Difference tables Polynomial Interpolation: Newton’s forward

Newton’s forward and backward interpolation formulas are used to construct polynomial interpolants for a given set of data points. These formulas are based on the concept of finite differences. Here’s an explanation of both methods:

1. Newton’s Forward Interpolation Formula:

Newton’s forward interpolation formula is used to approximate the value of a function at a point within the range of given data points. It’s particularly useful when the data points are equally spaced. The formula is derived using divided difference tables.

Let’s say we have
$1$

$n + 1$ data points $(x_{0}, y_{0}), (x_{1}, y_{1}), \dots, (x_{n}, y_{n})$ , where $x_{i}$ ‘s are equally spaced.

The forward difference table is constructed as follows:

The first column contains the given $x$ -values.
The second column contains the corresponding
$y$ -values.
The

The forward difference

$Δ^{k} y_{i}$ is defined as:

$Δ^{0} y_{i} = y_{i}$

$Δ^{k} y_{i} = Δ^{k - 1} y_{i + 1} - Δ^{k - 1} y_{i}$

The Newton’s forward interpolation formula is given by:

$P_{n} (x) = y_{0} + \sum_{k = 1 n} \frac{Δ ^{k} y ^{0}}{k !} (x - x_{0}) (x - x_{1}) \dots (x - x_{k - 1})$

Where:

$P_{n} (x)$ is the polynomial interpolant of degree
$y_{0}$ is the value of the function at the first data point.
$Δ^{k} y_{0}$ is the
$(x - x_{0}) (x - x_{1}) \dots (x - x_{k - 1})$ are the forward difference operators.

2. Newton’s Backward Interpolation Formula:

Newton’s backward interpolation formula is similar to the forward formula, but it’s used when the data points are equally spaced in the reverse direction (i.e., when

$x_{i}$ ‘s are decreasing).

The backward difference table is constructed similarly to the forward difference table, but we calculate backward differences instead.

The backward difference

$\nabla^{k} y_{i}$ is defined as:

$\nabla^{0} y_{i} = y_{i}$

$\nabla^{k} y_{i} = \nabla^{k - 1} y_{i} - \nabla^{k - 1} y_{i - 1}$

The Newton’s backward interpolation formula is given by:

$P_{n} (x) = y_{n} + \sum_{k = 1 n} \frac{\nabla ^{k} y ^{n}}{k !} (x - x_{n}) (x - x_{n - 1}) \dots (x - x_{n - k + 1})$

Where:

$P_{n} (x)$ is the polynomial interpolant of degree $n$ .
$y_{n}$ is the value of the function at the last data point.

$k$ -th backward difference at the last data point.
$(x - x_{n}) (x - x_{n - 1}) \dots (x - x_{n - k + 1})$ are the backward difference operators.

These formulas provide efficient ways to interpolate and approximate functions using polynomial interpolation, particularly when the data points are evenly spaced.