The Limitation of Looking Forward

In our last masterclass, we built a powerful tool for predicting the unknown: the Gregory-Newton Forward Interpolation formula. It's an efficient, "upgradable" way to create a polynomialA polynomial is a friendly type of math expression with variables raised to positive whole-number powers, like 3x² + 5x - 7. They are very easy for computers to calculate. that connects our data points. But it has a specific, inherent bias: it is most accurate near the beginning of our data set.

Imagine you are analyzing a dataset of a rocket's altitude over time, with data points recorded every second from $t=0$ to $t=10$. The forward method is the perfect tool to estimate the altitude at $t=0.5$. But what if you need to know the altitude at $t=9.5$? Using the forward formula, which is anchored at $t=0$, would be like trying to aim a target by looking at where you started your journey, not where you are now. It can be inaccurate and inefficient.

To solve this, we need a different perspective. We need a formula that is anchored at the end of our data and works backwards. This is the Gregory-Newton Backward Interpolation Formula, and it's the essential other half of our interpolation toolkit.

The Backward Difference: A New Perspective on the Same Data

The engine behind the backward formula is the backward difference operator, denoted by the symbol $\nabla$ (nabla). While the forward operator $\Delta$ looked ahead, the backward operator $\nabla$ looks behind.

The Backward Difference Operator ($\nabla$)

The definition is simple: the backward difference is the current value minus the previous value.

Just like with forward differences, we can apply this operator repeatedly to find higher-order differences:

• Second Backward Difference ($\nabla^2 y_i$): $\nabla^2 y_i = \nabla y_i - \nabla y_{i-1}$.
• Third Backward Difference ($\nabla^3 y_i$): $\nabla^3 y_i = \nabla^2 y_i - \nabla^2 y_{i-1}$.

The Difference Table Revisited

Here is the most important insight: a backward difference table contains the exact same numbers as a forward difference table. The only thing that changes is the notation and which diagonal of coefficients we use for our formula.

For example, the first forward difference $\Delta y_0 = y_1 - y_0$ is identical to the first backward difference at the next point, $\nabla y_1 = y_1 - y_0$. The table is the same; only our perspective changes. The interactive lab below makes this clear.

The Gregory-Newton Backward Difference Formula

The backward formula is constructed similarly to the forward one, but it starts from the last data point, $y_n$, and uses the coefficients from the backward-pointing diagonal (highlighted in red in our lab).

We define a normalized variable $s$, but this time, it's relative to the last point, $x_n$:

The Gregory-Newton Backward Difference Formula is then:

Notice the subtle but crucial difference in the binomial coefficientA term used in combinatorics, often written as 'n choose k'. Here, it's a way of creating polynomial terms of increasing degree. terms. Instead of $s(s-1)$, we now have $s(s+1)$, and so on. This change in sign is what allows the formula to properly build the polynomial by working backwards from the end of the data set.

A Strategic Choice: Forward, Backward, or Central?

With multiple formulas at our disposal, how do we choose which one to use? The decision is based entirely on where the point we want to estimate, $x$, lies within our dataset.

• Newton's Forward Formula: Uses the top diagonal of the table ($y_0, \Delta y_0, \dots$). It is most accurate for interpolating values of $x$ that are near the beginning of the data set (i.e., close to $x_0$).
• Newton's Backward Formula: Uses the bottom diagonal of the table ($y_n, \nabla y_n, \dots$). It is most accurate for interpolating values near the end of the data set (i.e., close to $x_n$).
• Central Difference Formulas (e.g., Stirling's Formula): These more advanced formulas use a zig-zag path through the *center* of the difference table. They provide the highest accuracy for interpolating values near the middle of the data set.

Choosing the right formula for the job is a key skill. If your data is a time series and you want to predict the near future, you would use the most recent data points and the backward formula. If you are trying to fill in a missing data point at the beginning of a test, you would use the forward formula.

Solving Numerical Problems: A Step-by-Step Guide

Problem 1: Using Newton's Backward Formula

Question: The following data represents the velocity of a rocket. Use a third-order Newton's backward difference polynomial to estimate the velocity at $t=27$ seconds.

Test Your Intuition!