Method of iteration

Introduction

Iterative approaches are useful for solving and finding the roots of nonlinear equations. It is a method for obtaining increasingly precise solutions to a linear equation through successive approximations. Such a method necessitates a large number of iterations of mathematical operations to arrive at a solution, and computers are frequently used in the process to make the task simple and efficient.

Now, for example of Method of Iteration, let’s take a quadratic equation:

a0 x2 + a1x + a2

Now using the Method of Iteration, the solution of the above equation will be,

Xk+1 = – a2+a0x k2/a1     where, k=0,1,2.. 

Or 

Xk+1 = – a2/a0x k2+a1     where, k=0,1,2.. 

Discussion

Historical remarks

There are no real records or beginnings of the Method of Iteration however, it is believed that the first iterative method for solving a linear issue was described in a letter sent by Gauss to one of his pupils. He also suggested that a 4-by-4 system of equations may be solved by repeatedly solving the component with the greatest residual.

Later, in the early 1950s, D.M.Young made some significant claims about the repetition approach, and his work finalized his idea. 

On the other hand, three swiss mathematicians Cornelius Lanczos, Magnus Hestenes, and Eduard Stiefel, were working on the conjugate gradient technique but in the very beginning, the nature and applicability were wrongly comprehended. Later in the 1970s, it was demonstrated that conjugacy-based approaches are effective for partial differential equations, particularly those of the elliptic type. 

Example of Method of Iteration

Using Method of Iteration, find the real root of the equation x3 + x2 = 1.

Solution:

Given equation is: – x3 + x2 = 1

We can rewrite it as x3 + x2 -1= 0

Now, let f(x) = x3 + x2 -1

Now taking the value x =0 

f(0) = -1 (positive)

Similarly at x =1

f(1) = 1 (negative)

Now we can say that the root value lies between 0 to 1

Now by solving the f(x)=0,

x3 + x2 -1= 0

x2 (x + 1) = 1

x2= 1/ (x + 1)

x= 1/ (x + 1)

pi(x) =  1/ (x + 1)

x1 = pi(x0) = 1/1+ 0.5 = 0.81649

x2 = pi(x1) = 1/1+ 0.81649 = 0.74196

x3 = pi(x2) = 1/1+ 0.74196 = 0.75767

x4= pi(x3) = 1/1+ 0.75767 = 0.75427

x5 = pi(x4) = 1/1+ 0.75427 = 0.75500

x6 = pi(x5) = 1/1+ 0.75500 = 0.75485

x7 = pi(x6) = 1/1+ 0.75485 = 0.75488

After observing the results, the difference between x6 and x7is very small, hence the root of the equation will be 0.75488.

Types of method of iteration

Methods of Iteration are categorized on the basis of the number of initial approximation value iterations. The two categories are mentioned below:

1. Bracketing iteration methods
2. Open end iteration methods

Bracketing iteration method

This technique of Method of Iteration is also known as interpolation methods. These procedures begin with two initial roots enclosed in brackets, then gradually reduce the width of the bracket until the desired result is obtained. In other words, a root is found when the interval is small enough. They typically use the intermediate value theorem, which asserts that if the endpoints of an interval have values with opposing signs, the function must have at least one root in the interval. As a result, they must begin with an interval such that the function takes opposite signs at the interval’s endpoints. In the case of polynomials, however, there are additional ways for determining the number of roots in an interval (Descartes’ rule of signs, Badan’s theorem, and Sturm’s theorem). They lead to efficient methods for polynomial real-root isolation, ensuring that all real roots are found with guaranteed accuracy.

This method is further subdivided into 2 more categories:

1. Bisection method
2. Regular-falsi method

Open-end iteration methods

Extrapolation procedures are what they’re called. This Method of Iteration starts with one or two initial roots that don’t require the use of brackets. Open-end iteration is a technique for predicting the value of a variable beyond the original observation range based on its connection with another variable. Extrapolation is similar to interpolation in that it provides estimates between known observations, but it is more uncertain and has a larger chance of generating nonsensical results. Extrapolation may also refer to expanding a technique based on the presumption that analogous procedures will be helpful in the future.

This method is further subdivided into three more categories

1. Newton Raphson method
2. Secant method
3. Muller’s method

Conclusion

Iteration methods are also frequently used to find approximate solutions to stationary and evolutionary problems that involve differential equations. These techniques work by transforming differential issues into integral problems. If one can show that the corresponding integral operator is subject to the Banach theorem’s criteria, then all of the general theory’s results apply to the case at hand.