function values), and subsequent columns contain the divided differences.

Here's an example difference table:

x	f(x)	f[x0, x1]	f[x0, x1, x2]	...
x0	f[x0]
x1	f[x1]	f[x0, x1]
x2	f[x2]	f[x1, x2]	f[x0, x1, x2]
...	...	...	...	...
xn	f[xn]	f[xn-1, xn]	f[xn-2, xn-1, xn]	...

Step 4: Formulate the Interpolating Polynomial The interpolating polynomial can be written as follows:

P(x) = f[x0] + f[x0, x1] * (x - x0) + f[x0, x1, x2] * (x - x0) * (x - x1) + ...

Each term in the polynomial incorporates divided differences to approximate the function's value at a given x.

Step 5: Evaluate the Interpolating Polynomial Once we have constructed the interpolating polynomial, we can use it to estimate the function's value at any desired point within the range of the data.

Keep in mind that while Newton's interpolation provides good approximations for points within the range of the given data, it may not be as accurate outside that range due to the effect of the Runge phenomenon. In such cases, other interpolation or approximation methods may be more suitable.