Question

Which method is commonly used to find roots of nonlinear equations?

• A. Euler’s method
• B. Runge–Kutta method
• C. Newton–Raphson method
• D. Gauss elimination method

Hint:

Newton–Raphson method has {quadratic convergence}, which makes it faster than many other numerical methods.

Answer 1

The Correct Option is C

Step 1: Understanding nonlinear equations.
A nonlinear equation is an equation of the form \[ f(x) = 0 \] where $f(x)$ is a nonlinear function. Such equations usually cannot be solved exactly using algebraic methods.
Step 2: Need for numerical methods.
Since analytical solutions are often not possible, numerical methods are used to approximate the roots of nonlinear equations with increasing accuracy.
Step 3: Basic idea of Newton–Raphson method.
The Newton–Raphson method is based on the idea of approximating a nonlinear function by its tangent line at a chosen point. The point where the tangent cuts the $x$-axis gives a better approximation of the root.
Step 4: Mathematical formula.
If $x_n$ is the current approximation of the root, the next approximation is given by: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] This process is repeated until the successive values converge to the actual root.
Step 5: Why Newton–Raphson is preferred.
The Newton–Raphson method converges very fast when the initial guess is close to the true root. This makes it one of the most efficient and commonly used root-finding techniques.
Step 6: Analysis of options.
(A) Euler’s method: Used for solving differential equations.
(B) Runge–Kutta method: Also used for differential equations.
(C) Newton–Raphson method: Correct — used for finding roots of nonlinear equations.
(D) Gauss elimination method: Used for solving systems of linear equations.
Step 7: Final conclusion.
The most commonly used numerical technique for finding roots of nonlinear equations is the \[ \boxed{\text{Newton–Raphson method}} \]