Scribd
Upload
110 views3 pages

Iterative Methods for Root Finding

This document discusses iterative methods for finding approximate roots of non-linear algebraic equations. It describes two categories of root finding methods: bracketing methods, which include the bisection, incremental search, and false position methods; and open methods, which include the Newton-Raphson and secant methods. The bisection method works by repeatedly bisecting the interval in which the root is known to exist, and the incremental search method finds an interval containing a root by incrementally changing the value of x until the function changes sign.

Uploaded by

MLNDG boys
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
110 views3 pages

Iterative Methods for Root Finding

This document discusses iterative methods for finding approximate roots of non-linear algebraic equations. It describes two categories of root finding methods: bracketing methods, which include the bisection, incremental search, and false position methods; and open methods, which include the Newton-Raphson and secant methods. The bisection method works by repeatedly bisecting the interval in which the root is known to exist, and the incremental search method finds an interval containing a root by incrementally changing the value of x until the function changes sign.

Uploaded by

MLNDG boys
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 3

ROOTS OF EQUATION

Iterative methods are used when we cannot find the solution of algebraic equation

by direct method. We find the approximate root of non-linear algebraic equation by

iterative methods. This approximate solution is close to exact solution but never equal

to exact solution.

Methods for Root Finding

Two Categories:

1. Bracketing Method

a. Bisection Method

b. Incremental Search Method

c. False Position Method

2. Open Method

a. Newton Raphson Method

b. Secant Method

Bisection Method

The bisection method is used to find the roots of a polynomial equation. It

separates the interval and subdivides the interval in which the root of the equation lies.

The principle behind this method is the intermediate theorem for continuous functions.

It works by narrowing the gap between the positive and negative intervals until it closes

in on the correct answer. This method narrows the gap by taking the average of the

positive and negative intervals. It is a simple method and it is relatively slow. The
bisection method is also known as interval halving method, root-finding method, binary

search method or dichotomy method.

Let us consider a continuous function “f” which is defined on the closed interval

[a, b], is given with f(a) and f(b) of different signs. Then by intermediate theorem, there

exists a point x belong to (a, b) for which f(x) =0.

Algorithms:

Step 1: Choose lower “a” and upper “b” guesses for the root such that the

function changes sign over the interval. This can be checked by ensuring

that 𝑓(𝑎) ∗ 𝑓(𝑏) < 0.

Step 2: An estimate of the root, 𝑋𝑂 , is determined by

𝑎+𝑏
𝑋𝑂 =
2

Step 3: Make the following evaluations to determine in which subinterval

the root lies:

a. If 𝑓(𝑎) ∗ 𝑓(𝑋𝑂 ) < 0, the root lies in the lower interval. Therefore,

set 𝑏 = 𝑋𝑂 and return to step 2.

b. If 𝑓(𝑎) ∗ 𝑓(𝑋𝑂 ) > 0, the root lies in the upper interval. Therefore,

set 𝑎 = 𝑋𝑂 and return to step 2.

c. If 𝑓(𝑎) ∗ 𝑓(𝑋𝑂 ) = 0, the root equals 𝑋𝑂 ; terminate the computation

Incremental Search Method

The incremental search method is a numerical method that is used when is

needed to find an interval of two values of ‘x’ that is meant to contain at least one root
Algorithms:

Step 1: Set the value of ∆𝑋 and 𝑋𝑖 and get 𝑓(𝑋𝑖 )

Step 2: Calculate the value of 𝑋𝑖 + ∆𝑋, then get 𝑓(𝑋𝑖 + ∆𝑋)

Step 3: Determine the sign of 𝑓(𝑋𝑖 ) ∗ 𝑓(𝑋𝑖 + ∆𝑋)

a. If positive, then 𝑋𝑖 will assume the value of 𝑋𝑖 + ∆𝑋, then repeat

step 2

b. If negative, then ∆𝑋 will assume a smaller value and repeat step

You might also like