Question

Which of the following is a central difference operator?

• A. \(\Delta f(x_i) = f\left(\frac{x_i + h}{2}\right) - f\left(\frac{x_i - h}{2}\right)\)
• B. \(\nabla f(x_i) = f(x_i + h) - f(x_i)\)
• C. \(f\left(x_i + \frac{h}{2}\right) - f\left(x_i - \frac{h}{2}\right)\)
• D. \(E f(x_i) = f(x_i + h)\)

Hint:

Exam Tip:
Difference operators:

• Forward difference: \(\Delta f(x) = f(x + h) - f(x)\).
• Backward difference: \(\nabla f(x) = f(x) - f(x - h)\).
• Central difference: \(\delta f(x) = f(x + h/2) - f(x - h/2)\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In numerical analysis, difference operators are used to approximate derivatives.
The central difference operator is defined as: \[ \delta f(x) = f\left(x + \frac{h}{2}\right) - f\left(x - \frac{h}{2}\right) \]

Step 2: Analyzing the Options:

• (A) \(\Delta f(x_i) = f\left(\frac{x_i + h}{2}\right) - f\left(\frac{x_i - h}{2}\right)\).
This is not the standard forward difference. The forward difference is \(\Delta f(x_i) = f(x_i + h) - f(x_i)\).
So, (A) is incorrect.
• (B) \(\nabla f(x_i) = f(x_i + h) - f(x_i)\).
This is the forward difference, not the backward difference. The backward difference is \(\nabla f(x_i) = f(x_i) - f(x_i - h)\).
So, (B) is incorrect.
• (C) \(f\left(x_i + \frac{h}{2}\right) - f\left(x_i - \frac{h}{2}\right)\).
This is exactly the central difference operator.
So, (C) is correct.
• (D) \(E f(x_i) = f(x_i + h)\).
This is the shift operator, not a difference operator.
So, (D) is incorrect.

Step 3: Final Answer:
Therefore, option (C) is correct.