Interpolation using Newton’s Forward and Backward Difference Formulae

Definition

Interpolation is the mathematical technique of estimating unknown values of a function that lie between known data points. Newton’s difference formulae are specifically designed for datasets where the independent variable (x) has a constant difference between consecutive points (equispaced data). Newton’s Forward Difference Formula is used for interpolating near the beginning of a table, while the Backward Difference Formula is used for values near the end of a table.

Main Content

1. Finite Differences and Operators

The Forward Difference operator ($\Delta$) is defined as $\Delta y_i = y_{i+1} - y_i$. It measures how much the value changes as we move forward in the table.
The Backward Difference operator ($\nabla$) is defined as $\nabla y_i = y_i - y_{i-1}$. It measures the change as we look at the previous data point.

2. Newton’s Forward Difference Formula

This formula is applied when we need to find $y$ at a value of $x$ that is closer to the top of the data table.
The formula is: $y_p = y_0 + p\Delta y_0 + \frac{p(p-1)}{2!} \Delta^2 y_0 + \frac{p(p-1)(p-2)}{3!} \Delta^3 y_0 + \dots$ where $p = \frac{x - x_0}{h}$.

3. Newton’s Backward Difference Formula

This formula is applied when we need to find $y$ at a value of $x$ that is closer to the bottom of the data table.
The formula is: $y_p = y_n + p\nabla y_n + \frac{p(p+1)}{2!} \nabla^2 y_n + \frac{p(p+1)(p+2)}{3!} \nabla^3 y_n + \dots$ where $p = \frac{x - x_n}{h}$.

Working / Process

1. Construct the Difference Table

List the known values of $x$ and $f(x)$ in columns.
Calculate successive differences (first order, second order, etc.) by subtracting the previous value from the current value until only one entry remains.

x    y    Δy    Δ²y    Δ³y
x0   y0
          Δy0
x1   y1         Δ²y0
          Δy1          Δ³y0
x2   y2         Δ²y1
          Δy2
x3   y3

2. Determine the Interpolation Point

Identify if your target $x$ is at the beginning or end of the data range.
If $x$ is near $x_0$ (top), use Forward Formula. If $x$ is near $x_n$ (bottom), use Backward Formula.
Calculate the value of $p$ using the formula $p = (x - x_{base}) / h$, where $h$ is the common difference between $x$ values.

3. Apply the Formula

Substitute the value of $p$ and the corresponding difference values (either the top diagonal for Forward or the bottom diagonal for Backward) into the respective Newton formula.
Solve the equation algebraically to find the estimated value $y_p$.

Advantages / Applications

Newton's formulae are highly efficient for computer algorithms because they rely on simple arithmetic operations rather than complex integration or differentiation.
They are widely used in engineering and physics to predict physical behavior (like temperature or pressure) at intermediate points when continuous data is unavailable.
These methods are excellent for constructing polynomial curves that pass through a specific set of discrete data points.

Summary

Interpolation using Newton’s difference formulae is a fundamental numerical method for estimating values within an equispaced dataset. The forward formula is primarily used for interpolation at the start of a series, while the backward formula serves the end of the series. Key terms to remember include the 'Forward Difference Operator' ($\Delta$), the 'Backward Difference Operator' ($\nabla$), the step size ($h$), and the ratio ($p$). It is a cornerstone of numerical analysis for predicting unknown trends from known experimental data.