Question

The convergence of which of the following method depends on initial assumed value

• A. Simpson Method
• B. Newton-Raphson Method
• C. Euler Method
• D. Runge-Kutta method

Hint:

Newton-Raphson is a local convergence method.
Always avoid choosing an initial guess \(x_0\) where the derivative of the function is zero (\(f'(x_0) = 0\)), as this causes the division term in the formula to fail.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Numerical methods are used to solve mathematical problems like integration, root-finding, and differential equations when analytical solutions are difficult to obtain.
Some numerical methods are iterative and require an initial guess (or initial assumed value) to start the computation.

Step 2: Detailed Explanation:
Let us examine the purpose and characteristics of each method:
- Simpson's Method: This is a technique for numerical integration (approximating definite integrals using parabolas). It does not require an initial guess or depend on iterative convergence.
- Newton-Raphson Method: This is an iterative root-finding algorithm used to solve non-linear equations of the form \(f(x) = 0\).
The formula is:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] Because the method uses local tangent lines, its convergence depends heavily on the initial starting value \(x_0\).
If \(x_0\) is chosen close to the actual root, the method converges quadratically.
If \(x_0\) is chosen poorly (for example, near a local extremum where \(f'(x_n) \approx 0\)), the method can fail, fluctuate, or diverge.
- Euler's Method and Runge-Kutta Method: These are step-by-step numerical methods used to solve ordinary differential equations (ODEs). While they require an initial value of the variable (initial condition), their stability and error propagation depend on the step size \(h\) rather than the "convergence to a root" depending on an assumed guess.
Therefore, the Newton-Raphson method is the one whose convergence depends on the initial assumed value.

Step 3: Final Answer:
The correct option is (B).