Iteration Method Algorithm and Flowchart

Iteration method, also known as the fixed point iteration method, is one of the most popular approaches to find the real roots of a nonlinear function. It requires just one initial guess and has a fast rate of convergence which is linear.

These algorithm and flowchart presented here and the iteration method itself are used to determine the real roots of functions in the form of an infinite series such as the geometric series, arithmetic series, Taylor’s series, and others.

Similar to the Newton-Raphson method, both the algorithm and flowchart in case of iteration method require one initial guess. And, just like the bisection method, this method has slow and linear convergence.

Iterative method is also referred to as an open bracket method or a simple enclosure method. It is based on modification approach to find the root of the function. Overall, it gives good accuracy just like the other methods.

A function f(x) = 0 is solved by the assumed initial guess. The fixed point iteration method algorithm/flowchart work in such as way that modifications alongside iteration are progressively continued with the newer and fresher approximations of the initial approximation.

Features of Iteration Method:

• No. of initial guesses – 1
• Type – open bracket
• Rate of convergence – fast
• Convergence – linear
• Approach – modification
• Accuracy – good
• Programming effort – easy

Iteration Method Algorithm:

1. Start
2. Read values of x0 and e.
   *Here x0 is the initial approximation
   e is the absolute error or the desired degree of accuracy, also the stopping criteria*
3. Calculate x1 = g(x0)
4. If [x1 – x0] <= e, goto step 6.
   *Here [ ] refers to the modulus sign*
5. Else, assign x0 = x1 and goto step 3.
6. Display x1 as the root.
7. Stop

Iteration Method Flowchart:

Also see,
Iteration Method C Program

This fixed point iteration method algorithm and flowchart comes to be useful in many mathematical formulations and theorems. Often, approximations and solutions to iterative guess strategies utilized in dynamic engineering problems are sought using this method.

Some of the well-known methods or derivations utilizing this method are Picard–Lindelöf theorem. Newton’s method, Runge-Kutta methods and Halley’s method.