Finite Difference Operators
Definitions, Identities, and Relationships
1. The 6 Fundamental Operators
Basic Definitions
Forward (\(\Delta\))
\(\Delta f(x) = f(x+h) – f(x)\)
Next – Current
Backward (\(\nabla\))
\(\nabla f(x) = f(x) – f(x-h)\)
Current – Previous
Shift (\(E\))
\(E f(x) = f(x+h)\)
Step Forward
Inv. Shift (\(E^{-1}\))
\(E^{-1} f(x) = f(x-h)\)
Step Back
Central (\(\delta\))
\(\delta f(x) = f(x+\frac{h}{2}) – f(x-\frac{h}{2})\)
Half Ahead – Half Back
Mean (\(\mu\))
\(\mu f(x) = \frac{1}{2}[f(x+\frac{h}{2}) + f(x-\frac{h}{2})]\)
Average of Halves
2. Derived Identities & Second Order
Important Formulas
Common Combinations
Sum of Operators
$$ \Delta + \nabla = 2\mu\delta $$
(2 × Mean × Central)
Difference of Operators
$$ \Delta – \nabla = \delta^2 $$
(Square of Central Difference)
Unity Product
$$ (1+\Delta)(1-\nabla) = 1 $$
Diff-Product Eq.
$$ \Delta – \nabla = \Delta\nabla $$
(Diff equals Product)
Second Order Formulas
Forward Sq. (\(\Delta^2\))
\(f(x+2h) – 2f(x+h) + f(x)\)
Backward Sq. (\(\nabla^2\))
\(f(x) – 2f(x-h) + f(x-2h)\)
Fundamental Relations:
\(E = 1 + \Delta\)
\(\nabla = 1 – E^{-1}\)
\(\delta = E^{1/2} – E^{-1/2}\)
\(\mu = \frac{1}{2}(E^{1/2} + E^{-1/2})\)
Relation to Calculus (D):
\(E = e^{hD}\)
\(\Delta = e^{hD} – 1\)
\(\delta = 2 \sinh(hD/2)\)
\(\mu = \cosh(hD/2)\)
\(E = 1 + \Delta\)
\(\nabla = 1 – E^{-1}\)
\(\delta = E^{1/2} – E^{-1/2}\)
\(\mu = \frac{1}{2}(E^{1/2} + E^{-1/2})\)
Relation to Calculus (D):
\(E = e^{hD}\)
\(\Delta = e^{hD} – 1\)
\(\delta = 2 \sinh(hD/2)\)
\(\mu = \cosh(hD/2)\)
Numerical Analysis Notes
Divided Difference Properties
1. Relation to Derivatives (Mean Value Theorem)
If a function \( f(x) \) is differentiable \( n \) times, the divided difference involving \( n+1 \) points relates to the \( n \)-th derivative.
\[ f[x_0, x_1, \dots, x_n] = \frac{f^{(n)}(\xi)}{n!} \]
Where:
- \(\xi\) is some unknown value inside the interval \((x_0, x_n)\).
- \(n!\) is the factorial of the order.
Implication: If \( f(x) \) is a polynomial of degree \( n \), the \( n \)-th divided difference is constant, and the \( (n+1) \)-th difference is zero.
2. Divided Difference of \( f(x) = \frac{1}{x} \)
For the reciprocal function, there is a specific pattern for any order \( k \).
\[ f[x_0, x_1, \dots, x_k] = \frac{(-1)^k}{x_0 \cdot x_1 \cdot \dots \cdot x_k} \]
Examples:
- 1st Order: \( f[x_0, x_1] = \frac{-1}{x_0 x_1} \)
- 2nd Order: \( f[x_0, x_1, x_2] = \frac{1}{x_0 x_1 x_2} \)
- 3rd Order: \( f[x_0, x_1, x_2, x_3] = \frac{-1}{x_0 x_1 x_2 x_3} \)