What are that iteration methods compare different iterative method?

What are that closed circuit methods equalise different iterative method?What are that loop-the-loop methods compare different iterative method?What are the cringle methods?An iterative method is a powerful device of solving and finding the roots of the non linear equations. It is a process that uses successive approximations to obtain more accurate solutions to a linear system at each step. Such a method involves a jumbo number of iterations of arithmetic operations to arrive at a solution for which the computers are very often used in its process to make the parturiency simple and efficient. Iteration means the act of repeating a process usually with the aim of approaching a desired goal or rear or result. Each repetition of the process is also called iteration and the results of one iteration are used as the starting point for the next iteration.For example, to sack the quadratic equation we may choose any one of the following iteration methods a0x2 +a1x+a2=0 a) Xk+1 = -a2 +a0xk2/a1, k=0,1,2b) Xk+1= -a2/a0xk+a1,k=0,1,2-Types of iteration methodsBased upon the number of initial approximation values iteration methods can be divided into two categories Bracketing iteration methods Open end iteration methodsBracketing iteration methodThese methods are also known as interpolation methods. Under these methods we start with two initial roots that in support, then systematically quash the width of the bracket until the desired solution is arrived at. There are two popular methods under this category Bisection method Regular_falsi methodOpen end iteration method these methods are known as extrapolation methods. Under these methods we start with one or two initial roots that do not need the bracket the root.These methods are various types Netwon_raphson method Secant method Mullers methodBisection, regular_falsi and netwon_raphson methods are under root finding algorithm.Root finding algoA root-finding algorithm is a numerical method, or algorithm, for findin g a value x such that f(x) = 0, for a given mold f. Such an x is called a root of the function f.Iteration method is obtain the initial approximation to the root is based upon the intermediate value theorem.This theorem is states that if f(x) is continuous function on some intervala,b and f(a).f(b)