Rate of convergence of iterative methods

The Newton-Raphson method and the Secant method are both iterative techniques used for finding the roots of a given function. While they serve the same purpose, they have different approaches and characteristics.

Newton-Raphson Method:

Algorithm:
- Start with an initial guess $x_{0}$ .
- At each iteration, compute the next approximation
  $x_{n + 1}$ using the formula:
  
  xn+1=xn−f′(xn)f(xn)
- Repeat the process until the desired level of accuracy is achieved or until convergence criteria are met.
Advantages:
- Generally converges faster than the Secant method, especially when the initial guess is close to the root and the function is well-behaved.
- Provides quadratic convergence for simple roots.
Disadvantages:
- Requires computation of both the function $f (x)$ and its derivative $f^{'} (x)$ , which may not always be available or easy to compute.
- Convergence may fail if the initial guess is far from the root or if the function has complex behavior near the root.

Secant Method:

Algorithm:
- Start with two initial guesses $x_{0}$ and $x_{1}$ .
- At each iteration, compute the next approximation $x_{n + 1}$ using the formula: $x_{n + 1} = x_{n} - f (x_{n}) \times \frac{x ^{n} - x ^{n - 1}}{f ( x ^{n} ) - f ( x ^{n - 1} )}$
- Repeat the process until the desired level of accuracy is achieved or until convergence criteria are met.
Advantages:
- Does not require computation of the derivative, making it applicable when the derivative is unavailable or difficult to compute.
- Can still achieve convergence, albeit at a slower rate compared to the Newton-Raphson method.
Disadvantages:
- Generally converges slower than the Newton-Raphson method, especially when the initial guesses are far from the root.
- Does not exhibit quadratic convergence; convergence rate is closer to linear.

Comparison:

Newton-Raphson method requires the derivative of the function, while the Secant method does not.
Newton-Raphson method typically converges faster when the initial guess is close to the root and the function is smooth.
Secant method is more versatile and applicable to a wider range of functions, especially when the derivative is hard to compute or unavailable.
Both methods require careful selection of initial guesses and may encounter convergence issues depending on the nature of the function and the chosen initial points.

the choice between the Newton-Raphson method and the Secant method depends on the specific requirements of the problem, including the availability of the derivative and the desired convergence rate.