How to Use Newton Raphson Method: Step-by-Step Process & Example
The concept of Newton Raphson Method is essential in mathematics and helps in solving real-world and exam-level problems efficiently.
Understanding Newton Raphson Method
Newton Raphson Method is an iterative numerical method used to find roots (solutions) of a real-valued function. The method starts with an initial guess and uses calculus, specifically derivatives, to improve the accuracy of the solution with each iteration. This concept is widely used in root-finding algorithms, numerical analysis, and engineering problem-solving.
Formula Used in Newton Raphson Method
The standard formula is: \( x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})} \) where:
- \( x_{n} \) is the current approximation
- \( f(x) \) is the function whose root is being determined
- \( f'(x) \) is the derivative of the function
Repeat the process until satisfactory accuracy is reached.
Here’s a helpful table to understand the Newton Raphson Method variables:
Newton Raphson Method Table
| Symbol | Meaning | Role |
|---|---|---|
| \( x_{n} \) | Current guess of root | Input |
| \( f(x_{n}) \) | Function value at guess | Used in update |
| \( f'(x_{n}) \) | Derivative at guess | Used in update |
| \( x_{n+1} \) | Next, improved approximation | Output/next step |
This table shows how Newton Raphson Method updates guesses step by step using calculus.
Step-by-Step Process for Newton Raphson Method
Apply the Newton Raphson Method as follows:
1. Choose an initial approximation (\( x_0 \)) for the root of the equation \( f(x) = 0 \).2. Calculate \( f(x_0) \) and \( f'(x_0) \).
3. Find the next approximation using the formula:
4. Use \( x_1 \) as the new guess; repeat steps 2 and 3 to find \( x_2 \), \( x_3 \), etc.
5. Continue this process until two successive guesses are close enough (difference is less than a chosen tolerance).
Worked Example – Solving a Problem
Example: Find a root of the equation \( x^3 - 2x - 5 = 0 \) using the Newton Raphson Method. Start with \( x_0 = 2 \).
1. Set \( f(x) = x^3 - 2x - 5 \), \( f'(x) = 3x^2 - 2 \), \( x_0 = 2 \).2. Calculate \( f(2) = 8 - 4 - 5 = -1 \) and \( f'(2) = 12 - 2 = 10 \).
3. Apply the formula:
4. Next, \( f(2.1) = (2.1)^3 - 2\times2.1 - 5 = 9.261 - 4.2 - 5 = 0.061 \)
\( f'(2.1) = 3 \times (2.1)^2 - 2 = 3 \times 4.41 - 2 = 13.23 - 2 = 11.23 \)
5. Update:
6. Repeat with \( x_2 \) for higher accuracy as needed.
Thus, the root is approximately 2.0946.
Common Applications of Newton Raphson Method
The Newton Raphson Method has many practical uses:
- Solving nonlinear algebraic equations in mathematics and engineering
- Finding square roots, cube roots, or reciprocal values numerically
- Electrical power flow analysis in engineering
- Used in scientific calculators and computer algorithms
- Optimization in machine learning and data science
Convergence & Limitations
Newton Raphson Method converges quickly (quadratic convergence) if the initial guess is close to the actual root and the function and its derivative are well-behaved near the root. It may fail or diverge if the derivative is zero or changes rapidly, or if the initial guess is far from the real solution.
Tip: Always check \( f'(x) \ne 0 \) and choose a good starting point.
Newton Method in Machine Learning
The Newton Method (a variant of Newton Raphson Method) is used in optimization problems, especially for finding minima in machine learning algorithms. Instead of solving \( f(x) = 0 \), it helps minimize loss functions by iteratively updating parameters using derivatives (and sometimes the Hessian matrix for multivariate functions).
Example code (Python pseudo):
x_new = x_old - f_prime(x_old) / f_double_prime(x_old)
This is fundamental for certain regression algorithms and neural network training.
Common Mistakes to Avoid
- Using a starting value where the derivative \( f'(x) \) is zero or nearly zero
- Not checking the function’s behavior near the guess (can cause divergence)
- Stopping too early, before reaching the desired accuracy
- Confusing Newton Raphson Method with the Bisection Method
Real-World Applications
The concept of Newton Raphson Method appears in engineering design calculations, financial modeling, physics simulations, and wherever finding accurate solutions to equations is needed. Vedantu helps students see how maths applies beyond the classroom by using such techniques in real-life scenarios like computer-aided design or optimization of resources.
We explored the idea of Newton Raphson Method, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts.
Explore More Topics
1. Taylor Series (Basis for the iterative formula)
2. Differentiation Formula (Directly used for derivatives)
3. Polynomial (Most root-finding examples use these)
4. Integration (Connects to calculus background)
5. Differential Equations for Class 12 (Root problems arise here)
6. Double Integral (Advanced uses in multivariable Newton method)
7. Bisection Method (Comparative method for root finding)
8. Linear Equations in One Variable (Foundational for simple roots)
9. Calculus (Overall mathematical background)
10. Partial Derivative (When using multiple variables)
FAQs on Newton Raphson Method Explained for Students
1. What is the Newton Raphson Method?
The Newton Raphson Method is a powerful numerical technique used to find approximate roots of nonlinear equations. It employs the function’s derivative to iteratively improve the accuracy of the root estimate by using the formula xn+1 = xn - f(xn) / f'(xn) until the desired precision is achieved.
2. What is the Newton Raphson Method formula?
The main formula for the Newton Raphson Method is:
xn+1 = xn - f(xn) / f'(xn)
where xn is the current approximation, f(xn) is the function value at xn, and f'(xn) is its derivative. This formula is repeatedly applied to obtain increasingly accurate root approximations.
3. How do you solve an equation using Newton Raphson Method?
To solve an equation using the Newton Raphson Method, follow these steps:
1. Choose an initial guess x0 close to the suspected root.
2. Calculate f(xn) and f'(xn).
3. Compute the next approximation using xn+1 = xn - f(xn) / f'(xn).
4. Repeat step 2 and 3 until the value converges within the required tolerance or decimal places.
This iterative process yields a precise root approximation.
4. Where is Newton Raphson Method used?
The Newton Raphson Method finds applications in:
• Solving nonlinear equations in calculus and engineering.
• Finding roots of polynomial and transcendental equations.
• Power flow analysis in electrical engineering.
• Optimization problems in machine learning.
• Programming implementations in languages like Python, MATLAB, and C.
It is widely preferred due to its rapid quadratic convergence under suitable conditions.
5. What are the conditions for convergence of Newton's method?
The Newton Raphson Method converges rapidly if:
• The function f(x) is differentiable near the root.
• The derivative f'(x) at the root is non-zero (f'(x) ≠ 0).
• The initial guess is sufficiently close to the actual root.
• The condition |f(x) × f''(x)| < |f'(x)|2 holds.
If these conditions fail, the method may diverge or fail to find the root.
6. Is Newton-Raphson Method in the formula book for exams?
The Newton Raphson Method is typically included in formula books or exam materials for board exams and competitive tests, especially in the numerical methods or calculus sections. Students are advised to learn the formula and understand the algorithm steps, as it is frequently tested for root-finding and iterative methods questions.
7. Why might Newton Raphson fail for some starting points?
The Newton Raphson Method can fail to converge or give incorrect roots if:
• The initial guess is far from the actual root.
• The derivative f'(x) is zero or very close to zero at the approximation point, causing division errors.
• The function is not well-behaved or discontinuous near the root.
• The root is a multiple root (repeated root), which slows convergence.
Therefore, choosing a careful initial guess and analyzing function behavior is important.
8. Why do students confuse Newton Raphson with the bisection method?
Students often confuse the Newton Raphson and bisection methods because both are iterative root-finding techniques. However:
• Newton Raphson uses the function’s derivative and converges faster (quadratic convergence).
• Bisection method only uses function values, is guaranteed to converge but more slowly (linear convergence).
• Newton Raphson requires careful initial guesses; bisection method requires an interval where function changes sign.
Understanding these fundamental differences helps clarify confusion.
9. What errors can occur if the derivative is zero at a guess?
If the derivative f'(x) is zero at an approximation point, the Newton Raphson Method can't compute the next iteration due to division by zero, resulting in:
• Calculation errors or undefined values.
• The method may stall or diverge.
• Incorrect root approximations.
This is why it is critical to check that f'(x) ≠ 0 at all steps to ensure smooth convergence.
10. Are there shortcut tricks for Newton Raphson on calculators?
While there are no universal shortcuts, calculators with programming or iterative function support can:
• Allow input of the Newton Raphson formula as a function to automate iterations.
• Use built-in root-finding features based on Newton’s method.
• Store derivatives or approximate them numerically.
Students should familiarize themselves with calculator functions to efficiently perform multiple iterations for exam or study purposes.
11. Does Newton Raphson work for all polynomial degrees?
The Newton Raphson Method can be applied to polynomials of any degree to find their roots. However:
• Convergence depends on the choice of the initial guess.
• For high-degree polynomials with multiple roots, convergence may be slow or fail.
• It works best when roots are simple (non-repeated) and well-separated.
Thus, while generally applicable, success depends on function characteristics and starting values.
12. How does Newton Raphson compare to fixed point iteration?
Newton Raphson and fixed point iteration both solve equations iteratively but differ:
• Newton Raphson uses the function’s derivative for faster (quadratic) convergence.
• Fixed point iteration uses a rearranged function g(x) and generally converges slower (linear).
• Newton Raphson can be unstable if derivative is zero or initial guess is poor.
• Fixed point iteration is simpler but may require careful function reformulation.
Choosing between them depends on the problem and convergence requirements.
