For the fixed point iteration method; re-arranging method, how can I describe ho the iterative formula works?
The fixed point iteration method is an unsystematic method of solving a frequently complex equation by iteration.
The advantage of the method is its simplicity. We do not need to calculate the first or higher derivative. Thus it would work even with tabular functions. Another advantage, which is common to most iterative methods, is that it is self correcting. If a minor calculation error is made, it will still converge to the correct answer at the end.
The disadvantage of the method is that it is unsystematic. We do not know if it will converge to a solution, and if it does, which solution. The way to obtain different solutions is to rearrange the iterative formula. Sometimes by rearranging, a different solution can be obtained.
For more information, see:
http://en.wikipedia.org/wiki/Fixed_point_iteration
Post if you need clarifications.
How can I say that the re-arranging method is less efficient with using the iterative formula than the Newton - Raphson Method?
The fixed point iteration method is less efficient (in general) because the efficiency is undefined and problem dependent.
The Newton-Raphson method depends on the derivative of the function. With very few exceptions, such as near asymptotes, convergence is second-order. In most cases, the number of digits of accuracy doubles with each iteration.
With fixed-point iteration, convergence is not even guaranteed, and not rapid, unless it "happens" to coincide with the Newton-Raphson formula.
In the fixed-point iteration method, the re-arranging technique is used to find a solution to a given equation in the form of x = g(x), where g(x) is a function. The concept behind this approach is to start with an initial guess for the solution and then iteratively improve the guess until it converges to the desired solution.
To describe how the iterative formula works, follow these steps:
1. Start with an initial guess value for x, denoted as x₀.
2. Apply the function g(x) to the initial guess, which gives g(x₀).
3. Update the guess value by assigning the result of g(x₀) back to x. This becomes the new guess, denoted as x₁: x₁ = g(x₀).
4. Repeat steps 2 and 3, using the latest guess value xᵢ to calculate the next guess xᵢ₊₁: xᵢ₊₁ = g(xᵢ).
5. Continue this process iteratively, calculating x₃, x₄, x₅, and so on, until the desired level of accuracy or the convergence criteria is met.
The iterative formula xᵢ₊₁ = g(xᵢ) allows you to update the guess value by plugging in the previous guess into the function g(x). This process continues until the sequence of guesses converges to the solution of the equation, meaning that successive guesses become increasingly close to each other.
It's important to note that for the method to work, the function g(x) must be chosen carefully to ensure convergence and meet certain conditions, such as being continuous and differentiable in a given interval. Additionally, the initial guess should be reasonably close to the actual solution for efficient convergence.