Finite Difference Calculus

Worked Example: Finite Differences of a Cubic

Consider the discrete polynomial f(n) = n³. We want to find the first forward difference Δf(n) at n = 2, and reconstruct f(2) using the Newton forward difference formula.

1. Polynomial Values: We compute f(0) = 0, f(1) = 1, f(2) = 8, f(3) = 27.
2. First Forward Differences (Δ): Δf(n) = f(n+1) - f(n).
   • Δf(0) = 1 - 0 = 1
   • Δf(1) = 8 - 1 = 7
   • Δf(2) = 27 - 8 = 19
   So, the first forward difference at n = 2 is exactly 19.
3. Continuous Derivative: By contrast, the continuous derivative is f'(n) = 3n². At n = 2, f'(2) = 12, which differs from the finite difference 19.
4. Higher Differences at 0: To use the Newton reconstruction, we need differences at n = 0.
   • Δf(0) = 1
   • Δ²f(0) = Δf(1) - Δf(0) = 7 - 1 = 6
   • Δ³f(0) = Δf(2) - 2Δf(1) + Δf(0) = 19 - 2(7) + 1 = 6
5. Newton Reconstruction: f(n) = Σ (n choose j) Δ^jf(0). For n = 2: f(2) = (2 choose 0)·0 + (2 choose 1)·1 + (2 choose 2)·6 = 1·0 + 2·1 + 1·6 = 0 + 2 + 6 = 8. This perfectly reconstructs the original f(2) = 8.

Experiment: Build Exact Difference Tables

Open the Finite Difference Calculus module to explore the differences of integer polynomials.

1. Select Polynomial degree 3, Integer n 2, and Difference order 1. Read the Forward difference as 19 and the Continuous derivative as 12. Observe the highlighted node in the difference triangle visualization.
2. Change the Difference order to 3. Notice the Forward difference is now 6. For f(n) = n³, the third difference is a constant 3! = 6.
3. Set the Difference order to 5. The Forward difference becomes 0, because differences of order greater than the polynomial degree vanish perfectly in discrete calculus.
4. Change the Polynomial degree to 4 and Difference order to 2 (keeping n = 2). The Forward difference is 110, while the Continuous derivative is 48.
5. Change the Polynomial degree to 0, Integer n to 0, and Difference order to 1. Notice the polynomial evaluates to the constant 1 even at n = 0, and the forward difference is 0.

Limitations

The module explicitly uses a limited set of small integers for the polynomial degree (up to 4) and evaluation point n (up to 5). While the principles of finite difference calculus apply to all integer polynomials, the exact arithmetic in standard JavaScript integers is bounded by Number.MAX_SAFE_INTEGER (2⁵³ - 1). Higher degrees or large n values would eventually cause precision loss. Furthermore, this algebraic framework does not represent the approximate error analysis (truncation errors) of numerical differentiation over small floating-point steps, focusing entirely on exact discrete differences.

References

The following resources provide a rigorous algebraic foundation for differences and Newton series representations.

• NIST Digital Library of Mathematical Functions, Section 3.3: Differences - Covers forward, backward, and central differences.
• NIST Digital Library of Mathematical Functions, Section 26.8: Combinatorial Sums - Explains the usage of binomial coefficients in polynomial expansions and difference calculus.