Newton Raphson Method

Newton Raphson Method Meaning

• Choosing an initial approximation
• Computing the function's value and its derivative at the approximation point
• Updating the approximation using the Newton Raphson formula
• Repeating the above steps until a satisfactory level of accuracy is achieved

How does Newton Raphson Method formula work?

 
    x_1 = x_0 - [(x_0)^3 - (x_0)^2 -1] / [3*(x_0)^2 - 2x_0]
       = 1 - [(1)^3 - (1)^2 -1] / [3*(1)^2 - 2*1]
       = 1.5

Analysing the Newton Raphson Method

Advantages and Disadvantages of Newton Raphson Method

• Speed: The Newton Raphson Method exhibits quadratic convergence, which essentially means that the number of correct digits approximately doubles with each iteration. Thus, the method rapidly converges towards the root, saving considerable computational effort.
• Flexibility: The method can handle a myriad of functions, as long as the function is differentiable within a specific interval.
• Accuracy: Due to its rapid convergence, the method often produces highly accurate results, making it a useful tool for precise calculations.

• Dependency on Initial Guess: If the initial guess is too far from the actual root or if it's a bad approximation, then the method could potentially diverge from the solution rather than converging.
• Sensitive to Derivative: If the derivative of the function at the initial guess is zero, the method fails as it leads to division by zero in the Newton Raphson formula.
• Complex Functions: For functions with complex roots, the method might fail to find the correct solution.

Conditions necessary for Newton Raphson Method

Deeper Look into Newton Raphson Method

Convergence Rate of Newton Raphson Method - What you need to know

An overview of the Newton Raphson Method Algorithm

1. Start with an initial guess \(x_0\) that's reasonably close to the true root.
2. Compute the value of the function \(f(x_0)\) and its derivative \(f'(x_0)\) at \(x_0\).
3. Apply the Newton Raphson formula to calculate the next approximation: \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\).
4. Iterate the above steps until the difference between the current approximation and the next approximation is less than a specified tolerance level, or until the maximum number of iterations is reached.

    x_n = 1
    for i in range(max_iterations):
        f_x = x_n ** 3 - x_n - 1
        df_x = 3 * x_n ** 2 - 1
        x_{n+1} = x_n - f_x / df_x
        if abs(x_{n+1} - x_n) < tolerance:
            break
        x_n = x_{n+1}
    

Exploring Practicality of Newton Raphson Method

Compelling Applications of Newton Raphson Method

• Engineering: Engineers regularly confront systems and equations that require exact solutions. The Newton Raphson Method provides an efficient way of estimating roots in electrical and control engineering, mechanical system designs, and geotechnical engineering. In power systems, for instance, it's often utilised in load flow studies to solve non-linear algebraic equations and estimate the state of the power network.
• Physics: In the field of Physics, the Newton Raphson Method forms the basis of various algorithms used to solve equations representing physical properties and behaviours. This includes applications within quantum mechanics and fluid dynamics. With the freedom to tweak the initial approximation, physicists can apply this technique creatively and freely.
• Economics and Statistics: Economists exploit this method to estimate parameters in econometric models, where identifying the maximum likelihood involves solving some equations iteratively. Similarly, in statistics, it's used to find roots in regression analysis and non-linear least squares methodologies.
• Environmental Science: The Newton Raphson Method is routinely used in environmental modelling, especially for handling non-linearity in the relationships between various environmental factors. This includes prediction models for air pollution, water quality, and climate change.

Case Studies Showcasing Real-Life Use of Newton Raphson Method

Newton Raphson Method in action: Experimental Lessons

Putting Newton Raphson Method to the Test: Real-World Examples

Step by Step Guide: How to Apply the Newton Raphson Method

    x_n = 1
    tolerance = 10 ** (-10)

    for i in range(100):
        f_x = x_n ** 3 - x_n - 1
        df_x = 3 * x_n ** 2 - 1
        x_{n+1} = x_n - f_x / df_x
        if abs(x_{n+1} - x_n) < tolerance:
            break
        x_n = x_{n+1}